Publication

Finite-time stabilization of affine nonlinear systems

2026

Abstract

The finite-time stabilization problem deals with the enforcement of a quantitative bound on the state trajectory of a dynamical system along a finite-time horizon. Sufficient conditions have been provided to solve this problem for different classes of nonlinear systems. However, by implying a certain degree of conservatism, these conditions are only sufficient. This work contributes to fill this gap, by giving a necessary and sufficient condition for the finite-time stabilization of affine nonlinear systems. First a necessary and sufficient condition available in literature to check finite-time stability (FTS) of nonlinear systems is extended to the case of continuously differentiable Lyapunov-like functions. Thanks to the obtained smoothness of the Lyapunov function and by a proper definition of a FTS Control Lyapunov Function, we show that it is possible to apply the Sontag’s universal formula to derive a necessary and sufficient condition to solve the stabilization problem for affine nonlinear systems.

Official source

https://infoscience.epfl.ch/entities/publication/73fef321-e611-465b-8a21-42e4a04b31aa

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/73fef321-e611-465b-8a21-42e4a04b31aa

About this result

Related concepts (31)

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.

Dynamical systems theory

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle.

Lyapunov function

In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.

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Finite-time stabilization of affine nonlinear systems

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