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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Bulletin of mathematical biology 39 (1977), S. 317-337 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper uses the Gompertzian model for the growth of a cancer cell population subject to losses due to the action of cycle nonspecific therapy for the determination of a chemotherapy program obtained from optimal control theory. Application of the analysis to control of the bone cancer IgG multiple myeloma is presented. The program obtained from optimal control theory is compared with clinical results.
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 3 (1969), S. 263-281 
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Bolza problem of the calculus of variations in modern control notation is extended in scope to include situations in which a number of subarcs occur in a variety of ways. The subarcs are allowed to be overlapping and/or separated. This allows for several subarcs to occur in the same interval of the independent variable and also admits subarcs which are separated by jumps in the independent and state variables. In addition, the differential constraining equations and the integral quantity to be extremized are permitted to have different form from subarc to subarc. The necessary conditions for the extended Bolza problem are obtained by examining a related functional. Whereas the optimizing conditions for the state and control variables for each subarc are given by the usual Euler equations, new conditions associated with the end points of the subarcs are derived using ordinary theory of maxima and minima. The results presented here can be applied to a wide range of space trajectory problems. For some special cases, the theory reduces to results previously obtained and recorded elsewhere. A number of sample problems illustrating the theory are presented. The examples include the problem of inserting two payloads into separate orbits with one vehicle having two upper stages ignited simultaneously and a two-vehicle, dual-rendezvous problem.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 6 (1970), S. 299-319 
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A complete set of necessary and sufficient conditions for selecting optimal endpoints for extremals obtained from the variational Bolza problem in control notation has been developed. The method used to obtain these conditions is based on a seldom used concept of performing a dichotomy on the general optimization problem. With this concept, the problem of Bolza is decomposed into two problems, the first of which involves the selection of optimal paths with the endpoints considered fixed. The second problem involves the selection of optimal endpoints with the paths between the endpoints taken to be stationary curves. The convenience of the dichotomy in deriving the necessary and sufficient conditions for endpoints lies in its simplicity and elementary character; well-known necessary and sufficient conditions from the theory of ordinary maxima and minima are used. An endpoint necessary condition is first obtained which is simply the well-known transversality condition. An additional condition is then developed which, together with the transversality condition, leads to a set of necessary and sufficient conditions for a given extremal to be locally optimal with respect to endpoint variations. While the second condition presented is akin to the classical focal-point condition, the result is new in form and is directly applicable to the optimal control problem. In addition, it is relatively simple to apply and is easy to implement numerically when an analytical solution is not possible. It should be useful in situations where the transversality conditions yield more than one choice for an optimal endpoint. An analytic solution for a simple geodetics problem is presented to illustrate the theory. A discussion of numerical implementation of the sufficiency conditions and its application to an orbit transfer example is also included.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Dynamics and control 1 (1991), S. 35-52 
    ISSN: 1573-8450
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract The Lorenz equations are well known for their ability to produce chaotic motion. We investigate here the Lorenz system subject to a control input. Two different controllers are the designed for this system, one based on linear methods and one based on nonlinear methods. The objective of the controller is to drive the system to one of the unstable equilibrium points associated with uncontrolled chaotic motion. Each controller is able to produce stable motion. However, the character of this motion may differ considerably, depending on adjustment of “gains” used in the controller. In particular, the motion may contain chaotic transients. It is possible to create a system with intermediate-term-senstive dependence on initial conditions, but with no such long-term dependence.
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  • 5
    ISSN: 1573-8450
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract Classical control theory has always been concerned with uncertain inputs. However, this concern has generally been implicit rather than explicit. Modern control methods tend to be more explicit. In particular, methods based on game theory and methods based on a Lyapunov type of analysis have been proposed. One question which seems to be largely ignored is: how do these various methods compare when applied to a typical control problem? Indeed what performance measures can we use to compare various control designs dealing with uncertain inputs? This paper addresses both of these questions. We define performance measures called the v-reachable set and the v-attractive set. The v-reachable set measures the extent that the uncertain input can drive the system away from the target set and the v-attractive set measures the extent that the uncertainty diminishes the size of the controllable set to the target. An ideal control law would be one in which the v-reachable set is the target set and the v-attractive set is the controllable set. Five different controllers based on five different design philosophies are applied to two different control systems. We find that there can be marked differences in the above performance measures. However using these performance measures as guide to choosing feedback parameters, the differences between the various design methods can be minimized. It is found that for these problems a modified game theoretic approach and the Lyapunov type of approach tend to provide the best designs.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Evolutionary ecology 1 (1987), S. 59-94 
    ISSN: 1573-8477
    Keywords: Lag load ; Red Queen ; ESS ; coevolution ; evolutionary rate ; predator coevolution ; competitor coevolution ; stasis ; punctuated equilibrium ; evolutionary constraints ; White Queen's Constraint ; Alice's Constraint ; bauplan ; fitness-generating function ; versatility ; guilds ; adaptive zones ; constraint surface ; genostasis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Summary The Red Queen principle states that a set of interacting species reaches an evolutionary equilibrium at which all their rates of coevolution exactly balance each other. The lag-load model, which is one way of searching for Red Queens, has, by itself, previously predicted that they do not exist. But this model has assumed that infinite maladaptedness is possible. The lag-load model is improved by assuming that once the lag load of all but one species is determined, so is that of the final species. This assumption eliminates the possibility of infinite maladaptedness. Its result is to allow the lag-load model to yield Red Queen coevolution. It does this whether or not speciation and extinction rates are included. Thus the lag-load model is harmonized with the earlier Red Queen model derived from studies of predation. Because of the intercorrelation of phenotypic traits, the predatory model concluded that the eventual stable rate of coevolution must be zero (except for intermittent bursts after some correlation or compromise is successfully broken). Another model that predicts stable coevolutionary rates of zero is that of evolutionarily stable strategies (ESS). Red Queen assumes that the more extreme a phenotypic trait is, the better it is, and that there are no constraints on the growth of such a phenotypic trait value. Such traits are the key to the Red Queen prediction of progressive coevolution. ESS models make no such assumptions. Eliminating unbounded traits from the model of predator-victim evolution changed its prediction from progressive coevolution to stasis. Before this paper, no model had dealt simultaneously with both unbounded and constrained traits. To handle both sorts of phenotypic traits at the same time in the same model, we abandoned lag load as a measure of evolutionary rate (lag loads do not uniquely determine phenotype). Instead, we used the traditional assumption that rate is proportional to the slope of the adaptive landscape. A model, relying on continuous evolutionary game theory, was developed and simulated under various conditions in two or three species sets, with up to five independent traits coevolving simultaneously. The results were: (1) there was always a set of equilibrium densities eventually achieved by coevolution; if the population interaction represented by this stable coevolutionary state is also stable, then the system should persist whether it evolves further or not; (2) whenever traits were present which were unbounded and best at their most extreme values, then a Red Queen emerged; (3) whenever traits were present which were correlated with each other or constrained below infinity, then an ESS emerged; (4) if both types were present, both results occurred: Red Queen in the unbounded traits and ESS in the constrained ones. Because unbounded traits may not exist, the Red Queen may have no domain. But the domain of ESS is real. ESS should lead to the evolutionary pattern called punctuated equilibrium. The changes in design rules which punctuate stasis should lead to an ever-expanding independence of traits from each other, i.e. to more and more refined differentiation. A single set of design rules which governs a set of species is called a fitness-generating function. Such functions may help to define the concepts of adaptive zone and ecological guild.
    Type of Medium: Electronic Resource
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  • 7
  • 8
    Publication Date: 1996-05-01
    Print ISSN: 0022-5193
    Electronic ISSN: 1095-8541
    Topics: Biology
    Published by Elsevier
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  • 9
    Publication Date: 1970-10-01
    Print ISSN: 0022-3239
    Electronic ISSN: 1573-2878
    Topics: Mathematics
    Published by Springer
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  • 10
    Publication Date: 1969-04-01
    Print ISSN: 0022-3239
    Electronic ISSN: 1573-2878
    Topics: Mathematics
    Published by Springer
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