American Institute of Mathematical Sciences

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

Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces

Thuy N. T. Nguyen 1,

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.
Keywords: semi-discrete system, Null-controllability, parabolic equation..
Mathematics Subject Classification: Primary: 93C20, 93B05, 93B40; Secondary: 65J1.
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), 240-276. 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), 5357-5397. 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), 601-661. 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), 218-241. 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), 429-484. 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, Spinger, London, 328 (2006), 171-198. doi: 10.1007/11583592_5.  Google Scholar

[7]

I. Ekeland and R. Temam, Convex Analysic and Variational Problems, Classics in Applied Mathematics 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 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), 163-190. 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), 3037-3078. 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), 141-185. doi: 10.1007/BF01442651.  Google Scholar

[11]

H. O. Fattorini and H. Frankowska, Necessary conditions for infinite dimensional problems, Mathematics of Control Signals, and Systems, 4 (1991), 41-67. doi: 10.1007/BF02551380.  Google Scholar

[12]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture notes, vol 34, Seoul National University, Korea, 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, 117. Cambridge University Press, Cambridge, 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), 407-438. 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), 597-609. 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, ccsd-00097806, 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, 74. Cambridge University Press, Cambridge, 2000.  Google Scholar

[18]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. 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., Calc. Var., 8 (2002), 827-862. 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, Spinger- Verlag, New York, 75 (1985), 410-427. 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), 895-908. doi: 10.1137/0329049.  Google Scholar

[22]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. 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, Ecole Polytechnique, VIII (1998), 1-22.  Google Scholar

[24]

A. Munch and E. Zuazua, Numerical approximation of the null controls for the heat equation through transmutation, Inverse Problems, 26 (2010), 085018, 39 pp. 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), 261-279. 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., 44, Springer-Verlag, 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, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar

[29]

E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics, Vuibert, Paris, 2005, 246 pp.  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), 523-563. 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), 469-513. 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), 201-237.  Google Scholar

[34]

E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Wave approximated by finite-difference method, SIAM Review, 47 (2005), 197-243. 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), 1389-1417.  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), 240-276. 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), 5357-5397. 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), 601-661. 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), 218-241. 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), 429-484. 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, Spinger, London, 328 (2006), 171-198. doi: 10.1007/11583592_5.  Google Scholar

[7]

I. Ekeland and R. Temam, Convex Analysic and Variational Problems, Classics in Applied Mathematics 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 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), 163-190. 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), 3037-3078. 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), 141-185. doi: 10.1007/BF01442651.  Google Scholar

[11]

H. O. Fattorini and H. Frankowska, Necessary conditions for infinite dimensional problems, Mathematics of Control Signals, and Systems, 4 (1991), 41-67. doi: 10.1007/BF02551380.  Google Scholar

[12]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture notes, vol 34, Seoul National University, Korea, 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, 117. Cambridge University Press, Cambridge, 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), 407-438. 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), 597-609. 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, ccsd-00097806, 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, 74. Cambridge University Press, Cambridge, 2000.  Google Scholar

[18]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. 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., Calc. Var., 8 (2002), 827-862. 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, Spinger- Verlag, New York, 75 (1985), 410-427. 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), 895-908. doi: 10.1137/0329049.  Google Scholar

[22]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. 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, Ecole Polytechnique, VIII (1998), 1-22.  Google Scholar

[24]

A. Munch and E. Zuazua, Numerical approximation of the null controls for the heat equation through transmutation, Inverse Problems, 26 (2010), 085018, 39 pp. 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), 261-279. 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., 44, Springer-Verlag, 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, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar

[29]

E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics, Vuibert, Paris, 2005, 246 pp.  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), 523-563. 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), 469-513. 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), 201-237.  Google Scholar

[34]

E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Wave approximated by finite-difference method, SIAM Review, 47 (2005), 197-243. 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), 1389-1417.  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, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052
[3]

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
[4]

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
[5]

Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469
[6]

Zimo Zhu, Gang Chen, Xiaoping Xie. Semi-discrete and fully discrete HDG methods for Burgers' equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021132
[7]

Sylvie Benzoni-Gavage, Pierre Huot. Existence of semi-discrete shocks. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 163-190. doi: 10.3934/dcds.2002.8.163
[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]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699
[10]

André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021024
[11]

Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015
[12]

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
[13]

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
[14]

Benzion Shklyar. Exact null-controllability of interconnected abstract evolution equations with unbounded input operators. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021124
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences