Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions

Affiliation(s)

Department of Mathematics, Faculty of Applied Sciences, Umm AL-Qura University, Makkah, KSA.

Department of Mathematics, Faculty of Science, Taibah University, Madina, KSA.

Department of Mathematics, Faculty of Applied Sciences, Umm AL-Qura University, Makkah, KSA.

Department of Mathematics, Faculty of Science, Taibah University, Madina, KSA.

ABSTRACT

In this paper, we consider 2 × 2 non-cooperative elliptic system involving Laplace operator defined on bounded, continuous and strictly Lipschitz domain of Rn. First we prove the existence and uniqueness for the state of the system under conjugation conditions; then we discuss the existence of the optimal control of boundary type with Neumann conditions, and we find the set of equations and inequalities that characterize it.

Cite this paper

A. Qamlo and B. Mohammed, "Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions,"*Intelligent Control and Automation*, Vol. 4 No. 3, 2013, pp. 280-286. doi: 10.4236/ica.2013.43032.

A. Qamlo and B. Mohammed, "Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions,"

References

[1] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer Verlag, Berlin, 1971. doi:10.1007/978-3-642-65024-6

[2] J. L. Lions, “Some Methods in the Mathematical Analysis of Systems and their Control,” Science Press, Beijing, 1981.

[3] I. V. Sergienko and V. S. Deineka, “Optimal Control of Distributed Systems with conjugation Conditions,” Spring er, 2005.

[4] G. M. Bahaa, “Optimal Control for Cooperative Parabolic Systems Governed by Schrodinger Operator with Control Constraints,” IMA Journal of Mathematical Control and Information, Vol. 24, No. 1, 2007, pp. 1-12. doi:10.1093/imamci/dnl001

[5] H. A. El-Saify, H. M. Serag and M. A. Shehata, “Time Optimal Control Problem for Cooperative Hyperbolic Systems Involving the Laplace Operator,” Journal of Dynamical and Control Systems, Vol. 15, No. 3, 2009, pp. 405-423. doi:10.1007/s10883-009-9067-y

[6] I. M. Gali and H. M. Serag, “Optimal Control of Coop erative Elliptic Systems Defined on Rn,” Journal of the Egyptian Mathematical Society, Vol. 3, 1995, pp. 33-39.

[7] A. H. Qamlo, “Distributed Control for Cooperative Hy perbolic Systems Involving Schrodinger Operator,” In ternational Journal of Dynamics and Control, Vol. 1, No. 1, 2013, pp. 54-59. doi:10.1007/s40435-013-0007-z

[8] A. H. Qamlo, “Optimality Conditions for Parabolic Sys tems with Variable Coefficients Involving Schrodinger Operators,” Journal of King Saud University—Science, 2013. doi:10.1016/j.jksus.2013.05.005

[9] H. M. Serag, “On Optimal Control for Elliptic Systems with Variable Coefficients,” Revista de Matemáticas Ap licadas, Vol. 19, 1998, pp. 37-41.

[10] H. M. Serag, “Distributed Control for Cooperative Sys tems Governed by Schrodinger Operator,” Journal of Discrete Mathematical Sciences & Cryptography, Vol. 3, No. 1-3, 2000, pp. 227-234. doi:10.1080/09720529.2000.10697910

[11] H. M. Serag, “Optimal Control of Systems Involving Schrodinger Operators,” International Journal of Intelli gent Control and Systems, Vol. 32, No. 3, 2004, pp. 154-157.

[12] H. M. Serag and A. H. Qamlo, “On Elliptic Systems In volving Schrodinger Operators,” The Mediterranean Jour nal of Measurement and Control, Vol. 1, No. 2, 2005, pp. 91-96.

[13] H. A. El-Saify, “Boundary Control for the Parabolic Op erator with an Infinite Number of Variables,” Functional Differential Systems and Related Topics, Vol. 3, 1983, pp. 79-82.

[14] H. A. El-Saify, “Boundary Control for the Hyperbolic Oper ator with an Infinite Number of Variables,” Journal In stitute Mathematics & Computer Science (Computer Sci ence Series), Vol. 1, No. 2, 1990, pp. 47-51.

[15] H. A. El-Saify, “Boundary Control Problem with an Infi nite Number of Variables,” International Journal of Ma thematics and Mathematical Science, Vol. 28, No. 1, 2001, pp. 57-62. doi:10.1155/S0161171201007128

[16] M. H. Hassan and H. M. Serag, “Boundary Control of Quasi Static Problem with Viscous Boundary Condi tions,” Indian Journal of Pure and Applied Mathematics, Vol. 31, No. 7, 2000, pp. 767-772.

[17] H. M. Serag and A. H. Qamlo, “Boundary Control for Non-Cooperative Elliptic Systems,” Advances in Model ing & Analysis, Vol. 38, No. 3, 2001, pp. 31-42.

[1] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer Verlag, Berlin, 1971. doi:10.1007/978-3-642-65024-6

[2] J. L. Lions, “Some Methods in the Mathematical Analysis of Systems and their Control,” Science Press, Beijing, 1981.

[3] I. V. Sergienko and V. S. Deineka, “Optimal Control of Distributed Systems with conjugation Conditions,” Spring er, 2005.

[4] G. M. Bahaa, “Optimal Control for Cooperative Parabolic Systems Governed by Schrodinger Operator with Control Constraints,” IMA Journal of Mathematical Control and Information, Vol. 24, No. 1, 2007, pp. 1-12. doi:10.1093/imamci/dnl001

[5] H. A. El-Saify, H. M. Serag and M. A. Shehata, “Time Optimal Control Problem for Cooperative Hyperbolic Systems Involving the Laplace Operator,” Journal of Dynamical and Control Systems, Vol. 15, No. 3, 2009, pp. 405-423. doi:10.1007/s10883-009-9067-y

[6] I. M. Gali and H. M. Serag, “Optimal Control of Coop erative Elliptic Systems Defined on Rn,” Journal of the Egyptian Mathematical Society, Vol. 3, 1995, pp. 33-39.

[7] A. H. Qamlo, “Distributed Control for Cooperative Hy perbolic Systems Involving Schrodinger Operator,” In ternational Journal of Dynamics and Control, Vol. 1, No. 1, 2013, pp. 54-59. doi:10.1007/s40435-013-0007-z

[8] A. H. Qamlo, “Optimality Conditions for Parabolic Sys tems with Variable Coefficients Involving Schrodinger Operators,” Journal of King Saud University—Science, 2013. doi:10.1016/j.jksus.2013.05.005

[9] H. M. Serag, “On Optimal Control for Elliptic Systems with Variable Coefficients,” Revista de Matemáticas Ap licadas, Vol. 19, 1998, pp. 37-41.

[10] H. M. Serag, “Distributed Control for Cooperative Sys tems Governed by Schrodinger Operator,” Journal of Discrete Mathematical Sciences & Cryptography, Vol. 3, No. 1-3, 2000, pp. 227-234. doi:10.1080/09720529.2000.10697910

[11] H. M. Serag, “Optimal Control of Systems Involving Schrodinger Operators,” International Journal of Intelli gent Control and Systems, Vol. 32, No. 3, 2004, pp. 154-157.

[12] H. M. Serag and A. H. Qamlo, “On Elliptic Systems In volving Schrodinger Operators,” The Mediterranean Jour nal of Measurement and Control, Vol. 1, No. 2, 2005, pp. 91-96.

[13] H. A. El-Saify, “Boundary Control for the Parabolic Op erator with an Infinite Number of Variables,” Functional Differential Systems and Related Topics, Vol. 3, 1983, pp. 79-82.

[14] H. A. El-Saify, “Boundary Control for the Hyperbolic Oper ator with an Infinite Number of Variables,” Journal In stitute Mathematics & Computer Science (Computer Sci ence Series), Vol. 1, No. 2, 1990, pp. 47-51.

[15] H. A. El-Saify, “Boundary Control Problem with an Infi nite Number of Variables,” International Journal of Ma thematics and Mathematical Science, Vol. 28, No. 1, 2001, pp. 57-62. doi:10.1155/S0161171201007128

[16] M. H. Hassan and H. M. Serag, “Boundary Control of Quasi Static Problem with Viscous Boundary Condi tions,” Indian Journal of Pure and Applied Mathematics, Vol. 31, No. 7, 2000, pp. 767-772.

[17] H. M. Serag and A. H. Qamlo, “Boundary Control for Non-Cooperative Elliptic Systems,” Advances in Model ing & Analysis, Vol. 38, No. 3, 2001, pp. 31-42.