# 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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture notes, vol 34, Seoul National University, Korea, 1996. [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. [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. [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. [16] S. Labbé and E. Trélat, Generalization of the finite difference method in distributions spaces, Preprint HAL, ccsd-00097806, 2006. [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. [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. [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. [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. [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. [22] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. [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. [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. [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. [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. [27] L. S. Pontryagin, et al., The Mathematical Theory of Optimal Processes, vol. 4. Interscience, 1962. [28] O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematical and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. [29] E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics, Vuibert, Paris, 2005, 246 pp. [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. [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. [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. [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. [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. [35] E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, III (2006), 1389-1417.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture notes, vol 34, Seoul National University, Korea, 1996. [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. [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. [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. [16] S. Labbé and E. Trélat, Generalization of the finite difference method in distributions spaces, Preprint HAL, ccsd-00097806, 2006. [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. [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. [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. [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. [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. [22] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. [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. [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. [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. [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. [27] L. S. Pontryagin, et al., The Mathematical Theory of Optimal Processes, vol. 4. Interscience, 1962. [28] O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematical and its Applications, 103. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. [29] E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics, Vuibert, Paris, 2005, 246 pp. [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. [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. [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. [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. [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. [35] E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, III (2006), 1389-1417.
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