---

Discontinuity, Nonlinearity, and Complexity 15(3) (2026) 331--340 | DOI:10.5890/DNC.2026.09.003

Khalil EL Kazoui, Hassan Ezzaki, Abed Boulouz

LAMA Laboratory, Department of Mathematics, Faculty of Sciences, Ibn Zohr University, 80000, Agadir, Morocco

Download Full Text PDF

 

Abstract

In this paper, we are concerned with the output stabilization of infinite dimensional bilinear systems with delayed observation. We first establish the well-posedness of such systems and then we provide sufficient conditions for strong and weak output stabilization. Through the analysis of the wave equation and beam equation, we demonstrate the applicability and effectiveness of our proposed methods.

References

1. [1]	Suh, H. and Bien, Z. (1980), Use of time-delay actions in the controller design, IEEE Transactions on Automatic Control, 25(3), 600-603.

2. [2]	Fridman, E. (2014), Introduction to time-delay systems: Analysis and control, Birkhäuser, 75.

3. [3]	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.

4. [4]	Nicaise, S. and 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.

5. [5]	Nicaise, S. and Valein, J. (2010), Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM: Control, Optimisation and Calculus of Variations, 16(2), 420-456.

6. [6]	Fridman, E., Nicaise, S., and Valein, J. (2010), Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM Journal on Control and Optimization, 48(8), 5028-5052.

7. [7]	Nicaise, S. and Pignotti, C. (2014), Stabilization of second-order evolution equations with time delay, Mathematics of Control, Signals, and Systems, 26, 563-588.

8. [8]	Hamidi, Z., Elazzouzi, A., and Ouzahra, M. (2024), Feedback stabilization of delayed bilinear systems, Differential Equations and Dynamical Systems, 32(1), 87-100.

9. [9]	Guo, B.Z. and Yang, K.Y. (2009), Dynamic stabilization of an Euler–Bernoulli beam equation with time delay in boundary observation, Automatica, 45(6), 1468-1475.

10. [10]	Guo, B.Z. and 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.

11. [11]	Guo, B.Z., Xu, C.Z., and Hammouri, H. (2012), Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation, ESAIM: Control, Optimisation and Calculus of Variations, 18(1), 22-35.

12. [12]	Mei, Z.D. and Guo, B.Z. (2019), Stabilization for infinite-dimensional linear systems with bounded control and time delayed observation, Systems & Control Letters, 134, 104532.

13. [13]	Bounit, H. and Hammouri, H. (1999), Feedback stabilization for a class of distributed semilinear control systems, Nonlinear Analysis: Theory, Methods and Applications, 37(8), 953-969.

14. [14]	Ezzaki, L. and Zerrik, E.H. (2021), Stabilisation of second-order bilinear and semilinear systems, International Journal of Control, 94(12), 3483-3490.

15. [15]	Berrahmoune, L. (1999), Exponential decay for distributed bilinear control systems with damping, Rendiconti del Circolo Matematico di Palermo, 48, 111-122.

16. [16]	Ball, J.M. and Slemrod, M. (1979), Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems, Communications on Pure and Applied Mathematics, 32(4), 555-587.

17. [17]	Ball, J.M. and Slemrod, M. (1979), Feedback stabilization of distributed semilinear control systems, Applied Mathematics and Optimization, 5(1), 169-179.

18. [18]	Berrahmoune, L. (1999), Stabilization and decay estimate for distributed bilinear systems, Systems & Control Letters, 36(3), 167-171.

19. [19]	Ouzahra, M. (2010), Exponential and weak stabilization of constrained bilinear systems, SIAM Journal on Control and Optimization, 48(6), 3962-3974.

20. [20]	Zerrik, E.H. and Ezzaki, H. (2018), Output stabilization of distributed bilinear systems, Control Theory and Technology, 16(1), 58-71.

21. [21]	Zerrik, E.H. and Ezzaki, L. (2019), An output stabilization of infinite dimensional semilinear systems, IMA Journal of Mathematical Control and Information, 36(1), 101-123.

22. [22]	Zerrik, E.H. and Ouzahra, M. (2005), Output stabilization for infinite-dimensional bilinear systems, International Journal of Applied Mathematics and Computer Science, 15(2), 187-195.

23. [23]	Pazy, A. (2012), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science & Business Media.

24. [24]	Tucsnak, M. and Weiss, G. (2009), Observation and Control for Operator Semigroups, Springer Science & Business Media.