Solution dependence on initial conditions in differential variational inequalities

Abstract

In the first part of this paper, we establish several sensitivity results of the solution x(t, ξ) to the ordinary differential equation (ODE) initial-value problem (IVP) dx/dt = f(x), x(0) =  ξ as a function of the initial value ξ for a nondifferentiable f(x). Specifically, we show that for $\Xi_T \equiv \{\,x(t,\xi^0): 0 \leq t \leq T\,\}$ , (a) if f is “B-differentiable” on $\Xi_T$ , then so is the solution operator x(t;·) at ξ⁰; (b) if f is “semismooth” on $\Xi_T$ , then so is x(t;·) at ξ⁰; (c) if f has a “linear Newton approximation” on $\Xi_T$ , then so does x(t;·) at ξ⁰; moreover, the linear Newton approximation of the solution operator can be obtained from the solution of a “linear” differential inclusion. In the second part of the paper, we apply these ODE sensitivity results to a differential variational inequality (DVI) and discuss (a) the existence, uniqueness, and Lipschitz dependence of solutions to subclasses of the DVI subject to boundary conditions, via an implicit function theorem for semismooth equations, and (b) the convergence of a “nonsmooth shooting method” for numerically computing such boundary-value solutions.