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Unformatted text preview: Internal nullcontrollability for a structurally damped beam equation, Version 4.0 Julian Edward and Louis Tebou * March 10, 2005 Abstract In this paper we study the nullcontrollability of a beam equa tion with hinged ends and structural damping, the damping depending on a positive parameter. We prove that this system is exactly null controllable in arbitrarily small time. This result is proven using a combination of Ingham type inequalities, adapted for complex frequencies, and exponential decay on various frequency bands. We then let the damping parameter tend to zero and we recover an earlier nullcontrollability result for the undamped beam equation. 1 Introduction and statement of main results Let T > 0, and let be an open subset of (0 , 1). Let > 0. Consider the controllability problem: given u ( x, 0) = u ( x ) H 2 (0 , 1) H 1 (0 , 1) , u t ( x, 0) = u 1 ( x ) L 2 (0 , 1) , can we find a control function f L 2 ( (0 ,T )) such that the solution u of the structurally damped beam equation u tt + u xxxx u xxt = f in (0 , 1) (0 ,T ) u (0 ,t ) = u (1 ,t ) = u xx (0 ,t ) = u xx (1 ,t ) = 0 in (0 ,T ) u ( x, 0) = u ( x ) , u t ( x, 0) = u 1 ( x ) , in (0 , 1) , (0) * Department of Mathematics, Florida International University, Miami FL 33199, emails edwardj@fiu.edu and teboul@fiu.edu 1 satisfies u ( x,T ) = u t ( x,T ) = 0 , a.e.x (0 , 1)? (1) The system (0) with f = 0 is the onedimensional version of the mathe matical model for linear elastic systems with structural damping introduced by Chen and Russell in [2]. Nullcontrollability in time for arbitrarily small T for the undamped case ( = 0)was proven in [14]. Various results for bound ary control in the undamped case can be found in Zuazuas survey in [8]. For a study of optimal boundary controllability of the damped plate equation, see [12]. Recently Lasiecka and Triggiani [6] studied the nullcontrollability of the abstract equation: w tt + Sw + S w t = u, w (0) = w ,w t (0) = w 1 , > , [1 / 2 , 1] . Here S is a strictly positive, selfadjoint unbounded operator with compact resolvent, and the control u is assumed to be distributed throughout (0 , 1). Although the operator S is more general than operator 2 appearing in our case, the authors did not assume that the control u is confined to a proper subset (0 , 1) as we do here; this makes our problem harder because we cannot use Bessels inequality as they do. Also, their controllability is not uniform in . Hansen [3] proved an estimate on functions biorthogonal to a certain family of exponential functions on [0 ,T ]. As an application, he stud ied the nullcontrollability of a damped vibrating rectangular plate equation subject to boundary control on one side, with < 2, but his methods will also apply to the vibrating beam equation with internal damping to prove boundary nullcontrollability in time T for any < 2. However, his proof, which uses a compactness argument, will not yield estimates on the control...
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This note was uploaded on 12/17/2011 for the course MATH DM 431C taught by Professor Dr.julianedward during the Fall '11 term at FIU.
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