ALBERT

Notes: Abstract This paper presents benchmark (BMD) calculations and additional regression analyses of data from a study in which scores from 26 scholastic and psychological tests administered to 237 6- and 7-year-old New Zealand children were correlated with the mercury concentration in their mothers' hair during pregnancy. The original analyses of five test scores found an association between high prenatal mercury exposure and decreased test performance, using category variables for mercury exposure. Our regression analyses, which utilized the actual hair mercury level, did not find significant associations between mercury and children's test scores. However, this finding was highly influenced by a single child whose mother's mercury hair level (86 mg/kg) was more than four times that of any other mother. When that child was omitted, results were more indicative of a mercury effect and scores on six tests were significantly associated with the mothers' hair mercury level. BMDs calculated from five tests ranged from 32 to 73 mg/kg hair mercury, and corresponding BMDLs (95% lower limits on BMDs) ranged from 17 to 24 mg/kg. When the child with the highest mercury level was omitted, BMDs ranged from 13 to 21 mg/kg, and corresponding BMDLs ranged from 7.4 to 10 mg/kg.

Notes: Abstract In this paper we consider the problems of modeling the tumor growth and optimize the chemotherapy treatment. A biologically based model is used with the goal of solving an optimization problem involving discrete delivery of antineoplastic drugs. Our model is formulated via compartmental analysis in order to take into account the cell cycle. The cost functional measures not only the final size of the tumor but also the total amount of drug delivered. We propose an algorithm based on the discrete maximum principle to solve the optimal drug schedule problem. Our numerical results show nice interpretations from the medical point of view.

Notes: Abstract We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.

Notes: Abstract We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.

Notes: Abstract We consider a production planning problem in a two-machine flowshop subject to breakdown and repair of machines and subject to nonnegativity and upper bound constraints on work-in-process. The objective is to choose machine production rates over time to minimize the long-run average inventory/backlog and production costs. For sufficiently large upper bound on the work-in-process, the problem is formulated as a stochastic dynamic program. We then establish a verification theorem and a partial characterization of the optimal control policy if it exists.

Notes: Abstract In this paper a modified form of Pontryagin's maximum principle is developed, which also holds for problems of optimal control with respect to functions of several independent variables.

Notes: Abstract We study two continuity concepts for set-valued maps that play central roles in quantitative stability analysis of optimization problems: Aubin continuity and Lipschitzian localization. We show that various inverse function theorems involving these concepts can be deduced from a single general result on existence of solutions to an inclusion in metric spaces. As applications, we analyze the stability with respect to canonical perturbations of a mathematical program in a Hilbert space and an optimal control problem with inequality control constraints. For stationary points of these problems, Aubin continuity and Lipschitzian localization coincide; moreover, both properties are equivalent to surjectivity of the map of the gradients of the active constraints combined with a strong second-order sufficient optimality condition.

Notes: Abstract In this paper we explain that various (possibly discontinuous) value functions for optimal control problem under state-constraints can be approached by a sequence of value functions for suitable discretized systems. The key-point of this approach is the characterization of epigraphs of the value functions as suitable viability kernels. We provide new results for estimation of the convergence rate of numerical schemes and discuss conditions for the convergence of discrete optimal controls to the optimal control for the initial problem.

Notes: Abstract An optimal routing policy is obtained for Flexible Manufacturing Systems (FMSs) with limited buffers at the work stations. This policy is used to effectively drive a robotic material handling system. The routing decisions are made by a supervising computer on a real-time basis in order to avoid any work station running out of inputs and to control the blocking of the material handling system. Using our model, general material handling times can be assumed. The optimal policy and several key performance measures are computed, following the problem formulation as a continuous-time, semi-Markovian decision process. Fast convergence and computational stability are ensured by the ergodic solution algorithm augmented to solve the functional equations of the renewal process. The solution algorithm was implemented, tested on an extensive range of problems regarding the structure and the performance of the optimal policy. Complex environments involving diverse processing times, as well as very limited buffer storage, were examined. The interaction between the allocation of buffer spaces to work stations, the structural properties of the optimal monotone (threshold-type) policy and the system performance are also investigated.

Notes: Abstract We consider first nonlinear systems of the form x=A(x)x+B(x)u together with a standard quadratic cost functional and replace the system by a sequence of time-varying approximations for which the optimal control problem can be solved explicitly. We then show that the sequence converges. Although it may not converge to a global optimal control of the nonlinear system, we also consider a similar approximation sequence for the equation given by the necessary conditions of the maximum principle and we shall see that the first method gives solutions very close to the optimal solution in many cases. We shall also extend the results to parabolic PDEs which can be written in the above form on some Hilbert space.

Notes: Abstract This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R3. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field.

Notes: Abstract A formal method of constructing the viscosity solutions for abstract nonlinear equations of Hamilton–Jacobi–Bellman (HJB) type was developed in the previous work of the author. A new advantage of this method (which was called an ‘nonlinear potentials’ method) is that it gives a possibility to choose at the first step an expected regularity of the solution and then – to construct this solution. This makes the whole procedure more simple because an analysis of regularity of viscosity solutions is usually the most complicated step. Nonlinear potentials method is a generalization of Krylov's approach to study HJB equations. In this article nonlinear potentials method is applied to elliptic degenerate HJB equations in Rd with variable coefficients.

Notes: Abstract In the paper we study the topological structure of the solution set of a class of nonlinear evolution inclusions. First we show that it is nonempty and compact in certain function spaces and that it depends in an upper semicontinuous way on the initial condition. Then by strengthening the hypothesis on the orientor field F(t, x), we are able to show that the solution set is in fact an R δ-set. Finally some applications to infinite dimensional control systems are also presented.

Notes: Abstract We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.

Notes: Abstract We consider a pair consisting of an optimization problem and its optimality function (P,θ), and define consistency of approximating problem-optimality function pairs, (P N ,θ N ) to (P,θ), in terms of the epigraphical convergence of the P N to P, and the hypographical convergence of the optimality functionsθ N toΛ. We then show that standard discretization techniques decompose semi-infinite optimization and optimal control problems into families of finite dimensional problems, which, together with associated optimality functions, are consistent discretizations to the original problems. We then present two types of techniques for using consistent approximations in obtaining an approximate solution of the original problems. The first is a “filter” type technique, similar to that used in conjunction with penalty functions, the second one is an adaptive discretization technique that can be viewed as an implementation of a conceptual algorithm for solving the original problems.

Notes: Abstract Joint Implementation (JI) calls for cooperation between industrialized and developing countries in the mitigation of greenhouse gas (GHG) emissions. However, a major concern of potential host countries is that, if they utilize their low-cost options for JI now, they will be left with only high cost options in the future, thereby penalizing them at a time when they may be obligated to mitigate GHGs themselves. This paper formalizes this hypothesis by utilizing an optimal control framework analogous to the Hotelling model of non-renewable resource extraction. The results are that cumulative abatement effects can impose costs on the future, but that they can be offset by technological change, market power, or compensation.

Notes: Abstract The present study examines the optimal recycling rate for municipal solid waste. First, an optimal control model is developed to account for the physical costs of recycling, the social costs of landfilling, and consumers' environmental preferences. Second, an optimal solution is simulated using waste disposal data from the Helsinki region in Finland. The benefits from recycling are included in the simulation using the results of a recent contingent valuation study. The results of the present research suggest that mandates for achieving 50% recycling in municipalities are not far-fetched and are both economically and environmentally justified.

Notes: Abstract The probabilities associated with global warming damage are likely to be continuously revised in the light of new information. Such revisions of probability are the defining characteristic of ambiguity, as opposed to risk. This paper examines how climate change ambiguity may affect optimal greenhouse gas emission strategies, via the decision maker's attitude towards anticipated changes of damage probabilities. Two conceptualizations of ambiguity are distinguished, according to the emphasis placed on the ambiguity of priors or on the ambiguity of news, respectively. It is shown that the way in which ambiguity is viewed and the attitude taken towards it have a substantial influence on the optimal emission trajectory.

Notes: Abstract We determined mercury in fish (perch Perca fluviatilis) from 26 Russian lakes in three regions over four years. The lakes ranged in size from 2 to 395,000 ha, in pH from 4.5 to 10.0, and in color from 3 to 190 hazen. Sixteen lakes were drainage lakes, with permanent outlets, and 10 were seepage lakes, with no permanent inlets or outlets. The lakes were generally located in forested regions with little or no human habitation in the watershed. The three regions were geologically distinct: Precambrian Shield granitic bedrock covered with thin soil; Triassic bedrock covered with thick glacial tills; and Triassic bedrock covered with thin sediments. At each lake water samples were collected and analyzed for pH, add neutralizing capacity (ANC), major cations, and anions. Dissolved mercury species were estimated with a thermodynamic equilibrium model (MINTEQA2). Mercury content of dorsal muscle varied from 0.04 to 1.0 μg/g wet weight, and was linearly related to calculated HgCH3Cl (r20.68, p〈0.001). Lake HgCH3Cl, in turn, was related to lake pH (r2=0.86, p〈0.001). Stepwise multiple regression selected lake HgCH3Cl and color as the factors most highly related to fish mercury content, with the model accounting for 75% of the variation.

Notes: Abstract A reactive plume model that includes atmospheric chemical reactions of mercury was developed. The model simulates advective transport with the mean wind flow; horizontal and vertical turbulent diffusion; gas phase; aqueous-phase and particulate chemistry; cloud microphysics; wet deposition and dry deposition. The model was applied to the simulation of clear sky, non-precipitating cloud and precipitating cloud scenarios. No significant mercury chemistry occurs in the absence of droplets. In clouds, Hg(II) is reduced to Hg(0) with more reduction taking place in precipitating clouds than in non-precipitating clouds.

Notes: Abstract A denuder-based method for sampling and separating gaseous and participate mercury in the air is described. Two different denuder configurations developed in Vilnius, Lithuania (silver) and in Gothenburg, Sweden (gold) are compared. Data were acquired at different sampling locations around the cities of Vilnius and Gothenburg. The concentration of particulate Hg was found to be 0.04 to 0.40 ng m−3 in the Vilnius region, and 0.11 to 0.57 ng m−3 in the Gothenburg region. Intel-calibration results for the silver and gold denuders are presented. The results obtained by the two different denuder configurations and sampling set-ups display satisfactory agreement.

Notes: Abstract The concentration of mercury has been determined in assimilation organs of forest trees from 10 main industrial regions of Slovakia, four mountain forests and one military area and compared with concentration of mercury from 1356 permanent monitoring plots of Slovakia. The mercury concentration ranges for 2 yr old needles of Picea abies Karst. were (in mg kg−1): 1.249–4.402 (Rudnany iron ore mines), 0.013–0.749 (nine other industrial regions), 0.021–0.737 (four mountain forests) and 0.053–0.538 (military area). The mercury content in the soil (0–5 cm) from a mercury smelting plant ranged from 9.9 to 130 mg kg−1, and the moss Pleurozium schreberi contained 3.8–9.1 mg kg−1. The values obtained were compared with the available literature data.

Notes: Abstract Increasing mercury contents are reported from freshwater systems and fish in northern Europe and North America. Mercury input from soils is a major source with the leaching being affected by increased atmospheric mercury deposition compared to pre-industrial times and by other environmental conditions such as acid rain. The results of a mathematical model-calculation of vertical inorganic Hg(II) leaching in a Scandinavian iron-humus podzol under different atmospheric input rates of mercury are presented. Leaching under background rain conditions was calculated to be considerably stronger than under acid rain conditions. Increasing fractions of deposited soluble or solute atmospheric mercury were leached from the O f(h)-horizon with decreasing soil content of soluble mercury under acid rain conditions; this effect was less pronounced under background rain conditions. The steady state concentrations of soluble mercury of the upper soil horizons were calculated and compared with the actual concentrations of total (= soluble + insoluble mercury) and extractable (= estimate of soluble) mercury measured in these horizons. The results indicate that even if the deposition of airborne mercury to soil is strongly reduced, the total mercury content of the soil decreases only slowly. It may take decades or even centuries before a new steady state concentration of total mercury is established in the soil. The decrease of the mercury concentration in the O f(h)-horizon is probably largely dependent on the turnover of organic matter, binding most of the deposited airborne mercury in an insoluble form. Hence, present day mercury leaching is likely to be dominated by mercury deposited during former times and temporarily retained in an insoluble form in the organic matter.

Notes: Abstract The water-mobilizability of mercury from contaminated floodplain soils of the river Elbe in Northern Germany was evaluated by batch extraction experiments. It was shown that only a small amount of the total mercury present (about 1% per extraction) can be mobilized by water. This mercury is transported entirely in the form of a complex bound to humic acids (HA); particulates and fulvic acids (FA) did not seem to contribute to the process. It could not be removed from the HA even at pH 1, indicating an extremely strong complexation e.g. by sulfur-containing ligands. Furthermore, the influence of pH on the mobilization was investigated. It was found that in the range of natural pH-values, there was no observable effect of pH on the mobilization of either mercury or dissolved organic carbon (DOC). This surprising finding is explained by an unexpectedly high buffering capacity of the humics, both in the acidic and in the alkaline region. Only at extreme pH-values there was deviation from this behaviour. In contrast to other heavy metals, the amount of mobilized mercury decreases at pH 〈 3; and at pH 〉 12, an increased mobilization of mercury was observed because the humics are mobilized completely, accompanied by the total amount of mercury bound to them.

Notes: Abstract As described by Jemelov and Ramel (1995), the Scientific Committee on Problems of the Environment (SCOPE) sponsored an investigation of Hg in ecosystems with special emphasis on tropical regions. In these regions small-scale gold mining activities have occupied about 10 million people worldwide who use Hg for extracting gold.

Notes: Abstract Total gaseous mercury (TGM) and rainwater were collected on board of two research vessels (F. S. ALKOR and R.V. BELGICA) positioned 200 km apart in the center of the North Sea during the North Sea Experiment, September 1991. On the F. S. ALKOR (up-wind ship) TGM concentrations ranged from 0.7 to 2.6 ng·m−3 with an average of 1.5 ng m−3 and on the R. V. BELGICA (down-wind ship) TGM ranged from 0.7 to 1.9 ng·m−3 with an average of 1.2 ng·m−3. An average 20% decrease is observed from the up-wind to the down wind ship. which may largely be affected by entrainment into the free troposphere. An overall removal (entrainment) velocity of 0.95 cm·s−1 was calculated for the whole experiment. The average removal velocity was 0.5 cm·s−1 for dry periods and varied between 1 to 5 cm·s−1 during rain events. Rainwater concentrations varied between 5 and 25 ng·1−1. Based on these data an annual wet deposition flux of 1.08 ng Hg cm−2 yr1− was estimated for the North Sea.

Notes: Abstract Total mercury (Hg) and methylmercury (MeHg) concentrations were determined in sediments and in the polychaete worm Nereis diversicolor at 13 stations of a brackish water intertidal mudflat of the Scheldt estuary. Hg and MeHg concentrations in sediments ranged from 144 to 1192 ng g−1 dw and from 0.8 to 6 ng g−1 dw, respectively. Both Hg and MeHg concentrations increased with an increase of organic matter (OM) content and fine grain fraction. In contrast, Hg accumulation by N. diversicolor was significantly (p 〈 0.05) higher at stations with sandy sediments (mean value: 125 ng g−1 dw) than at stations with muddy sediments (mean value, 80 ng g−1), probably because Hg availability for bioaccumulation at muddy stations was reduced by high OM content of the muddy sediments. MeHg accounted for an average of 0.7% of the total Hg in sediments and 18% of the total Hg in N. diversicolor. Seasonal variations significantly affected Hg concentrations in sediments and MeHg in N. diversicolor. Total Hg concentrations in sediments were significantly (p 〈 0.05) higher in autumn and winter than in spring and summer whereas MeHg concentrations were lowest in winter compared to the other seasons. On the other hand, total Hg concentrations in the worms were lowest in spring whereas MeHg concentrations were significantly (p 〈 0.01) higher in spring and summer than in autumn and winter.

Notes: Abstract The international reference lake sediment, LKSD-4, was used to compare Hg, organic C and Zn extracted from its ‘soluble organic’ phase by two commonly used reagents: 0.1 M Na4P2O7 solution at pH 10 and 0.5 M NaOH solution at pH 12. While recoveries of Hg and Zn by 0.1 M Na4P2O7 are not affected by changes in sample weight to reagent volume ratio (W/V) or contact time, those by NaOH show a marked dependency. In general, the NaOH leach extracts more organic C and Hg from LKSD-4 but less Zn. Over the range of conditions studied, the NaOH-based method extracted 4.7–9.8% C, 27–103 ng g−1 Hg and 19–69 μg g−1 Zn from LKSD-4, compared to 2.3–2.8% C, 17–24 ng g−1 Hg and 64–72 μg g−1 Zn by the Na4P2O7 leach. Clearly, different groups of organic substances are being dissolved by these two reagents and therefore a comparison of data from different laboratories becomes meaningless. This paper suggests that more research is needed into the exactNature of metal-organic associations extracted by selective leaches and into associated artifacts of extraction such as readsorption phenomena.

Notes: We studied mercury concentrations and amounts in tissues of 19 starved young Common Tern chicks (median age 4 days) and in eggs from the same colony. Concentrations and burden were similar between eggs and newly hatched chicks. Mercury concentrations were highest in down, which contained at least 38% of the body mercury. The mercury burden of the whole body and of the tissues as well as the concentration in down increased with age and body mass, indicating the importance of down as an elimination pathway. Conversion ratios between mercury concentrations in tissues and the whole chick body varied according to the contamination level.

Notes: Abstract Measurements of the differential tolerance between enzyme genotypes and shifts in allozyme frequencies in populations from contaminated habitats have prompted the use of allozymes as markers of population-level toxicant effects. However, such studies often do not consider other factors that influence allele frequencies, including natural clines, migration, the intensity and specificity of selection and toxicant-induced genetic bottlenecks. In addition, selection components other than survival are not included. Consequently, the associated conclusions remain speculative. To assess this approach rigorously, a simulation study was conducted with the mosquitofish (Gambusia holbrooki) GPI-2 locus. Laboratory studies have shown the GPI-238/38 homozygote at this locus to be less tolerant than other genotypes during acute exposure to mercury. The GPI-2100/100 genotype has also been shown to have a reproductive disadvantage at lower mercury concentrations. Simple and then more complex models were used to quantify the relative effects of viability selection, random genetic drift and migration on the GPI-238 allele frequency. Simulations were also performed to assess the contribution of sexual and fecundity selection. A simple population model suggested that viability selection plays a greater role than does mortality-driven, genetic drift in the decrease of the sensitive allele under the conditions of this study. A more complex, stochastic model indicated that no significant mortality-driven drift was taking place in this system. In both models, migration mitigated the effect of selection. Sexual and fecundity selection had little effect on the allele frequencies in these simulations. We conclude that, provided the system under study is clearly understood, shifts in allele frequency can indicate the population-level effects of pollutants.

Notes: Abstract This paper presents the optimal control variational principle for Perzyna model which is one of the main constitutive relation of viscoplasticity in dynamics. And it could also be transformed to solve the parametric quadratic programming problem. The FEM form of this problem and its implementation have also been discussed in the paper.

Notes: Abstract This paper will seek the optimal control and corresponding trajectories of the singularly perturbed nonlinear state regulator problem. Under appropriate hypotheses, it will be possible to complete an asymptotic solution which is uniformly valid when σ→0.

Notes: Abstract A simple technique based on historical records of tonnage and grade of ore produced provides a means for calculating how much of a mineral product will be available in the future at various average grades. Estimates made on this basis are independent of geologic considerations or changing economic and political factors, although they are based on mining history, which was largely determined by these factors. The relatively minor element, mercury, was used for the test case reported here, but the method has been found applicable to forecasts of resources for other mineral products. Mercury resources available in ore in which the average grade is as low as 0.1% are estimated to be 53 ×10 6 kg (1.5 ×10 6 flasks) for the United States and 1551 ×10 6 kg (45 ×10 6 flasks) for the world; this amount is more than adequate to meet predicted demand to the year 2000. The expectable price of mercury in 1978 dollars at this 0.1% grade is projected to be $58.75 per kg ($2,025 per flask), but at a 10% annual inflation rate, it would be more than $12,000 per flask. To satisfy just the projected U.S. demand for mercury by 2000, the price is calculated to be $48.96 per kg ($1,688 per flask) in 1978 dollars at an average annual grade of 0.12%.

Notes: Abstract Generalized semi-Markov processes (GSMPs) and stochastic Petri nets (SPNs) are generally regarded as performance models (as opposed to logical models) of discrete event systems. Here we take the view that GSMPs and SPNS are essentially automata (generators) driven by input sequences that determine the timing of events. This view combines the deterministic, logical aspects and the stochastic, timed aspects of the two models. We focus on two conditions, (M) and (CX) (which we previously developed to study monotonicity and convexity properties of GSMPs), and the antimatroid and lattice structure they imply for the language generated by a GSMP or SPN. We illustrate applications of these structural properties in the areas of derivative estimation, simulation variance reduction, parallel simulation, and optimal control.

Notes: Abstract Most of the results to date in discrete event supervisory control assume a “zero-or-infinity” structure for the cost of controlling a discrete event system, in the sense that it costs nothing to disable controllable events while uncontrollable events cannot be disabled (i.e., their disablement entails infinite cost). In several applications however, a more refined structure of the control cost becomes necessary in order to quantify the tradeoffs between candidate supervisors. In this paper, we formulate and solve a new optimal control problem for a class of discrete event systems. We assume that the system can be modeled as a finite acylic directed graph, i.e., the system process has a finite set of event trajectories and thus is “terminating.” The optimal control problem explicitly considers the cost of control in the objective function. In general terms, this problem involves a tradeoff between the cost of system evolution, which is quantified in terms of a path cost on the event trajectories generated by the system, and the cost of impacting on the external environment, which is quantified as a dynamic cost on control. We also seek a least restrictive solution. An algorithm based on dynamic programming is developed for the solution of this problem. This algorithm is based on a graph-theoretic formulation of the problem. The use of dynamic programming allows for the efficient construction of an “optimal subgraph” (i.e., optimal supervisor) of the given graph (i.e., discrete event system) with respect to the cost structure imposed. We show that this algorithm is of polynomial complexity in the number of vertices of the graph of the system.

Notes: Abstract A continuous time dynamic model of discrete scheduling problems for a large class of manufacturing systems is considered in the present paper. The realistic manufacturing based on multi-level bills of materials, flexible machines, controllable buffers and deterministic demand profiles is modeled in the canonical form of optimal control. Carrying buffer costs are minimized by controlling production rates of all machines that can be set up instantly. The maximum principle for the model is studied and properties of the optimal production regimes are revealed. The solution method developed rests on the iterative approach generalizing the method of projected gradient, but takes advantage of the analytical properties of the optimal solution to reduce significantly computational efforts. Computational experiments presented demonstrate effectiveness of the approach in comparison with pure iterative method.

Notes: Abstract This paper discusses dynamic methods for solving a class of multi-project scheduling problems in which rates of job performances are controllable and resources such as money, energy or manpower per time unit, are renewable and continuously divisible. The objective is to complete the projects as close to the common due date as possible. Two different ways of imposing sequential precedence relations between project jobs are explored by formulating two dynamic models and studying their relationships on the optimal solution. Efficient time-decomposition algorithms for finding either globally optimal schedules or lower bound guided near-optimal solutions are suggested and computationally tested.

Notes: Abstract We propose a modeling framework for a class of hybrid systems which arise in many manufacturing environments and study related optimal control problems. In this framework, discrete entities have a state characterized by a temporal component whose evolution is described by event-driven dynamics, and a physical component whose evolution is described by time-driven dynamics. As a first step towards developing an optimal control theory for such hybrid systems, we formulate a problem consisting of a single-stage manufacturing process and use calculus of variations techniques to obtain structural properties and an explicit algorithm for deriving optimal policies.

Notes: Abstract This paper is concerned with the problem of production planning in a flexible manufacturing system consisting of a single or parallel failure-prone machines producing a number of different products. The objective is to choose the rates of production of the various products over time in order to meet their demands at the minimum long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost. It is shown using the vanishing discount approach for the average cost problem that the Hamilton-Jacobi-Bellman equation in terms of directional derivatives has a solution consisting of the minimal average cost and the so-called potential function. The result helps in establishing a verification theorem, and in specifying an optimal control policy in terms of the potential function. The results settle a hitherto open problem as well as generalize known results.

Notes: Abstract A control system described by a nonlinear equation of parabolic type is considered in the situation where there may be no global solution. A particular optimal control problem subject to state constraints is studied. A proof of the existence of an optimal control is presented. The penalty method is used to obtain necessary conditions for optimal control. A proof of the convergence of this method is given. The successive approximation method is used to obtain an approximate solution for the conditions derived.

Notes: Abstract The standard assumption in bioeconomic resource models is that optimal policies maximize the present value of economic surplus to society. This assumption implies that regulatory agencies should not be concerned with the distributional consequences of management strategies. Both contemporary welfare-theoretic and rent-seeking approaches suggests distributional issues are important in designing resource management policies. This paper explores resource management when the managing agency has preferences defined over the economic welfare of various groups with a direct economic interest in the use of resources. Policy schemes consistent with this approach are derived and compared with standard results.

Notes: Abstract This paper studies how inclusion of many sources, sinks and reservoirs -- a comprehensive approach -- affects climate policy, compared with a control merely of CO2. Two questions of particular importance arise in such an analysis. One is how to aggregate the emissions of different climate gases, and the other is how to include all relevant measures in the analysis. To aggregate gases properly, an intertemporal analysis should be carried out. To assure that all relevant measures are included, we suggest that certain measures to reduce emissions of greenhouse gases are specified explicitly and evaluated together with indirect measures, such as carbon charges. A numerical analysis based on an optimal control model indicates that direct measures may play an important role in the design of climate policy, especially for the control of the emissions of greenhouse gases other than CO2. Similar to other studies of the time-path for abatement efforts, the bulk of abatement should be taken by the end of the planning period. This result is significantly strengthened if gases with short life-times in the atmosphere, such as methane, are subject to control.

Notes: Abstract Increasing mercury contents are reported from freshwater systems and fish in northern Europe and North America. Mercury input from soils is a major source with the leaching being affected by increased atmospheric mercury deposition compared to pre-industrial times and by other environmental conditions such as acid rain. The results of a mathematical model-calculation of vertical inorganic Hg(II) leaching in a Scandinavian iron-humus podzol under different atmospheric input rates of mercury are presented. Leaching under background rain conditions was calculated to be considerably stronger than under acid rain conditions. Increasing fractions of deposited soluble or solute atmospheric mercury were leached from the O f(h)-horizon with decreasing soil content of soluble mercury under acid rain conditions; this effect was less pronounced under background rain conditions. The steady state concentrations of soluble mercury of the upper soil horizons were calculated and compared with the actual concentrations of total (= soluble + insoluble mercury) and extractable (= estimate of soluble) mercury measured in these horizons. The results indicate that even if the deposition of airborne mercury to soil is strongly reduced, the total mercury content of the soil decreases only slowly. It may take decades or even centuries before a new steady state concentration of total mercury is established in the soil. The decrease of the mercury concentration in the O f(h)-horizon is probably largely dependent on the turnover of organic matter, binding most of the deposited airborne mercury in an insoluble form. Hence, present day mercury leaching is likely to be dominated by mercury deposited during former times and temporarily retained in an insoluble form in the organic matter.

Notes: Abstract Different species of mercury have different physical/chemical properties and thus behave quite differently in air pollution control equipment and in the atmosphere. In general, emissions of mercury from coal combustion sources are approximately 20–50% elemental mercury (Hg°) and 50–80% divalent mercury (Hg(II)), which may be predominantly HgCl2. Emissions of mercury from waste incinerators are approximately 10–20% Hg° and 75–85% Hg(II). The partitioning of mercury in flue gas between the elemental and divalent forms may be dependent on the concentration of particulate carbon, HCl and other pollutants in the stack emissions. The emission of mercury from combustion facilities depends on the species in the exhaust stream and the type of air pollution control equipment used at the source. Air pollution control equipment for mercury removal at combustion facilities includes activated carbon injection, sodium sulfide injection and wet lime/limestone flue gas desulfurization. While Hg(II) is water-soluble and may be removed from the atmosphere by wet and dry deposition close to combustion sources, the combination of a high vapor pressure and low water-solubility facilitate the long-range transport of Hg° in the atmosphere. Background mercury in the atmosphere is predominantly Hg°. Elemental mercury is eventually removed from the atmosphere by dry deposition onto surfaces and by wet deposition after oxidation to water- soluble, divalent mercury.

Notes: Abstract Different species of mercury have different physical/chemical properties and thus behave quite differently in air pollution control equipment and in the atmosphere. In general, emissions of mercury from coal combustion sources are approximately 20–50% elemental mercury (Hg°) and 50–80% divalent mercury (Hg(II)), which may be predominantly HgCl2. Emissions of mercury from waste incinerators are approximately 10–20% Hg° and 75–85% Hg(II). The partitioning of mercury in flue gas between the elemental and divalent forms may be dependent on the concentration of particulate carbon, HCl and other pollutants in the stack emissions. The emission of mercury from combustion facilities depends on the species in the exhaust stream and the type of air pollution control equipment used at the source. Air pollution control equipment for mercury removal at combustion facilities includes activated carbon injection, sodium sulfide infection and wet lime/limestone flue gas desulfurization. While Hg(II) is water-soluble and may be removed from the atmosphere by wet and dry deposition close to combustion sources, the combination of a high vapor pressure and low water-solubility facilitate the long-range transport of Hg° in the atmosphere. Background mercury in the atmosphere is predominantly Hg°. Elemental mercury is eventually removed from the atmosphere by dry deposition onto surfaces and by wet deposition after oxidation to water-soluble, divalent mercury.

Notes: Abstract Sediment cores were collected from lakes in the Devils Lake Basin in North Dakota to determine if mercury (Hg) accumulation chronologies from sediment-core data are good indicators of variations in Hg accumulation rates in saline lakes. Sediment cores from Creel Bay and Main Bay, Devils Lake were selected for detailed analysis and interpretation. The maximum Hg concentration in the Creel Bay core was 0.15 micrograms per gram at 8 to 9 centimeters. The maximum Hg concentration in the Main Bay core was 0.07 micrograms per gram at 5 to 7 centimeters. The general decreases in Hg concentrations with depth are attributed to historic variations in atmospheric Hg deposition rate. Hg stratigraphies combined with210 Pb and 137Cs dating analyses yield Hg chronologies that indicate a general increase in Hg accumulation rates in Devils Lake since the middle of the 19th century. Mean modern Hg accumulation rates in Creel Bay were 4.9 nanograms per square centimeter per year, and rates in Main Bay were 1.8 nanograms per square centimeter per year. Mean preindustrial Hg accumulation rates in Creel Bay were 1.2 nanograms per square centimeter per year, and rates in Main Bay were 1.6 nanograms per square centimeter per year. Relatively low Hg concentrations in recent sediments in the Devils Lake Basin, along with similarities in Hg accumulation rates between lakes in the Devils Lake Basin and other lakes in the northern interior of North America, indicate that local sources of Hg are not important sources of Hg. Results of the study indicate that accurate Hg chronologies are discernible in sediment cores collected from saline lakes. However, spatial and temporal variations in lake level and water chemistry common to saline lakes make interpretation of radioisotopic and geochemical chronologies difficult. Hg geochemistry in Devils Lake, and presumably in other saline lakes, is dynamic. The results of this study indicate that the absolute amount of sediment transported to Devils Lake, along with the associated Hg and total organic carbon, and the distribution of sedimentation patterns in Devils Lake may be affected by changing lake levels.

Notes: Abstract The San Francisco Estuary Regional Monitoring Program for Trace Substances (RMP) began in 1993 and is sponsored by 74 local, state, and federal agencies and companies through their discharge or Bay use permits. The RMP monitors water, sediment, toxicity, and bivalve bioaccumulation at 25 sites in the Bay that are considered to represent "background" conditions. Several major environmental issues have been identified by the RMP. Polychlorinated biphenyls and mercury were often above water quality guidelines, and often occurred in fish tissues above U.S. Environmental Protection Agency (U.S. EPA) screening values. Concentrations do not appear to be decreasing, suggesting continuing inputs. Episodes of aquatic toxicity often occurred following runoff events that transport contaminants into the Bay from urbanized and agricultural portions of the watershed. Sediment toxicity occurred throughout the Bay, and has been correlated with concentrations of specific contaminants (chlordanes, polynuclear aromatic hydrocarbons) at some locations; mixtures of contaminants were probably also important. Since the RMP does not monitor all ecosystem components, assessments of the overall condition of the Bay cannot be made. However, in terms of contamination, the RMP samples suggest that the South Bay, and North Bay sites are moderately contaminated.

Notes: Abstract Samples of fish and sediments collected from waters within 10 km of the Chernobyl Nuclear Power Plant were analysed for radiocaesium and other atmospherically transported pollutants (lead (Pb) and mercury (Hg)) were measured in fish tissues. For comparison, fish muscle was also sampled from a less-contaminated area of the Kiev Reservoir and from a hatchery in Kiev. In sediments, 137Cs was the major gamma-emitting isotope, with concentrations of over 100 Bq g−1 in canals adjacent to the reactor and in ponds that were downwind during the accident. The radiocaesium distributions appeared non-normal, were very patchy and could vary by over 100% in samples collected metres to tens of metres apart. Fish muscle radiocaesium from ponds within 10 km of the power plant was in the range of 6–192 Bq g−1. The fish muscle radiocaesium concentrations were highest in ponds from the downwind sites and were correlated with the sediment radiocaesium concentrations. The lead and mercury concentrations in fish were relatively low (medians 〈0.8 μg Hg per g dry mass and 〈150 ng Pb per g dry mass), suggesting little contamination from lead applied to the burning reactor after the accident. The radiocaesium levels in fish in areas close to the reactor continue to be high enough to cause health concerns to humans that might utilize these resources.

Notes: Abstract Two small drainage lakes of similar surface area (0.4–0.6 km 2 ) and depth, and situated in close proximity to one another (about 9 km apart) in the Muskoka-Haliburton district of rural Ontario, Canada were chosen to investigate the effects of differing pH on Hg accumulation in fish and other biota. Blue Chalk Lake is circumneutral (pH = 6.3–6.9) with an average dissolved organic carbon (DOC) concentration of 2.69 mg/L; Plastic Lake is acidic (pH = 5.2–5.6) with a similar DOC level (2.72 mg/L). Neither lake is an impoundment, and neither receives inputs from point-source urban or industrial waste emissions. Surficial (top 4 cm) sediments and various aquatic biota, including crayfish (Orconectes virilis)], and 5 species of fish [brown bullhead (Ictalurus nebulosus), creek chub (Semotilus atromaculatus), pumpkinseed sunfish (Lepomis gibbosus), yellow perch (Perca flavenscens) and rock bass (Ambloplites rupestris)] were sampled for Hg analysis. Only creek chub and pumpkinseed occurred in both lakes. Significant size (length) vs. Hg-concentration relationships were observed for brown bullhead, pumpkinseed, yellow perch and rock bass, but not for crayfish nor for creek chub in either lake. Rock bass had the highest mean Hg concentration (1.02 ± 0.98 μg/g dry wt.). Pumpkinseed from Plastic Lake had the highest overall rate of accumulation with increasing size. Several individual pumpkinseed and rock bass sampled from the acidic Plastic Lake had Hg concentrations equal to or greater than levels reported to be hazardous to breeding piscivorous birds, such as common loons (Gavia immer), whereas no fish from Blue Chalk approached this threshold (0.3 μg/g wet wt). This study illustrates the importance of within-lake parameters, particularly pH, and species trophic level, in determining Hg accumulation in fish in lakes of similar size that receive similar atmospheric Hg inputs.

Notes: Abstract The influence of humic acid (HA) on the transport and transformation of mercury (Hg) in soil was studied. No available Hg could be detected (〈2.5 μg kg−1) in alluvial soil when the content of HA-carbon (HA-C) was higher than 0.2 g kg−1 although a large amount of Hg (8 μg kg−1) was applied to the soil. The available Hg decreased with the increase of HA in purple soil (r=0.735). There are significant correlations between HA concentration and organic Hg in the tested soils (r=0.974 for the purple soil and r=0.979 for the alluvial soil). The increase of HA results in decrease of Hg absorbed by plant from the soil. A loss of Hg from soil caused by microbes was observed.

Notes: Abstract The influence of humic acid (HA) on the transport and transformation of mercury (Hg) in soil was studied. No available Hg could be detected (〈2.5 μg kg-1) in alluvial soil when the content of HA-carbon (HA-C) was higher than 0.2 g kg-1 although a large amount of Hg (8 μg kg-1) was applied to the soil. The available Hg decreased with the increase of HA in purple soil (r=0.735). There are significant correlations between HA concentration and organic Hg in the tested soils (r=0.974 for the purple soil and r=0.979 for the alluvial soil). The increase of HA results in decrease of Hg absorbed by plant from the soil. A loss of Hg from soil caused by microbes was observed.

Notes: Abstract As described by Jernelov and Ramel (1995), the Scientific Committee on Problems of the Environment (SCOPE) sponsored an investigation of Hg in ecosystems with special emphasis on tropical regions. In these regions small-scale gold mining activities have occupied about 10 million people worldwide who use Hg for extracting gold.

Notes: Abstract Total gaseous mercury (TGM) and rainwater were collected on board of two research vessels (F. S. ALKOR and R.V. BELGICA) positioned 200 km apart in the center of the North Sea Experiment, September 1991. On the F.S. ALKOR (up-wind ship) TGM concentrations ranged from 0.7 to 2.6 ng·m-3 with an average of 1.5 ng·m-3 and on the R. V. BELGICA (down-wind ship) TGM ranged from 0.7 to 1.9 ng·m-3 with an average of 1.2 ng·m-3. An average 20% decrease is observed from the up-wind to the downwind ship, which may largely be affected by entrainment into the free troposphere. An overall removal was 0.5 cm·s-1 for dry periods and varied between 1 to 5 cm·s-1 during rain events. Rainwater concentrations varied between 5 and 25 ng·l-1. Based on these data an annual wet deposition flux of 1.08 ng Hg cm-2 yr1- was estimated for the North Sea.

Notes: Abstract Sediment cores were collected from lakes in the Devils Lake Basin in North Dakota to determine if mercury (Hg) accumulation chronologies from sediment-core data are good indicators of variations in Hg accumulation rates in saline lakes. Sediment cores from Creel Bay and Main Bay, Devils Lake were selected for detailed analysis and interpretation. The maximum Hg concentration in the Creel Bay core was 0.15 micrograms per gram at 8 to 9 centimeters. The maximum Hg concentration in the Main Bay core was 0.07 micrograms per gram at 5 to 7 centimeters. The general decreases in Hg concentrations with depth are attributed to historic variations in atmospheric Hg deposition rate. Hg stratigraphies combined with210Pb and137Cs dating analyses yield Hg chronologies that indicate a general increase in Hg accumulation rates in Devils Lake since the middle of the 19th century. Mean modern Hg accumulation rates in Creel Bay were 4.9 nanograms per square centimeter per year, and rates in Main Bay were 1.8 nanograms per square centimeter per year. Mean preindustrial Hg accumulation rates in Creel Bay were 1.2 nanograms per square centimeter per year, and rates in Main Bay were 1.6 nanograms per square centimeter per year. Relatively low Hg concentrations in recent sediments in the Devils Lake Basin, along with similarities in Hg accumulation rates between lakes in the Devils Lake Basin and other lakes in the northern interior of North America, indicate that local sources of Hg are not important sources of Hg. Results of the study indicate that accurate Hg chronologies are discernible in sediment cores collected from saline lakes. However, spatial and temporal variations in lake level and water chemistry common to saline lakes make interpretation of radioisotopic and geochemical chronologies difficult. Hg geochemistry in Devils Lake, and presumably in other saline lakes, is dynamic. The results of this study indicate that the absolute amount of sediment transported to Devils Lake, along with the associated Hg and total organic carbon, and the distribution of sedimentation patterns in Devils Lake may be affected by changing lake levels.

Notes: Abstract The planned restoration of the Kissimmee River ecocystem will backfill approximately 35 km of flood control canal (C-38) that cuts through the meandering river channel, re-establish natural flow patterns, and restore the river/floodplain ecosystem. Water quality monitoring, including nutrients, total suspended solids (TSS), turbidity, dissolved oxygen (DO), and mercury, was conducted during a pilot `test fill' project to determine if soil disturbance during canal backfilling would negatively impact these water quality constituents. Surface water nutrient concentrations varied little between sites. Generally, highest concentrations occurred prior to construction, with lowest concentrations occurring during and after construction. During construction, TSS concentrations increased at sites immediately upstream, downstream, and adjacent to the construction area. Increased turbidity was generally restricted to areas immediately upstream and downstream of the test plug, with maximum levels occurring during the initial construction phase. Some downstream increases in turbidity were observed; however, impacts were short-term, lasting less than 24 h. Depresssed DO levels (〈2 mg/l) were observed upstream of the test plug following completion of the initial plug across C-38. Dissolved oxygen levels remained low for approximately 6 weeks, with no apparent ecological impacts. Total mercury (HgT) within canal sediment ranged from 9.2–180 ng/g and methylmercury concentrations ranged from 0.037–0.708 ng/g. Concentration of total mercury and total methylmercury (MeHgT) in the backfill material were much lower than concentrations in the canal sediment. No significant change in aqueous HgT concentrations occurred over the sampling period, although construction-induced turbidity could have temporarily caused a slightly elevated concentration immediately downstream of the construction site. Methylmercury concentrations in the water column ranged from 0.033–0.518 ng/l. No significant differences in mean MeHgT concentrations occured between sites or between sampling dates, except at one downstream site where MeHgT declined significantly over the sampling period.

Notes: Abstract Approximately 250 000 kg of mercury was lost towater and soils at the U.S. Dept. of Energy Y-12 Plantin Oak Ridge, Tennessee in the 1950s and early 1960s. A creek originating within the plant receivedcontinuous inputs of waterborne mercury, predominantlyas dissolved inorganic mercury, from groundwater,streambed contamination, and sump and process waterdischarges to the contaminated storm sewer network.These produce aqueous total mercury concentrations of1–2 μg L-1 in the upper reaches of the stream,decreasing to about 0.1–0.2 μg L-1 in its lowerreaches. A program to reduce mercury concentrationsin the creek identified specific sources (buildingsumps, contaminated springwater seeps, foundationdrains, and contaminated piping) and rerouted wateraround contaminated portions of the drain system orcollected and treated mercury-contaminated waterbefore discharging it. As a result, waterbornemercury concentrations in the creek and total mercuryloading were reduced from 1.8 μg L-1 to0.6 μg L-1 and 100 to 20 g d-1, respectively, in the last 5 yr.Mean mercury concentrations in fish nearest sourceareas in the creek headwaters decreased at roughly thesame rate as waterborne total mercury concentrationsover the past five years, but at the facility boundarydownstream the decline in mercury bioaccumulation wasmuch less. At sites 5–15 km farther downstream, nodecrease was evident. Dissolved methylmercury tendedto increase with distance downstream in a patterninverse to that noted for its dissolved inorganicmercury precursor.Improvements in water quality and modification ofweirs to allow the passage of fish have resulted inthe establishment of large populations of fish inmercury-contaminated headwater areas previously devoidof fish. It may be that the accumulation, retention,and eventual downstream transport of this reservoir ofbiologically incorporated methylmercury has acted tobuffer against expected reductions in mercury in fishat downstream sites.

Notes: Abstract Mercury (Hg) in the aquatic ecosystem of Clear Lake has been documented since the 1970s when fishes were found to have elevated levels of toxic methyl mercury (meHg). Mining practices at the Sulphur Bank Mercury Mine (active intermittently from 1872–1957) along the shoreline of Clear Lake included the bulldozing of waste rock and overburden ore into the shallow nearshore regions of the lake and the creation of steeply sloped piles of waste rock at the water's edge. This process, plus erosion of the waste rock piles, resulted in the accumulation of an estimated 100 metric tons of Hg in Clear Lake. A monitoring program to assess Hg in Clear Lake was established in 1992, and conducted continuously from 1994. Drought conditions in California had persisted for ca. 6 yrs prior to 1992, when the U.S. Environmental Protection Agency (USEPA) remediated the steeply sloped eroding waste rock piles, which appeared to reduce sediment Hg concentrations significantly. In April 1995, a white flocculent material was observed in Clear Lake adjacent to the mine and has been observed every year since, leading to the discovery of ongoing acid mine drainage (AMD), low pH fluids high in Hg and extremely high in sulfate. AMD is now believed to be the most likely cause of elevated meHg in Clear Lake. The discovery of this source of meHg production in Clear Lake, which will significantly influence remedial options, was only made possible by implementation of a diligent monitoring program.

Notes: Abstract We prove the maximum principle for an optimal control problem governed by the system $$y'(t) + A(t)y(t) = f(t,y(t),u(t)),{\text{ }}u(t) \in U(t), $$ with state constraint $$(y(0),y(T)) \in C \subset H \times H $$ , under three different hypotheses: (H1) C is a convex set with nonempty interior; (H2) $$C = \{ y_0 \} \times C_{0,} {\text{ with }}C_0 $$ a convex set with nonempty interior in H and the evolution system satisfying compactness hypotheses; (H3) the periodic case $$y(0) = y(T)$$ , with the evolution system satisfying compactness hypotheses. We do not assume the controls to be bounded. We give some examples for distributed control problems.

Notes: Abstract We consider a production planning problem for a dynamic jobshop producing a number of products and subject to breakdown and repair of machines. The machine capacities are assumed to be finite-state Markov chains. As the rates of change of the machine states approach infinity, an asymptotic analysis of this stochastic manufacturing systems is given. The analysis results in a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. The long-run average cost for the original problem is shown to converge to the long-run average cost of the limiting problem. The convergence rate of the long-run average cost for the original problem to that of the limiting problem together with an error estimate for the constructed asymptotic optimal control is established.

Notes: Abstract This work deals with the necessary conditions of optimality for some optimal control problems governed by elliptic variational inequalities. Boundary control and state constrained problems are considered. The techniques used are based on those in Ref. 1 and a new penalty functional is defined in this paper.

Notes: Abstract An important and numerous literature argues that nonconcavity (often convexity with respect to the state) of the Hamiltonian leads to multiple steady states, instability, and a threshold. This threshold property provides a powerful paradigm to explain history dependency and hysteresis. This paper shows that economically relevant properties (in particular, multiple steady states and thresholds) are possible in strict concave models too. Two corresponding necessary conditions with intuitive economic interpretation are derived.

Notes: Abstract Existence theorems for optimal control problems in Banach spaces are stated and proved.

Notes: Abstract This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. A modified quasilinearization algorithm is developed. Its main property is the descent property in the performance indexR, the cumulative error in the constraints and the optimality conditions. Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of the scaling factor (or stepsize) α in the system of variations. The stepsize is determined by a one-dimensional search on the performance indexR. Since the first variation δR is negative, the decrease inR is guaranteed if α is sufficiently small. Convergence to the solution is achieved whenR becomes smaller than some preselected value. In order to start the algorithm, some nominal functionsx(t),u(t), π and nominal multipliers λ(t), ρ(t), μ must be chosen. In a real problem, the selection of the nominal functions can be made on the basis of physical considerations. Concerning the nominal multipliers, no useful guidelines have been available thus far. In this paper, an auxiliary minimization algorithm for selecting the multipliers optimally is presented: the performance indexR is minimized with respect to λ(t), ρ(t), μ. Since the functionalR is quadratically dependent on the multipliers, the resulting variational problem is governed by optimality conditions which are linear and, therefore, can be solved without difficulty. To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized timet, defined in such a way that the extremal arc has a normalized time length Δt=1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time τ at which the terminal boundary is reached is regarded to be a component of the parameter π being optimized. The present general formulation differs from that of Ref. 3 because of the inclusion of the nondifferential constraints to be satisfied everywhere over the interval 0⩽t⩽1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) there are problems involving state equality constraints which can be reduced to the present scheme through suitable transformations, and (iii) there are some problems involving inequality constraints which can be reduced to the present scheme through the introduction of auxiliary variables. Numerical examples are presented for the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.

Notes: Abstract This paper summarizes recent advances in the area of gradient algorithms for optimal control problems, with particular emphasis on the work performed by the staff of the Aero-Astronautics Group of Rice University. The following basic problem is considered: minimize a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. First, the sequential gradient-restoration algorithm and the combined gradient-restoration algorithm are presented. The descent properties of these algorithms are studied, and schemes to determine the optimum stepsize are discussed. Both of the above algorithms require the solution of a linear, two-point boundary-value problem at each iteration. Hence, a discussion of integration techniques is given. Next, a family of gradient-restoration algorithms is introduced. Not only does this family include the previous two algorithms as particular cases, but it allows one to generate several additional algorithms, namely, those with alternate restoration and optional restoration. Then, two modifications of the sequential gradient-restoration algorithm are presented in an effort to accelerate terminal convergence. In the first modification, the quadratic constraint imposed on the variations of the control is modified by the inclusion of a positive-definite weighting matrix (the matrix of the second derivatives of the Hamiltonian with respect to the control). The second modification is a conjugate-gradient extension of the sequential gradient-restoration algorithm. Next, the addition of a nondifferential constraint, to be satisfied everywhere along the interval of integration, is considered. In theory, this seems to be only a minor modification of the basic problem. In practice, the change is considerable in that it enlarges dramatically the number and variety of problems of optimal control which can be treated by gradient-restoration algorithms. Indeed, by suitable transformations, almost every known problem of optimal control theory can be brought into this scheme. This statement applies, for instance, to the following situations: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with equality constraints on the time rate of change of the state, (iv) problems with control inequality constraints, (v) problems with state inequality constraints, and (vi) problems with inequality constraints on the time rate of change of the state. Finally, the simultaneous presence of nondifferential constraints and multiple subarcs is considered. The possibility that the analytical form of the functions under consideration might change from one subarc to another is taken into account. The resulting formulation is particularly relevant to those problems of optimal control involving bounds on the control or the state or the time derivative of the state. For these problems, one might be unwilling to accept the simplistic view of a continuous extremal arc. Indeed, one might want to take the more realistic view of an extremal arc composed of several subarcs, some internal to the boundary being considered and some lying on the boundary. The paper ends with a section dealing with transformation techniques. This section illustrates several analytical devices by means of which a great number of problems of optimal control can be reduced to one of the formulations presented here. In particular, the following topics are treated: (i) time normalization, (ii) free initial state, (iii) bounded control, and (iv) bounded state.

Notes: Abstract In a recent paper by Scott and Jefferson, the optimal control of the service rate for a single-server queue with limited waiting space is treated by the maximum principle. We show that their control policies are necessarily suboptimal. Characterizations for optimal control are derived and used to obtain corresponding optimal trajectories in both nonsingular and singular regions.

Notes: Abstract Three-dimensional minimum-time 180° turns of a fighter aircraft are computed for several initial velocitiesV 0 and altitudesh 0. It is shown that the optimum turns consist of split -S maneuvers forV 0≦V 10, three-dimensional maneuvers forV 10〈V 0〈V 20, and half-loops forV 0≧V 20, withV 10,V 20 being functions of altitude or thrust/weight ratio.

Notes: Abstract In this paper, two linear-quadratic systems are shown to be asymptotically decoupled. We obtain the steady-state term and the first-order transient term of optimal controls. We also obtain an estimate of the rate at which the first-order transient term diminishes and approaches zero.

Notes: Abstract This paper provides a dynamic programming approach to the maximum principle for the optimal control of systems with distributed parameters. The process of the systems under consideration is governed by a partial differential equation.

Notes: Abstract An algorithm for finding agood solution for a multiple criteria optimal control problem is given. The criteria are assumed to be ordered according to their importance to the decision-maker. The algorithm consists of successive solutions of single criterion optimal control problems. Other criteria are taken into account by adding constraints to the problem in a systematic manner.

Notes: Abstract This paper describes a collection of parallel optimal control algorithms which are suitable for implementation on an advanced computer with the facility for large-scale parallel processing. Specifically, a parallel nongradient algorithm and a parallel variablemetric algorithm are used to search for the initial costate vector that defines the solution to the optimal control problem. To avoid the computational problems sometimes associated with simultaneous forward integration of both the state and costate equations, a parallel shooting procedure based upon partitioning of the integration interval is considered. To further speed computations, parallel integration methods are proposed. Application of this all-parallel procedure to a forced Van der Pol system indicates that convergence time is significantly less than that required by highly efficient serial procedures.

Notes: Abstract When dealing with the time-optimal problem for linear control systems, there may be a difference between optimal open-loop control and corresponding synthesized feedback control, since in the latter case one is led to allow for generalized (Filippov) solutions. In this note, it is shown that the set of two-dimensional linear control systems with a convex polyhedron as control domain, which exhibit such paradoxical behavior (completely characterized by Brunovský), has a nonempty interior, in a natural and appropriately defined topology on the space of all such linear control systems.

Notes: Abstract Consideration is given to continuous-time, parameter-dependent optimal control problems with state-variable jump discontinuities atN variable interior times. A maximum principle involving known costate jump conditions is stated and is proved by transforming the problem into a standard Mayer control problem. An illustrative example for fisheries management is included.

Notes: Abstract Elliptic control problems with a quadratic cost functional require the solution of a system of two elliptic boundary-value problems. We propose a fast iterative process for the numerical solution of this problem. The method can be applied to very special problems (for example, Poisson equation for a rectangle) as well as to general equations (arbitrary dimensions, general region). Also, nonlinear problems can be treated. The work required is proportional to the work taken by the numerical solution of a single elliptic equation.

Notes: Abstract To find the optimal control of chemical processes, Pontryagin's minimum principle can be used. In practice, however, one is not only interested in the optimal solution, which satisfies the restrictions on the control, the initial and terminal conditions, and the process parameters. It is also important to known how the optimal control and the minimum value of the objective function change, due to small variations in all the restrictions and the parameters. It is shown how to determine the effect of these variations directly from the optimal solution. This saves computer time, compared with the more traditional sensitivity analysis based on computing the optimal control for every single variation considered. The theory is applied to a chemical process.

Notes: Abstract The deterministic linear-system, quadratic-cost optimal control problem is considered when the only state information available is a partial linear observation of the initial statex 0. Thus, it is only known that the initial condition belongs to a particular linear variety. A control function is found which is optimal, in the sense (roughly) that (i) it can be computed using available information aboutx 0 and (ii) no other control function which can be found using that information gives lower cost than it does for every initial condition that could have given rise to the information. The optimal control can be found easily from the conventional Riccati equation of optimal control. Applications are considered in the presence of unknown exponential disturbances and to the case with a sequence of partial state observations.

Notes: Abstract In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the gradient phase. It is shown that the Lagrange multipliers associated with the gradient phase not only solve the auxiliary minimization problem of the gradient phase, but are also endowed with a supplementary optimality property: they minimize the error in the optimality conditions, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.

Notes: Abstract In this paper, we study the approximation by the penalty method of a control problem governed by a pseudo-parabolic equation with a noncoercive control functional and with control and state constraints. The existence of solutions to the penalized problems is established. In addition, the convergence of the penalized problems to the solution, the Lagrange multipliers, and the minimum value of the original problem is studied. The results apply to Sobolev and parabolic equations as well.

Notes: Abstract In Refs. 1–3, existence results have been obtained for optimal control problems whose state equations are described by certain nonlinear integral equations of Urysohn type. We generalize and synthesize these results by formulating a general lower closure result from which the results of Refs. 1–3 are shown to follow. In the course of this, we also present a novel and rather abstract treatment of existence problems for variable-time optimal control, quite in the spirit of Ref. 4.

Notes: Abstract In this paper, the Nerlove-Arrow model of optimal dynamic advertising policies is generalized by assuming a general probability distribution of the “forgetting time,” rather than the exponential one. A control problem with integrodifferential equations of motion is defined for which the transitory and steady-state properties of the optimal advertising policy are examined. The effects of assumptions like IHR-distributions and DHR-distributions, the existence of an upper bound for the forgetting time, etc., are explained. It is shown that there are two (in the case of an exponential distribution even three) different current-value adjoint functions associated with the problem, and relations between the two (three) are established. Also provided is a sensitivity analysis.

Notes: Abstract The purpose of this paper is to draw a detailed comparison between Newton's method, as applied to discrete-time, unconstrained optimal control problems, and the second-order method known as differential dynamic programming (DDP). The main outcomes of the comparison are: (i) DDP does not coincide with Newton's method, but (ii) the methods are close enough that they have the same convergence rate, namely, quadratic. The comparison also reveals some other facts of theoretical and computational interest. For example, the methods differ only in that Newton's method operates on a linear approximation of the state at a certain point at which DDP operates on the exact value. This would suggest that DDP ought to be more accurate, an anticipation borne out in our computational example. Also, the positive definiteness of the Hessian of the objective function is easy to check within the framework of DDP. This enables one to propose a modification of DDP, so that a descent direction is produced at each iteration, regardless of the Hessian.

Notes: Abstract Using a recent result of Castaing and Clauzure on the lower semicontinuity of integral functionals, we prove the existence of an optimal control for a broad class of nonlinear infinite-dimensional control systems. An example of a distributed parameter system is worked in detail.

Notes: Abstract This paper presents two demonstrably convergent, first-order, differential dynamic programming algorithms for the solution of optimal control problems with constraints on the control, but without constraints on the trajectory or the terminal state. The second of these algorithms can be used on more difficult problems than the first one, but it is correspondingly more complex.

Notes: Abstract In this paper, we present a convergent extension of the first-order strong-variational algorithm by Mayne and Polak (Ref. 1) for solving optimal control problems with control constraints to delay systems. Although the algorithm is similar to the one presented in Ref. 1, the proof of convergence is different, since the differential dynamic techniques used by Mayne and Polak are not applicable.

Notes: Abstract This paper is concerned with optimal flight trajectories in the presence of windshear. The penetration landing problem is considered with reference to flight in a vertical plane, governed by either one control (the angle of attack, if the power setting is predetermined) or two controls (the angle of attack and the power setting). Inequality constraints are imposed on the angle of attack, the power setting, and their time derivatives. The performance index being minimized measures the deviation of the flight trajectory from a nominal trajectory. In turn, the nominal trajectory includes two parts: the approach part, in which the slope is constant; and the flare part, in which the slope is a linear function of the horizontal distance. In the optimization process, the time is free; the absolute path inclination at touchdown is specified; the touchdown velocity is subject to upper and lower bounds; and the touchdown distance is subject to upper and lower bounds. Three power setting schemes are investigated: (S1) maximum power setting; (S2) constant power setting; and (S3) control power setting. In Scheme (S1), it is assumed that, immediately after the windshear onset, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant; in this scheme, the only control is the angle of attack. In Scheme (S2), it is assumed that the power setting is held at a constant value, equal to the prewindshear value; in this scheme, the only control is the angle of attack. In Scheme (S3), the power setting is regarded as a control, just as the angle of attack. Under the above conditions, the optimal control problem is solved by means of the primal sequential gradient-restoration algorithm (PSGRA). Numerical results are obtained for several combinations of windshear intensities and initial altitudes. The main conclusions are given below with reference to strong-to-severe windshears. In Scheme (S1), the touchdown requirements can be satisfied for relatively low initial altitudes, while they cannot be satisfied for relatively high initial altitudes; the major inconvenient is excess of velocity at touchdown. In Scheme (S2), the touchdown requirements cannot be satisfied, regardless of the initial altitude; the major inconvenient is defect of horizontal distance at touchdown. In Scheme (S3), the touchdown requirements can be satisfied, and the optimal trajectories exhibit the following characteristics: (i) the angle of attack has an initial decrease, which is followed by a gradual, sustained increase; the largest value of the angle of attack is attained near the end of the shear; in the aftershear region, the angle of attack decreases gradually; (ii) initially, the power setting increases rapidly until maximum power setting is reached; then, maximum power setting is maintained in the shear region; in the aftershear region, the power setting decreases gradually; (iii) the relative velocity decreases in the shear region and increases in the aftershear region; the point of minimum velocity occurs at the end of the shear; and (iv) depending on the windshear intensity and the initial altitude, the deviations of the flight trajectory from the nominal trajectory can be considerable in the shear region; however, these deviations become small in the aftershear region, and the optimal flight trajectory recovers the nominal trajectory. A comparison is shown between the optimal trajectories of Scheme (S3) and the trajectories arising from alternative guidance schemes, such as fixed controls (fixed angle of attack, coupled with fixed power setting) and autoland (angle of attack controlled via path inclination signals, coupled with power setting controlled via velocity signals). The superiority of the optimal trajectories of Scheme (S3) is shown in terms of the ability to meet the path inclination, velocity, and distance requirements at touchdown. Therefore, it is felt that guidance schemes based on the properties of the optimal trajectories of Scheme (S3) should prove to be superior to alternative guidance schemes, such as the fixed control guidance scheme and the autoland guidance scheme.

Notes: Abstract In this paper, we consider a class of infinite-horizon discounted optimal control problems with nonsmooth problem data. A maximum principle in terms of differential inclusions with a Michel type transversality condition is given. It is shown that, when the discount rate is sufficiently large, the problem admits normal multipliers and a strong transversality condition holds. A relationship between dynamic programming and the maximum principle is also given.

Notes: Abstract The problem of control in the presence of unknown but limited disturbance for a discrete-time linear system with polyhedral input and state bounds is investigated. Two problems are considered: that of reaching an assigned target set in the state space; and that of keeping the state in a given region using the available controls. In both cases, a solution is given via linear programming. A computational procedure for the control synthesis is proposed which can be implemented to obtain a feedback control.

Notes: Abstract This paper addresses a finite-horizon profit maximization three-machine replacement problem. More precisely, a model is formulated allowing for preventive maintenance to slow down machine quality and profit reduction caused by obsolescence, to determine the timing of replacing an existing machine by another available machine with improved technology. This decision is considered under uncertainty regarding the introduction time of a machine with a not-yetachieved technology. Given an exponential probability distribution function of the introduction time, the optimality of a bang-bang nonincreasing preventive maintenance control is shown. Moreover, subproblems maximizing the expected discounted profit are analyzed. Closed-form solutions are provided to compare machines of different technologies and to derive an analytical sensitivity analysis concerned with many issues related to the problem. The results are not necessarily intuitive and simple. For example, different relationships between the planning horizon and the preventive maintenance switching time are presented for the three-machine problem versus the single-machine problem. The focus of this paper is on the formulation and the analytical analysis of the problem rather than on its computational aspects.

Notes: Abstract For the ergodic control problem with degenerate diffusions, the existence of an optimal solution is established for various interesting classes of solutions.

Notes: Abstract A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.

Notes: Abstract This paper deals with the problem of designing multirate-output contrlleers for sampled-dataH ∞-optimal control of linear continuous-time systems. Two formulations of the problem are studied. In the first, the intersample behavior of the disturbance and the controlled output signals is not considered, whereas in the second the continuous-time nature of these signals is taken into account. It is shown that, in both cases and unter appropriate conditions, it is plausible to reduce the repective initial problem to an associated discrete-timeH ∞-optimization problem for which a fictitious static state feedback controller is to be designed. This fact has a beneficial influence on the theoretical and numerical complexity of the problem, since only one algebraic Riccati equation is to be solved here, as compared to two algebraic Riccati equations needed in known techniques concerning theH ∞-optimization problem with dynamic measurement feedback.

Notes: Abstract We present necessary conditions of optimality for an infinitehorizon optimal control problem. The transversality condition is derived with the help of stability theory and is formulated in terms of the Lyapunov exponents of solutions to the adjoint equation. A problem without an exponential factor in the integral functional is considered. Necessary and sufficient conditions of optimality are proved for linear quadratic problems with conelike control constraints.

Notes: Abstract We study polynomial ordinary differential systems $$\dot M(t) = QM - M(M'QM){\text{, }}M(0) = M_0 ,t \geqslant 0,$$ whereQ≥0 is an n×n matrix and M(t) is an n×k matrix. It is proven that, as t grows to infinity, the solution M(t) tends to a limit BU, where U is a k×k orthogonal matrix and B is an n×k matrix whose columns are k pairwise orthogonal, normalized eigenvectors of Q. Moreover, for almost every M 0, these eigenvectors correspond to the k maximal eigenvalues of Q; for an arbitrary Q with independent columns, we provide a procedure of computing B by employing elementary matrix operations on M 0. This result is significant for the study of certain neural network systems, and in this context it shows that M(∞) provides a principal component analyzer.

Notes: Abstract We consider a general optimization problem which is an abstract formulation of a broad class of state-constrained optimal control problems in relaxed form. We describe a generalized mixed Frank–Wolfe penalty method for solving the problem and prove that, under appropriate assumptions, accumulation points of sequences constructed by this method satisfy the necessary conditions for optimality. The method is then applied to relaxed optimal control problems involving lumped as well as distributed parameter systems. Numerical examples are given.

Notes: Abstract An optimal control problem with four linear controls describing a sophisticated concern model is investigated. The numerical solution of this problem by combination of a direct collocation and an indirect multiple shooting method is presented and discussed. The approximation provided by the direct method is used to estimate the switching structure caused by the four controls occurring linearly. The optimal controls have bang-bang subarcs as well as constrained and singular subarcs. The derivation of necessary conditions from optimal control theory is aimed at the subsequent application of an indirect multiple shooting method but is also interesting from a mathematical point of view. Due to the linear occurrence of the controls, the minimum principle leads to a linear programming problem. Therefore, the Karush–Kuhn–Tucker conditions can be used for an optimality check of the solution obtained by the indirect method.

Notes: Abstract In this paper, we develop a new method to approximate the solution to the Hamilton–Jacobi–Bellman (HJB) equation which arises in optimal control when the plant is modeled by nonlinear dynamics. The approximation is comprised of two steps. First, successive approximation is used to reduce the HJB equation to a sequence of linear partial differential equations. These equations are then approximated via the Galerkin spectral method. The resulting algorithm has several important advantages over previously reported methods. Namely, the resulting control is in feedback form and its associated region of attraction is well defined. In addition, all computations are performed off-line and the control can be made arbitrarily close to optimal. Accordingly, this paper presents a new tool for designing nonlinear control systems that adhere to a prescribed integral performance criterion.

Notes: Abstract In this paper, using a Razumikhin-type approach, the stabilization of a class of uncertain nonlinear systems with time-varying delay is considered. The proposed controller is based on a specific optimal control problem. Global asymptotic stability is guaranteed for the proposed control if some algebraic condition is met. An example illustrates the use of the main result.

Notes: Abstract This paper deals with the optimal control of a one-machine two-product manufacturing system with setup changes, operating in a continuous time dynamic environment. The system is deterministic. When production is switched from one product to the other, a known constant setup time and a setup cost are incurred. Each product has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a feedback control problem. The objective is to minimize the total backlog, inventory and setup costs incurred over a finite horizon. The optimal solution provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady state, the optimal cyclic schedule is determined. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region, the optimal control policy is determined analytically.

Notes: Abstract The history of atmosheric mercury inputs to remote arctic regions can be measured in lake sediment cores using lead-210 chronology. In this investigation, total mercury deposition is measured in sediments from Imitavik and Annak Lakes on the Belcher Islands in southeastern Hudson Bay, an area in the southern Canadian Arctic with no history of local industrial or agricultural sources of contamination. Both lakes received background and atmospheric inputs of mercury while Annak also received mercury from raw domestic sewage from the Hamlet of Sanikiluaq, a growing Inuit community of about 550 established in the late 1960's. Results from Imitavik show that anthropogenic mercury inputs, apparently transported through the atmosphere, began to appear in the mid-eighteenth century, and continued to the 1990's. Annak had a similar mercury history until the late 1960's when disposal of domestic sewage led to increased sediment and contaminant accumulation. The high input of mercury to Annak confirms that Sanikiluaq residents are exposed to mercury through native food sources.

Notes: Abstract A modeling system has been developed to simulate regional environmental exposure to mercury due to atmospheric deposition of mercury to watersheds. The atmospheric fate and transport of mercury is simulated using a comprehensive three-dimensional Eulerian model, the Trace Element Atmospheric Model (TEAM). The aquatic chemistry and bioaccumulation of mercury in fish are simulated using a model of mercury cycling in a lake/watershed system, the Regional Mercury Cycling Model (R-MCM). Fish consumption was derived from a review of available surveys. Previous work focused on an assessment of the environmental and inter-individual variability in key input data (Seigneur et al., 1997a). We address here the uncertainties associated with critical model input variables (e.g., atmospheric deposition velocities, precipitation rate, limnological characteristics). A probabilistic assessment is conducted to propagate the uncertainties in the input data through the modeling system and develop a probability distribution of the human mercury dose that reflects these uncertainties. The standard deviation of the distribution of the calculated human dose is about 50% of the mean value. For the example considered here (i.e., Park Lake in Michigan, U.S.A.), 80% of the uncertainty in the human dose was due to uncertainties in the speciation of mercury air emissions, pH and temperature of the lake, burial velocity of the sediments, and rate of fish consumption.