Optimization of discrete and differential inclusions with multiple delays

This paper studies a new class of optimal control problems involving discrete and differential inclusions with multiple delays and state constraints. Under a suitable regularity condition, optimality conditions for problems with two delays are analysed. By applying a discretization approach in the form of Euler-Lagrange type inclusions, sufficient optimality conditions for problems with multiple delays are established. In such an adjoint inclusion, to each delay parameter corresponds one absolutely continuous function. The transition from a discrete problem to a discrete-approximate problem is achieved through equivalence relations and the associated locally adjoint mappings (LAMs). These relations enable the derivation of optimality conditions for discrete-approximate problems with multiple delays. Finally, by passing to the limit, sufficient optimality conditions for differential inclusions with multiple delays are obtained. In particular, applications of the developed results to first-order linear optimal control problems with single delays are presented.