Theoretical and Numerical Indirect Stabilization of Coupled Hyperbolic Systems with a Delay Term in the Internal Feedback

Abstract

The focal point of this paper is to theoretically investigate and numerically validate the effect of time delay on the exponential stabilization of a class of coupled hyperbolic systems with a delay term in the internal feedback. The class in question consists of two strongly coupled wave equations featuring delayed and non-delayed damping terms on the first wave equation. Through a standard change of variables and semigroup theory, we address the well-posedness of the considered coupled system. Thereon, based on some observability inequalities, we derive sufficient conditions guaranteeing the exponential decay of a suitable energy. On the other hand, from the numerical point of view, we validate the theoretical results in one-dimensional domains based on a suitable numerical approximation obtained through the Finite Difference Method. More precisely, we construct a discrete numerical scheme, which in addition to being converging and conditionally stable, preserves the energy decay property of its continuous counterpart. The theoretical analysis and the implementation of our developed numerical scheme assert the effect of the time-delayed damping on the exponential stability of strongly coupled wave equations.

Keywords

Stabilization / Hyperbolic systems / Delay feedbacks / Numerical analysis / 93B07 / 35L05 / 93C05 / 93D15 / 65M06 / 93C20

References

[1] Adimy M, Crauste F. Global stability of a partial differential equation with distributed delay due to cellular replication. Nonlinear Anal. Theory Methods Appl., 2003, 54(8): 1469-1491

[2] Akil M, Badawi H, Nicaise S, Wehbe A. Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface. Math. Methods Appl. Sci., 2021, 44(8): 6950-6981

[3] Akil M, Badawi H, Wehbe A. Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay. Commun. Pure Appl. Anal., 2021, 20(9): 2991-3028

[4] Alabau-Boussouira F. Indirect boundary stabilization of weakly coupled hyperbolic systems. SIAM J. Control. Optim., 2002, 41: 511-541

[5] Alabau-Boussouira F. A two level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM J. Control. Optim., 2003, 42(3): 871-906

[6] Alabau-Boussouira, F.: A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE’s by a reduced number of controls. Adv. Differ. Equ. 18, 1005–1072 (2013)

[7] Alabau-Boussouira F, Léautaud M. Indirect controllability of locally coupled wave-type systems and applications. J. Math. Pures Appl., 2013, 99: 544-576

[8] Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic Press, New York (2014)

[9] Ammari, K., Chentouf, B., Smaoui, N.: Well-posedness and stability of a nonlinear time-delayed dispersive equation via the fixed point technique: a case study of no interior damping. Math. Methods Appl. Sci. 45(8), 4555–4566 (2022)

[10] Ammari, K., Chentouf, B., Smaoui, N.: Note on the stabilization of a vibrating string via a switching time-delay boundary control: a theoretical and numerical study. SeMA 80(4), 647–662 (2023)

[11] Ammari, K., Nicaise, S.: Stabilization of Elastic Systems by Collocated Feedback. Springer, Cham (2014)

[12] Ammari K, Nicaise S, Pignotti C. Feedback boundary stabilization of wave equations with interior delay. Syst. Control Lett., 2010, 59(10): 623-628

[13] Avdonin S, Rivero AC, de Teresa L. Exact boundary controllability of coupled hyperbolic equations. Int. J. Appl. Math. Comput. Sci., 2013, 23(4): 701-710

[14] Barbu, V.: Partial Differential Equations and Boundary Value Problems. Kluwer Academic Publishers, Dordrecht (1998)

[15] Bardos C, Lebeau G, Rauch J. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control. Optim., 1992, 30: 1024-1065

[16] Benhassi, E.M.A., Ammari, K., Boulite, S., Maniar. L.: Exponential energy decay of some coupled second order systems. Semigroup Forum 86, 362–382 (2013)

[17] Bennour A, Ammar Khodja F, Teniou D. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evol. Equ. Control Theory, 2017, 6: 487-516

[18] Cazenave T, Haraux A. An Introduction to Semilinear Evolution Equations, 1998, Oxford, Oxford University Press

[19] Chen G. Control and stabilization for the wave equation in a bounded domain. SIAM J. Contol. Optim., 1979, 17(1): 66-81

[20] Coron JM. Control and Nonlinearity, 2007, Providence, American Mathematical Society

[21] D’Ancona P, Georgiev V, Kubo H. Weighted decay estimates for the wave equation. J. Differ. Equ., 2001, 177(1): 146-208

[22] Gerbi S, Kassem C, Mortada A, Wehbe A. Exact controllability and stabilization of locally coupled wave equations: theoretical results. Z. Anal. Anwend., 2021, 40(1): 67-96

[23] Gerbi, S., Kassem, C., Mortada, A., Wehbe, A.: Numerical study of the stabilization of 1D locally coupled wave equations. Z. Anal. Anwend. 40(2), 131–151 (2021)

[24] Komornik V. Decay estimates for the wave equation with internal damping. Int. Ser. Numer. Math., 1994, 118: 253-266

[25] Koumaiha, M., Toufaily, L., Wehbe, A.: Boundary observability and exact controllability of strongly coupled wave equations. Discrete Contin. Dyn. Syst. Ser. S 15, 1269–1305 (2022)

[26] Liu K. Locally distributed control and damping for the conservative systems. SIAM J. Cont. Optim., 1997, 35(5): 1574-1590

[27] Liu WJ, Zuazua E. Decay rates for dissipative wave equations. Ric. Mat., 1999, 48(240): 61-75

[28] Liu, Z., Rao, B.: A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete Contin. Dyn. Syst. 23, 399–414 (2009)

[29] Mokhtari, Y., Khodja, F.A.: Boundary controllability of two coupled wave equations with space-time first-order coupling in 1-D. J. Evol. Equ. 22(2), 31 (2022)

[30] Nicaise S. Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications. Rend. Mat. Appl., 2003, 23: 83-116

[31] Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Cont. Optim., 2006, 45(5): 1561-1585

[32] Nicaise, S., Valein, J.: Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM Control Optim. Calc. Var. 16(2), 420–456 (2010)

[33] Oliveira RL, Oquendo HP. Stability and instability results for coupled waves with delay term. J. Math. Phys., 2020, 61(7 071505

[34] Pignotti C. A note on stabilization of locally damped wave equations with time delay. Syst. Control Lett., 2012, 61(1): 92-97

[35] Rebiai, SE., Sidi Ali, F.Z.: Exponential stability of compactly coupled wave equations with delay terms in the boundary feedbacks. In: Pötzsche, C., Heuberger, C., Kaltenbacher, B., Rendl, F. (eds) System Modeling and Optimization. CSMO 2013. IFIP Advances in Information and Communication Technology, vol. 443, pp. 278–284. Springer, Berlin, Heidelberg (2014)

[36] Silga R, Kyelem BA, Bayili G. Indirect boundary stabilization with distributed delay of coupled multi-dimensional wave equations. Ann. Univ. Craiova Math., 2022, 49(1): 15-34

[37] Tébou L. Stabilization of the wave equation with localized nonlinear damping. J. Differ. Equ., 1998, 145(2): 502-524

[38] Wehbe A, Youssef W. Indirect locally internal observability and controllability of weakly coupled wave equations. Differ. Equ. Appl., 2011, 3: 449-462

[39] Willard, S.: General Topology. Courier Corporation, New York (2012)

[40] Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12(4), 770–785 (2006)

[41] Xu, G.Q., Zhang, L.: Uniform stabilization of 1-D coupled wave equations with anti-dampings and joint delayed control. SIAM J. Cont. Optim. 58(6), 3161–3184 (2020)

[42] Zuazua E. Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ., 1990, 15(2): 205-235

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