Controllability Problem for the String Equation with External Load with the use of Atomic Functions
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CONTROLLABILITY PROBLEM FOR THE STRING EQUATION WITH EXTERNAL LOAD WITH THE USE OF ATOMIC FUNCTIONS UDC 517.958+517.977.5
Ye. M. Donchik
Abstract. The necessary and sufficient conditions for the approximate controllability and null-controllability by atomic functions are obtained for the string equation controlled by an external load on a rectangular domain. The function of external load is presented as a scalar product of atomic functions and controls. The controls that solve these problems are found explicitly. Keywords: wave equation, controllability problem, control of external load, Fourier transform, function of bounded variation. INTRODUCTION Controllability problems in partial differential equations arise in the analysis of various physical processes such as wave ones and draw attention of many mathematicians (see, for example, [1–6] and the bibliography in [4, 5]). For example, Lions [1] outlined the fundamentals of the systematization of optimal control problems for partial differential equations. Il’in [2] considered the two-endpoint boundary control of vibrations as well as one-point with the fixed second endpoint in terms of the generalized solution of a wave equation with finite energy. Zuazua [4] analyzed the conditions of e-controllability and null controllability in space L2 for a series of control problems in partial differential equations. REVIEW OF THE AVAILABLE STUDIES Fardigola and Sklyar [5] considered the controllability problem for a wave equation in the Sobolev space H ls with the control function in the form of the boundary condition ¶ 2 w ( x, t ) ¶ 2 w ( x, t ) = 0 , x > 0 , t Î ( 0, T ) , ¶x 2 ¶t 2 with the control w( 0, t ) = -u( t ), t Î ( 0, T ) . They also obtained the necessary and sufficient conditions for approximate and null controllability. Khalina [6] analyzed the string equation in the Sobolev space, where the control functions appear in the boundary conditions w ( 0, t ) = u 0 ( t ), w ( p , t ) = u p ( t ), t Î ( 0, T ) . Fardigola [7] considered the Helmholtz equation ¶ 2 w ( x, t ) ¶ 2 w ( x, t ) 2 - q w ( x, t ) , x > 0 , t Î ( 0, T ) , q ³ 0 , = ¶x 2 ¶t 2 with the control functions in the form of the boundary condition w x ( 0, t ) = u( t ), t Î ( 0, T ) . Curtain and Pritchard [8] analyzed the strong stabilization of infinite-dimensional systems to improve perturbation. Lasiecka and Triggiani [9] studied the regularity of structurally damping problems with pointwise and boundary control. Y. You [10] considered the exact controllability (null-controllability) for the Petrovsky equation in a bounded domain. Academy of Interior Forces, Kharkov, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, January–February, 2014, pp. 111–124. Original article submitted April 5, 2013. 98
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2014 Springer Science+Business Media New York
However, in some cases, the solution of controllability problems, where controls appear in the boundary conditions, involve difficulties in the practical implementation of the solution proc
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