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<p>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&#xa0;(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.</p>

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Finite-time stabilization of affine nonlinear systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月18日

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  • Gianmaria De Tommasi,
  • Federico Fiorenza,
  • Adriano Mele,
  • Alfredo Pironti

摘要

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.