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Numerical Construction of Nonsmooth Control Lyapunov Functions

  • Robert Baier
  • Philipp BraunEmail author
  • Lars Grüne
  • Christopher M. Kellett
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2227)

Abstract

Lyapunov’s second method is one of the most successful tools for analyzing stability properties of dynamical systems. If a control Lyapunov function is known, then asymptotic stabilizability of an equilibrium of the corresponding dynamical system can be concluded without the knowledge of an explicit solution of the dynamical system. Whereas necessary and sufficient conditions for the existence of nonsmooth control Lyapunov functions are known by now, constructive methods to generate control Lyapunov functions for given dynamical systems are not known up to the same extent. In this paper we build on previous work to compute (control) Lyapunov functions based on linear programming and mixed integer linear programming. In particular, we propose a mixed integer linear program based on a discretization of the state space where a continuous piecewise affine control Lyapunov function can be recovered from the solution of the optimization problem. Different to previous work, we incorporate a semiconcavity condition into the formulation of the optimization problem. Results of the proposed scheme are illustrated on the example of Artstein’s circles and on a two-dimensional system with two inputs. The underlying optimization problems are solved in Gurobi (2016, http://www.gurobi.com).

Keywords

Control Lyapunov functions Mixed integer programming Dynamical systems 

AMS Subject Classifications

93D30 90C11 93D05 93D20 

Notes

Acknowledgements

The authors acknowledge the numerical experiments of Sigurdur Hafstein who calculated a control Lyapunov function for the special case of Artstein’s circles. His approach with mixed-integer programming differs from the presented work but certainly was one source of motivation to develop another approach based on local semiconcavity.

The authors, P. Braun, L. Grüne and C. M. Kellett, are supported by the Australian Research Council (Grant number: ARC-DP160102138).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Robert Baier
    • 1
  • Philipp Braun
    • 1
    • 2
    Email author
  • Lars Grüne
    • 1
  • Christopher M. Kellett
    • 2
  1. 1.University of BayeuthChair of Applied MathematicsBayreuthGermany
  2. 2.University of NewcastleSchool of Electrical Engineering and ComputingCallaghanAustralia

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