Performance bounds for optimal control of polynomial systems: A convex optimization based approach

An approach for nonlinear optimal control of polynomial systems is considered in this paper. We relax the HJB equation to HJB inequalities and consider solutions of the resulting inequalities in order to compute an upper bound and a lower bound on the cost function. Computation of both the upper bound and lower bound can be cast as robust SDPs, which can be efficiently solved by the existing numerical tools. The idea is based on representation of the given system in a linear-like form. Our approach can be applied to search for polynomial solutions of any degree of the HJB inequalities. Suboptimal controllers are obtained in terms of the solutions of the HJB inequalities.