Finite-Time Stabilization and Optimal Feedback Control

Abstract

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability have been developed in the literature using continuous Lyapunov functions. In this paper, we develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finitetime stability and finite-time stabilization. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a differential inequality involving fractional powers. This Lyapunov function can clearly be seen to be the solution to a partial differential equation that corresponds to a steady-state form of the Hamilton-Jacobi-Bellman equation, and hence, guaranteeing both finite-time stability and optimality.

Implications of dissipativity, inverse optimal control, and stability margins for nonlinear stochastic feedback regulators

“…Extensions of this framework for exploring connections between optimal finite‐time stabilization and finite‐time stabilization for stochastic dynamical systems are currently under development. The proposed framework can also allow us to further explore connections with stochastic inverse optimal control, stochastic dissipativity, and stability margins for finite‐time stabilizing regulators that minimize a derived cost functional involving subquadratic terms.…”

Nonlinear–nonquadratic optimal and inverse optimal control for stochastic dynamical systems

“…Alternatively, when the nonlinear–nonquadratic performance measure involves terms of order x p , where p <2, then we have a subquadratic cost criterion, which pays close attention to the system state near the origin. In this case, the optimal controller is sublinear and, hence, exhibits finite settling time behavior .…”

Finite‐region stabilization via dynamic output feedback for 2‐D Roesser models

“…Finite‐time stability plays vital roles in many practical applications, for instance, the problem of not exceeding some given bounds for the state trajectories, when there exist some saturation elements in the control loop, or the problem of controlling the trajectory of a spacecraft from an initial point to a final point in a prescribed time interval. With the in‐depth of research on FTS theory for 1‐D systems, many interesting results have come into play, see previous studies . Among them, Amato et al used a two‐step procedure (state feedback design followed by observer synthesis) in the finite‐time stabilization problem for 1‐D continuous‐time linear systems.…”

“…With the in-depth of research on FTS theory for 1-D systems, many interesting results have come into play, see previous studies. [16][17][18][19][20][21][22][23] Among them, Amato et al 21 used a two-step procedure (state feedback design followed by observer synthesis) in the finite-time stabilization problem for 1-D continuous-time linear systems.…”