Well-posedness and optimal control of a nonsmooth fractional dynamical system

We consider a nonsmooth nonlinear system governed by a set-valued maximal monotone operator in Hilbert spaces. The novelty brought by our study is the involvement of the Caputo fractional derivative. In our paper, we aim to achieve three primary objectives. The first one is to show how a reduction approach with fractional differential equations allows us to establish the unique solvability of the initial value problem and give some important properties of the solution as the continuous dependence with respect to the data. The second objective is to prove the existence of optimal solution to an optimal control problem involving the nonsmooth fractional dynamical system in finite dimensions. The third aim is to put into practice our obtained results in the analysis of a non-regular electrical circuit, including a non-regular electronic component such as a Zener diode, as well as a fractional inductor. This application constitutes an additional contribution to our paper.