Optimality conditions for controls of semilinear evolution systems with mixed constraints (Q1355010)

The control systems are described by semilinear equations \[ y'(t)= Ay(t)+ f(t,y(t),u(t)) \] in a Banach space \(X\); \(A\) is a semigroup generator and \(f:[0,T]\times X\times U\to X\), where \(U\) is a metric space in which the control variable takes values. The problem includes state-dependent constraints \(u(t)\in\Gamma(t,y(t))\) on the controls, state constraints \(y(t)\in Q(t)\) and an initial-terminal value constraint \(h(y(0),y(T))\in\Omega\). The functional to be minimized is \[ J(y(\cdot),u(\cdot))= \int^T_0 f^0(t,y(t),u(t))dt, \] with \(f^0:[0,T]\times X\times U\to\mathbb{R}\). The object of this paper is to obtain necessary conditions for optimal controls of the type of Pontryagin's maximum principle. For another slant on this type of problem and other references, see the reviewer [J. Optimization Theory Appl. 88, No. 1, 25-59 (1996; Zbl 0843.49015)].