Non-structural controllability of linear elastic systems with structural damping

This paper proves that any initial condition in the energy space for the plate equation with square root damping ζ̈ − ρ∆ζ̇ +∆2ζ = u on a smooth bounded domain, with hinged boundary conditions ζ = ∆ζ = 0, can be steered to zero by a square integrable input function u supported in arbitrarily small time interval [0, T ] and subdomain. As T tends to zero, for initial states with unit energy norm, the norm of this u grows at most like exp(Cp/T p) for any real p > 1 and some Cp > 0. Indeed, this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non-structural controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on the subdomain allowing to apply the control transmutation method, this estimate is improved into p = 1 and the dependence of Cp on the subdomain is made explicit. These results are analogous to the optimal ones known for the heat flow. A wide variety of dissipative linear elastic control systems may be represented by a second-order differential equation in a Hilbert space: ζ̈(t) +Dζ̇(t) + Sζ(t) = Bu(t), t ∈ R+ (1) where each dot denotes a derivative with respect to the time variable t and the function ζ represents the evolution of the system under the action of the input function u. The structural vibration modes of the conservative system represented by (1) with B = D = 0 are prescribed by the positive self-adjoint operator S. This ideal system is perturbed by a dissipative mechanism prescribed by the positive self-adjoint operator D. The system is actuated through a control mechanism prescribed by the operator B (possibly unbounded to take into account trace operators prescribing the boundary value of distributed states). Throughout this paper, controllability will always mean the ability of steering any initial state (z(0), ż(0)) to zero over a finite time by some appropriate input function u (i.e. exact controllability to zero or null controllability). This paper concerns the specific dissipative mechanism D = S with α ∈ (0, 1) called structural damping, which generalizes the square root damping model α = 1/2 introduced in [CR82]: “The basic property of structural damping, which is said to be consistent with empirical studies, is that the amplitudes of the normal modes of vibration are attenuated at rates which are proportional to the oscillation frequencies”. This model was also studied under the name “proportional damping” (cf. [Bal90]). The quite different case α = 1 is known as “Kelvin-Voigt” damping. When B is the identity and α ∈ (0, 1], this is the first class of parabolic-like control models considered in [LT98, Tri03] with the extra assumption that S has compact resolvent, dispensed with in [AL03]. This paper focuses on the cost of fast controls as in [Tri03, AL03]. The controllability results known for these systems hold for a control time which can be chosen as small as wished. This asymptotic is referred to as fast control. The cost Date: December 14, 2005. 2000 Mathematics Subject Classification. 93B05, 49J20, 34K35, 74K20.