Sensitivity Analysis and Robust Optimal Control for Coupled Evolution Inclusions with State-Dependent Maximal Monotone Operators

We consider a class of strongly coupled nonsmooth systems consisting of a semilinear evolution inclusion and a differential inclusion governed by state-dependent maximal monotone operators. Our main contributions are fourfold. First, we collect the well-posedness, compactness, and Painlev\'e--Kuratowski continuity properties of the parameterized solution map required for the subsequent optimization analysis. Second, for Bolza-type optimization over the solution set, we prove the existence of optimal pairs, establish continuity properties of the value function, and derive upper semicontinuity of the optimal-solution map. Third, we study fixed-parameter optimal control, simultaneous control-parameter design, min--max robust control, and Hurwicz-type compromise control under parameter uncertainty, and we establish existence results for each formulation. Fourth, we report numerical experiments for sweeping-type systems that illustrate the sensitivity and robustness phenomena predicted by the theory.