Stabilization of the transmission Schrodinger equation with boundary time-varying delay
Authors: Moumen Latifa , Rebiai Salah-Eddine 
Published: January 2025

We consider a system of transmission of the Schrodinger equation with Neumann feedback control that contains a time-varying delay term and that acts on the exterior boundary. Using a suitable energy function and a suitable Lyapunov functional, we prove under appropriate assumptions that the solutions decay exponentially.
MSC: 35Q93, 93D15
REFERENCES[1] Allag, I., & Rebiai, S. (2014). Well-posedness, regularity and exact controllability for the problem of transmission of the Schrödinger equation. Quarterly of Applied Mathematics, 72(1), 93-108.‏. Search in Google Scholar   Digital Object Identifier MathSciNet[2] Bayili, G., Aissa, A. B., & Nicaise, S. (2020). Same decay rate of second order evolution equations with or without delay. Systems & Control Letters, 141, 104700.‏. Search in Google Scholar   Digital Object Identifier[3] Cavalcanti, M. M., Corrêa, W. J., Lasiecka, I., & Lefler, C. (2016). Well-posedness and uniform stability for nonlinear Schrödinger equations with dynamic/Wentzell boundary conditions. Indiana University Mathematics Journal, 1445-1502.‏. Search in Google Scholar   Article view[4] Chen, H., Xie, Y., & Genqi, X. (2019). Rapid stabilisation of multi-dimensional Schrödinger equation with the internal delay control. International Journal of Control, 92(11), 2521-2531. Search in Google Scholar   Digital Object Identifier[5] Cardoso, F., & Vodev, G. (2010). Boundary stabilization of transmission problems. Journal of mathematical physics, 51(2).‏ Search in Google Scholar   Digital Object Identifier[6] Cui, H. Y., Han, Z. J., & Xu, G. Q. (2016). Stabilization for Schrödinger equation with a time delay in the boundary input. Applicable Analysis, 95(5), 963-977.‏. Search in Google Scholar   Digital Object Identifier[7] Cui, H., Xu, G., & Chen, Y. (2019). Stabilization for Schrödinger equation with a distributed time delay in the boundary input. IMA Journal of Mathematical Control and Information, 36(4), 1305-1324.‏. Search in Google Scholar   Digital Object Identifier[8] Datko, R. (1988). Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization, 26(3), 697-713.‏. Search in Google Scholar   Digital Object Identifier[9] Datko, R., Lagnese, J., & Polis, M. (1986). An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM Journal on Control and Optimization, 24(1), 152-156.‏. Search in Google Scholar   Digital Object Identifier[10] Guo, B. Z., & Mei, Z. D. (2019). Output feedback stabilization for a class of first-order equation setting of collocated well-posed linear systems with time delay in observation. IEEE Transactions on Automatic Control, 65(6), 2612-2618.‏. Search in Google Scholar   Digital Object Identifier[11] Guo, B. Z., & Yang, K. Y. (2010). Output feedback stabilization of a one-dimensional Schrödinger equation by boundary observation with time delay. IEEE Transactions on Automatic Control, 55(5), 1226-1232.‏. Search in Google Scholar   Digital Object Identifier[12] Kato, T. (1985). Abstract differential equations and nonlinear mixed problems (p. 89). Pisa: Scuola normale superiore.‏ Search in Google Scholar[13] Kato, T. (2011). Linear and quasi-linear equations of evolution of hyperbolic type. In Hyperbolicity: Lectures given at the Centro Internazionale Matematico Estivo (CIME), held in Cortona (Arezzo), Italy, June 24–July 2, 1976 (pp. 125-191). Berlin, Heidelberg: Springer Berlin Heidelberg.. Search in Google Scholar   Digital Object Identifier[14] B. Kellogg (1972) Properties of solutions of elliptic boundary value problems, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations edited by A. K. Aziz, Academic Press, New York, 47-81.[15] Lasiecka, I., Triggiani, R., & Zhang, X. (2000). Nonconservative wave equations with unobserved Neumann BC: global uniqueness and observability in one shot. Contemporary Mathematics, 268, 227-326.‏. Search in Google Scholar   Article View[16] Machtyngier, E., & Zuazua, E. (1994). Stabilization of the Schrodinger equation. Portugaliae Mathematica, 51(2), 243-256.‏. Search in Google Scholar   Article view[17] Nicaise, S., & Rebiai, S. E. (2011). Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback. Portugaliae Mathematica, 68(1), 19-39.‏. Search in Google Scholar   Digital Object Identifier[18] Nicaise, S., & Pignotti, C. (2006). Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM Journal on Control and Optimization, 45(5), 1561-1585.‏.  Search in Google Scholar   Digital Object Identifier[19] Nicaise, S., Pignotti, C., & Valein, J. (2011). Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems-Series S, 4(3), 693-722. Search in Google Scholar   Article view[20] Rebiai, S. E., & Ali, F. S. (2016). Uniform exponential stability of the transmission wave equation with a delay term in the boundary feedback. IMA Journal of Mathematical Control and Information, 33(1), 1-20.‏. Search in Google Scholar   Digital Object Identifier[21] A.E. Taylor and D.C. Lay. (1980). Introduction to Functional Analysis. John Wiley and Sons, New York-Chichester-Brisbane Book View[22] Xu, G. Q., Yung, S. P., & Li, L. K. (2006). Stabilization of wave systems with input delay in the boundary control. ESAIM: Control, optimisation and calculus of variations, 12(4), 770-785. Search in Google Scholar   Digital Object Identifier[23] K.Y. Yang and C.Z. Yao (2013) Stabilization of one-dimensional Schrodinger equation with variable coefficient under delayed boundary output feedback. Asian J. Control, 15, 1531-1537.  Search in Google Scholar  Digital Object Identifier
Communicated Editor: Pr. Baowei FengManuscript received Dec 26, 2023; revised Feb 23, 2024; accepted Mar 10, 2024; published May 19, 2024.