Control of an equation by maximum principle

In this paper, some results, which are related to well posedness, controllability and optimal control of a beam equation, are presented. In order to obtain the optimal control function, maximum principle is employed. Performance index function is defined as quadratic functional of displacement and velocity and also includes a penalty in terms of control function. The solution of the control problem is formulated by using Galerkin expansion. Obtained results are given in the table and graphical forms.

Keywords:

Beam, optimal control,

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F. H. Clarke, Maximum principle under minimal hypotheses, SIAM J. Control Optimization, 14(1976), 1078-1091.
H. F. Guliyev, K. S. Jabbarova, The exact controllability problem for the second order linear hyperbolic equation, Differential Equations and Control Processes,(2010).
E. B. Lee, A sufficient condition in the theory of optimal control, SIAM Journal on Control, 1(1963), 241-245.
K. Yildirim, I. Kucuk, Active piezoelectric vibration control for a Timoshenko beam, Journal of the Franklin Institute, (2015).
Barnes, E. A., Necessary and sufficient optimality conditions for a class of distributed parameter control systems, SIAM Journal on Control, 9(1971), 62-82.
Kucuk, I., Yildirim, K., Sadek, I., Adali, S., Optimal control of a beam with Kelvin-Voigt damping subject to forced vibrations using a piezoelectric patch actuator, Journal of Vibration and Control, (2013).
Kucuk, I., Yildirim, K., Necessary and Sufficient Conditions of Optimality for a Damped Hyperbolic Equation in One Space Dimension, Abstract and Applied Analysis, Art. ID 493130(2014), 10 pages.
Kucuk, I., Yildirim, K., Adali, S., Optimal piezoelectric control of a plate subject to time-dependent boundary moments and forcing function for vibration damping, Computers and Mathematics with Applications, 69(2015), 291-303.
Sadek, I., Necessary and sufficient conditions for the optimal control of distributed parameter systems subject to integral constraints,Journal of the Franklin Institute, 325(1988), 565-583.
Das, S., Vishal, K., Gupta, P. K., Yildirim, A., An approximate analytical solution of time-fractional telegraph eqaution, Applied Mathematics and Computation, 217(2011), 7405-7411.
Koshlyakov, N. S., Smirnov, M. M., Gliner, E. B., Differential Equations of Mathematical Physics, North-Holland Publishing Company, Amsterdam(1964).
Zachmaonoglou, E. C., Thoe, D.W., Intoduction to Partial Differential equations with applications, Dover Publ., New York(1986).