Pontryagin's maximum principle in optimal control theory
type: Item
instance of: scholarly article
title: Pontryagin's maximum principle in optimal control theory
zbMATH Open document ID: 0945.49017
DOI: 10.1007/BF02365017
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-10-12T00:00:00Z
review text: 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). The author does not introduce any original results but brings together valuable views on this central result in the optimal control theory. He formulates the most general control problem: \(\dot x= f(x,u,t),\;t\in[t_1,t_2]\); \(u=(u_1,u_2)\), \(u_2(t)\in U_2(t)\) \(\dot\forall t,\) \(R(x,u_1,t)\leq 0\), \(G(x,t)\leq 0,\) \(K_1(p)\leq 0,\) \(K_2(p)=0\), \(p=(x_1,x_2,t_1,t_2)\), \(x_1=x(t_1)\), \(x_2=x(t_2),\) \(J=J(p,u)=K_0(p)+\int_{t_1}^{t_2} f^0(x,u,t) dt\to\min.\) The weak maximum principle is formulated firstly for the simplest problem \[ \dot x=f(x,u,t),\quad t\in[t_1,t_2];\quad u(t)\in U(t)\;\dot \forall t,\quad J(p,u)\to \min. \] The adjoint equation has the form \(\;\dot \psi = -\frac {\partial H}{\partial x}(t,\psi), \) the transversality conditions are \[ \psi(t_1^*)=\lambda^0\frac{\partial K_0}{\partial x_1}(p^*); \quad \psi(t_2^*)=-\lambda^0\frac{\partial K_0}{\partial x_2}(p^*), \] \[ \begin{aligned} \text{ess } \varlimsup_{t\to t_s^*} \sup_{u\in U(x_s^*,t)} & H\biggl( x_s^*,u,t,(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial x_s}(p^*)\biggl)+(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial t_s}(p^*)\geq 0;\\ \text{ess } \varliminf_{t\to t_s^*} \sup_{u\in U(x_s^*,t)} &H\biggl( x_s^*,u,t,(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial x_s}(p^*)\biggr)+(-1)^{s-1}\lambda^0\frac {\partial K_0}{\partial t_s}(p^*)\leq 0; \end{aligned} \] \(s=1,2.\) The condition of the maximum for the Pontryagin function \[ H(t,u^*(t))=\sup_{\substack{ u\in U(t)}} H(t,u) \] holds for almost every \(t\), where \[ H(t,u)=h(x^*(t),u,t,\psi(t),\lambda^0)=\langle f(x,u,t),\psi\rangle-\lambda^0f^0(x,u,t), \quad \lambda^0\geq 0. \] Two methods of proving the maximum principle are described. One is based on the penal…
reviewed by: Igor Bock
Mathematics Subject Classification ID: 49K15
zbMATH DE Number: 1348490
zbMATH Keywords: optimal control
zbMATH Keywords: systems of ordinary differential equations
zbMATH Keywords: Pontryagin maximum principle
author: A. V. Arutyunov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02365017
OpenAlex ID: W2056103632
description: scientific article; zbMATH DE number 1348490