Results 141 to 150 of about 661 (188)

The basic Pontryagin maximum principle

1991
Abstract In this chapter we state the Pontryagin maximum principle (PMP) in its simplest form and use it to solve some simple examples. Extensions to a less restricted class of problems are discussed in Chapter 7, but the proof of the PMP is postponed to Chapter 9.
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An Elementary Proof of the Pontryagin Maximum Principle

Vietnam Journal of Mathematics, 2020
The subject is the standard control problem for systems of ODE \begin{gather*} \begin{aligned} \text{minimize} & \quad \ell_0(x(0), x(T)) \\ \text{subject to} & \quad x'(t) = f(t, x(t), u(t)) \quad (u(t) \in U) \end{aligned} \\ \ell_j(x(0), x(T)) \le 0 \quad j = 1,\dots ,l \, , \quad \ell_j(x(0), x(T)) = 0 \quad j = l+1,\dots ,r \, . \end{gather*} If \(
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Pontryagin-Type Stochastic Maximum Principle and Beyond

2021
The main purpose of this chapter is to derive first order necessary conditions, i.e., Pontryagin-type maximum principle for optimal controls of general nonlinear stochastic evolution equations in infinite dimensions, in which both the drift and the diffusion terms may contain the control variables, and the control regions are allowed to be nonconvex ...
Qi Lü, Xu Zhang
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8 The Pontryagin Maximum Principle

1967
Publisher Summary This chapter presents a reformulation of the proof of the Pontryagin maximum principle and applies these techniques to give a proof of the bang-bang principle. The objective in reformulating the proof of the Pontryagin maximum principle is to emphasize one central fact.
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PONTRYAGIN'S MAXIMUM PRINCIPLE AND OPTIMAL CONTROL

1961
Abstract : The mathematical formulation of the most general problem of Optimal Control can be considered as a problem of Mayer subjected to unilateral constraints, i.e., to certain restrictions expressible in terms of inequalities. Results of the classical calculus of variations in their usual forms cannot give a general solution to this problem ...
H. Halkin, I. Fluegge-Lotz
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On Discrete Analogues of Pontryagin's Maximum Principle†

International Journal of Control, 1965
ABSTRACT A discrete form of Pontryagin's Maximum Principle recently proposed by a number of authors, is shown to be fallacious and a corresponding correct but weaker result is derived. Certain classes of problem are identified for which the original strongor result is valid.
R. JACKSON, F. HORN
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On certain minimax problems and Pontryagin’s maximum principle

Calculus of Variations and Partial Differential Equations, 2009
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Pontryagin’s Maximum Principle via Singular Perturbations

2003
Necessary optimality conditions for the time optimal control under pointwise state constraints are derived. The system considered consists of a parabolic equation coupled with an ordinary differential equation in a Banach space. Systems of this type typically arise in modelling population dynamics in a contaminated environment.
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Pontryagin's maximum principle in optimal control theory

Journal of Mathematical Sciences, 1999
The author investigates the Pontryagin's maximum principle which occured first time in the monograph ``The mathematical theory of optimal processes'' by \textit{L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze} and \textit{E. F. Mishchenko} (1961; Zbl 0102.31901).
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Optimal Control Problem. Pontryagin maximum Principle

2000
We study the following general optimal control problem: $$\dot x = f(x,u,t)\quad t \in [{t_1},{t_2}],{t_1} < {t_{2,}}$$ (1.1) $$u = ({u_1},{u_2}),\quad {u_2}(t) \in {U_2}(t)\forall t,$$ (1.2) $$R(x,{u_1},t) \leqslant 0$$ (1.3) $$G(x,t) \leqslant 0,$$ (1.4) $${K_1}(p) \leqslant 0,\quad {K_2}(p) = 0,$$ (1.5)
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