Local convergence of the steepest descent method in Hilbert spaces (Q703669)

The authors select a broad class of functionals defined on an infinite dimensional Hilbert space (the Sobolev space of all absolutely continuous functions on an interval [0,T] whose derivative lies in the space of square integrable functions on (0,T)), and for this class , the local convergence of the steepest descent method is proved, under no convexity, monotonicity or coerciveness requirements. The functionals under consideration are assumed to satisfying a Palais -Smale-type condition and having a locally Lipschitz continuous gradient. This class includes a wide class of functionals appearing in classical calculus of variations. The approach is based on the solutions of an autonomous ordinary differential equations of first order having in the right hand side the gradient of the functional.