Existence of solutions for a class of nonlinear control problems (Q1823451)

We consider problems of control and problems of optimal control, monitored by an abstract equation of the form \(Ex=N_ ux\) in a finite interval [0,T]; here, x is the state variable with values in a reflexive Banach space; u is the control variable with values in a metric space; E is linear and monotone; and \(N_ u\) is nonlinear of the Nemitsky type. Thus, by well-known devices, the results apply also to parabolic partial differential equations in a cylinder [0,T]\(\times G\), \(G\subset {\mathbb{R}}^ n\), with Cauchy data for \(t=0\) and Dirichlet or Neumann conditions on the lateral surface of the cylinder. We prove existence theorems for solutions and existence theorems for optimal solutions, by reduction to a theorem of \textit{N. Kenmochi} [Hiroshima Math. J. 1, 435-443 (1971; Zbl 0262.47043)] for reflexive Banach spaces.