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Unformatted text preview: Internal null-controllability for a structurally damped beam equation, Version 4.0 Julian Edward and Louis Tebou * March 10, 2005 Abstract In this paper we study the null-controllability 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 null-controllability result for the undamped beam equation. 1 Introduction and statement of main results Let T > 0, and let ω be an open subset of (0 , 1). Let ρ > 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 ( ω × (0 ,T )) such that the solution u of the structurally damped beam equation u tt + u xxxx- ρu xxt = χ ω 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 [email protected] and [email protected] 1 satisfies u ( x,T ) = u t ( x,T ) = 0 , a.e.x ∈ (0 , 1)? (1) The system (0) with f = 0 is the one-dimensional version of the mathe- matical model for linear elastic systems with structural damping introduced by Chen and Russell in . Null-controllability in time for arbitrarily small T for the undamped case ( ρ = 0)was proven in . Various results for bound- ary control in the undamped case can be found in Zuazua’s survey in . For a study of optimal boundary controllability of the damped plate equation, see . Recently Lasiecka and Triggiani  studied the null-controllability of the abstract equation: w tt + Sw + ρS α w t = u, w (0) = w ,w t (0) = w 1 ,ρ > ,α ∈ [1 / 2 , 1] . Here S is a strictly positive, self-adjoint 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 Δ 2 appearing in our case, the authors did not assume that the control u is confined to a proper subset ω ⊂ (0 , 1) as we do here; this makes our problem harder because we cannot use Bessel’s inequality as they do. Also, their controllability is not uniform in ρ . Hansen  proved an estimate on functions biorthogonal to a certain family of exponential functions on [0 ,T ]. As an application, he stud- ied the null-controllability of a damped vibrating rectangular plate equation subject to boundary control on one side, with ρ < 2, but his methods will also apply to the vibrating beam equation with internal damping to prove boundary null-controllability in time T for any ρ < 2. However, his proof, which uses a compactness argument, will not yield estimates on the control...
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