On the Existence of Solutions Characterized by Riccati Equations to Infinite-Time Horizon Nonlinear Optimal Control Problems

摘要

In a recent survey paper presented by the author during the 17th IFAC World Congress in Seoul, South Korea, in 2008, the theory developed to date on State-Dependent Riccati Equation (SDRE) control has been reviewed, discussing issues on existence of solutions as well as optimality and stability properties associated with SDRE controllers. In this study, existence of solutions associated with general infinite-time horizon nonlinear optimal control problems for nonlinear regulation of input-affine systems is considered and examined in detail, providing a link between the Hamilton-Jacobi-Bellman equation, Lagrangian manifolds and solutions characterized by Riccati equations, using stable manifold theory. The motivation for characterization of solutions to nonlinear optimal control problems by Riccati equations, in particular by symmetric positive-definite solutions, is also justified in hopes of providing a sound theoretical basis for existence of solutions of SDRE controls under very mild conditions.