Optimal control, well-posedness and sensitivity analysis for a class of generalized evolutionary systems (Q6614466)

The authors study a system governed by the following fractional differential variational-hemivariational inequality in a reflexive Banach space \(E\): \N\[\N\left\{ \begin{array}{l} ^CD_t^\alpha x(t) \in Ax(t) + F(t,x(t),u(t))\quad \mbox{a.e.}\,\,t \in I := [0,b], \\\Nu(t) \in SOL(K; g(t,x(t),\cdot0, J(x(t),\cdot), J(x(t),\cdot),\phi,h)\quad \mbox{a.e.}\,\,t \in I,\\\Nx(0) = x_0. \end{array} \right.\N\]\NHere \(^CD_t^\alpha,\) \(0 < \alpha \leq 1\) denotes the Caputo fractional derivative, \(A\) is an infinitesimal generator of a compact and uniformly bounded \(C_0\)-semigroup in \(E,\) \(F \colon I \times E \times U \multimap E,\) where \(U\) is a reflexive Banach space, is a multimap with convex closed values. For a nonempty convex closed subset \(K \subset U\), the notation \(SOL(K;g(t,x(t),\cdot),J(x(t),\cdot),\phi,h)\) stands for the solution set of the mixed variational-hemivariational inequality of the form\N\[\N\langle g(t,x(t),u(t)),v - u(t)\rangle_U + J^0(x(t),\gamma u(t);\gamma(v - u(t))) + \phi(v,u(t)) \geq \langle h,v - u(t)\rangle_U\N\]\Nfor all \(v \in K\), where \(J^0\) denotes the Clarke directional derivative of a locally Lipschitz function \(J\).\N\NThe authors demonstrate the nonemptiness and compactness of the solution set to the above problem. The existence of an optimal control is proved and the well-posedness of the problem, including the existence, uniqueness and stability of solutions is studied and a sensitivity analysis of the problem related to multiparameters is explored. An application to a fractional heat equation is considered.