An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator
Author(s)
Ahlam Hasan Qamlo
Affiliation(s)
Department of Mathematics, Faculty of Applied sciences, Umm AL-Qura University, Makkah, KSA.
Department of Mathematics, Faculty of Applied sciences, Umm AL-Qura University, Makkah, KSA.
ABSTRACT
In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.
In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.
KEYWORDS
Hyperbolic Systems, Schrödinger Operator, Boundary Control Problem, Boundary Observation, Cooperative
Hyperbolic Systems, Schrödinger Operator, Boundary Control Problem, Boundary Observation, Cooperative
Cite this paper
Qamlo, A. (2014) An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator. Intelligent Control and Automation, 5, 262-271. doi: 10.4236/ica.2014.54028.
Qamlo, A. (2014) An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator. Intelligent Control and Automation, 5, 262-271. doi: 10.4236/ica.2014.54028.
References
[1] Serag, H.M. (2000) Distributed Control for Cooperative Systems Governed by Schrödinger Operator. Journal of Discrete Mathematical Sciences and Cryptography, 3, 227-234.
http://dx.doi.org/10.1080/09720529.2000.10697910 [2] Bahaa, G.M. (2007) Optimal Control for Cooperative Parabolic Systems Governed by Schrodinger Operator with Control Constraints. IMA Journal of Mathematical Control and Information, 24, 1-12.
http://dx.doi.org/10.1093/imamci/dnl001 [3] Qamlo, A.H. (2013) Distributed Control for Cooperative Hyperbolic Systems Involving Schrödinger Operator. International Journal of Dynamics and Control, 1, 54-59.
http://dx.doi.org/10.1007/s40435-013-0007-z [4] Qamlo, A.H. (2013) Optimality Conditions for Parabolic Systems with Variable Coefficients Involving Schrödinger Operators. Journal of King Saud University—Science, 26, 107-112.
http://dx.doi.org/10.1016/j.jksus.2013.05.005 [5] Serag, H.M. (2004) Optimal Control of Systems Involving Schrödinger Operators. International Journal of Control and Intelligent Systems, 32, 154-157.
http://dx.doi.org/10.2316/Journal.201.2004.3.201-1319 [6] Serag, H.M. and Qamlo, A.H. (2005) On Elliptic Systems Involving Schrödinger Operators. The Mediterranean Journal of Measurement and Control, 1, 91-96. [7] Bahaa, G.M. (2006) Boundary Control for Cooperative Parabolic Systems Governed by Schrodinger Operator. Differential Equations and Control Processes, 1, 79-88. [8] Bahaa, G.M. and Qamlo, A.H. (2013) Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions. Intelligent Control and Automation, 4, 280-286.
http://dx.doi.org/10.4236/ica.2013.43032 [9] Bahaa, G.M. and Qamlo, A.H. (2013) Boundary Control Problem for Infinite Order Parabolic System with Time Delay and Control Constraints. European Journal of Scientific Research, 104, 392-406. [10] Serag, H.M. and Qamlo, A.H. (2001) Boundary Control for Non-Cooperative Elliptic Systems. Advances in Modeling & Analysis, 38, 31-42. [11] Lions, J.L. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Springer Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-65024-6 [12] Fleckinger, J. (1981) Estimates of the Number of Eigenvalues for an Operator of Schrödinger Type. Proceedings Royal Society Edinburgh Section A: Mathematics, 89, 355-361. [13] Fleckinger, J. (1994) Method of Sub-Suber Solutions for Some Elliptic System Defined on Rn. Preprint UMR MIP. Universite Toulouse 3. France.
[1] Serag, H.M. (2000) Distributed Control for Cooperative Systems Governed by Schrödinger Operator. Journal of Discrete Mathematical Sciences and Cryptography, 3, 227-234.
http://dx.doi.org/10.1080/09720529.2000.10697910 [2] Bahaa, G.M. (2007) Optimal Control for Cooperative Parabolic Systems Governed by Schrodinger Operator with Control Constraints. IMA Journal of Mathematical Control and Information, 24, 1-12.
http://dx.doi.org/10.1093/imamci/dnl001 [3] Qamlo, A.H. (2013) Distributed Control for Cooperative Hyperbolic Systems Involving Schrödinger Operator. International Journal of Dynamics and Control, 1, 54-59.
http://dx.doi.org/10.1007/s40435-013-0007-z [4] Qamlo, A.H. (2013) Optimality Conditions for Parabolic Systems with Variable Coefficients Involving Schrödinger Operators. Journal of King Saud University—Science, 26, 107-112.
http://dx.doi.org/10.1016/j.jksus.2013.05.005 [5] Serag, H.M. (2004) Optimal Control of Systems Involving Schrödinger Operators. International Journal of Control and Intelligent Systems, 32, 154-157.
http://dx.doi.org/10.2316/Journal.201.2004.3.201-1319 [6] Serag, H.M. and Qamlo, A.H. (2005) On Elliptic Systems Involving Schrödinger Operators. The Mediterranean Journal of Measurement and Control, 1, 91-96. [7] Bahaa, G.M. (2006) Boundary Control for Cooperative Parabolic Systems Governed by Schrodinger Operator. Differential Equations and Control Processes, 1, 79-88. [8] Bahaa, G.M. and Qamlo, A.H. (2013) Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions. Intelligent Control and Automation, 4, 280-286.
http://dx.doi.org/10.4236/ica.2013.43032 [9] Bahaa, G.M. and Qamlo, A.H. (2013) Boundary Control Problem for Infinite Order Parabolic System with Time Delay and Control Constraints. European Journal of Scientific Research, 104, 392-406. [10] Serag, H.M. and Qamlo, A.H. (2001) Boundary Control for Non-Cooperative Elliptic Systems. Advances in Modeling & Analysis, 38, 31-42. [11] Lions, J.L. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Springer Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-65024-6 [12] Fleckinger, J. (1981) Estimates of the Number of Eigenvalues for an Operator of Schrödinger Type. Proceedings Royal Society Edinburgh Section A: Mathematics, 89, 355-361. [13] Fleckinger, J. (1994) Method of Sub-Suber Solutions for Some Elliptic System Defined on Rn. Preprint UMR MIP. Universite Toulouse 3. France.