Sufficient optimality conditions in minimax control problems
type: Item
instance of: scholarly article
title: Sufficient optimality conditions in minimax control problems
zbMATH Open document ID: 0765.49015
author: A. F. Tarakanov
published in: Automation and Remote Control
publication date: 1992-09-26T00:00:00Z
review text: The following minimax control problem with terminal constraints is considered: find a control \(u^*(\cdot)\in D\), defined on the segment \(\bigl[t_ 0,\max_{1\leq j\leq N} t_ j\bigr]\), for which \[ \max_{1\leq j\leq N} F_ j(u^*(\cdot),t_ j)=\min_{{u(\cdot)\in D\atop t_ j\geq t_ o}}\max_{1\leq j\leq N} F_ j(u(\cdot),t_ j), \] where \(F_ j(u(\cdot),t_ j)=\int^{t_ j}_{t_ 0} f^ 0_ j(x_ j(t),u(t),t)dt\) and \(x_ j(\cdot)\) is a solution of the equation \(x_ j(t)=f_ j(x_ j(t),u(t),t)\), \(t\in [t_ 0,t_ j],\) \(1\leq j\leq N\), with the condition \(x_ j(t_ 0)=x_ 0\), \(x_ j(t_ j)\in S_ j\), \(1<j<N\), \(u(t)\in U\subseteq E\). The set of admissible controls \(D\) is the set of bounded measurable functions \(u(\cdot)\) with values in a given set \(U\), such that there exist \(t_ j: x_ j(t_ j)\in S_ j\), \(1<j<N\). The method of penalty functions is used to reduce this problem to a variational one. The assumptions on \(f_ j\), \(f^ 0_ j\), on the objective sets \(S_ j\) and on the set \(U\) are dictated by the sufficient conditions under which the method of penalty functions is valid. In addition to them it is required that the functions \(f_ j\) and \(f^ 0_ j\) are continuously differentiable with respect to \(t\). With the use of dynamic programming ideas local sufficient optimality conditions, supplementing the necessary conditions, are obtained. An example of control design is presented.
reviewed by: G.Yu.Garnaeva
Mathematics Subject Classification ID: 49K35
zbMATH DE Number: 56394
zbMATH Keywords: minimax control
zbMATH Keywords: terminal constraints
zbMATH Keywords: penalty functions
zbMATH Keywords: dynamic programming
zbMATH Keywords: local sufficient optimality conditions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 56394