Optimization of a Class of Distributed Parameter Systems with Deviating Argument

In this paper we consider an optimal control problem described by a system of nonlinear first order hyperbolic partial differential equations with deviating argument, including integral inequality constraints. The control variables are assumed to be measurable, with the corresponding state variables in Lp. We introduce the adjoint equations, derive an integral representation of the increments of the functionals involved, and use separation theorems of functional analysis to obtain new necessary optimality conditions in the form of the Pontryagin maximum principle