# American Institute of Mathematical Sciences

March  2015, 20(2): 613-640. doi: 10.3934/dcdsb.2015.20.613

## Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces

 1 Université d'Orléans, Bâtiment de Mathématiques (MAPMO), B.P. 6759, 45067 Orléans cedex 2

Received  December 2012 Revised  February 2014 Published  January 2015

We address in this work the minimization of the $L^q$-norm $(q>2)$ of semidiscrete controls for parabolic equation. As shown in [15], under the main approximation assumptions that the discretized semigroup is uniformly analytic and that the degree of unboundedness of control operator is lower than 1/2, uniform controllability is achieved in $L^2$ for semidiscrete approximations for the parabolic systems. The main goal of this paper is to overcome the limitation of [15] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform controllability property also holds in $L^q \ (q>2)$ even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls in $L^q\ (q>2)$ is provided. An example of application is implemented for the one-dimensional heat equation with Dirichlet boundary control.
Citation: Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613
##### References:
 [1] F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators and uniform controllability of semi discretized parabolic equations,, J. Math. Pur. Appl., 93 (2010), 240.  doi: 10.1016/j.matpur.2009.11.003.  Google Scholar [2] F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators in arbitrary dimension and applications,, SIAM J. Control Optim, 48 (2010), 5357.  doi: 10.1137/100784278.  Google Scholar [3] F. Boyer, F. Hubert and J. Le Rousseau, Uniform null-controllability properties for space/time-discretized parabolic equations,, Numer. Math, 118 (2011), 601.  doi: 10.1007/s00211-011-0368-1.  Google Scholar [4] J. Bramble, A. Shatz, V. Thomee and L. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations,, SIAM J. Num. Anal., 14 (1977), 218.  doi: 10.1137/0714015.  Google Scholar [5] C. Carthel, R. Glowinski and J. L. Lions, On exact and approximate Boundary Controllabilities for the heat equation: a numerical approach,, J. Optimal. Theory Appl., 82 (1994), 429.  doi: 10.1007/BF02192213.  Google Scholar [6] Y. Chitour and E. Trélat, Controllability of partial differential equations, Advanced topics in control systems theory,, Lecture Notes in Control and Inform. Sci, 328 (2006), 171.  doi: 10.1007/11583592_5.  Google Scholar [7] I. Ekeland and R. Temam, Convex Analysic and Variational Problems,, Classics in Applied Mathematics 28, (1999).  doi: 10.1137/1.9781611971088.  Google Scholar [8] S. Ervedoza and J. Valein, On the observability of abstract time-discrete linear parabolic equations,, Revista Matematica Complutense, 23 (2010), 163.  doi: 10.1007/s13163-009-0014-y.  Google Scholar [9] S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems,, Journal of Functional Analysis, 254 (2008), 3037.  doi: 10.1016/j.jfa.2008.03.005.  Google Scholar [10] H. O. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems,, Appl. Math. Optim., 15 (1987), 141.  doi: 10.1007/BF01442651.  Google Scholar [11] H. O. Fattorini and H. Frankowska, Necessary conditions for infinite dimensional problems,, Mathematics of Control Signals, 4 (1991), 41.  doi: 10.1007/BF02551380.  Google Scholar [12] A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations,, Lecture notes, vol 34 (1996).   Google Scholar [13] R. Glowinski, J. L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach,, Encyclopedia of Mathematical and its Applications, (2008).  doi: 10.1017/CBO9780511721595.  Google Scholar [14] J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation,, M2AN Math. Model. Number. Anal., 33 (1999), 407.  doi: 10.1051/m2an:1999123.  Google Scholar [15] S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control system,, Systems and Control Letters, 55 (2006), 597.  doi: 10.1016/j.sysconle.2006.01.004.  Google Scholar [16] S. Labbé and E. Trélat, Generalization of the finite difference method in distributions spaces,, Preprint HAL, (2006).   Google Scholar [17] I. Lasiecka and R. Triggiani, Control theory for partial differential equation: Continuous and approximation theories. I. Abstract parabolic systems,, Encyclopedia of Mathematics and its Applications, (2000).   Google Scholar [18] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335.  doi: 10.1080/03605309508821097.  Google Scholar [19] L. Leon and E. Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation,, ESAIM Control Optim., 8 (2002), 827.  doi: 10.1051/cocv:2002025.  Google Scholar [20] X. Li and Y. L. Yao, Maximum Principle of distributed parameter systems with time lags,, Lecture Notes in Control and Information Sciences, 75 (1985), 410.  doi: 10.1007/BFb0005665.  Google Scholar [21] X. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems,, SIAM J. Control Optim., 29 (1991), 895.  doi: 10.1137/0329049.  Google Scholar [22] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar [23] A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1-D heat equation,, Sem. EDP, VIII (1998), 1.   Google Scholar [24] A. Munch and E. Zuazua, Numerical approximation of the null controls for the heat equation through transmutation,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085018.  Google Scholar [25] M. Negreanu and E. Zuazua, Uniform boundary controllability of discre 1-D wave equation. Optimization and control of distributed systems,, Systems Control Lett., 48 (2003), 261.  doi: 10.1016/S0167-6911(02)00271-2.  Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Appl. Math. Sci., (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] L. S. Pontryagin, et al., The Mathematical Theory of Optimal Processes,, vol. 4. Interscience, (1962).   Google Scholar [28] O. Staffans, Well-posed Linear Systems,, Encyclopedia of Mathematical and its Applications, (2005).  doi: 10.1017/CBO9780511543197.  Google Scholar [29] E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics,, Vuibert, (2005).   Google Scholar [30] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhauser Advanced Texts Basler Lehrbucher, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar [31] E. Zuazua, Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square,, J. Math. Pures Appl., 78 (1999), 523.  doi: 10.1016/S0021-7824(98)00008-7.  Google Scholar [32] E. Zuazua, Controllability of the partial differential equations and its semi-discrete approximations,, Discrete Contin. Dyn. Syst., 8 (2002), 469.  doi: 10.3934/dcds.2002.8.469.  Google Scholar [33] E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for 1-D wave equation,, Rendiconti di Matematica VIII, 24 (2004), 201.   Google Scholar [34] E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Wave approximated by finite-difference method,, SIAM Review, 47 (2005), 197.  doi: 10.1137/S0036144503432862.  Google Scholar [35] E. Zuazua, Control and numerical approximation of the wave and heat equations,, International Congress of Mathematicians, III (2006), 1389.   Google Scholar

show all references

##### References:
 [1] F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators and uniform controllability of semi discretized parabolic equations,, J. Math. Pur. Appl., 93 (2010), 240.  doi: 10.1016/j.matpur.2009.11.003.  Google Scholar [2] F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators in arbitrary dimension and applications,, SIAM J. Control Optim, 48 (2010), 5357.  doi: 10.1137/100784278.  Google Scholar [3] F. Boyer, F. Hubert and J. Le Rousseau, Uniform null-controllability properties for space/time-discretized parabolic equations,, Numer. Math, 118 (2011), 601.  doi: 10.1007/s00211-011-0368-1.  Google Scholar [4] J. Bramble, A. Shatz, V. Thomee and L. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations,, SIAM J. Num. Anal., 14 (1977), 218.  doi: 10.1137/0714015.  Google Scholar [5] C. Carthel, R. Glowinski and J. L. Lions, On exact and approximate Boundary Controllabilities for the heat equation: a numerical approach,, J. Optimal. Theory Appl., 82 (1994), 429.  doi: 10.1007/BF02192213.  Google Scholar [6] Y. Chitour and E. Trélat, Controllability of partial differential equations, Advanced topics in control systems theory,, Lecture Notes in Control and Inform. Sci, 328 (2006), 171.  doi: 10.1007/11583592_5.  Google Scholar [7] I. Ekeland and R. Temam, Convex Analysic and Variational Problems,, Classics in Applied Mathematics 28, (1999).  doi: 10.1137/1.9781611971088.  Google Scholar [8] S. Ervedoza and J. Valein, On the observability of abstract time-discrete linear parabolic equations,, Revista Matematica Complutense, 23 (2010), 163.  doi: 10.1007/s13163-009-0014-y.  Google Scholar [9] S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems,, Journal of Functional Analysis, 254 (2008), 3037.  doi: 10.1016/j.jfa.2008.03.005.  Google Scholar [10] H. O. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems,, Appl. Math. Optim., 15 (1987), 141.  doi: 10.1007/BF01442651.  Google Scholar [11] H. O. Fattorini and H. Frankowska, Necessary conditions for infinite dimensional problems,, Mathematics of Control Signals, 4 (1991), 41.  doi: 10.1007/BF02551380.  Google Scholar [12] A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations,, Lecture notes, vol 34 (1996).   Google Scholar [13] R. Glowinski, J. L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach,, Encyclopedia of Mathematical and its Applications, (2008).  doi: 10.1017/CBO9780511721595.  Google Scholar [14] J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation,, M2AN Math. Model. Number. Anal., 33 (1999), 407.  doi: 10.1051/m2an:1999123.  Google Scholar [15] S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control system,, Systems and Control Letters, 55 (2006), 597.  doi: 10.1016/j.sysconle.2006.01.004.  Google Scholar [16] S. Labbé and E. Trélat, Generalization of the finite difference method in distributions spaces,, Preprint HAL, (2006).   Google Scholar [17] I. Lasiecka and R. Triggiani, Control theory for partial differential equation: Continuous and approximation theories. I. Abstract parabolic systems,, Encyclopedia of Mathematics and its Applications, (2000).   Google Scholar [18] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335.  doi: 10.1080/03605309508821097.  Google Scholar [19] L. Leon and E. Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation,, ESAIM Control Optim., 8 (2002), 827.  doi: 10.1051/cocv:2002025.  Google Scholar [20] X. Li and Y. L. Yao, Maximum Principle of distributed parameter systems with time lags,, Lecture Notes in Control and Information Sciences, 75 (1985), 410.  doi: 10.1007/BFb0005665.  Google Scholar [21] X. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems,, SIAM J. Control Optim., 29 (1991), 895.  doi: 10.1137/0329049.  Google Scholar [22] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar [23] A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1-D heat equation,, Sem. EDP, VIII (1998), 1.   Google Scholar [24] A. Munch and E. Zuazua, Numerical approximation of the null controls for the heat equation through transmutation,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085018.  Google Scholar [25] M. Negreanu and E. Zuazua, Uniform boundary controllability of discre 1-D wave equation. Optimization and control of distributed systems,, Systems Control Lett., 48 (2003), 261.  doi: 10.1016/S0167-6911(02)00271-2.  Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Appl. Math. Sci., (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] L. S. Pontryagin, et al., The Mathematical Theory of Optimal Processes,, vol. 4. Interscience, (1962).   Google Scholar [28] O. Staffans, Well-posed Linear Systems,, Encyclopedia of Mathematical and its Applications, (2005).  doi: 10.1017/CBO9780511543197.  Google Scholar [29] E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics,, Vuibert, (2005).   Google Scholar [30] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhauser Advanced Texts Basler Lehrbucher, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar [31] E. Zuazua, Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square,, J. Math. Pures Appl., 78 (1999), 523.  doi: 10.1016/S0021-7824(98)00008-7.  Google Scholar [32] E. Zuazua, Controllability of the partial differential equations and its semi-discrete approximations,, Discrete Contin. Dyn. Syst., 8 (2002), 469.  doi: 10.3934/dcds.2002.8.469.  Google Scholar [33] E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for 1-D wave equation,, Rendiconti di Matematica VIII, 24 (2004), 201.   Google Scholar [34] E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Wave approximated by finite-difference method,, SIAM Review, 47 (2005), 197.  doi: 10.1137/S0036144503432862.  Google Scholar [35] E. Zuazua, Control and numerical approximation of the wave and heat equations,, International Congress of Mathematicians, III (2006), 1389.   Google Scholar
 [1] Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control & Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037 [2] Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020052 [3] Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011 [4] Thuy N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control & Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203 [5] Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469 [6] Sylvie Benzoni-Gavage, Pierre Huot. Existence of semi-discrete shocks. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 163-190. doi: 10.3934/dcds.2002.8.163 [7] Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699 [8] Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029 [9] Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $1$-$d$ coupled wave equations. Mathematical Control & Related Fields, 2019  doi: 10.3934/mcrf.2020015 [10] Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 [11] Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761 [12] Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639 [13] Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263 [14] Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695 [15] Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031 [16] Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020 [17] Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001 [18] Debayan Maity. On the null controllability of the Lotka-Mckendrick system. Mathematical Control & Related Fields, 2019, 9 (4) : 719-728. doi: 10.3934/mcrf.2019048 [19] Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143 [20] J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136

2019 Impact Factor: 1.27

## Metrics

• PDF downloads (32)
• HTML views (0)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]