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

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