Lyapunov Functions by Interpolating Numerical Quadrature: Proof of Convergence

Lyapunov functions for nonlinear systems, whose dynamics are defined by ordinary differential equations, are computed by solving linear programming feasibility problems in the CPA method. Further, the CPA method is constructive and can generate a Lyapunov function on any compact subset of the basin of attraction of an asymptotically stable equilibrium. Instead of solving the linear programming feasibility problem, one can use converse theorems to determine a candidate solution and then verify the constraints of the feasibility problem. This procedure has the advantage of being usually much faster. Further, a partial solution to the feasibility problem that violates the constraints in some areas can be analyzed, whereas a solver either generates a feasible solution or assures that a feasible solution does not exist. In this paper we prove that the numerical quadrature of numerically integrated solutions will deliver a feasible solution to the linear programming problem, given that the time horizon is large enough and the time steps are small enough in the numerical integration and quadrature. Further, the relevant theorems are general enough to allow for considerable flexibility in the particular implementation as they cover a wider range of numerical methods both for integration and quadrature.