Optimal control of polynomial systems with performance bounds: A convex optimization approach

This paper is concerned with an approach for a nonlinear optimal control of polynomial systems. The Hamilton-Jacobi-Bellman (HJB) equation is relaxed into HJB inequalities. Both an upper bound and a lower bound on the cost function, as well as a suboptimal controller, can be computed from solutions of the resulting inequalities. Solving the HJB inequalities can be cast as state-dependent matrix inequalities (SDMIs), whose derivation is based on representation of the given polynomial system in a linear-like form. The resulting SDMI for the upper-bound computation is nonconvex in the decision variables, and hence an iterative procedure is proposed to deal with the nonconvexity. On the other hand, the resulting SDMI for the lower-bound computation can be written as a state-dependent linear matrix inequality (SDLMI), which is a convex optimization problem solvable by existing numerical tools. A numerical example is provided to illustrate the proposed approach.