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

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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 and 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.
[2]	V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer Netherlands, 2012. doi: 10.1007/978-94-007-2247-7.
[3]	V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Inc. San Diego, 1993.
[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.
[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.
[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.
[7]	J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, American Mathematical Society, USA, 136 (2007). doi: 10.1090/surv/136.
[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.
[9]	V. N. Do, Controllability of semilinear systems, Journal of Optimization Theory and Applications, 65 (1990), 41-52. doi: 10.1007/BF00941158.
[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.
[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.
[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.
[13]	T. Karite and A. Boutoulout, Regional constrained controllability for parabolic semilinear systems, International Journal of Pure and Applied Mathematics, 113 (2017), 113-129.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001.
[22]	J. L. Lions, Optimal Control of Systems Governed Partial Differential Equations, Springer-Verlag, New York, 1971.
[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.
[24]	J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 1, 2, Dunod, Paris, 1968.
[25]	J. L. Lions, Contrôlabilité exacte perturbations et stabilisation des systèmes distribués, Tome 1, contrôlabilité exacte, Masson, Paris, 1988.
[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.
[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.
[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.
[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.
[30]	E. Zerrik and F. Ghafrani, Minimum energy control subject to output constraints: Numerical approach, IEE Proc-Control Theory Appl, 149 (2002), 105-110.
[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.

show all references

References:

[1]	G. Aronsson, Global controllability and bang-bang steering of certain nonlinear systems, SIAM J. Control, 11 (1973), 607-619.
[2]	V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer Netherlands, 2012. doi: 10.1007/978-94-007-2247-7.
[3]	V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Inc. San Diego, 1993.
[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.
[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.
[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.
[7]	J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, American Mathematical Society, USA, 136 (2007). doi: 10.1090/surv/136.
[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.
[9]	V. N. Do, Controllability of semilinear systems, Journal of Optimization Theory and Applications, 65 (1990), 41-52. doi: 10.1007/BF00941158.
[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.
[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.
[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.
[13]	T. Karite and A. Boutoulout, Regional constrained controllability for parabolic semilinear systems, International Journal of Pure and Applied Mathematics, 113 (2017), 113-129.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001.
[22]	J. L. Lions, Optimal Control of Systems Governed Partial Differential Equations, Springer-Verlag, New York, 1971.
[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.
[24]	J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 1, 2, Dunod, Paris, 1968.
[25]	J. L. Lions, Contrôlabilité exacte perturbations et stabilisation des systèmes distribués, Tome 1, contrôlabilité exacte, Masson, Paris, 1988.
[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.
[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.
[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.
[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.
[30]	E. Zerrik and F. Ghafrani, Minimum energy control subject to output constraints: Numerical approach, IEE Proc-Control Theory Appl, 149 (2002), 105-110.
[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.

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