Partial-state stabilization and optimal feedback control

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization. Partial asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the Hamilton-Jacobi-Bellman equation, and hence, guaranteeing both partial stability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic controller synthesis to nonlinear-nonquadratic optimal partial-state stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying systems with quadratic and nonlinear nonquadratic cost functionals are also provided. An illustrative numerical example is presented to demonstrate the efficacy of the proposed linear and nonlinear partial stabilization framework.