On the optimal control of systems evolving with state suprema

This paper studies optimal control processes governed by a specific family of systems described by functional differential equations (FDEs) involving the sup-operator. Systems evolving with the state suprema constitute a useful abstraction for various models of technological and biological processes. The specific theoretic framework incorporates state suprema in the right hand side of the initially given differential equation and finally leads to a FDE with the state-dependent delays. We study a class of nonlinear FDE-featured optimal control problems (OCPs) in the presence of some additional control constraints. Our aim is to develop implementable first-order optimality conditions for the retarded OCPs under consideration. We use the celebrated Lagrange approach and prove a variant of the Pontryagin-like Minimum Principle for the given OCPs. Moreover, we discuss a computational approach to the main dynamic optimization problems and also consider a possible application of the developed methodology to the Maximum Power Point Tracking (MPPT) control of solar energy plants.