Finite-time partial stability and stabilization, and optimal feedback control

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control design for finite-time partial stability and finite-time, partial-state stabilization. Finite-time partial 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, satisfies a differential inequality involving fractional powers, and can clearly be seen to be the solution to the steady-state form of the Hamilton-Jacobi-Bellman equation guaranteeing both partial stability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic controller synthesis to nonlinear-nonquadratic optimal finite-time, partial-state stabilization. In addition, we specialize our results to address the problem of optimal finite-time control for nonlinear time-varying systems. Finally, we develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the finite-time, partial-state stabilization problem and use this result to address finite-time, partial-state stabilizing sublinear controllers that minimize a derived performance criterion involving subquadratic terms.