Convergence analysis of a class of nonsmooth gradient systems

Wenlian Lu, Jun Wang

Research output: Contribution to journalArticlepeer-review

29 Citations (Scopus)


In this paper, we present new results on convergence of a class of nonlinear dynamical systems modeled by the gradients of nonsmooth cost functions. This class of systems arises from neural-network research and can be regarded as a generalization of existing neural-network models. Using the recently developed Łojasiewicz gradient inequality, the convergence analysis of this gradient-like differential inclusion is given. Without assuming the smoothness of the cost function or analyticity of the activation function, we prove the output convergence of the differential system. In addition, for the piecewise analytic activation function with positive first-order derivative, we prove the state convergence. Furthermore, we also discuss the relationship between the convergence rate and the location of terminal limit point. Numerical examples are provided to illustrate the theoretical results and present the goal-seeking capability of the systems.

Original languageEnglish
Pages (from-to)3514-3527
Number of pages14
JournalIEEE Transactions on Circuits and Systems I: Regular Papers
Issue number11
Publication statusPublished - 2008
Externally publishedYes


  • Absolute stability
  • Differential inclusion
  • Nonsmooth cost function
  • Łojasiewicz inequality


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