A Coupled Nonsmooth Dynamical System: Global Well-Posedness, Stability and Sensitivity Analysis

This paper studies a coupled nonsmooth dynamical system in which a semilinear evolution equation is coupled with an implicit algebraic law governed by the normal cone to a time-dependent convex set.

The main difficulty is that the algebraic variable is not given by an explicit feedback, but must be recovered from a state-dependent normal cone relation.

Using a transformed variable, we recast each frozen algebraic law as a variational inequality over a convex set.

Under a strongly pseudomonotone, Lipschitz continuous hypothesis, this frozen problem has a unique algebraic response and admits a sensitivity estimate for the state-to-control map without a projection-contraction argument.

We also give verifiable sufficient conditions for this hypothesis, including a weighted strongly monotone construction and a standard Lipschitz-smallness condition, and point out that the framework allows strongly pseudomonotone nonmonotone frozen operators.

The coupled system is then reduced to a semilinear evolution equation with a single state variable.

By combining $C_0$-semigroup estimates with a Bielecki fixed-point argument, we prove global existence and uniqueness of mild solution pairs on finite time intervals and establish Hadamard well-posedness through explicit continuous-dependence estimates.

We further derive an incremental exponential stability criterion in the dissipative semigroup regime and prove continuity of the parameter-to-solution map with respect to the initial datum and an external parameter.

A reduced contact-mechanics example illustrates the variational inequality formulation and an explicit projection residual that can be used as a starting point for projection- and Newton-type inner solvers after discretization.