Finite-Time Stabilization and Optimal Feedback Control

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 stabilit...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2016-04, Vol.61 (4), p.1069-1074
Main Authors: Haddad, Wassim M., L'Afflitto, Andrea
Format: Article
Language:English
Subjects:
Adaptive control
Asymptotic stability
Closed loop systems
Convergence
differential inequalities
Dynamical systems
Feedback control
Finite-time stability
finite-time stabilization
Hamilton-Jacobi-Bellman theory
Lyapunov functions
Lyapunov methods
Mathematical analysis
Nonlinear dynamical systems
Nonlinear dynamics
optimal control
Optimization
Stability
Stabilization
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Summary: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 technical note, we develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite-time 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.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2015.2454891