Optimal singular control for nonlinear semistabilization

The singular optimal control problem for asymptotic stabilization has been extensively studied in the literature. In this paper, the optimal singular control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Two approaches are presented to address the nonlinear semistable singular control problem. Namely, we solve the nonlinear semistable singular control problem by using the cost-to-go function to cancel the singularities in the corresponding Hamilton-Jacobi-Bellman equation. For this case, we show that the minimum value of the singular performance measure is zero. In the second approach, we provide a framework based on the concepts of state-feedback linearization and feedback equivalence to solve the singular control problem for semistabilization of nonlinear dynamical systems. For this approach, we also show that the minimum value of the singular performance measure is zero. A numerical example is presented to demonstrate the efficacy of the proposed singular semistabilization frameworks.