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December  2021, 11(4): 555-566. doi: 10.3934/naco.2020055

Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach

Touria Karite 1,, and  Ali Boutoulout 2,

1. 

Department of Electrical engineering & Informatics, National School of Applied Sciences of Fez, Sidi Mohamed Ben Abdellah University, Route d'Imouzzer, BP 72, Fez, MA
2. 

TSI Team, Department of Mathematics, Faculty of Sciences, Moulay Ismail University, BP 11201, Avenue Zitoune, Meknes, MA

* Corresponding author: Touria Karite

Received  February 2019 Revised  October 2020 Published  December 2021 Early access  November 2020

The aim of this paper is to study the problem of constrained controllability for distributed parabolic linear system evolving in spatial domain $ \Omega $ using the Reverse Hilbert Uniqueness Method (RHUM approach) introduced by Lions in 1988. It consists in finding the control $ u $ that steers the system from an initial state $ y_{_{0}} $ to a state between two prescribed functions. We give some definitions and properties concerning this concept and then we resolve the problem that relays on computing a control with minimum cost in the case of $ \omega = \Omega $ and in the regional case where $ \omega $ is a part of $ \Omega $.

Keywords: Distributed systems, parabolic systems, regional controllability, Constrained controllability, RHUM approach.
Mathematics Subject Classification: Primary: 93B05; 93C05; 93C20; Secondary: 34H05; 49J20.
Citation: Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 555-566. doi: 10.3934/naco.2020055
References:
[1]

G. Aronsson, Global controllability and bang-bang steering of certain nonlinear systems, SIAM J. Control, 11 (1973), 607-619.   Google Scholar

[2]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer Netherlands, 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar

[3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Inc. San Diego, 1993.   Google Scholar
[4]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser Boston, Second Edition, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

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M. Bergounioux, A penalization method for optimal control of elliptic problems with state constraints, SIAM J. Control Optim., 30 (1992), 305-323.  doi: 10.1137/0330019.  Google Scholar

[6]

J. F. Bonnans and E. Casas, On the choice of the function space for some state constrained control problems, Numer. Funct. Anal. Optim., 4 (1984-1985), 333-348.  doi: 10.1080/01630568508816197.  Google Scholar

[7]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, American Mathematical Society, USA, 136 (2007). doi: 10.1090/surv/136.  Google Scholar

[8]

R. F. Curtain and H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, Springer-Verlag New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[9]

V. N. Do, Controllability of semilinear systems, Journal of Optimization Theory and Applications, 65 (1990), 41-52.  doi: 10.1007/BF00941158.  Google Scholar

[10]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optimization, 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[11]

A. El Jai and A. J. Pritchard, Regional controllability of distributed systems, In: Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems. Lecture Notes in Control and Information Sciences (eds. R. F. Curtain, A. Bensoussan and J. L. Lions), Springer, Berlin, Heidelberg, 185 (1993), 326–335. doi: 10.1007/BFb0115033.  Google Scholar

[12]

A. El Jai, A. J. Pritchard, M. C. Simon and E. Zerrik, Regional controllability of distributed systems, International Journal of Control, 62 (1995), 1351-1365.  doi: 10.1080/00207179508921603.  Google Scholar

[13]

T. Karite and A. Boutoulout, Regional constrained controllability for parabolic semilinear systems, International Journal of Pure and Applied Mathematics, 113 (2017), 113-129.   Google Scholar

[14]

T. Karite and A. Boutoulout, Regional boundary controllability of semi-linear parabolic systems with state constraints, Int. J. Dynamical Systems and Differential Equations, 8 (2018), 150-159.  doi: 10.1504/IJDSDE.2018.089105.  Google Scholar

[15]

T. Karite, A. Boutoulout and F. Z. El Alaoui, Some numerical results of regional boundary controllability with output constraints, In Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016 (eds. Klingenberg C., Westdickenberg M.), Springer Proceedings in Mathematics & Statistics, Springer, 237 (2018), 111–122.  Google Scholar

[16]

T. Karite, A. Boutoulout and F. Z. El Alaoui, Regional enlarged controllability of semilinear systems with constraints on the gradient: Approaches and simulations, J. Control Autom. Electr. Syst., 30 (2019), 441-452.   Google Scholar

[17]

T. Karite, A. Boutoulout and D. F. M. Torres, Enlarged controllability of riemann–liouville fractional differential equations, Journal of Computational and Nonlinear Dynamics, 13 (2018), 090907-1. Google Scholar

[18]

T. Karite, A. Boutoulout and D. F. M. Torres, Enlarged controllability and optimal control of sub-diffusion processes with caputo fractional derivatives, Progr. Fract. Differ. Appl., 6 (2020), 1-14.  doi: 10.1186/s13662-015-0593-5.  Google Scholar

[19]

I. Kazufumi and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, SIAM, 2008. doi: 10.1137/1.9780898718614.  Google Scholar

[20]

I. Lasiecka, State constrained control problems for parabolic systems: regularity of optimal solutions, Appl. Math. Optim., 6 (1980), 1-29.  doi: 10.1007/BF01442881.  Google Scholar

[21]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[22]

J. L. Lions, Optimal Control of Systems Governed Partial Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

[23]

J. L. Lions, Sur la contrôlabilité exacte élargie, In Partial Differential Equations and the Calculus of Variations (eds. F. Colombini and al.), Springer Science+Business Media, New York, 1989.  Google Scholar

[24]

J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 1, 2, Dunod, Paris, 1968.  Google Scholar

[25]

J. L. Lions, Contrôlabilité exacte perturbations et stabilisation des systèmes distribués, Tome 1, contrôlabilité exacte, Masson, Paris, 1988.  Google Scholar

[26]

D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica J., 36 (2000), 789-814.  doi: 10.1016/S0005-1098(99)00214-9.  Google Scholar

[27]

B. S. Mordukhovich, Optimization and feedback design of state-constrained parabolic systems, , In Mathematics Research Reports, Paper 52, (2007), Availabe at http://digitalcommons.wayne.edu/math_reports/52.  Google Scholar

[28]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, , In Mathematics Research Reports, paper 72, (2010), http://digitalcommons.wayne.edu/math_reports/72. doi: 10.1080/00036811003735840.  Google Scholar

[29]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, Applicable Analysis, 90 (2011), 1075-1109.  doi: 10.1080/00036811003735840.  Google Scholar

[30]

E. Zerrik and F. Ghafrani, Minimum energy control subject to output constraints: Numerical approach, IEE Proc-Control Theory Appl, 149 (2002), 105-110.   Google Scholar

[31]

E. Zerrik, F. Ghafrani and M. Raissouli, An extended controllability problem with minimum energy, Journal of Mathematical Sciences, 161 (2009), 344-354.  doi: 10.1007/s10958-009-9558-0.  Google Scholar

show all references

References:
[1]

G. Aronsson, Global controllability and bang-bang steering of certain nonlinear systems, SIAM J. Control, 11 (1973), 607-619.   Google Scholar

[2]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer Netherlands, 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar

[3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Inc. San Diego, 1993.   Google Scholar
[4]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser Boston, Second Edition, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[5]

M. Bergounioux, A penalization method for optimal control of elliptic problems with state constraints, SIAM J. Control Optim., 30 (1992), 305-323.  doi: 10.1137/0330019.  Google Scholar

[6]

J. F. Bonnans and E. Casas, On the choice of the function space for some state constrained control problems, Numer. Funct. Anal. Optim., 4 (1984-1985), 333-348.  doi: 10.1080/01630568508816197.  Google Scholar

[7]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, American Mathematical Society, USA, 136 (2007). doi: 10.1090/surv/136.  Google Scholar

[8]

R. F. Curtain and H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, Springer-Verlag New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[9]

V. N. Do, Controllability of semilinear systems, Journal of Optimization Theory and Applications, 65 (1990), 41-52.  doi: 10.1007/BF00941158.  Google Scholar

[10]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optimization, 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[11]

A. El Jai and A. J. Pritchard, Regional controllability of distributed systems, In: Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems. Lecture Notes in Control and Information Sciences (eds. R. F. Curtain, A. Bensoussan and J. L. Lions), Springer, Berlin, Heidelberg, 185 (1993), 326–335. doi: 10.1007/BFb0115033.  Google Scholar

[12]

A. El Jai, A. J. Pritchard, M. C. Simon and E. Zerrik, Regional controllability of distributed systems, International Journal of Control, 62 (1995), 1351-1365.  doi: 10.1080/00207179508921603.  Google Scholar

[13]

T. Karite and A. Boutoulout, Regional constrained controllability for parabolic semilinear systems, International Journal of Pure and Applied Mathematics, 113 (2017), 113-129.   Google Scholar

[14]

T. Karite and A. Boutoulout, Regional boundary controllability of semi-linear parabolic systems with state constraints, Int. J. Dynamical Systems and Differential Equations, 8 (2018), 150-159.  doi: 10.1504/IJDSDE.2018.089105.  Google Scholar

[15]

T. Karite, A. Boutoulout and F. Z. El Alaoui, Some numerical results of regional boundary controllability with output constraints, In Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016 (eds. Klingenberg C., Westdickenberg M.), Springer Proceedings in Mathematics & Statistics, Springer, 237 (2018), 111–122.  Google Scholar

[16]

T. Karite, A. Boutoulout and F. Z. El Alaoui, Regional enlarged controllability of semilinear systems with constraints on the gradient: Approaches and simulations, J. Control Autom. Electr. Syst., 30 (2019), 441-452.   Google Scholar

[17]

T. Karite, A. Boutoulout and D. F. M. Torres, Enlarged controllability of riemann–liouville fractional differential equations, Journal of Computational and Nonlinear Dynamics, 13 (2018), 090907-1. Google Scholar

[18]

T. Karite, A. Boutoulout and D. F. M. Torres, Enlarged controllability and optimal control of sub-diffusion processes with caputo fractional derivatives, Progr. Fract. Differ. Appl., 6 (2020), 1-14.  doi: 10.1186/s13662-015-0593-5.  Google Scholar

[19]

I. Kazufumi and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, SIAM, 2008. doi: 10.1137/1.9780898718614.  Google Scholar

[20]

I. Lasiecka, State constrained control problems for parabolic systems: regularity of optimal solutions, Appl. Math. Optim., 6 (1980), 1-29.  doi: 10.1007/BF01442881.  Google Scholar

[21]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[22]

J. L. Lions, Optimal Control of Systems Governed Partial Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

[23]

J. L. Lions, Sur la contrôlabilité exacte élargie, In Partial Differential Equations and the Calculus of Variations (eds. F. Colombini and al.), Springer Science+Business Media, New York, 1989.  Google Scholar

[24]

J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 1, 2, Dunod, Paris, 1968.  Google Scholar

[25]

J. L. Lions, Contrôlabilité exacte perturbations et stabilisation des systèmes distribués, Tome 1, contrôlabilité exacte, Masson, Paris, 1988.  Google Scholar

[26]

D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica J., 36 (2000), 789-814.  doi: 10.1016/S0005-1098(99)00214-9.  Google Scholar

[27]

B. S. Mordukhovich, Optimization and feedback design of state-constrained parabolic systems, , In Mathematics Research Reports, Paper 52, (2007), Availabe at http://digitalcommons.wayne.edu/math_reports/52.  Google Scholar

[28]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, , In Mathematics Research Reports, paper 72, (2010), http://digitalcommons.wayne.edu/math_reports/72. doi: 10.1080/00036811003735840.  Google Scholar

[29]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, Applicable Analysis, 90 (2011), 1075-1109.  doi: 10.1080/00036811003735840.  Google Scholar

[30]

E. Zerrik and F. Ghafrani, Minimum energy control subject to output constraints: Numerical approach, IEE Proc-Control Theory Appl, 149 (2002), 105-110.   Google Scholar

[31]

E. Zerrik, F. Ghafrani and M. Raissouli, An extended controllability problem with minimum energy, Journal of Mathematical Sciences, 161 (2009), 344-354.  doi: 10.1007/s10958-009-9558-0.  Google Scholar

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