Optimal control of distributed semilinear systems with essentially bounded measurable control functions

This paper examines an optimal control problem governed by a class of semilinear systems, with controls taking values in L-infinity (Omega), where Omega is an open bounded set of R-n (n >= 1). The problem consists in minimizing a cost functional over a convex bounded set of admissible controls. By means of compactness assumptions, sufficient conditions for the existence of optimal controls are formulated. Then, first-order optimality conditions are derived, using the Gateaux derivative of the cost functional. Numerical examples of two partial differential equations, a fractional diffusion equation and a wave equation, are provided to illustrate the obtained results.