Minimization with differential inequality and equality constraints applied to complete Lyapunov functions

Meshfree collocation in reproducing kernel Hilbert spaces is an established method to solve generalized interpolation problems such as PDEs. It can be formulated as an optimization problem with equality constraints. In this paper, we consider optimization problems with both inequality and equality constraints for general linear operators, and develop a general theory of discretizing such problems. The unique solution of these discretized problems is obtained using quadratic optimization, and we show that the solutions of the discretized problems strongly converge to the unique solution of the original problem. The general theory is applied to compute complete Lyapunov functions for autonomous ordinary differential equations.