In stability theory and nonlinear control, Massera's lemma, named after José Luis Massera, deals with the construction of the Lyapunov function to prove the stability of a dynamical system.[1] The lemma appears in (Massera 1949, p. 716) as the first lemma in section 12, and in more general form in (Massera 1956, p. 195) as lemma 2. In 2004, Massera's original lemma for single variable functions was extended to the multivariable case, and the resulting lemma was used to prove the stability of switched dynamical systems, where a common Lyapunov function describes the stability of multiple modes and switching signals.

Massera's original lemma

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Massera’s lemma is used in the construction of a converse Lyapunov function of the following form (also known as the integral construction)

 

for an asymptotically stable dynamical system whose stable trajectory starting from  

The lemma states:

Let   be a positive, continuous, strictly decreasing function with   as  . Let   be a positive, continuous, nondecreasing function. Then there exists a function   such that

  •   and its derivative   are class-K functions defined for all t ≥ 0
  • There exist positive constants k1, k2, such that for any continuous function u satisfying 0 ≤ u(t) ≤ g(t) for all t ≥ 0,
 

Extension to multivariable functions

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Massera's lemma for single variable functions was extended to the multivariable case by Vu and Liberzon.[2]

Let   be a positive, continuous, strictly decreasing function with   as  . Let   be a positive, continuous, nondecreasing function. Then there exists a differentiable function   such that

  •   and its derivative   are class-K functions on  .
  • For every positive integer  , there exist positive constants k1, k2, such that for any continuous function   satisfying
  for all  ,  
we have
 
 

References

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  • Massera, José Luis (1949), "On Liapounoff's conditions of stability", Annals of Mathematics, Second Series, 50 (3): 705–721, doi:10.2307/1969558, JSTOR 1969558, MR 0035354
  • Massera, José Luis (1956), "Contributions to stability theory", Annals of Mathematics, Second Series, 64 (1): 182–206, doi:10.2307/1969955, JSTOR 1969955, MR 0079179
  • Massera, José Luis; Schäffer, Juan Jorge (1966), Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Boston, MA: Academic Press, MR 0212324

Footnotes

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  1. ^ Khalil, H.K. (2001), Nonlinear Systems, Prentice Hall, ISBN 978-0-13-067389-3
  2. ^ Vu, L.; Liberzon, D. (2005), "Common Lyapunov functions for families of commuting nonlinear systems", Systems & Control Letters, 54 (5): 405–416, CiteSeerX 10.1.1.590.5565, doi:10.1016/j.sysconle.2004.09.006.