Exponential convergence and instability of solutions of uncertain dynamical systems (Q869823)

A system of ordinary differential equations \[ \dot x= f(t,x,\alpha),\quad x\in\mathbb{R}^n \] with initial conditions \(x(t_0)= x_0\) is considered in this paper. The parameter \(\alpha\) represents deterministic uncertainties; it is assumed to belong to a compact set \({\mathcal S}\subset\mathbb{R}^d\). Let \(r(\alpha)> 0\) be a function such that \[ \lim_{\|\alpha\|\to 0} r(\alpha)= r_0\quad\text{and}\quad \lim_{\|\alpha\|\to\infty} r(\alpha)= +\infty. \] The aim of the paper is to provide conditions for either exponential stability or instability of the moving set \(A(r)= \{x\in\mathbb{R}^n:\| x\|= r(\alpha)\}\) for each \(\alpha\in{\mathcal S}\). The approach is based on the method of matrix-valued Lyapunov functions \[ V(t,x,\eta)= \eta^T U(t,x)\eta, \] where \(U\) is a matrix-valued continuously differentiable function and \(\eta\) is a vector. The results are illustrated by examples.