Step-guided strategies for control problems with uncertain dynamics
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 \(U\).
scientific article; zbMATH DE number 91859
scientific article