Global asymptotic stability of a forced Newtonian system with dissipation
The authors consider the second order system (1) \(u'' + cu' + Au + \nabla G(u) = p(t)\) where \(u \in \mathbb{R}^n\), \(c > 0\), \(A\) is a semidefinite positive symmetric matrix, \(G \in C^2 (\mathbb{R}^n, \mathbb{R})\). In addition they assume that there exist constants \(a,b\) such that \(a I_n \leq D^2G (\xi) \leq bI_n \forall \xi \in \mathbb{R}^n\). Their main results read as follows: Theorem 1. Assume that \(a,b\) satisfy \(a + \lambda_1 (A) > 0\), \(b < a + c^2 + 2c \sqrt {a + \lambda_1 (A)}\), where \(\lambda_1 (A)\) denotes the smallest eigenvalue of \(A\). Then (1) has a unique bounded solution that is globally exponentially asymptotically stable. Moreover, if \(p(t)\) is \(T\)-periodic then such a solution is also \(T\)-periodic. Theorem 2. Assume that \(\nabla G\) is bounded and \(p\) admits a decomposition of the form \(p = p^* + p^{**}\) with \(\sup \{|\int^t_0 \pi p^* (s)ds |: t \in \mathbb{R}\} < \infty\); \(p^{**}\) is bounded and there exist constants \(\alpha, \delta > 0\) such that \[ \liminf_{|\xi |\to \infty, \xi \in K_\delta} {\bigl( \xi, \nabla G (\xi) - p^{**} (t) \bigr) \over |\xi |} > \alpha \] uniformly with respect to \(t\). Then (1) has a bounded solution. Moreover, (i) if there exists a monotone function \(\varphi : (0, \infty) \to (0, \infty)\) such that \(\lim_{r \to \infty} \varphi (r) = 0\) and for each \(r > 0\) we have \(\varphi (r) I_n \leq D^2 G (\xi) \leq c^2 I_n\) \(\forall \xi \in \mathbb{R}^n\) with \(|\xi |\leq r\), then such solution is globally exponentially asymptotically stable (in particular it is the unique bounded solution); (ii) if \(p(t)\) is \(T\)-periodic then such solution is also \(T\)-periodic.
globally exponentially asymptotically stable
scientific article; zbMATH DE number 861862