On an effective method of pursuit in linear discrete games with different types of constraints on controls (Q1095057)

The paper discusses a discrete time linear pursuit-evasion problem according to the approach of \textit{L. S. Pontryagin} [Sov. Math., Dokl. 8, 769-771 (1967; Zbl 0157.163)]. The motion of the state vector \(z\in {\mathbb{R}}^ n\) is described by \(z(k+1)=Az(k)-Bu(k)+Cv(k)\); \(z(0)=z_ 0\), \(k=0,1,2,...\). The controls \(u(k)\in {\mathbb{R}}^ p\), \(v(k)\in {\mathbb{R}}^ q\) are assumed to satisfy the conditions \(\sum \| u(k)\|^ 2\leq \rho^ 2\), \(v(k)\in Q_ k\), where \(\rho >0\) and \(Q_ k\) are given subsets of \({\mathbb{R}}^ q\). Termination at time \(k=k_ 1\) is defined as \(z(k_ 1)\in M_ 1=M_ 1+M_ 2\), where \(M_ 1\) is a given subspace of \({\mathbb{R}}^ n\) and \(M_ 2\) is a subset of its orthogonal complement. The central problem is the computation of u(k) at each k when the values of v(s) are known for \(s\in N(k)\), where N(k) is a given subset which does not contain future values of the discrete time. A theorem and a corollary, both rather technical, are given and they give sufficient conditions for termination. Two worked out examples are given.