Equivalence of Lp Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups

Xueli, Song ; Jigen, Peng

Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 882-889

Résumé

${{L}_{p}}$ 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.

Détail

DOI : 10.4153/CMB-2011-070-0

Mots-clés : 34D05, 47H20, exponentially stable, Lp stable, nonlinear Lipschitzian semigroups

Bibliographie

[1] [1] Arendt, W., Batty, C. J. K., Hieber, M., and Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Probems. Monographs in Mathematics 96. Birkhäuser Verlag, Basel, 2001. Google Scholar

[2] [2] Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leiden, 1976. Google Scholar

[3] [3] Belleni-Morante, A. and Mcbride, A. C., Applied Nonlinear Semigroups: An Introduction. JohnWiley & Sons, Chichester, 1998. Google Scholar

[4] [4] Datko, R., Extending a theorem of A. M. Liapunov to Hilbert space. J. Math. Anal. Appl. 32(1970), 610–616. Google Scholar | DOI

[5] [5] Engel, K. J. and Nagel, R., One-parameter Semigroups of Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer-Verlag, New York, 2000. Google Scholar

[6] [6] Fattorini, H. O., The Cauchy Prolem. Encyclopedia of Mathematics and its Applications18. Addison-Wesley, Reading, MA, 1983. Google Scholar

[7] [7] Goldsein, J. A., Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs, Oxford University Press, New York, 1985. Google Scholar

[8] [8] Hahn, W., Stability of Motion. Die Grundlehren der mathematischen Wissenschaften 138. Springer-Verlag, New York, 1967. Google Scholar

[9] [9] Ichikawa, A., Stability and optimal control of stochastic evolution equations. In: Distributed Parameter Control Systems: Theory and Application, Chapter 5, Pergamon Press, New York, 1982. Google Scholar

[10] [10] Ichikawa, A., Equivalence of L stability and exponential stability for a class of nonlinear semigroups. Nonlinear Anal. 8(1984), no. 7, 805–815. Google Scholar | DOI

[11] [11] Kobayshi, Y. and Tanaka, N., Semigroups of Lipschitz operators. Adv. Differential Equations 6(2001), no. 5, 613–640. Google Scholar

[12] [12] Kobayshi, Y. and Tanaka, N., Convergence and approximation of semigroups of Lipschitz operators. Nonlinear Anal. 61(2005), no. 5, 781–821. Google Scholar | DOI

[13] [13] Miyadera, I., Nonlinear Semigroups. Translations of Mathematical Monographs 109. American Mathematical Society, Providence, RI, 1992. Google Scholar

[14] [14] van Neerven, J. M. A. A., The Asymptotic Behviour of Semigroups of Linear Operators. Operator Theory: Advances and Applications 88. Birkhäuser Verlag, Basel, 1996. Google Scholar

[15] [15] Pazy, A., On the applicability of Lyapunov's theorem in Hilbert space. SIAM J. Math. Anal. 3(1972), 291–294. Google Scholar | DOI

[16] [16] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44. Springer-Verlag, New York, 1983. Google Scholar

[17] [17] Peng, J. G., Theoretical Researchs on Nonlinear Lipschitz Opertors and their Applications. (in Chinese), Ph.D. dissertation, Xi’an Jiaotong University, 1998. Google Scholar

[18] [18] Peng, J. G. and Xu, Z. B., On asymptotic behaviours of nonlinear semigroup of Lipschitz operators with applications.(in Chinese), Acta Math. Sinica 45(2002), no. 6, 1099–1106. Google Scholar

[19] [19] Peng, J. G. and Xu, Z. B., A novel dual approach to nonlinear semigroups of Lipschitz operators. Trans. Amer. Math. Soc. 357(2005), no. 1, 409–424. Google Scholar | DOI

[20] [20] Zabczyk, J., Remarks on the control of discrete-time distributed parameter systems. SIAM J. Control 12(1974), 721–735. Google Scholar | DOI