Abstract
In this paper, we study the null controllability for general backward and forward linear stochastic parabolic equations with dynamic boundary conditions. For this purpose, new Carleman estimates for the solutions of both forward and backward stochastic adjoint problems are established.
Similar content being viewed by others
References
Barbu V, Răşcanu A, Tessitore G. Carleman estimates and controllability of linear stochastic heat equations. Appl Math Optim. 2003;47:97–120.
Chae D, Imanuvilov O Yu, Kim SM. Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J Dyn Control Syst 1996;2:449–483.
Chueshov I, Schmalfuss B. Parabolic stochastic partial differential equations with dynamical boundary conditions. Diff Integr Equations. 2004;17:751–80.
Da Prato G, Zabczyk J. Stochastic equations in infinite dimensions. Cambridge University Press; 2014.
Fernández-Cara E, González-Burgos M, Guerrero S, Puel JP. Null controllability of the heat equation with boundary Fourier conditions: the linear case. ESAIM Control Optim Calc Var. 2006;12:442–65.
Fernández-Cara E, Guerrero S. Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J Control Optim. 2006;45:1395–446.
Fursikov A. Optimal control of distributed systems. Theory and applications: American Mathematical Society; 1999.
Fursikov A, Imanuvilov OY. Controllability of evolution equations, vol. 34. Lecture notes series. Research Institute of Mathematics: Seoul national university; 1996.
Imanuvilov OY, Yamamoto M. Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinearparabolic equations. Publications of the Research Institute for Mathematical Sciences. 2003;39:227–74.
Khoutaibi A, Maniar L. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evol Equ Control Theory. 2020;9:535–59.
Lebeau G, Robbiano L. Contrôle exact de l’équation de la chaleur. Communications in Partial Differential Equations. 1995;20:335–56.
Lions JL. Some aspects of the optimal control of distributed parameter systems SIAM J Control Optim 1972.
Liu X. Global Carleman estimate for stochastic parabolic equations, and its application. ESAIM Control Optim Calc Var. 2014;20:823–39.
Liu X, Yu Y. Carleman estimates of some stochastic degenerate parabolic equations and application. SIAM J Control Optim. 2019;57:3527–52.
Lü Q. Some results on the controllability of forward stochastic heat equations with control on the drift. J Funct Anal. 2011;260:832–51.
Lü Q, Zhang X. Mathematical control theory for stochastic partial differential equations. Springer; 2021.
Maniar L, Meyries M, Schnaubelt R. Null controllability for parabolic equations with dynamic boundary conditions. Evol Equ Control Theory. 2017;6:381–407.
Pardoux E, Peng S. Adapted solution of a backward stochastic differential equation. Syst Control Lett. 1990;14:55–61.
Tang S, Zhang X. Null controllability for forward and backward stochastic parabolic equations. SIAM J Control Optim. 2009;48:2191–216.
Barbu V, Bonaccorsi S, Tubaro L. Stochastic parabolic equations with nonlinear dynamical boundary conditions. J Math Anal Appl. 2015;427:484–98.
Yan Y. Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications. J Math Anal Appl. 2018;457:248–72.
Zabczyk J. Mathematical control theory. Springer; 2020.
Zhang X. Unique continuation for stochastic parabolic equations. Diff Integr Equations. 2008;21:81–93.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Baroun, M., Boulite, S., Elgrou, A. et al. Null Controllability for Stochastic Parabolic Equations with Dynamic Boundary Conditions. J Dyn Control Syst 29, 1727–1756 (2023). https://doi.org/10.1007/s10883-023-09656-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-023-09656-y
Keywords
- Null controllability
- Linear stochastic parabolic equations
- Observability
- Global Carleman estimates
- Dynamic boundary conditions