Necessary conditions for infinite-dimensional control problems (Q1176538)

Let \(V\) be a complete metric space, \(E\) a Hilbert space, \(f: V\to E\), \(f_ 0: V\to \mathbb{R}\) continuous functions, and \(Y\subset E\). The authors consider the following nonlinear programming problem: find \(u_ 0\in V\) such that \(f(u_ 0)\in Y\) and \(f_ 0(u_ 0)=\min\{f_ 0(u): u\in V, f(u)\in Y\}\). They obtain necessary conditions for minima of Kuhn-Tucker type. As an application the authors derive Pontryagin's maximum principle for a class of control problems for systems described by semilinear operator differential equations in Hilbert space. Convergence properties of sequences of near-optimal controls for such systems are studied too.