In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these systems are proved, and the formulation of the control problem for different observation functions is discussed.
References
[1]
Lions, J.L. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Springer Verlag, Berlin, Band 170. https://doi.org/10.1007/978-3-642-65024-6
[2]
Kotarski, W., El-Saify, H.A. and Shehata, M.A. (2007) Time Optimal Control of Parabolic Lag Systems with an Infinite Number of Variables. Journal of the Egyptian Mathematical Society, 15, 21-34.
[3]
Qamlo, A.H. (2008) Boundary Control for Cooperative Systems Involving Parabolic Operators with an Infinite Number of Variables. Advances in Differential Equations and Control Processes, 2, 135-151.
[4]
Qamlo, A.H. (2009) Distributed Control of Cooperative Systems Involving Hyperbolic Operators with an Infinite Number of Variables. International Journal of Functional Analysis, Operator Theory and Applications, 1, 115-128.
[5]
Qamlo, A.H. and Bahaa, G.M. (2013) Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions. Intelligent Control and Automation, 4, 280-286. https://doi.org/10.4236/ica.2013.43032
[6]
Qamlo, A.H. (2013) Distributed Control for Cooperative Hyperbolic Systems Involving Schrödinger Operator. International Journal of Dynamics and Control, 1, 54-59. https://doi.org/10.1007/s40435-013-0007-z
[7]
Qamlo, A.H. (2013) Optimality Conditions for Parabolic Systems with Variable Coefficients Involving Schrödinger Operators. Journal of King Saud University— Science, 26, 107-112. https://doi.org/10.1016/j.jksus.2013.05.005
[8]
Qamlo, A.H. (2014) An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator. Intelligent Control and Automation, 5, 262-271. https://doi.org/10.4236/ica.2014.54028
[9]
Qamlo, A.H., Serag, H.M. and Al-Zahrani, E. (2015) Optimal Control for Non- Cooperative Parabolic Systems with Conjugation Conditions. European Journal of Scientific Research, 131, 215-226.
[10]
Serag, H.M. and Qamlo, A.H. (2005) On Elliptic Systems Involving Schrödinger operators. The Mediterranean Journal of Measurement and Control, 1, 91-96.
[11]
Serag, H.M. (2007) Distributed Control for Cooperative Systems Involving Parabolic Operators with an Infinite Number of Variables. IMA Journal of Mathematical Control and Information, 24, 149-161. https://doi.org/10.1093/imamci/dnl018
[12]
Bandaliyev, R.A., Guliyev, V.S., Mamedov, I.G. and Rustamov, Y.I. (2019) Optimal Control Problem for Bianchi Equation in Variable Exponent Sobolev Spaces. Journal of Optimization Theory and Applications, 180, 303-320. https://doi.org/10.1007/s10957-018-1290-9
[13]
Qamlo, A.H. and Bedaiwi, G.M. (2017) Distributed Control for 2×2 Coupled Infinite Order Hyperbolic Systems. Advances in Differential Equations and Control Processes, 18, 201-227. https://doi.org/10.17654/DE018040201
[14]
Gali, I.M. and El-Saify, H.A. (1983) Control of System Governed by Infinite Order Equation of Hyperbolic Type. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, Vol. 3, 99-103.
[15]
Gali, I.M., El-Saify, H.A. and El-Zahaby, S.A. (1983) Distributed Control of a System Governed by Dirichlet and Neumann Problems for Elliptic Equations of Infinite Order. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, Vol. 3, 83-87.
[16]
Bahaa, G.M. and Tharwat, M.M. (2012) Time-Optimal Control of (n × n) Infinite Order Parabolic System with Time Lags Given in Integral Form. Journal of Information and Optimization Sciences, 33, 233-258. https://doi.org/10.1080/02522667.2012.10700145
[17]
Bahaa, G.M. and El-Gohany, E. (2013) Optimal Control Problem for (n × n) Infinite Order Parabolic Systems. International Journal of Applied Mathematics & Statistics, 36, 26-41.
[18]
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.
[19]
El-Saify, H.A. (2006) Optimality Conditions for Infinite Order Hyperbolic Differential System. Journal of Dynamical and Control Systems, 12, 313-330. https://doi.org/10.1007/s10450-006-0002-1
[20]
Kotarski, W. and Bahaa, G.M. (2005) Optimal Control Problem for Infinite Order Hyperbolic System with Mixed Control-State Constraints. European Journal of Control, 11, 149-159. https://doi.org/10.3166/ejc.11.150-156
[21]
Kotarski, W. and Bahaa, G.M. (2007) Optimality Conditions for Infinite Order Hyperbolic Problem with Non-Standard Functional and Time Delay. Journal of Information and Optimization Sciences, 28, 315-334. https://doi.org/10.1080/02522667.2007.10699746
[22]
Dubinskii, J.A. (1976) Non-Triviality of Sobolev Spaces of Infinite Order for a Full Euclidean Space and a Torus. Matiematiczeskii Sbornik, 100, 436-446.
[23]
Dubinskii, J.A. (1981) About One Method for Solving Partial Differential Equations. Doklady Akademii Nauk SSSR, 258, 780-784.
[24]
Dubinskii, J.A. (1986) Sobolev Spaces in Infinite Order and Differential Equations. Mathematics and Its Applications. East European Series 3, Springer, Berlin, 1-157.
