Step-guided strategies for control problems with uncertain dynamics
type: Item
instance of: scholarly article
title: Step-guided strategies for control problems with uncertain dynamics
zbMATH Open document ID: 0774.93040
DOI: 10.1016/0895-7177(91)90043-7
published in: Mathematical and Computer Modelling
publication date: 1993-01-16T00:00:00Z
review text: The paper deals with both upper and lower bounds for the value function \[ V(t_ 0,x_ 0)=\inf_{u\in U}\{\sup\{g[x(.,t_ 0,x_ 0,u,v(.))]: v(t)\in Q\}\} \] of the problem: \(\min g[x(.)]\) subject to Equation \(\dot x(t)=f_ 1(t,x(t),u(t))-f_ 2(t,x(t),v(t))\), \(t_ 0\leq t\leq T\), with the uncertain parameter \(v(t)\) belonging to the compact set \(Q\). Here \(U\) is the space of all step-guided strategies \(u\) [see the author, Proc. 28th IEEE Conference on Decision\& Control, Tampa/FL, 2449-2450 (1989)] of the form of a triplet \((u(t,x,w),\Delta,w_ R(.))\), where \(\Delta\) is a finite partition of \([t_ 0,T]\), \(u: [t_ 0,T]\times R^ n\times R^ n\mapsto P\), \(P\) is compact and the values of \(w_ R(t)\) are instantly selected by the controller according to a nonanticipating rule \(R\). The above mentioned bounds are given in terms of solutions of simplified unperturbed control problems \(\min g[w(.)]\) subject to Equation \(\dot w(t)=f_ 3(t,w(t),h(t))\) where the control \(h(t)\) may be any Lebesgue measurable selection from a compact set \(H\). Possible situations between \(f_ 1(t,x,P)\), \(f_ 2(t,x,Q)\) and \(f_ 3(t,x,H)\) that may happen in the context of relationships between the influence of \(u\) on the system versus \(v\), are covered by the four theorems which provide not only tools to find the bounds on \(V(t_ 0,x_ 0)\) in each situation but also constructive procedures for finding \(\varepsilon\)- optimal control policies in \(U\).
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 91A23
Mathematics Subject Classification ID: 93C73
zbMATH DE Number: 91859
zbMATH Keywords: unperturbed control
zbMATH Keywords: Lebesgue measurable selection
author: Leszek S. Zaremba
MaRDI profile type: Publication
full work available at URL: 0895-7177(91)90043-7
OpenAlex ID: W1994857080
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description: scientific article; zbMATH DE number 91859