The SILP-relaxation method in optimal control: General boundary conditions. II (Q686845)

[For part I see the author, ibid. 11, No. 1, 143-151 (1992; Zbl 0764.49019).] The author studies the following control problem: Minimize the integral \(\int^ T_ 0 r(t,x,u)dt\) subject to the following constraints: \(\dot x=g(t,x,u)\) almost everywhere over \([0,T]\) in the sense of Carathéodory; \(u(y)\in U\), \(x(t)\in X\), where \(U\subset\mathbb{R}^ m\) is a compact control domain, \(X\subset\mathbb{R}^ n\) is a closed connected set; \(x(\cdot)\) is an \(n\)-vector of an absolutely continuous state function; \(u(\cdot)\) is an \(m\)-vector of a bounded measurable control function; \(r\) and \(g\) are Lipschitz over \([0,T]\times X\times U\); \(x(T)= C\cdot x(0)\), where \(C\) 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. A similar control problem was investigated by the author in part I of this paper and is thus a natural continuation. It moreover discusses approximation properties and gives as an example the numerical treatment of a nice geometric extremal problem by \textit{J. Focke} [Acta Math. Acad. Sci. Hung. 20, 39-68 (1969; Zbl 0174.25304)].