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  • numerical methods  (12)
  • Collagen  (11)
  • Springer  (23)
  • American Association for the Advancement of Science
  • 1990-1994
  • 1970-1974  (23)
  • 1950-1954
  • 1974  (23)
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  • Springer  (23)
  • American Association for the Advancement of Science
Years
  • 1990-1994
  • 1970-1974  (23)
  • 1950-1954
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  • 1
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    Calcified tissue international 14 (1974), S. 327-331 
    ISSN: 1432-0827
    Keywords: Bone ; Piezoelectricity ; Collagen ; Aging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract The piezoelectric constant of mature and immature bone (defined herein) has been measured in an effort to determine whether it varies with age. It was found that the average value of the piezoelectric constantd 14 of femur from three week old calves was 58% of the value of femur from three year old bulls. The results were interpreted to indicate qualitative differences in the corresponding collagen matrices. Mature human tibia from males ranging in age from 21 to 53 years of age showed a small but significant increase ind 14 with age. Some data concerning diseased human bone, and well-preserved human bone excavated in Peru are also presented.
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  • 2
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    Calcified tissue international 15 (1974), S. 325-328 
    ISSN: 1432-0827
    Keywords: Cementum ; Collagen
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract A method for the isolation of dental cementum is described. Amino acid analysis of the insoluble matrix of cementum, and chromatography of the CNBr peptides derived therefrom, strongly suggest that this material is mainly collagen with the chain composition α1[(I)]2α2.
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  • 3
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    Calcified tissue international 16 (1974), S. 109-127 
    ISSN: 1432-0827
    Keywords: Dentinogenesis ; Diphosphonates ; Calcification ; Collagen ; Electron Microscopy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract The effects of high doses of ethylene-1-hydroxy-1,1-diphosphonate (EHDP) were investigated at the light microscopic and subcellular level. The administration of EHDP at a concentration of 7.5–10 mg P/kg body weight/day over a short period of time resulted in complete inhibition of crystal formation in predentin and pre-enamel. An increased predentin width was observed and within newly-formed predentin areas the formation ofcollagen fibrils was grossly disturbed. In addition, fine precipitates appeared in the ground substance. The presence of unusual thread-like elements within specific bodies in the cytoplasm of the odontoblastic processes may be indicative of an interference by EHDP in e.g. the synthesis of precollagen. The possibility of an inhibition by EHDP of the extracellular aggregation of collagen molecules is also discussed. EHDP further inhibited crystal formation within dentinal globules. Functioning ameloblasts were grossly affected in EHDP-treated rats, and it is suggested that this is related to an inhibition of crystal formation in pre-enamel.
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  • 4
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    Calcified tissue international 16 (1974), S. 153-155 
    ISSN: 1432-0827
    Keywords: Galactose ; Glucose-6-phosphate dehydrogenase ; Collagen ; Galactose-1-phosphate
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract The bone tissue levels of galactose-1-P have been found to be a function of the concentration of galactose in blood.In vitro, the glucose-6-phosphate dehydrogenase (bone isoenzyme) was competitively inhibited by galactose-1-P. Galactose or galactitol were found ineffective. Therefore, the previously reported depressed activity of bone glucose-6-phosphate dehydrogenase in galactose-fed rats should be interpreted as mediated by galactose-1-P. When compared with their controls, galactose-fed rats showed a significantly lower growth rate which was reflected in a lower bone collagen content and in a lower percentage of salt soluble collagen.
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  • 5
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    Calcified tissue international 16 (1974), S. 209-217 
    ISSN: 1432-0827
    Keywords: Mucopolysaccharides ; Collagen ; Calvarium ; Development
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract Mucopolysaccharides (MPS) and collagen have been analyzed in different types of cartilage and some other tissues at various stages of prenatal and postnatal human development. The concentration of the chondroitin sulfate-type MPS increased from the 38th embryonal day, the 6-sulfate exceeding in amount the 4-sulfate. The increase in collagen concentration was accompanied by a decrease in the proportion of neutral salt-soluble collagen. There was no accumulation of chondroitin sulfates in the calvaria. A maximal concentration of MPS of a low molecular weight occurred in the skin and in the ear pinna after the 10th prenatal week. Postnatally, the MPS-concentration in the iliac crest and in the tibial articular cartilages decreased gradually, the highest values being reached in the newborn; the collagen-concentration showed an approximately reciprocal course.
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  • 6
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    Calcified tissue international 16 (1974), S. 239-250 
    ISSN: 1432-0827
    Keywords: EHDP ; Bone ; Collagen ; Histology ; Bone chemical analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract The purpose of this study was to determine the short-term effects of various systemic doses of disodium ethane-1-hydroxy-1,1-diphosphonate (EHDP) on bone organic matrix and to relate these effects to the corresponding dose-related changes in bone mineral. EHDP was administered daily by subcutaneous injection at doses of 0.25, 2.5 and 40 mg/kg body weight for periods of one and two weeks. At both time intervals, rat tibiae were quantitatively analyzed for mineral content (ash, calcium and phosphorus) and for organic matrix content (matrix weight, nitrogen and certain amino acids). The latter data were correlated with semiquantitative histological analyses of the tibiae. Results of this study demonstrate that the short term effects of EHDP on bone chemistry and histology are variable and depend on the systemic dose and the duration of treatment. Systemic doses of 0.25 and 2.5 mg/kg EHDP following daily administration for one week resulted in transitory decreases in bone mineral content compared to controls. Following two weeks of treatment, both of these dose levels resulted in increased bone mineral content and, in addition, the 2.5 mg/kg dose resulted in tibiae which contained more organic matrix compared to control bones. In contrast to the low dose effects, a high systemic dose of EHDP—e.g. 40 mg/kg administered daily for 1 or 2 weeks—appears to act solely by inhibiting mineralization of newly-formed matrix.
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  • 7
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    Calcified tissue international 14 (1974), S. 31-48 
    ISSN: 1432-0827
    Keywords: Bone ; Cartilage ; Calcification ; Collagen ; Phosphate
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé Une méthode pour l'étude de la croissance des os longs de foetus de rat, en culture d'organe, dans un milieu chimiquement défini, a été mise au point. Les extrémités cartilagineuses et les parties centrales de l'os sont analysées séparément pour leur croissance et minéralisation en étudiant leur contenu en collagène, calcium et phosphate, poids sec, et incorporation de proline marquée en hydroxyproline. La croissance et la minéralisation des parties centrales osseuses sont plus lentes dans un milieu chimiquement défini qu'in vivo. La croissance peut être accélérée en ajoutant au milieu des acides aminés non essentiels, de l'albumine ou du sérum. Les extrémités cartilagineuses présentent une augmentation plus importante en poids et contenu en collagène que les parties centrales et l'adjonction de diverses substances a moins d'effet sur la croissance. La croissance et la minéralisation des parties centrales sont augmentées en élevant la concentration du milieu en phosphate de 1.5 à 4.5 mM, avec ou sans adjonction de sérum ou d'albumine. A une concentration faible de calcium (0.5 mM), la croissance et la minéralisation des parties centrales sont arrêtées. A une concentration faible en magnésium (0.5 mM), la minéralisation est augmentée, mais la croissance est arrêtée.
    Abstract: Zusammenfassung Es wird eine Methode beschrieben, mit welcher das Wachstum der Röhrenknochen von Rattenembryos in einem chemisch bestimmten Medium in Organkultur untersucht werden kann. Die Knorpelenden und Knochenschäfte wurden gesondert auf Wachstum und Mineralisation geprüft, indem Collagen-, Calcium- und Phosphatgehalt, das Trockengewicht und der Einbau von markiertem Prolin in Hydroxyprolin gemessen wurden. Wachstum und Mineralisation des Knochenschaftes waren langsamer in einem chemisch bestimmten Medium als in vivo. Das Wachstum konnte beschleunigt werden, indem dem Medium nicht-essentielle Aminosäuren, Albumin oder Serum beigegeben wurden. Die Knorpelenden zeigten eine viel stärkere Zunahme an Gewicht und Collagengehalt als die Schäfte, und Anreicherung des Mediums hatte weniger Wirkung auf ihr Wachstum. Das Wachstum und die Mineralisation der Knochenschäfte nahmen zu, wenn die Phosphatkonzentration im Medium zwischen 1,5 und 4,5 mM erhöht wurde, und zwar unabhängig davon, ob dem Medium Serum oder Albumin beigegeben wurde oder nicht. Bei niederer Calciumkonzentration (0,5 mM) im Medium wurden Wachstum und Mineralisation der Knochenschäfte beeinträchtigt. Bei niedriger Magnesiumkonzentration (0,5 mM) wurden die Mineralisation erhöht, das Wachstum hingegen gehemmt.
    Notes: Abstract A method for studying the growth of fetal rat long bones in a chemically defined medium in organ culture is described. Cartilage ends and bone shafts were analyzed separately for growth and mineralization by measuring the collagen, calcium, and phosphate content, dry weight, and incorporation of labeled proline into hydroxyproline. Growth and mineralization of the bone shaft were slower in a chemically defined medium thanin vivo. Growth could be enhanced by supplementation of the medium with non-essential amino acids, albumin or serum. Cartilage ends showed a greater increase in weight and collagen content than the shafts, and medium supplements had less effect on their growth. Bone shaft growth and mineralization were enhanced by increasing medium phosphate concentration over a range of 1.5 to 4.5 mM whether or not the medium was supplemented with serum or albumin. At a low medium calcium concentration (0.5 mM) bone shaft growth and mineralization were impaired. At a low magnesium concentration (0.5 mM) mineralization was enhanced, but growth was impaired.
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  • 8
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    Calcified tissue international 14 (1974), S. 177-194 
    ISSN: 1432-0827
    Keywords: Collagen ; Dentine ; Lathyrism ; Cross-linking
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Description / Table of Contents: Résumé Le collagène de l'os et des tissus mous des animaux lathyrique est plus soluble que celui d'animaux normaux, en utilisant des méthodes d'extraction classiques. Pour vérifier s'il existe und phénomène similaire au niveau du collagène dentinaire lathyrique, la solubilitè due collagène matricielle dentinaire est étudiée dans deux solvants. De jeunes rats mâles adultes sont soumis à un régime lathyrogénique pendant 50 jours, période pendant laquelle la majeure partie du collagène dentinaure des incisives semble être formée. Le collagène dentinaire décalcifié est préparé par dialyse d'incisives nettoyées avec de l'acide acétique 0.5M. et la matrice non-dialysable est extraite par le même solvant, suivi d'hydrochlorure de guanidine 5M. Le collagène dentinaure des animaux et lathyriques est insoluble dans l'acide acétique 0.5M: l'extraction par l'hydrochlorure de guanidine solubilise environ 20% du collagène lathyrique, ainsi qu'une certaire quantité de protéine non-callagénique, contrairement au collagène de l'os et des tissus mous d'animaux lathyriques, où de plus grandes quantités de collagène peuvent être solubilisées par ce procédé. Le fractionnement de l'extrait par hydrochlorure de guanidine des animaux lathyriques à l'aide de la chromatographie à cellulose de carboxyméthyle donne un rapport anormalement élevé des composés α1 et α2, qui est nettement plus grand que le rapport 2∶1 de l'os de rat Lythyrique. Ces résultats montrent des différences nettes entre les collagènes osseux et dentinaire.
    Abstract: Zusammenfassung Es ist bekannt, daß das Collagen in Knochen und Weichgeweben von lathyritischen Tieren besser löslich ist als dasjenige normaler Tiere, wenn traditionalle Extraktionsmethoden verwendet werden. Um festzustellen, ob sich das lathyritische Dentincollagen ähnlich verhalte, wurde die Löslichkeit des Collagens der Dentinmatrix in zwei Lösungsmitteln untersucht. Junge erwachsene männliche Ratten erhielten während 50 Tagen eine lathyrogene Nahrung; es konnte nachgewiesen werden, daß während dieser Zeit das meiste Collagen der Dentin-Matrix der Schneidezähne völlig umgebaut wurde. Calciumfreies Dentinmatrix-Collagen wurde erhalten, indem gereinigte Schneidezähne ausgiebig mit 0,5 M Essigsäure dialysiert wurden; die nicht-dialysierbare Matrix wurde weiter mit demselben Lösungsmittel und dann mit 5M Guanidinhydrochlorid extrahiert. Das Collagen der Dentinmatrix von normalen und lathyritischen Tieren war in 0,5 M Essigsäure unlöslich; bei der Extraktion mit Guanidinhydrochlorid lösten sich etwa 20% des lathyritischen Matrix-Collagens sowie ein Teil nicht-collagenen Proteins, im Gegensatz zum Collagen der Weichgewebe und Knochen von lathyritischen Tieren, bei welchen mit diesem Verfahren größere Mengen von Collagen gelöst werden können. Die Fraktionierung des guanidinhydrochlorid-Extraktes von lathyritischen Tieren mittels Carboxymehtyl-Cellulose-Chromatographie zeigte ein unerwartet hohes Verhältnis der α1 und α2 zueinander; es war beträchtlich höher als das 2∶1-Verhältnis, das bei lathyritischen Rattenknochen nachgewiesen wurde. Diese Befunde zeigen signifikante Unterschiede zwischen Knochen- und Dentincollagen auf.
    Notes: Abstract The collagen in bone and soft tissues of lathyritic animals is known to be more soluble than that of normal animals when traditional methods of extraction are employed. To ascertain whether a similar phenomenon occures in lathyritic dentine collagen, the solubility of dentine matrix collagen was investigated in two solvents. Young adult male rats were maintained on a lathyrogenic diet for 50 days, during which time the major part of the dentine matrix collagen of the incisors was shown to have turned over completely. Decalcified dentine matrix collagen was prepared by exhaustively dialysing cleaned incisors against 0.5M acetic acid, and the non-dialysable matrix was further extracted with the same solvent, and then with 5 M guanidine hydrochloride. The dentine matrix collagen of both normal and lathyritic animals proved to be insoluble in 0.5M acetic acid; guanidine hydrochloride extraction solubilized approximately 20% of the lathyritic matrix collagen as well as a quantity of non-collagen protein, in contrast to the soft tissue and bone collagens of lathyritic animals in which greater amounts of collagen may be solubilized by this procedure. Fractionation of the guanidine hydrochloride extract from the lathyritic animals by carboxymethyl cellulose chromatography revealed an unexpectedly high ratio of α1 to α2 components considerably greater than the 2∶1 ratio demonstrated for lathyritic rat bone. These data provide evidence for significant differences between bone and dentine collagens.
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  • 9
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    Calcified tissue international 15 (1974), S. 77-80 
    ISSN: 1432-0827
    Keywords: Osteoclasts ; Resorption ; Collagen ; Radioautography
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract Six litter mate Wistar rats were injected intraperitoneally with 2 μCi/g body weight3H-proline at the age of 15 and 20 days. Six to seven weeks later resorption of cementum, dentine and bone was caused experimentally by luxation of the first right mandibular molar. The animals were killed 10 days later in glutaraldehyde perfusion fixation and 5 μ serial sections of the mandibles were processed for radioautography. Cementum, dentine and bone showed bands of labelling corresponding to the stage of odontogenesis and osteogenesis at the time of injection. Osteoclasts were found in contact with labelled cementum, dentine and bone but the number of grains over osteoclasts was insignificant.
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  • 10
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    Calcified tissue international 15 (1974), S. 125-132 
    ISSN: 1432-0827
    Keywords: Bone ; Collagen ; Formic Acid ; Crosslinkages ; Cyanogen Bromide
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Abstract Bovine and chicken bone collagens have been solubilized and presumably denatured (gelatin) by treatment of demineralized, powdered tissue with 70% formic acid. Short periods of extraction such as four hours at 30°, conditions commonly used during cyanogen bromide cleavage of collagen, solubilized 50% and 15% of the chicken and bovine bone collagens respectively. Treatment of the tissues with sodium borohydride partially inhibited the extraction of collagen from chicken bone, but had little effect on the extraction of calf bone collagen. The heterogeneity of the bone gelatin from both species on disc electrophoretic analysis suggested that peptide bonds had been cleaved in some of the collagen chains during exposure to formic acid, thus facilitating the solubilization of the bone collagen as the gelatin. Analysis of the collagen extracted from chicken bone for reducible crosslinks indicated that a large proportion of these bonds had remained intact, in contrast to a previous finding that most of these crosslinks were destroyed in bone gelatin extracted by 4 M CaCl2, at pH 7.0. The stability of the major reducible crosslinks in bone collagen to severe acid conditions may explain in part some of its unique properties, such as its failure to swell or be solubilized in dilute acid, which distinguish it from soft tissue collagens.
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  • 11
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    Cell & tissue research 152 (1974), S. 543-550 
    ISSN: 1432-0878
    Keywords: Dentine ; Collagen ; Microfibrils ; Mineralization ; Ion beam thinning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine
    Notes: Summary Transmission electron microscopy (TEM) samples of dentine were prepared by ion beam thinning using a liquid nitrogen cooled holder, 4 to 5 kV Ar+, 70–100 μA beam. The samples were fresh, unfixed, and not subjected to mechanical deformation or dissolution. They were examined at 80 or 100 kV in the TEM, when contrasts were due entirely to the mineral component. Parallel pairs of elongated electron dense features extended for distances exceeding the collagen D period, appearing as electron dense circles in “transverse section”. These images are interpreted as indicating the deposition of mineral around collagen microfibrils. The repeat distance between close packed pairs of dark lines ranged from 38 Å upwards. There was little regular packing of the electron dense circles (or dots) seen in “transverse section”.
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  • 12
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    Journal of optimization theory and applications 13 (1974), S. 74-93 
    ISSN: 1573-2878
    Keywords: Stochastic control problems ; numerical methods ; perturbation methods ; suboptimal control ; closed-loop control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A numerical technique is described for solving approximately certain small-noise stochastic control problems. The method uses quantities computable from the optimal solution to the corresponding deterministic control problem. Numerical results are given for a two-dimensional linear regular problem with saturation and a time-optimal problem.
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  • 13
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    Journal of optimization theory and applications 13 (1974), S. 179-185 
    ISSN: 1573-2878
    Keywords: Optimal control theory ; functional differential equations ; numerical methods ; hereditary processes ; contraction mapping principle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This note considers the optimal control of a system represented by nonlinear differential-integral equations with a general cost function. A second-order iterative method of solution based on the fixed-point contraction mapping principle is proposed.
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  • 14
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    Journal of optimization theory and applications 13 (1974), S. 164-178 
    ISSN: 1573-2878
    Keywords: Two-point boundary-value problems ; calculus of variations ; numerical methods ; differential equations ; computing methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the parameter variation method, a scalar parameterk, kε[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method. The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables).
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  • 15
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    Journal of optimization theory and applications 13 (1974), S. 635-659 
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; numerical methods ; unconstrained minimization ; function minimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with the problem of investigating the properties and comparing the methods of nonlinear programming. The steepest-descent method, the method of Davidon, the method of conjugate gradients, and other methods are investigated for the class of essentially nonlinear valley functions.
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  • 16
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    Journal of optimization theory and applications 14 (1974), S. 393-403 
    ISSN: 1573-2878
    Keywords: Augmented penalty function ; penalty function methods ; method of multipliers ; numerical methods ; state variable constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Three augmented penalty function algorithms are tested and compared with an ordinary penalty function algorithm for two demonstration optimal control problems. Although the augmented penalty function is quite helpful in solving control problems with terminal state constraints, the convergence can be improved significantly by providing systematic increases in the penalty constant.
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    Journal of optimization theory and applications 14 (1974), S. 529-556 
    ISSN: 1573-2878
    Keywords: Calculus of variations ; optimal control ; computing methods ; numerical methods ; boundary-value problems ; modified quasilinearization algorithm ; nondifferential constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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.
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  • 18
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    Journal of optimization theory and applications 13 (1974), S. 218-255 
    ISSN: 1573-2878
    Keywords: Calculus of variations ; optimal control ; computing methods ; numerical methods ; gradient methods ; seqential gradient-restoration algorithm ; restoration algorithm ; boundary-value problems ; bounded control problems ; bounded state problems ; nondifferential constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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. The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functionsx(t),u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, and the stepsize of the restoration phase by a one-dimensional search on the constraint errorP. If α g is the gradient stepsize and α r is the restoration stepsize, the gradient corrections are ofO(α g ) and the restoration corrections are ofO(α r α g 2). Therefore, for α g sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalÎ at the end of any complete gradient-restoration cycle is smaller than the functionalI at the beginning of the cycle. 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. 4 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) problems involving state equality constraints can be reduced to the present scheme through suitable transformations, and (iii) problems involving inequality constraints can be reduced to the present scheme through suitable transformations. The latter statement applies, for instance, to the following situations: (a) problems with bounded control, (b) problems with bounded state, (c) problems with bounded time rate of change of the state, and (d) problems where some bound is imposed on an arbitrarily prescribed function of the parameter, the control, the state, and the time rate of change of the state. Numerical examples are presented for both the fixed-final-time case and the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.
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  • 19
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    Journal of optimization theory and applications 14 (1974), S. 263-270 
    ISSN: 1573-2878
    Keywords: Two-point boundary-value problems ; differential equations ; Newton-Raphson methods ; computing methods ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A method based on matching a zero of the right-hand side of the differential equations, in a two-point boundary-value problem, to the boundary conditions is suggested. Effectiveness of the procedure is tested on three nonlinear, two-point boundary-value problems.
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  • 20
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 13 (1974), S. 303-318 
    ISSN: 1573-2878
    Keywords: Differential games ; closed-loop controls ; numerical methods ; optimal strategies ; zero-sum games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a method for generating nearoptimal closed-loop solutions to zero-sum perfect information differential games with and without the final time explicitly specified, and with and without control constraints. This near-optimal closed-loop solution is generated by periodically updating the solution to the two-point boundary-value problem obtained by the application of the necessary conditions for a saddle-point solution. The resulting updated open-loop control is then used between updating intervals. Three examples are presented to illustrate the application of this method.
    Type of Medium: Electronic Resource
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  • 21
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 13 (1974), S. 620-634 
    ISSN: 1573-2878
    Keywords: Mathematical programming ; function minimization ; method of dual matrices ; computing methods ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Ref. 2, four algorithms of dual matrices for function minimization were introduced. These algorithms are characterized by the simultaneous use of two matrices and by the property that the one-dimensional search for the optimal stepsize is not needed for convergence. For a quadratic function, these algorithms lead to the solution in at mostn+1 iterations, wheren is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at mostn+2. In this paper, the above-mentioned algorithms are tested numerically by using five nonquadratic functions. In order to investigate the effects of the stepsize on the performances of these algorithms, four schemes for the stepsize factor are employed, two corresponding to small-step processes and two corresponding to large-step processes. The numerical results show that, in spite of the wide range employed in the choice of the stepsize factor, all algorithms exhibit satisfactory convergence properties and compare favorably with the corresponding quadratically convergent algorithms using one-dimensional searches for optimal stepsizes.
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  • 22
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 13 (1974), S. 519-537 
    ISSN: 1573-2878
    Keywords: Mathematical programming ; function minimization ; method of dual matrices ; computing methods ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the method of dual matrices for the minimization of functions is introduced. The method, which is developed on the model of a quadratic function, is characterized by two matrices at each iteration. One matrix is such that a linearly independent set of directions can be generated, regardless of the stepsize employed. The other matrix is such that, at the point where the first matrix fails to yield a gradient linearly independent of all the previous gradients, it generates a displacement leading to the minimal point. Thus, the one-dimensional search is bypassed. For a quadratic function, it is proved that the minimal point is obtained in at mostn + 1 iterations, wheren is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at mostn + 2. Three algorithms of the method are presented. A reverse algorithm, which permits the use of only one matrix, is also given. Considerations pertaining to the applications of this method to the minimization of a quadratic function and a nonquadratic function are given. It is believed that, since the one-dimensional search can be bypassed, a considerable amount of computational saving can be achieved.
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  • 23
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 14 (1974), S. 233-250 
    ISSN: 1573-2878
    Keywords: Time-optimal control ; decomposition methods ; two-point boundary-value problems ; trajectories ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A decomposition technique is presented for minimum-time trajectories which are characterized by intermediate constraints and discontinuities. The optimization of such multiple are trajectories is usually a formidable task. One optimization method, trajectory decomposition, breaks the original trajectory at points of discontinuity into separate arcs and then optimizes each are subject to prescribed boundary conditions. This constitutes a first level of control. Each first-level solution is evaluated by a second-level controller, which iteratively specifies new are boundary conditions in order to achieve an optimum solution. Unfortunately, this two-level method cannot be applied directly to minimum-time trajectories. The two-level trajectory decomposition method is extended here to a three-level technique for treating the minimum-time trajectory. The first level again optimizes each are for specified intervention parameters. The new second level, the time interface controller, exploits certain homogeneity properties to satisfy time transversality conditions at all boundaries and to couple the first-level solution arcs in time. The third level, the state interface controller, satisfies state transversality conditions at the arc junctions iteratively while driving the trajectory to its optimum. The new three-level procedure represents a feasible decomposition because each solution trajectory in the iterative sequence is physically realizable. The technique also offers a decentralization of control effort and reduction of initial-value sensitivities. An example problem is formulated.
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