Sufficient optimality conditions in minimax control problems
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.
scientific article; zbMATH DE number 56394