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Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups
  Published:2011-04-14
 Printed: Dec 2012
Canadian Mathematical Bulletin
  • Song Xueli,
    Department of Mathematics and Information Science, Chang'an University, Xi'an 710064, China
  • Peng Jigen,
    Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China
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Abstract

$L_p$ stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is $L_p$ stable for some $p>0$. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones.
Keywords: exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroups exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroups
MSC Classifications: 34D05, 47H20 show english descriptions Asymptotic properties
Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 34D05 - Asymptotic properties
47H20 - Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]
 
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