Indirect boundary stabilization with distributed delay of coupled multi-dimensional wave equations

Roland Silga, Bila Adolphe Kyelem, Gilbert Bayili

Abstract


In this article, our main concern is the study of the effect of a distributed time-delay in boundary stabilization of a strongly coupled multi-dimensional wave equations. We will establish that the system with time-delay inherits the same exponential decay rate from the corresponding one without delay

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M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis 54 (2003), no. 9, 1469-1491. https://doi.org/10.1016/S0362-546X(03)00197-4

M. Adimy, F.n Crauste, and A. Abdllaoui, Asymptotic behaviour of a discrete maturity structured system of hematopoietic stem cell dynamics with several delays, Mathematical Modelling of Natural Phenomena 1 (2006), no. 1, 1-22. https://doi.org/10.1051/mmnp:2008001

F.A. Khodja and A. Bader, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control and Optimization 39 (2001), no. 4, 1833-1851. https://doi.org/10.1137/S0363012900366613

C.D. Benchimol, A note on weak stabilization of contraction semigroups, SIAM J. Control Optim. 16 (1978), 373-379.

L. Berezansky and E. Braverman, Oscillation properties of a logistic equation with distributed delay, Nonlinear Analysis: Real World Applications 4 (2003), no. 3, 1-19. https://doi.org/10.1016/S1468-1218(02)00010-X

S.R. Foguel, Powers of contraction in Hilbert space, Pacific J. Math. 13 (1963), no. 1, 551-561.

K. Gu, J. Chen, and V.L. Kharitonov, Stability of Time-Delay Systems, Control Engineering, Birkhäuser Boston, 2003.

K. Gu, An improved stability criterion for systems with distributed delays, International Journal of Robust and Non-linear Control 13 (2003), no. 7, 819-831. https://doi.org/10.1002/rnc.847

Q.-L. Han, Stability criteria for a class of linear neutral systems with time-varying discrete and distributed delays, IMA Journal of Mathematical Control and Information 20 (2003), no. 4, 371-386. https://doi.org/10.1093/imamci/20.4.371

F.L. Huang, Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Berlin 35 (1985), 585-603.

J.-L. Lions and E. Magenes, Problémes aux limites non-homogénes et applications, vol. 1, Dunod, Paris, 1968.

S. Long and D. Xu, Global exponential stability of impulsive dynamical systems with distributed delays, Electronic Journal of Qualitative Theory of Differential Equations 10 (2007), no. 4, 1-13.

W. Michiels, S. Mondié, D. Roose, and M. Dambrine, The effect of approximating distributed delay control laws on stability, In: (S.I. Niculescu, K. Gu, editors) Advances in Time-Delay Systems, 207-222, Springer Berlin Heidelberg, 2004.

C.I. Morarescu, S.I. Niculescu, and W. Michiels, Asymptotic stability of some distributed delay systems: an algebraic approach, 13th IFAC Workshop on Control Application of Optimisation, April 2006, Cachan, France. Retrieved at https://hal.archives-ouvertes.fr/hal-02272259/

U. Mnz and F. Allgwer, $L^2$-gain based controller design for linear systems with distributed delays and rational delay kernels, IFAC Proceedings Volumes 40 (2007), no. 9, 77-82.

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006), no. 5, 1561-1585. https://doi.org/10.1137/060648891

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential and Integral Equations 21 (2008), no. 9-10, 935-958.

J. Pruss, On the spectrum of C_0-semigroups, Trans. Amer. Math. Soc. 284 (1984), 847-857.

J.-P. Richard, Time delay systems: an overview of some recent advances and open problems, Science direct automatica 39 (2003), no. 10, 1667-1694. https://doi.org/10.1016/S0005-1098(03)00167-5

R. Sipahi, F. Atay, and S.I. Niculescu, Stability of traffic flow behaviour with distributed delays modelling the memory effects of the drivers, SIAM Journal of Applied Mathematics 68 (2007), no. 1, 738-759. https://doi.org/10.1137/060673813

Y. Suh, H.-J. Kang, and Y. Ro, Stability condition of distributed delay systems based on an analytic solution to Lyapunov functional equations, Asian Journal of Control 8 (2006), no. 3, 91-96. https://doi.org/10.1111/j.1934-6093.2006.tb00258.x

B. Sz-Nagy and C. Foias, Analyse Harmonique des opérateurs de l'espace de Hilbert, Masson Paris, 1967.

L. Toufayli, Stabilisation polynomiale et et contrôlabilité exacte des équations des ondes par des contrôles indirects et dynamiques, PhD thesis, Université de Strasbourg, 2013.

S.H.R. Vadrevu and P. Rao, Global stability in chemostat models involving time delays and wall growth, Nonlinear Analysis-real World Applications 5 (2004), no. 2, 141-158. https://doi.org/10.1016/S1468-1218(03)00022-1

E.I. Verriest, Stability of systems with distributed delays, In: Preprints of the IFAC Conference on System, Structure and Control, July 1995, Nantes, France, 294-299.

E.I. Verriest, Linear systsems with rational distributed delay: Reduction and stability, 1999 European Control Conference (ECC) (1999), 3637-3642. https://10.23919/ECC.1999.7099895




DOI: https://doi.org/10.52846/ami.v49i1.1430