A Solution Method, a Necessary and Sufficient Condition for the Existence of the Stabilizing Solution and the Structure of the Hamilton-Jacobi Equation

Abstract

In this paper the Hamilton-Jacobi equation in nonlinear control theory is analyzed using symplectic geometry. First, a solution method of the Hamilton-Jacobi equation is outlined and shown to be a natural extension of the well-known theory of the Riccati equation. Second, a necessary and sufficient condition for the existence of the stabilizing solution is proposed. Finally, the structure of solutions of the Hamilton-Jacobi equation, such as the maximal and minimal solution, is clarified. To this end, the theory of the nonlinear Lyapunov equation is developed.