Optimal Control of Second Order Sweeping Processes with Discrete and Differential Inclusions

We discuss the problem of optimal control theory given by second order sweeping processes with discrete and differential inclusions. The main problem is to derive sufficient optimality conditions for second-order sweeping processes with differential inclusions. By using first and second order difference operators in a continuous problem we associate the second order sweeping processes with a discrete-approximate problem. On the basis of the discretization method in the form of Euler-Lagrange inclusions, optimality conditions for discrete approximate inclusions and transversality conditions are obtained. The establishment of Euler-Lagrange type adjoint inclusions is based on the presence of equivalence relations for locally adjoint mappings. To demonstrate the results obtained, a second-order sweeping process with a "linear" differential inclusion is considered.