Boundary Control Problem of Infinite Order Distributed Hyperbolic Systems Involving Time Lags

Affiliation(s)

Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.

Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.

ABSTRACT

Various optimal boundary control problems for linear infinite order distributed hyperbolic systems involving constant time lags are considered. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the Neumann problem with the quadratic performance functional are derived.

Various optimal boundary control problems for linear infinite order distributed hyperbolic systems involving constant time lags are considered. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the Neumann problem with the quadratic performance functional are derived.

Cite this paper

B. Mohamed, "Boundary Control Problem of Infinite Order Distributed Hyperbolic Systems Involving Time Lags,"*Intelligent Control and Automation*, Vol. 3 No. 3, 2012, pp. 211-221. doi: 10.4236/ica.2012.33024.

B. Mohamed, "Boundary Control Problem of Infinite Order Distributed Hyperbolic Systems Involving Time Lags,"

References

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[2] G. Knowles, “Time-Optimal Control of Parabolic Systems with Boundary Conditions Involving Time Delays,” Journal of Optimization Theory and Applications, Vol. 25, No. 4, 1978, pp. 563-574. doi:10.1007/BF00933521

[3] K. H. Wong, “Optimal Control Computation for Parabolic Systems with Boundary Conditions Involving Time Delays,” Journal of Optimization Theory and Applications, Vol. 53, No. 3, 1987, pp. 475-507. doi:10.1007/BF00938951

[4] A. Kowalewski, “Optimal Control of Hyperbolic System with Boundary Condition Involving a Time-Varying Lag,” Proceedings of the IMACS/IFAC International Symposium on Modeling and Simulation of DPS, Hiroshima, 6-9 October 1987, pp. 462-467.

[5] A. Kowalewski, “Boundary Control of Hyperbolic Systems with Boundary Condition Involving a Time Delay,” Analysis and Optimization of Systems, Vol. 111, 1988, pp. 507-518. doi:10.1007/BFb0042240

[6] A. Kowalewski, “Optimal Control of Distributed Hyperbolic System with Boundary Condition Involving a Time Lag,” Automatic Remote Control, XXXIII, 1988, pp. 537-545.

[7] A. Kowalewski, “Optimal Control of Distributed Parabolic System Involving Time Lags,” IMA Journal of Mathematical Control and Information, Vol. 7, No. 4, 1990, pp. 375-393. doi:10.1093/imamci/7.4.375

[8] A. Kowalewski, “Optimal Control of Parabolic Systems with Time-Varying Lags,” IMA Journal of Mathematical Control and Information, Vol. 10, No. 2, 1993, pp. 113-129. doi:10.1093/imamci/10.2.113

[9] A. Kowalewski, “Optimal Control of a Distributed Parabolic Systems with Multiple Time-Varying Lags,” International Journal of Control, Vol. 69, No. 3, 1998, pp. 361-381. doi:10.1080/002071798222712

[10] A. Kowalewski, “Optimization of Parabolic Systems with Deviating Arguments,” International Journal of Control, Vol. 72, No. 11, 1999, pp. 947-959. doi:10.1080/002071799220498

[11] A. Kowalewski and J. Duda, “On Some Optimal Control Problem for a Parabolic System with Boundary Condition Involving a Time-Varying Lag,” IMA Journal of Mathematical Control and Information, Vol. 9, No. 2, 1992, pp. 131-146. doi:10.1093/imamci/9.2.131

[12] A. Kowalewski and J. Duda, “An Optimization Problem Time Lag Distributed Parabolic Systems,” IMA Journal of Mathematical Control and Information, Vol. 21, No. 1, 2004, pp. 15-31. doi:10.1093/imamci/21.1.15

[13] W. Kotarski, H. A. El-Saify and G. M. Bahaa, “Optimal Control of Parabolic Equation with an Infinite 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] W. Kotarski and G. M. Bahaa, “Optimality Conditions for Infinite Order Hyperbolic Problem with Non-Standard Functional and Time Delay,” Journal of Information & Optimization Sciences, Vol. 28, No. 6, 2007, pp. 315-334.

[15] H. A. El-Saify, “Optimal Control for n × n Parabolic System Involving Time Lag,” IMA Journal of Mathematical Control and Information, Vol. 22, No. 3, 2005, pp. 240250. doi:10.1093/imamci/dni011

[16] H. A. El-Saify, “Optimal Boundary Control Problem for n × n Infinite Order Parabolic Lag System,” IMA Journal of Mathematical Control and Information, Vol. 23, No. 4, 2006, pp. 433-445. doi:10.1093/imamci/dni065

[17] J. A. Dubinskii, “Sobolev Spaces of Infinite Order and the Behavior of Solution of Some Boundary Value Problems with Unbounded Increase of the Order of the Equation,” Mathematics of the USSR-Sbornik, Vol. 27, No. 2, 1975, p. 143. doi:10.1070/SM1975v027n02ABEH002506

[18] J. A. Dubinskii, “Non-Triviality of Sobolev Spaces of Infinite Order for a Full Euclidean Space and a Torus,” Mathematics of the USSR-Sbornik, Vol. 100, 1976, pp. 436-446.

[19] J. A. Dubinskii, “Sobolev Spaces of Infinite Order and Differential Equations,” Springer-Verlag, New York, 1986.

[20] J. L. Lions and E. Magenes, “Non-Homogeneous Boundary Value Problem and Applications,” Springer-Verlag, New York, 1972.

[21] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer-Verlag, New York, 1971.

[22] H. A. El-Saify and G. M. Bahaa, “Optimal Control for n × n Hyperbolic Systems Involving Operators of Infinite Order,” Mathematica Slovaca, Vol. 52, No. 4, 2002, pp. 409-424.

[23] G. M. Bahaa, “Quadratic Pareto Optimal Control of Parabolic Equation with State-Control Constraints and Infinite Number of Variables,” IMA Journal of Mathematical Control and Information, Vol. 20, No. 2, 2003, pp. 167-178. doi:10.1093/imamci/20.2.167

[24] G. M. Bahaa, “Time-Optimal Control Problem for Infinite Order Parabolic Equation with Control Constraints,” Differential Equations and Control Processes. The Electronic Journal, Vol. 4, 2005, pp. 64-81. http://www.neva.ru/journal

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

[26] G. M. Bahaa, “Quadratic Pareto Optimal Control for Boundary Infinite Order Parabolic Equation with StateControl Constraints,” AMO-Advanced Modeling and Optimization, Vol. 9, 2007, pp. 37-51.

[27] G. M. Bahaa, “Optimal Control Problems of Parabolic Equations with an Infinite 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

[28] G. M. Bahaa and W. Kotarski, “Optimality Conditions for n × n Infinite Order Parabolic Coupled Systems with Control Constraints and General Performance Index,” IMA Journal of Mathematical Control and Information, Vol. 25, No. 1, 2008, pp. 49-57. doi:10.1093/imamci/dnm003

[29] W. Kotarski and G. M. Bahaa, “Optimal Control Problem for Infinite Order Hyperbolic System with Mixed Control-State Constraints,” European Journal of Control, Vol. 11, No. 2, 2005, pp. 150-156. doi:10.3166/ejc.11.150-156

[30] W. Kotarski, H. A. El-Saify and G. M. Bahaa, “Optimal Control Problem for a Hyperbolic System with Mixed Control-State Constraints Involving Operator of Infinite Order,” International Journal of Pure and Applied Mathematics, Vol. 1, No. 3, 2002, pp. 239-252.

[31] A. Kowalewski, “Time-Optimal Control of Infinite Order Hyperbolic Systems with Time Delays,” International Journal of Applied Mathematics and Computer Science, Vol. 19, No. 4, 2009, pp. 597-608. doi:10.2478/v10006-009-0047-x

[32] A. Kowalewski and A. Krakowiak, “Time-Optimal Boundary Control of Infinite Order Parabolic System with Time Lags,” International Journal of Applied Mathematics and Computer Science, Vol. 18, No. 2, 2008, pp. 189-198. doi:10.2478/v10006-008-0017-8

[1] P. K. C. Wang, “Optimal Control of Parabolic Systems with Boundary Conditions Involving Time Delays,” SIAM Journal on Control, Vol. 13, No. 2, 1975, pp. 274-293. doi:10.1137/0313016

[2] G. Knowles, “Time-Optimal Control of Parabolic Systems with Boundary Conditions Involving Time Delays,” Journal of Optimization Theory and Applications, Vol. 25, No. 4, 1978, pp. 563-574. doi:10.1007/BF00933521

[3] K. H. Wong, “Optimal Control Computation for Parabolic Systems with Boundary Conditions Involving Time Delays,” Journal of Optimization Theory and Applications, Vol. 53, No. 3, 1987, pp. 475-507. doi:10.1007/BF00938951

[4] A. Kowalewski, “Optimal Control of Hyperbolic System with Boundary Condition Involving a Time-Varying Lag,” Proceedings of the IMACS/IFAC International Symposium on Modeling and Simulation of DPS, Hiroshima, 6-9 October 1987, pp. 462-467.

[5] A. Kowalewski, “Boundary Control of Hyperbolic Systems with Boundary Condition Involving a Time Delay,” Analysis and Optimization of Systems, Vol. 111, 1988, pp. 507-518. doi:10.1007/BFb0042240

[6] A. Kowalewski, “Optimal Control of Distributed Hyperbolic System with Boundary Condition Involving a Time Lag,” Automatic Remote Control, XXXIII, 1988, pp. 537-545.

[7] A. Kowalewski, “Optimal Control of Distributed Parabolic System Involving Time Lags,” IMA Journal of Mathematical Control and Information, Vol. 7, No. 4, 1990, pp. 375-393. doi:10.1093/imamci/7.4.375

[8] A. Kowalewski, “Optimal Control of Parabolic Systems with Time-Varying Lags,” IMA Journal of Mathematical Control and Information, Vol. 10, No. 2, 1993, pp. 113-129. doi:10.1093/imamci/10.2.113

[9] A. Kowalewski, “Optimal Control of a Distributed Parabolic Systems with Multiple Time-Varying Lags,” International Journal of Control, Vol. 69, No. 3, 1998, pp. 361-381. doi:10.1080/002071798222712

[10] A. Kowalewski, “Optimization of Parabolic Systems with Deviating Arguments,” International Journal of Control, Vol. 72, No. 11, 1999, pp. 947-959. doi:10.1080/002071799220498

[11] A. Kowalewski and J. Duda, “On Some Optimal Control Problem for a Parabolic System with Boundary Condition Involving a Time-Varying Lag,” IMA Journal of Mathematical Control and Information, Vol. 9, No. 2, 1992, pp. 131-146. doi:10.1093/imamci/9.2.131

[12] A. Kowalewski and J. Duda, “An Optimization Problem Time Lag Distributed Parabolic Systems,” IMA Journal of Mathematical Control and Information, Vol. 21, No. 1, 2004, pp. 15-31. doi:10.1093/imamci/21.1.15

[13] W. Kotarski, H. A. El-Saify and G. M. Bahaa, “Optimal Control of Parabolic Equation with an Infinite 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] W. Kotarski and G. M. Bahaa, “Optimality Conditions for Infinite Order Hyperbolic Problem with Non-Standard Functional and Time Delay,” Journal of Information & Optimization Sciences, Vol. 28, No. 6, 2007, pp. 315-334.

[15] H. A. El-Saify, “Optimal Control for n × n Parabolic System Involving Time Lag,” IMA Journal of Mathematical Control and Information, Vol. 22, No. 3, 2005, pp. 240250. doi:10.1093/imamci/dni011

[16] H. A. El-Saify, “Optimal Boundary Control Problem for n × n Infinite Order Parabolic Lag System,” IMA Journal of Mathematical Control and Information, Vol. 23, No. 4, 2006, pp. 433-445. doi:10.1093/imamci/dni065

[17] J. A. Dubinskii, “Sobolev Spaces of Infinite Order and the Behavior of Solution of Some Boundary Value Problems with Unbounded Increase of the Order of the Equation,” Mathematics of the USSR-Sbornik, Vol. 27, No. 2, 1975, p. 143. doi:10.1070/SM1975v027n02ABEH002506

[18] J. A. Dubinskii, “Non-Triviality of Sobolev Spaces of Infinite Order for a Full Euclidean Space and a Torus,” Mathematics of the USSR-Sbornik, Vol. 100, 1976, pp. 436-446.

[19] J. A. Dubinskii, “Sobolev Spaces of Infinite Order and Differential Equations,” Springer-Verlag, New York, 1986.

[20] J. L. Lions and E. Magenes, “Non-Homogeneous Boundary Value Problem and Applications,” Springer-Verlag, New York, 1972.

[21] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer-Verlag, New York, 1971.

[22] H. A. El-Saify and G. M. Bahaa, “Optimal Control for n × n Hyperbolic Systems Involving Operators of Infinite Order,” Mathematica Slovaca, Vol. 52, No. 4, 2002, pp. 409-424.

[23] G. M. Bahaa, “Quadratic Pareto Optimal Control of Parabolic Equation with State-Control Constraints and Infinite Number of Variables,” IMA Journal of Mathematical Control and Information, Vol. 20, No. 2, 2003, pp. 167-178. doi:10.1093/imamci/20.2.167

[24] G. M. Bahaa, “Time-Optimal Control Problem for Infinite Order Parabolic Equation with Control Constraints,” Differential Equations and Control Processes. The Electronic Journal, Vol. 4, 2005, pp. 64-81. http://www.neva.ru/journal

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

[26] G. M. Bahaa, “Quadratic Pareto Optimal Control for Boundary Infinite Order Parabolic Equation with StateControl Constraints,” AMO-Advanced Modeling and Optimization, Vol. 9, 2007, pp. 37-51.

[27] G. M. Bahaa, “Optimal Control Problems of Parabolic Equations with an Infinite 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

[28] G. M. Bahaa and W. Kotarski, “Optimality Conditions for n × n Infinite Order Parabolic Coupled Systems with Control Constraints and General Performance Index,” IMA Journal of Mathematical Control and Information, Vol. 25, No. 1, 2008, pp. 49-57. doi:10.1093/imamci/dnm003

[29] W. Kotarski and G. M. Bahaa, “Optimal Control Problem for Infinite Order Hyperbolic System with Mixed Control-State Constraints,” European Journal of Control, Vol. 11, No. 2, 2005, pp. 150-156. doi:10.3166/ejc.11.150-156

[30] W. Kotarski, H. A. El-Saify and G. M. Bahaa, “Optimal Control Problem for a Hyperbolic System with Mixed Control-State Constraints Involving Operator of Infinite Order,” International Journal of Pure and Applied Mathematics, Vol. 1, No. 3, 2002, pp. 239-252.

[31] A. Kowalewski, “Time-Optimal Control of Infinite Order Hyperbolic Systems with Time Delays,” International Journal of Applied Mathematics and Computer Science, Vol. 19, No. 4, 2009, pp. 597-608. doi:10.2478/v10006-009-0047-x

[32] A. Kowalewski and A. Krakowiak, “Time-Optimal Boundary Control of Infinite Order Parabolic System with Time Lags,” International Journal of Applied Mathematics and Computer Science, Vol. 18, No. 2, 2008, pp. 189-198. doi:10.2478/v10006-008-0017-8