Existence of SDRE stabilizing feedback

The state-dependent Riccati equation (SDRE) approach to nonlinear system stabilization relies on representing a nonlinear system's dynamics in a manner to resemble linear dynamics, but with state-dependent coefficient matrices that can then be inserted into state-dependent Riccati equations to generate a feedback law. Although stability of the resulting closed-loop system need not be guaranteed a priori, simulation studies have shown that the method can often lead to suitable control laws. In this note, we consider the nonuniqueness of state-dependent representations. In particular, we show that if there exists any stabilizing feedback leading to a Lyapunov function with star-convex level sets, then there always exists a representation of the dynamics such that the SDRE approach is stabilizing. The main tool in the proof is a novel application of the S-procedure for quadratic forms.