Optimality of Distributed Control for *n × n* Hyperbolic Systems with an Infinite Number of Variables

Affiliation(s)

Department of Mathematics, Faculty of Applied sciences, Umm AL-Qura University, Makkah, Saudi Arabia.

Department of Mathematics, Faculty of Applied sciences, Umm AL-Qura University, Makkah, Saudi Arabia.

ABSTRACT

In this paper, we study the existence of solutions for 2*l* order (*n ×* *n*) cooperative systems governed by Dirichlet and Neumann problems
involving hyperbolic operators with an infinite number of variables and with
variable coefficients. The necessary and sufficient conditions for optimality
of the distributed control with constraints are obtained and the set of
inequalities that defining the optimal control of these systems are also
obtained.

Cite this paper

A. Qamlo, "Optimality of Distributed Control for*n × n* Hyperbolic Systems with an Infinite Number of Variables," *Advances in Pure Mathematics*, Vol. 3 No. 6, 2013, pp. 598-608. doi: 10.4236/apm.2013.36077.

A. Qamlo, "Optimality of Distributed Control for

References

[1] G. M. Bahaa, “Optimal Control Problems of Parabolic Equations with an Infnite Number of Variables and with Equality Constraints,” IMA Journal of Mathematical Control and Information, Vol. 25, No. 1, 2008, pp. 37-48. doi:10.1093/imamci/dnm002

[2] G. M. Bahaa and F. El-Shatery, “Optimal Control for n × n Coupled Parabolic Systems with Control-Constrained and Infinite Number of Variables,” International Journal of Mathematics & computation, Vol. 19, No. 2, 2013, pp. 106-118.

[3] G. M. Bahaa and F. El-Shatery, “Optimal Control for n × n Coupled Elliptic Systems with Control-Constrained and Infinite Number of Variables,” International Journal of Applied Mathematics & Statistics, Vol. 39, No. 9, 2013, pp. 93-104.

[4] H. A. El-Saify, “On Optimal Control for the Hyperbolic Operator with an Infinite Number of Variables,” Journal of Information and Optimization Sciences, Vol. 4, No. 3, 1983, pp. 249-253.

[5] H. A. El-Saify, H. M. Serag and G. M. Bahaa, “On Optimal Control for n × n Elliptic System Involving Operators with an Infinite Number of Variables,” AMSE, Vol. 37, No. 4, 2000, pp. 47-61.

[6] I. M. Gali and H. A. El-Saify, “Optimal Control of a System Governed by Hyperbolic Operator with an Infinite Number of Variables,” Journal of Mathematical Analysis and Applications, Vol. 85, No. 1, 1982, pp. 24-30. doi:10.1016/0022-247X(82)90023-3

[7] I. M. Gali and H. A. El-Saify, “Distributed Control of a System Governed by Dirichlet and Neumann Problems for a Self-Adjoint Elliptic Operator with an Infinite Number of Variables,” Journal of Optimization Theory and Applications, Vol. 39, No. 2, 1983, pp. 293-298. doi:10.1007/BF00934534

[8] W. Kotarski, “Optimal Control of a System Governed by a Parabolic Equation with an Infinite Number of Variables,” Journal of Optimization Theory and Applications, Vol. 60, No. 1, 1989, pp. 33-41. doi:10.1007/BF00938797

[9] A. H. Qamlo, “Boundary Control for Cooperative Systems Involving Parabolic Operators with an Infinite Number of Variables,” Advances in Differential Equations and Control Processes, Vol. 2, No. 2, 2008, pp. 135-151.

[10] A. H. Qamlo, “Distributed Control of Cooperative Systems Involving Hyperbolic Operators with an Infinite Number of Variables,” International Journal of Functional Analysis, Operator Theory and Applications, Vol. 1, No. 2, 2009, pp. 115-128.

[11] H. M. Serag, “Distributed Control for Cooperative Systems Involving Parabolic Operators with an Infinite Number of Variables,” IMA Journal of Mathematical Control and Information, Vol. 24, No. 2, 2007, pp. 149-161. doi:10.1093/imamci/dnl018

[12] W. Kotarski, “Optimal Control of a System Governed by a Parabolic Equation with an Infinite Number of Variables and Time Delay,” Journal of Optimization Theory and Applications, Vol. 63, No. 1, 1989, pp. 57-67. doi:10.1007/BF00940731

[13] W. Kotarski, H. A. El-Saify and G. M. Bahaa, “Optimal Control of Parabolic Equation with an Infnte Number of Variables for Non Standard Functional and Time Delay,” IMA Journal of Mathematical Control and Information, Vol. 19, No. 4, 2002, pp. 461-476. doi:10.1093/imamci/19.4.461

[14] G. M. Bahaa, “Time-Optimal Control Problem for Parabolic Equations with Control Constraints and Infinite Number of Variables,” IMA Journal of Mathematical Control and Information, Vol. 22, No. 3, 2005, pp. 364-375. doi:10.1093/imamci/dni033

[15] 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

[16] W. Kotarski, H. A. El-Saify and M. A. Shehata, “Time Optimal Control of Parabolic Lag Systems with an Infinite Number of Variables,” Journal of the Egyptian Mathematical Society, Vol. 15, No. 1, 2007, pp. 21-34.

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

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

[19] H. M. Serag, “Optimal Control of Systems Involving Schrodinger operators,” Control and Intelligent Systems, Vol. 32, No. 3, 2004, pp. 154-157. doi:10.2316/Journal.201.2004.3.201-1319

[20] H. M. Serag and A. H. Qamlo, “On Elliptic Systems Involving Schrodinger Operators,” Mediterranean Journal of Measurement and Control, Vol. 1, No. 2, 2005, pp. 91-96.

[21] H. M. Serag and A. H. Qamlo, “Maximum Principle and Existence of Solutions for Non Necessarily Cooperative Systems Involving Schrodinger Operators,” Mathematica Slovaca, Vol. 58, No. 5, 2008, pp. 567-580. doi:10.2478/s12175-008-0094-z

[22] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer Verlag, Berlin, 1971.

[23] Ju. M. Berezanskii, “Self-Adjiontness of Elliptic Operators with an Infnite Number of Variables,” Ukrainian Mathematical Journal, Vol. 27, 1975, pp. 729-742.

[24] Ju. M. Berezanskii and I. M. Gali, “Positive Definite Functions of an Infinite Number of Variables in a Layer,” Ukrainian Mathematical Journal, Vol. 24, No. 4, 1972, pp. 435-464.

[1] G. M. Bahaa, “Optimal Control Problems of Parabolic Equations with an Infnite Number of Variables and with Equality Constraints,” IMA Journal of Mathematical Control and Information, Vol. 25, No. 1, 2008, pp. 37-48. doi:10.1093/imamci/dnm002

[2] G. M. Bahaa and F. El-Shatery, “Optimal Control for n × n Coupled Parabolic Systems with Control-Constrained and Infinite Number of Variables,” International Journal of Mathematics & computation, Vol. 19, No. 2, 2013, pp. 106-118.

[3] G. M. Bahaa and F. El-Shatery, “Optimal Control for n × n Coupled Elliptic Systems with Control-Constrained and Infinite Number of Variables,” International Journal of Applied Mathematics & Statistics, Vol. 39, No. 9, 2013, pp. 93-104.

[4] H. A. El-Saify, “On Optimal Control for the Hyperbolic Operator with an Infinite Number of Variables,” Journal of Information and Optimization Sciences, Vol. 4, No. 3, 1983, pp. 249-253.

[5] H. A. El-Saify, H. M. Serag and G. M. Bahaa, “On Optimal Control for n × n Elliptic System Involving Operators with an Infinite Number of Variables,” AMSE, Vol. 37, No. 4, 2000, pp. 47-61.

[6] I. M. Gali and H. A. El-Saify, “Optimal Control of a System Governed by Hyperbolic Operator with an Infinite Number of Variables,” Journal of Mathematical Analysis and Applications, Vol. 85, No. 1, 1982, pp. 24-30. doi:10.1016/0022-247X(82)90023-3

[7] I. M. Gali and H. A. El-Saify, “Distributed Control of a System Governed by Dirichlet and Neumann Problems for a Self-Adjoint Elliptic Operator with an Infinite Number of Variables,” Journal of Optimization Theory and Applications, Vol. 39, No. 2, 1983, pp. 293-298. doi:10.1007/BF00934534

[8] W. Kotarski, “Optimal Control of a System Governed by a Parabolic Equation with an Infinite Number of Variables,” Journal of Optimization Theory and Applications, Vol. 60, No. 1, 1989, pp. 33-41. doi:10.1007/BF00938797

[9] A. H. Qamlo, “Boundary Control for Cooperative Systems Involving Parabolic Operators with an Infinite Number of Variables,” Advances in Differential Equations and Control Processes, Vol. 2, No. 2, 2008, pp. 135-151.

[10] A. H. Qamlo, “Distributed Control of Cooperative Systems Involving Hyperbolic Operators with an Infinite Number of Variables,” International Journal of Functional Analysis, Operator Theory and Applications, Vol. 1, No. 2, 2009, pp. 115-128.

[11] H. M. Serag, “Distributed Control for Cooperative Systems Involving Parabolic Operators with an Infinite Number of Variables,” IMA Journal of Mathematical Control and Information, Vol. 24, No. 2, 2007, pp. 149-161. doi:10.1093/imamci/dnl018

[12] W. Kotarski, “Optimal Control of a System Governed by a Parabolic Equation with an Infinite Number of Variables and Time Delay,” Journal of Optimization Theory and Applications, Vol. 63, No. 1, 1989, pp. 57-67. doi:10.1007/BF00940731

[13] W. Kotarski, H. A. El-Saify and G. M. Bahaa, “Optimal Control of Parabolic Equation with an Infnte Number of Variables for Non Standard Functional and Time Delay,” IMA Journal of Mathematical Control and Information, Vol. 19, No. 4, 2002, pp. 461-476. doi:10.1093/imamci/19.4.461

[14] G. M. Bahaa, “Time-Optimal Control Problem for Parabolic Equations with Control Constraints and Infinite Number of Variables,” IMA Journal of Mathematical Control and Information, Vol. 22, No. 3, 2005, pp. 364-375. doi:10.1093/imamci/dni033

[15] 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

[16] W. Kotarski, H. A. El-Saify and M. A. Shehata, “Time Optimal Control of Parabolic Lag Systems with an Infinite Number of Variables,” Journal of the Egyptian Mathematical Society, Vol. 15, No. 1, 2007, pp. 21-34.

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

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

[19] H. M. Serag, “Optimal Control of Systems Involving Schrodinger operators,” Control and Intelligent Systems, Vol. 32, No. 3, 2004, pp. 154-157. doi:10.2316/Journal.201.2004.3.201-1319

[20] H. M. Serag and A. H. Qamlo, “On Elliptic Systems Involving Schrodinger Operators,” Mediterranean Journal of Measurement and Control, Vol. 1, No. 2, 2005, pp. 91-96.

[21] H. M. Serag and A. H. Qamlo, “Maximum Principle and Existence of Solutions for Non Necessarily Cooperative Systems Involving Schrodinger Operators,” Mathematica Slovaca, Vol. 58, No. 5, 2008, pp. 567-580. doi:10.2478/s12175-008-0094-z

[22] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer Verlag, Berlin, 1971.

[23] Ju. M. Berezanskii, “Self-Adjiontness of Elliptic Operators with an Infnite Number of Variables,” Ukrainian Mathematical Journal, Vol. 27, 1975, pp. 729-742.

[24] Ju. M. Berezanskii and I. M. Gali, “Positive Definite Functions of an Infinite Number of Variables in a Layer,” Ukrainian Mathematical Journal, Vol. 24, No. 4, 1972, pp. 435-464.