Well-Posedness and General Decay of Solutions for the Heat Equation with a Time Varying Delay Term


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Authors: A. BRAIK, A. BENIANI AND Y. MILOUDI

DOI: 10.46793/KgJMat2202.267B

Abstract:

We consider the nonlinear heat equation in a bounded domain with a time varying delay term

                  ∫
                     t
ut + Δ2u  −  J (t)    g (t− s )Δ2u  (s)ds + αK   (t)u + βK  (t)u (t −  τ(t)) =  0,
                    0
with initial conditions. By introducing suitable energy and Lyapunov functionals, under some assumptions, we then prove a general decay result of the energy associated of this system under some conditions.

Keywords:

Heat equation, time varying delay, energy decay, Lyapunov functional, global existence, viscoelastic term.



References:

[1]   C. T. Abdallah, P.  ˜
D   orato, J. Benites-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory system, American Control Conference, San Francisco, 1993, 3106–3107.

[2]   S. V. C. ˜H  uang, An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays, SIAM J. Sci. Comput. 25 (2004), 1608–1632.

[3]   S. V. C. ˜
H   uang, Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays, Front. Math. China 4 (2009), 63–87.

[4]   S. Y. C. Q. Xu and L. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var. 12 (2006), 770–785.

[5]   R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim. 26 (1988), 697–713.

[6]   B. Feng, General decay for a viscoelastic wave equation with strong time-dependent delay, Bound. Value Probl. (2017), 1–11.

[7]   M. Ferhat and A. Hakem, On convexity for energy decay rates of a viscoelastic wave equation with a dynamic boundary and nonlinear delay term, Facta Univ. Ser. Math. Inform. 30 (2015), 67–87.

[8]   E. Fridman and Y. Orlov, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica 45 (2009), 194–201.

[9]   R. R. H. ˜L  ogemann and G. Weiss, Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop, SIAM J. Control Optim. 34 (1996), 572–600.

[10]   W. Liu, General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term, Taiwanese J. Math. 17 (2013), 2101–2115.

[11]   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), 1561–1585.

[12]   S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, SIAM J. Control Optim. 21 (2008), 9–10.

[13]   S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations 41 (2011), 1–20.

[14]   S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. B 21 (2011), 693–722.

[15]   R. Rebarber and S. Townley, Robustness with respect to delays for exponential stability of distributed parameter systems, SIAM J. Control Optim. 37 (1998), 230–244.

[16]   J. B. S. ˜B   ernard and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete Contin. Dyn. Syst. Ser. B 1(2) (2001), 233–256.

[17]   S. A. Messaoudi, General decay of solutions of a weak viscoelastic equation, Arab. J. Sci. Eng. 36 (2011), 1569–1579.

[18]   J. V. S. Ñicaise and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S (2009), 559–581.

[19]   I. Suh and Z. Bien, Use of time delay action in the controller design, IEEE Trans. Automat. Control 25 (1980), 600–603.

[20]   J. R. T. ˜
C   araballo and L. Shaikhet, Method of lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl. (2007), 1130–1145.

[21]   T. X. Wang, Stability in abstract functional-differential equations. Part II. Applications, J. Math. Anal. Appl. 186 (1994), 835–861.

[22]   S. Wu and S. Gan, Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations, Comput. Math. Appl. 55 (2008), 2426–2443.

[23]   Q. C. Zhong, Robust control of time-delay systems, Springer, London, 2006.