Properties of attainable sets of evolution inclusions and control systems of subdifferential type

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and the evolution operators that are the compositions of a linear operator and the subdifferentials of a time-dependent proper convex lower semicontinuous function. Alongside the initial inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are the compositions of the same linear operator and the subdifferentials of the Moreau–Yosida regularizations of the initial function. We demonstrate that the attainable set of the initial inclusion as a multivalued function of time is the time uniform limit of a sequence of the attainable sets of the approximating inclusions in the Hausdorff metric. We obtain similar results for evolution control systems of subdifferential type with mixed constraints on control. As application we consider an example of a control system with discontinuous nonlinearities containing some linear functions of the state variables of the system.