IJPAM: Volume 113, No. 1 (2017)

Title

REGIONAL ENLARGED CONTROLLABILITY FOR
PARABOLIC SEMILINEAR SYSTEMS

Authors

Touria Karite$^1$, Ali Boutoulout$^2$
$^{1,2}$TSI Team, MACS Laboratory
Institute of Sciences
Moulay Ismail University
Meknes, MOROCCO

Abstract

The aim of this paper is to study the problem of the enlarged controllability for distributed parabolic semilinear systems evolving in spatial domain $\Omega$. It consists in finding the control $u$ with minimum cost that steers the system from the initial state $y_{_{0}}$ to a state between two prescribed levels $l_{1}$ and $l_{2}$.

We give some definitions and properties concerning this concept and then we use two approaches to calculate the control $u$, the first one is based on the sub-differential method and the second one on the Lagrangian multiplier method. An algorithm is obtained and it's simulated numerically.

History

Received: November 30, 2016
Revised: January 17, 2017
Published: February 28, 2017

AMS Classification, Key Words

AMS Subject Classification: 93B05, 93C20
Key Words and Phrases: distributed systems, parabolic systems, regional controllability, Lagrangian approach, sub-differential method, semilinear systems, Uzawa algorithm

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Bibliography

1
J.P. Aubin, L'analyse Non Linéaire et ses Motivations Économiques, Dunod, 1984.

2
W.M. Bian, Constrained controllability of some Non-linear systems, Applicable Analysis, 72, No. 1-2 (1999), 57-73, doi: https://doi.org/10.1080/00036819908840730.

3
A. Boutoulout, A. Kamal and S.A.O. Beinane, Regional Controllability of Semi-Linear Distributed Parabolic Systems: Theory and Simulation, Intelligent Control and Automation, Vol, No. 3 (2012), 146-158, doi: https://doi.org/10.4236/ica.2012.32017.

4
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York (1991), doi: https://doi.org/10.1007/978-1-4612-3172-1.

5
N. Carmichael and M.D. Quinn, Fixed-Point Methods in Nonlinear Control, Lecture notes in Control and Information Sciences, Springer-Verlag, Berlin Heidelberg (1984), doi: https://doi.org/10.1007/BFb0005643.

6
R.F. Curtain and H. Zwart, An introduction to infinite dimensional linear systems theory, Springer-Verlag, New York (1995), doi: https://doi.org/10.1007/978-1-4612-4224-6.

7
A. El Jai, A.J. Pritchard, M.C. Simon and E. Zerrik, Regional controllability of distributed parameter systems, International Journal of Control, 62, (1995), 1351-1365, doi: https://doi.org/10.1080/00207179508921603.

8
A. El Jai and A.J. Pritchard, Sensors and actuators in distributed systems analysis, Wiley, New York (1988).

9
K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York (1999), doi: https://doi.org/10.1007/b97696.

10
C. Fabre, P.J. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy Soc. Edinburgh 125 A, (1995), 31-61, doi: https://doi.org/10.1017/S0308210500030742.

11
M. Fortin, and R. Glowinski, Augmented Lagrangian Methods: Applications to the numerical solution of boundary-value problems, 15, North-Holland, Amsterdam (1983).

12
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York (1981), doi: https://doi.org/10.1007/BFb0089647.

13
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications 1, Dunod, Paris (1968).

14
A. Matei, Weak solvability via Lagrange multipliers for two frictional contact models, Mathematics and Mechanics of Solids, 21, No. 7 (2014), 826–841, doi: https://doi.org/10.1177/1081286514541577.

15
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York (1983), doi: https://doi.org/10.1007/978-1-4612-5561-1.

16
R. Tyrrell Rockafellar, Lagrange multipliers and optimality, SIAM Review, 35, No. 2 (1993), 183-238, doi: https://doi.org/10.1137/1035044.

17
K. Yosida, Functional analysis, Springer-Verlag, Berlin Heidelberg New York (1980), doi: https://doi.org/10.1007/978-3-642-61859-8.

18
E. Zeidler, Applied functional analysis : Applications to mathematical physics, Springer-Verlag, New York (1995), doi: https://doi.org/10.1007/978-1-4612-0815-0.

19
E. Zerrik, A. El Jai and A. Boutoulout, Actuators and regional boundary controllability of parabolic system, Int. J. Syst. Sci., 31, No. 1 (2000), 73-82, doi: https://doi.org/10.1080/002077200291479.

20
X. Zhang, Exact controllability of semilinear evolution systems and its application, Journal of Optimization Theory and Applications, 107, No. 2 (2000), 415-432, doi: https://doi.org/10.1023/A:1026460831701.

How to Cite?

DOI: 10.12732/ijpam.v113i1.11 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 113
Issue: 1
Pages: 113 - 129


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