A dynamical approach for the stability of second order dissipative systems.
The nonlinear initial value problem \[ u''(t)+\lambda u'(t)+ \nabla\phi(u(t))= 0,\;t\geq 0,\qquad u(0)= u_0,\qquad u'(0)= u_1 \] in a real Hilbert space \(H\) is considered, where \(\phi: H\to\mathbb{R}\). In particular, the question of whether \(u(t)\) approaches a critical point of \(\phi\) as \(t\to\infty\) is addressed. This question is motivated by the problem of numerically minimizing a function \(\phi\). First, global existence and uniqueness are verified under the assumptions that \(\phi\) is continuously differentiable, \(\phi\) is bounded from below and \(\nabla\phi\) is Lipschitz continuous on the bounded subsets of \(H\). In Theorem 3.1, it is proven that \(u(t)\) converges to a minimizer of \(\phi\) weakly and \(\phi(u(t))\) converges to a minimum of \(\phi\) as \(t\to\infty\). Theorem 3.4 proves under the additional assumption \(\phi\) is strongly convex that \(u(t)\) is norm convergent to the unique global minimizer of \(\phi\). Theorem 3.6 demonstrates that Theorem 3.1 generalizes a result due to \textit{R. E. Bruck} [J. Funct. Anal. 18, 15--26 (1975; Zbl 0319.47041)]. Theorem 3.8 shows that under the assumptions that \(\phi\) is a Morse function (a differentiability-type assumption) and trajectories are precompact, \(u(t)\) converges to a critical point of \(\phi\). Finally, Theorem 4.1 demonstrates that if \(H= \mathbb{R}^n\) and if \(\phi\) is analytic, then \(u(t)\) converges to a critical point of \(\phi\).
scientific article; zbMATH DE number 1836169