# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2022004
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## A semigroup approach to stochastic systems with input delay at the boundary

 Department of Mathematics, Faculty of Sciences Agadir, Ibn Zohr University, Hay Dakhla, BP. 8106, 80000–Agadir, Morocco

Received  July 2021 Revised  December 2021 Early access February 2022

This work focuses on the well-posedness of abstract stochastic linear systems with boundary input delay and unbounded observation operators. We use product spaces and a semigroup approach to reformulate such delay systems into free-delay distributed stochastic systems with unbounded control and observation operators. This gives us the opportunity to use the concept of admissible control and observation operators as well as the concept of Yosida extensions to prove the existence and uniqueness of the solution process and provide an estimation of the observation process in relation to initial conditions and control process. As an example, we consider a stochastic Schrödinger system with input delay.

Citation: S. Hadd, F.Z. Lahbiri. A semigroup approach to stochastic systems with input delay at the boundary. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022004
##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9. [2] V. Barbu, Partial Differential Equations and Boundary Value Problems, Mathematics and its Applications, 441. Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-94-015-9117-1. [3] G. Bayili, S. Nicaise and R. Silga, Rational energy decay rate for the wave equation with delay term on the dynamical control, J. Math. Anal. Appl., 495 (2021), Paper No. 124693, 17 pp. doi: 10.1016/j.jmaa.2020.124693. [4] H. Cui, G. Xu and Y. Chen, Stabilization for Schrödinger equation with a distributed time delay in the boundary input, IMA J. Math. Control Inform., 36 (2019), 1305-1324.  doi: 10.1093/imamci/dny030. [5] R. J. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. [6] G. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [7] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [8] G. Fabbri and B. Goldys, An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise, SIAM J. Control Optim., 48 (2009), 1473-1488.  doi: 10.1137/070711529. [9] F. Gozzi and F. Masiero, Stochastic optimal control with delay in the control II: Verification theorem and optimal feedbacks, SIAM J. Control Optimi., 55 (2017), 3013-3038.  doi: 10.1137/16M1073637. [10] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math, 13 (1987), 213-229. [11] B. Z. Guo and Z. C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systems Control Lett., 54 (2005), 1135-1142.  doi: 10.1016/j.sysconle.2005.04.008. [12] S. Hadd, Unbounded perturbation of $C_0$-semigroups on Bananch spaces and applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7. [13] S. Hadd, An evolution equation approach to nonautonomous linear systems with state, input, and output delays, SIAM J. Control Optim., 45 (2006), 246-272.  doi: 10.1137/040612178. [14] S. Hadd, A. Idrissi and A. Rhandi, The regular linear systems associated to the shift semigroups and application to control delay systems, Math. Control Signals Systems, 18 (2006) 272–291. doi: 10.1007/s00498-006-0002-4. [15] S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete Contin. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703. [16] Z.-J. Han and G.-Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM Control Optim. Calc. Var., 17 (2011), 552-574.  doi: 10.1051/cocv/2010009. [17] V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, Chichester, 1994. [18] F. Z. Lahbiri and S. Hadd, A functional analytic approach to infinite dimensional stochastic linear systems, SIAM J. Control Optim., 59 (2021), 3762-3786.  doi: 10.1137/21M1389869. [19] F. Lamoline and J. Winkin, Well-posedness of boundary controlled and observed stochastic Port-Hamiltonian systems, IEEE Trans. Aut. Control, 65 (2020), 4258-4264.  doi: 10.1109/TAC.2019.2954481. [20] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. [21] Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM J. Control and Optim., 53 (2015), 3457-3482.  doi: 10.1137/151002605. [22] Q. Lü and X. Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Springer, Switzerland AG., 2021. [23] Q. Lü and X. Zhang, A concise introduction to control theory for stochastic partial differential equations, Mathematical Control and Related Fields. doi: 10.3934/mcrf.2021020. [24] M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs, 31. Princeton University Press, 2005. [25] D. Salamon, Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351. [26] O. J. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. [27] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäser, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [28] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math, 65 (1989), 17-43.  doi: 10.1007/BF02788172. [29] G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.  doi: 10.1137/0327028. [30] G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484. [31] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.

show all references

##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9. [2] V. Barbu, Partial Differential Equations and Boundary Value Problems, Mathematics and its Applications, 441. Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-94-015-9117-1. [3] G. Bayili, S. Nicaise and R. Silga, Rational energy decay rate for the wave equation with delay term on the dynamical control, J. Math. Anal. Appl., 495 (2021), Paper No. 124693, 17 pp. doi: 10.1016/j.jmaa.2020.124693. [4] H. Cui, G. Xu and Y. Chen, Stabilization for Schrödinger equation with a distributed time delay in the boundary input, IMA J. Math. Control Inform., 36 (2019), 1305-1324.  doi: 10.1093/imamci/dny030. [5] R. J. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. [6] G. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [7] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [8] G. Fabbri and B. Goldys, An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise, SIAM J. Control Optim., 48 (2009), 1473-1488.  doi: 10.1137/070711529. [9] F. Gozzi and F. Masiero, Stochastic optimal control with delay in the control II: Verification theorem and optimal feedbacks, SIAM J. Control Optimi., 55 (2017), 3013-3038.  doi: 10.1137/16M1073637. [10] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math, 13 (1987), 213-229. [11] B. Z. Guo and Z. C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systems Control Lett., 54 (2005), 1135-1142.  doi: 10.1016/j.sysconle.2005.04.008. [12] S. Hadd, Unbounded perturbation of $C_0$-semigroups on Bananch spaces and applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7. [13] S. Hadd, An evolution equation approach to nonautonomous linear systems with state, input, and output delays, SIAM J. Control Optim., 45 (2006), 246-272.  doi: 10.1137/040612178. [14] S. Hadd, A. Idrissi and A. Rhandi, The regular linear systems associated to the shift semigroups and application to control delay systems, Math. Control Signals Systems, 18 (2006) 272–291. doi: 10.1007/s00498-006-0002-4. [15] S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete Contin. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703. [16] Z.-J. Han and G.-Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM Control Optim. Calc. Var., 17 (2011), 552-574.  doi: 10.1051/cocv/2010009. [17] V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, Chichester, 1994. [18] F. Z. Lahbiri and S. Hadd, A functional analytic approach to infinite dimensional stochastic linear systems, SIAM J. Control Optim., 59 (2021), 3762-3786.  doi: 10.1137/21M1389869. [19] F. Lamoline and J. Winkin, Well-posedness of boundary controlled and observed stochastic Port-Hamiltonian systems, IEEE Trans. Aut. Control, 65 (2020), 4258-4264.  doi: 10.1109/TAC.2019.2954481. [20] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. [21] Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM J. Control and Optim., 53 (2015), 3457-3482.  doi: 10.1137/151002605. [22] Q. Lü and X. Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Springer, Switzerland AG., 2021. [23] Q. Lü and X. Zhang, A concise introduction to control theory for stochastic partial differential equations, Mathematical Control and Related Fields. doi: 10.3934/mcrf.2021020. [24] M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs, 31. Princeton University Press, 2005. [25] D. Salamon, Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351. [26] O. J. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. [27] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäser, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [28] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math, 65 (1989), 17-43.  doi: 10.1007/BF02788172. [29] G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.  doi: 10.1137/0327028. [30] G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484. [31] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.
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