Nonlinear optimal control: alternatives to Hamilton-Jacobi equation

The methods of frozen Riccati equations and nonlinear matrix inequalities (NLMIs) offer certain computational advantages over Hamilton-Jacobi equations (HJE), but may fail to be optimal. The frozen Riccati method uses a non-unique state-dependent linear representation to reduce the HJE to a state-dependent Riccati equation. While there usually exists some choices that recover the optimality, it may be difficult to find. The NLMI computation for analysis and synthesis is shown to be as hard as Lyapunov stability analysis; a finite difference approximation scheme is proposed to solve the NLMIs.