Marius Tucsnak

Stabilization of Second Order Evolution Equations by a Class of Unbounded Feedbacks

In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties. Mathematics Subject Classification. 93B52, 93D15, 93B07. Received October 19, 2000. Revised February 19, 2001.

Stabilization of Bernoulli--Euler Beams by Means of a Pointwise Feedback Force

We study the energy decay of a Bernoulli--Euler beam which is subject to a pointwise feedback force. We show that both uniform and nonuniform energy decay may occur. The uniform or nonuniform decay depends on the boundary conditions. In the case of nonuniform decay in the energy space we give explicit polynomial decay estimates valid for regular initial data. Our method consists of deducing the decay estimates from observability inequalities for the associated undamped problem via sharp trace regularity results.

HOW TO GET A CONSERVATIVE WELL-POSED LINEAR SYSTEM OUT OF THIN AIR. PART II. CONTROLLABILITY AND STABILITY ∗

Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from

Boundary Stabilization for the von Karman Equations

{\cal D}(A_0^{1/2})

Simultaneous Exact Controllability and Some Applications

(with the norm

Decay rates for a beam with pointwise force and moment feedback

\|z\|_{1/2}^2=\langle...

Global existence for the full von Kármán system

The boundary stabilization of a nonlinear plate model is studied. The equations used take in consideration the in-plane accelerations and the rotary inertia of the cross sections. Applying linear feedbacks, the authors obtain the exponential decay of the energy.

On a theorem of Ingham

We study the exact controllability of two systems by means of a common finite-dimensional input function, a property called simultaneous exact controllability. Most of the time we consider one system to be infinite-dimensional and the other finite-dimensional. In this case we show that if both systems are exactly controllable in time T 0 and the generators have no common eigenvalues, then they are simultaneously exactly controllable in any time T >T0. Moreover, we show that similar results hold for approximate controllability. For exactly controllable systems we characterize the reachable subspaces corresponding to input functions of class H1 and H2. We apply our results to prove the exact controllability of a coupled system composed of a string with a mass at one end. Finally, we consider an example of two infinite-dimensional systems: we characterize the simultaneously reachable subspace for two strings controlled from a common end. The result is obtained using a recent generalization of a classical inequality of Ingham.

Solving Inverse Source Problems Using Observability. Applications to the Euler-Bernoulli Plate Equation

Abstract. We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0,π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π=p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler–Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.

Motion of a rigid body in a viscous fluid

We consider here the full system of dynamic von Kármán equations, taking into account the in-plane acceleration terms, which is a model for the vibrations of a nonlinear elastic plate. We prove global existence and uniqueness of strong solutions for this system with various boundary conditions possibly including feedback terms which are useful for stabilization purposes.