Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems
Mathematics Letters
Volume 3, Issue 1, February 2017, Pages: 1-11
Received: Dec. 17, 2016; Accepted: Dec. 30, 2016; Published: Mar. 30, 2017
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Hussein El-Saify, Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
Mohammed Shehata, Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia
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In this paper, a minimum time problem for n×n co-operative hyperbolic systems involving Laplace operator and with time-delay is considered. First, the existence of a unique solution of such hyperbolic system with time-delay is proved. Then necessary conditions of a minimum time control are derived in the form of maximum principle. Finally the bang-bang principle and the approximate controllability conditions are investigated.
Time-Optimal Control Problem, Co-operative Systems, Hyperbolic Systems with Time Delay, Approximate Controllability, Bang-Bang Principle
To cite this article
Hussein El-Saify, Mohammed Shehata, Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems, Mathematics Letters. Vol. 3, No. 1, 2017, pp. 1-11. doi: 10.11648/
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This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
J. L. Lions, Optimal control of systems governed by partial differential equations, Springer-verlag, Band 170, (1971).
J. L. Lions and E. Magenes, Non homogeneous boundary value problem and applications. Spring-Verlage, New York. I, II, (1972).
J.-L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.
P. K. C. Wang, Time-optimal control of time-lag systems with time-lag control. Journal of Mathematical Analysis and Applications Vol. 52,No, 3 366- 378, (1975).
G. Knowles, Time optimal control of parabolic systems with boundary condition involving time delays Journal of Optimiz.Theor. Applics,25, (1978), 563-574 .
H. O. Fattorini. The Time Optimal Problem for Distributed Control of Systems Described by the Wave Equation. In: Aziz, A. K., Wingate, J. W., Balas, M. J. (eds.): Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York, San Francisco, London (1957 ).
W. Krabs, On Time-Minimal Distributed Control of Vibrating Systems Governed by an Abstract Wave Equation. AppI. Math. and Optim. 13. ( 1985 ), 137-149.
H. A. El-Saify, H. M. Serag and M. A. Shehata, Time-optimal control for co-operative hyperbolic systems Involving Laplace operator. Journal of Dynamical and Control systems. 15, 3, (2009), 405-423.
M. A. Shehata, Some time-optimal control problems for co-operative hyperbolic systems with distributed or boundary controls. Journal of Mathematical Sciences: Advances and Applications. vol 18, No 1-2, (2012), 63-83.
M. A. Shehata, Time -optimal control problem for co-operative parabolic systems with control in initial conditions, Advances in Pure Mathematics Journal , 3, No 9A, (2013), 38-43.
M. A. Shehata, Dirichlet Time-Optimal Control of Co-operative Hyperbolic Systems Advanced Modeling and Optimization Journal. Volume 16, Number 2, (2014), 355-369.
Byung Soo Lee, Mohammed Shehata, Salahuddin , Time -optimal control problem for co-operative parabolic systems with strong constraint control in initial conditions, Journal of Science and Technology, Vol. 4 No. 11, (2014).
R. A. Adams, Sobolev Spaces. Academic Prees, New York. (1975).
J. Fleckinger, J. Herna'ndez and F. DE. The'lin, On the existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems. Rev. R. Acad. Cien. Serie A. Mat. 97, 2 (2003), 461-466.
A. Friedman, Optimal control for parabolic variational inequalities. SIAM Journal of Control and Optimization , 25, 482-497, (1987).
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