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Arc-length-based Lyapunov tests for convergence and stability with applications to systems having a continuum of equilibria

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Abstract

In this paper, fundamental relationships are established between convergence of solutions, stability of equilibria, and arc length of orbits. More specifically, it is shown that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium. Next, arc-length-based Lyapunov tests are derived for convergence and stability. These tests do not require the Lyapunov function to be positive definite. Instead, these results involve an inequality relating the right-hand side of the differential equation and the Lyapunov function derivative. This inequality makes it possible to deduce properties of the arc length function and thus leads to sufficient conditions for convergence and stability. Finally, it is shown that the converses of all the main results hold under additional assumptions. Examples are included to illustrate how our results are particularly suited for analyzing stability of systems having a continuum of equilibria.

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Authors and Affiliations

  1. TCS Innovation Labs Hyderabad, Tata Consultancy Services Limited, Deccan Park, Hi Tech City, Madhapur, Hyderabad, 500081, India

    Sanjay P. Bhat

  2. Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI, 48109-2140, USA

    Dennis S. Bernstein

Authors
  1. Sanjay P. Bhat
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  2. Dennis S. Bernstein
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Correspondence to Sanjay P. Bhat.

Additional information

Preliminary versions of the results of this paper appeared in the proceedings of the American Control Conference, 1999 and 2003.

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Bhat, S.P., Bernstein, D.S. Arc-length-based Lyapunov tests for convergence and stability with applications to systems having a continuum of equilibria. Math. Control Signals Syst. 22, 155–184 (2010). https://doi.org/10.1007/s00498-010-0054-3

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  • DOI: https://doi.org/10.1007/s00498-010-0054-3

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