Necessary conditions of asymptotic stability for unilateral dynamical systems (Q1780774)

The authors present necessary conditions of asymptotic stability of the trivial stationary solution for a general class of unilateral dynamical systems involving the subdifferential \(\partial \varphi\) of a proper, convex, lower semicontinuous function \(\varphi:R^n\to\;R\cup\{+\infty\}\) and a nonlinear term expressed by means of a continuous operator \(F\) on \(R^n\). Precisely, defining the mapping \(\Lambda(x)=x-P_\varphi(x-F(x))\), where \(P_\varphi=(id+\partial\varphi)^{-1}\), it is shown that if \(O\) is an isolated zero of \(\Lambda\) and is asymptotically stable, then there exists \(\rho_0>0\) such that \(\deg(\Lambda,B_\rho,O)=1\), \(\forall \rho\in (0,\rho_0]\). Here \(\deg(\Lambda,B_\rho,O)\) denotes the Brouwer degree of \(\Lambda\) with respect to the open ball \(B_\rho=\{x\in R^n:\;\| x\| <\rho\}\) and \(O\). A special attention is payed to the situation where \(\varphi\) is the indicator function of a closed, convex set, and in particular to the complementarity dynamical systems. Several examples and applications are discussed.