The second-order in time continuous Newton method (Q2761421)

The authors deal with the analysis of the following dissipative second-order dynamical system NEWLINE\[NEWLINE\ddot x(t)+ \nabla^2\phi(x(t)) \dot x(t)+ \nabla\phi(x(t))= 0NEWLINE\]NEWLINE with the initial condition \((x(0),\dot x(0))= (x_0,\dot x_0)\in H\times H\) in a Hilbert space \(H\). Under weak conditions on \(\phi\), it is shown that every solution exists for \(t\geq 0\), the energy NEWLINE\[NEWLINEE(t)=\textstyle{{1\over 2}} |\dot x(t)+ \nabla\phi(x(t))|^2+ \phi(x(t))NEWLINE\]NEWLINE is a Lyapunov functional and \(\lim_{t\to\infty} \nabla\phi(x(t))= 0\).NEWLINENEWLINENEWLINEMoreover, for Morse-functions \(\phi: H\to\mathbb{R}\) or when \(\phi\) is convex, the convergence of the evolutions towards an equilibrium is shown.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00036].