A note on optimal control of a class of single input nonlinear systems

The main contribution of this paper is to develop a general methodology to solve a class of optimal nonlinear control problems with a single input based on a state-dependent Riccati equation. For a polynomial system of order n the paper proposes a formula for the dependence of the cost-to-go function on one of the variables, which then leads to a state-dependent Riccati equation that is linear in in the remaining unknowns. Furthermore, it is shown that the optimal cost-to-go function is also a Lyapunov function that can be used to prove stability of the closed-loop system. The relevance of the proposed methodology is illustrated in several examples for which analytical solutions are found, including the Van der Pol oscillator, a mass-spring system, and a nonlinear system in strict form.