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Quadratic Lyapunov functions for linear systems

  • Y. V. Venkatesh (a1)

The paper deals with the existence of a quadratic Lyapunov function V = x′P(t)x for an exponentially stable linear system with varying coefficients described by the vector differential equation The derivative dV/dt is allowed to be strictly semi-(F) and the locus dV/dt = 0 does not contain any arc of the system trajectory. It is then shown that the coefficient matrix A(t) of the exponentially stable system is not identically equal to a unit matrix multiplied by a scalar. The result subsumes that of Lehnigk(1).

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(1)Lehnigk S. H.Quadratic forms as Liapunov functions for linear differential equations with real constant coefficients. Proc. Cambridge Philos. Soc. 61 (1965), 883888.
(2)Malkin I. G.On the construction of Lyapunov functions for systems of linear equations. Prikl. Mat. Meh. 16 (1952), 239242.
(3)Coddington B. A. and Levinson N.Theory of ordinary differential equations (McGraw Hill; New York, 1955).
(4)Hahn W.Theory and applications of Lyapunov's direct method (Prentice-Hall, Englewood Cliffs; N.J. 1963).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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