Optimization of discrete and differential inclusions of Lagrange Problem involving different delays

In this paper, optimality conditions for the Lagrange Problem with differential inclusions(DFIs) involving different delays and boundary conditions are obtained. The velocity in the handled problem depends not only on the state at this instant, but also upon the history of the trajectory until this instant. We use the term Locally adjoint mapping(LAM) to determine adjoint discrete inclusions. The difficulty in the paper appears in the discretization method since the velocity depends also on two different states in the past. Some equivalence relations are applied to derive necessary and sufficient conditions for the discrete approximation problem. Passing to the limit, sufficient optimality conditions are established for the Lagrange optimal problem described by DFIs involving different delays. Similar results for the non-convex delayed Lagrange problem are obtained. We hope that with this study we will contribute to the development of optimal theory with delayed differential inclusions. In optimization problems done so far, as far as we know, conditions for derivatives depending on a single delay are considered, while in our study, the velocity depends on both the present and two different instants in the past of the state.