Existence and dependence on a parameter of solutions of a nonlinear two point boundary value problem

Hadlock, Charles R.

doi:10.1016/0022-0396(73)90064-8

Cited by 28 publications

(5 citation statements)

References 6 publications

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“…Then for each g(e) E " sufficiently small, which is K + 1 times continuously differentiable, there exists a solution X"(-, e), yL(., e), U"(', e) of (5) on 0<= '<--_ T/e, satisfying p"(0, e)= c(e). Moreover, XL(", e), Y'(', e), U'(", e) are K + 1 times continuously differentiable with respect to e, and satisfy decay condition (6) for some C > 0, > 0 and for all 0 <--<-T/e. In addition, if Xd ('), Y('), Ud (') represents the solution of (17), while for 1 <_-k _-< K, X('), Y('), U(') represents the solution of (18), then Our proof of Theorem 5.1 will follow by an appropriate interpretation of Hoppensteadt's Lemma 2 [8].…”

Section: L(t)r(t) + F(t)s(t) + F(t)v(t) +L(t)mentioning

confidence: 98%

“…A recent paper which treats a problem closer to our own is that of Hadlock [6]. He also treats a singularly perturbed two-point boundary value problem, although it does not involve a controller.…”

mentioning

confidence: 97%

“…Hypotheses essentially the same as (20) have previously been used by numerous other authors: Flatto and Levinson [4], Hadlock [6], Hoppensteadt [8], Levin 10], Chang 1] and Chang and Coppel [2].…”

mentioning

confidence: 98%

“…Moreover, we require that there exist some positive constants C, 6 such that for all -, 0 =< r =< T/e, (6) IXL(z, e)l+lyL(.r, e)I+[uL(z, e)[<=Ce-L For now, we will leave the choice of (e) in (5d) unspecified. Our solution to this stage consists of the sum (x*(t, e), y*(t, e), u*(t, e))+(X(", e), Y"(', e), U"(', e)).…”

mentioning

confidence: 99%

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Singular Perturbations of Two-Point Boundary Value Problems Arising in Optimal Control

Freedman¹,

Kaplan²

1976

SIAM J. Control Optim.

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This paper considers a two-point boundary value problem which arises from an application of the Pontryagin maximal principle to some underlying optimal control problem. The system depends singularly upon a small parameter, e. It is assumed that there exists a continuous solution of the system when e 0, known as the reduced solution. Conditions are given under which there exists an "outer solution", and "left and right boundary-layer solutions" whose sum constitutes a solution of the system which degenerates uniformly on compact sets to the reduced solution. The principal tool used in the proof is a Banach space implicit function theorem.

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Section: L(t)r(t) + F(t)s(t) + F(t)v(t) +L(t)mentioning

confidence: 98%

mentioning

confidence: 97%

mentioning

confidence: 98%

mentioning

confidence: 99%

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Singular Perturbations of Two-Point Boundary Value Problems Arising in Optimal Control

Freedman¹,

Kaplan²

1976

SIAM J. Control Optim.

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“…An application to nuclear reactors was reported in Reddy and Sannuti (1975). Asymptotic expansions and their validity were investigated by Hadlock (1970Hadlock ( ,1973, , Sannuti (1974aSannuti ( ,b,1975, Freedman and Granoff (1976), Freedman and Kaplan (1976), Kurina (1977), Vasileva and Dmitriev (1980), Vasileva and Faminskaya (1981) . A methodology similar to that of Sections 5 and 7 was developed by Chow (1979) .…”

Section: Composite Feedback Control Of Nonlinear Systemsmentioning

confidence: 99%

Applications of Singular Perturbation Techniques to Control Problems

Kokotović¹

1984

SIAM Rev.

386

126

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This paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years. The first three sections are devoted to the standard model and its time-scale, stability and controllability properties. The next two sections deal with linear-quadratic optimal control and one with cheap (near-singular) control. Then the composite control and trajectory optimization are considered in two sections, and stochastic control in one section. The last section returns to the problem of modeling, this time in the context of large scale systems. The bibliography contains more then 250 titles. D IS T R I B U T I O N / A V A I L A B I L I T Y O F A B S T R A C T U N C L A S S I F I E D / U N L I M I T E D S S A M E A S R PT. □ O T I C U S E R S □ 21. A B S T R A C T S E C U R I T Y C L A S S I F I C A T I O NUnclassified S E C U R I T Y C L A S S I F I C A T I O N O F T H IS P A G E S E C U R IT Y C L A S S IF IC A T IO N O F TH IS P A G E S E C U R I T Y C L A S S I F I C A T I O N O F T H I S P A G E APPLICATIONS OF SINGULAR PERTURBATION TECHNIQUES

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A Boundary value technique for solving singularly perturbed, fixed end‐point optimal control problems

Kadalbajoo

Singh

1988

Optim Control Appl Methods

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Existence and dependence on a parameter of solutions of a nonlinear two point boundary value problem

Cited by 28 publications

References 6 publications

Singular Perturbations of Two-Point Boundary Value Problems Arising in Optimal Control

Singular Perturbations of Two-Point Boundary Value Problems Arising in Optimal Control

Applications of Singular Perturbation Techniques to Control Problems

A Boundary value technique for solving singularly perturbed, fixed end‐point optimal control problems

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