The SILP-relaxation method in optimal control. I: General boundary conditions (Q1203063)

The author studies the following control problem: Minimize the integral \(\int^ T_ 0r(t,x,u)dt\) subject to the following constraints: \(\dot x=g(t,x,u)\) almost everywhere on \([0,T]\), \(u(t)\in U\), \(x(t)\in X\), where \(U\subset\mathbb{R}^ m\) is a compact control domain; \(X\subset\mathbb{R}^ n\) is a closed connected set: \(r(t,\xi,\nu)\) is summable over \([0,T]\) for all \(\xi\in X\), \(\nu\in U\); \(r(t,\cdot,\cdot)\) is continuous over \(X\times U\) for all \(t\in[0,T]\); \(g(\cdot,\cdot,\cdot)\) is continuous over \([0,T]\times X\times U\). The boundary conditions are given in one of the forms \(x_ k(0)=\zeta^ 0_ k\), \(k\in K\), or \(x_ \ell(T)=\zeta^ T_ \ell\), \(\ell\in L\), resp. \(x(T)=C\times (0)\), where \(K\) and \(L\) are subsets of \(\{1,\dots,n\}\) and \(C\) is a regular \((n,n)\) matrix. The above problem is assumed to be consistent, that is, there exists at least one admissible pair \((x,u)\) which satisfies all the above constraints. The author investigates the above problem, applying the relaxation method, a measure-theoretical approach for the treatment of the classical control problems. The theoretical consideration is illustrated by a minimum fuel travel problem.