×

Existence theorems and necessary conditions for a class of time optimal control problems from a general formulation of the minimum effort problems. (English) Zbl 0774.49007

Summary: Let \(X_ t\) (depending on the parameter \(t\)), \(Y\) and \(Z\) be normed linear spaces, \(T_ t: D(T_ t)\to Y\), \(S_ t: D(S_ t)\to Z\) linear operators with \(D(T_ t)\subseteq X_ t\), \(D(S_ t)\subseteq X_ t\), \(\Omega_ t\subseteq X_ t\) a convex set containing the zero element \(\theta\), and \(J\) a real-valued convex function defined on \(X_ t\times Y\) such that (i) \(J(x,y)\geq 0\) for \((x,y)\in X_ t\times Y\). (ii) \(J(\theta,\theta)=0\). (ii) \(J(x,y)\to +\infty\) as \((\| x\|^ 2+\| y\|^ 2)^{1/2}\to +\infty\). Consider the mapping \(\overline T_ t: (X_ t\times Y)\to (Y\times Z)\) defined by \(\overline T_ t(u,y)=(T_ t u+ y, S_ t u)\) where \((u,y)\in X_ t\times Y\). Let \((\xi,n)\in Y\times Z\) with \(n\in S_ t[\text{Core}_{T_ t}\Omega_ t\cap D(S_ t)]\) for some \(t\in (0,\infty)\). The problem is to determine \(u_ t\in [\Omega_ t\cap D(S_ t)\cap D(T_ t)]\) such that \(\overline T_ t(u_ t,y)=(\xi,n)\), \(S_ t u_ t=n\) and \(t\) is minimum.
The above-mentioned problem is analyzed by using the classical techniques of functional analysis. Existence problems are considered for a certain class of closed linear operators. The optimal solutions are characterized in terms of the adjoint operators. These results are applicable to linear minimum effort problems, constrained variational problems, optimal control of distributive systems and certain ill-posed variational problems.

MSC:

49J27 Existence theories for problems in abstract spaces
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
PDFBibTeX XMLCite