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This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check...

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.

We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.

Let $A_0$ be a possibly unbounded positive operator on the Hilbert space $H$, which is boundedly invertible. Let $C_0$ be a bounded operator from $𝒟\left(A_0^{\frac{1}{2}}\right)$ to another Hilbert space $U$. We prove that the system of equations $$\ddot{z}\left(t\right)+A_{0}z\left(t\right)+\frac{1}{2}{C}_0^{*}{C}_0\dot{z}\left(t\right)=C_0^{*}u\left(t\right)$$ $$y\left(t\right)=-C_0\dot{z}\left(t\right)+u\left(t\right),$$ determines a well-posed linear system with input $u$ and output $y$. The state of this system is $$x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \dot{z}\left(t\right)\end{array}\right]\in 𝒟\left(A_0^{\frac{1}{2}}\right)\times H=X,$$ where $X$ is the state space. Moreover, we have the energy identity $${\parallel x\left(t\right)\parallel }_X^2-{\parallel x\left(0\right)\parallel }_X^2={\int }_0^T{\parallel u\left(t\right)\parallel }_U^2\mathrm{d}t-{\int }_0^T{\parallel y\left(t\right)\parallel }_U^2\mathrm{d}t.$$ We show that the system described above is isomorphic...

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.

Let be a possibly unbounded positive operator on the Hilbert space , which is boundedly invertible. Let be a bounded operator from $𝒟\left(A_0^{\frac{1}{2}}\right)$ to another Hilbert space . We prove that the system of equations $$\ddot{z}\left(t\right)+A_{0}z\left(t\right)+\frac{1}{2}{C}_0^{*}{C}_0\dot{z}\left(t\right)=C_0^{*}u\left(t\right)$$ $$y\left(t\right)=-C_0\dot{z}\left(t\right)+u\left(t\right),$$ determines a well-posed linear system with input and output . The state of this system is $$x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \dot{z}\left(t\right)\end{array}\right]\in 𝒟\left(A_0^{\frac{1}{2}}\right)\times H=X,$$ where is the state space. Moreover, we have the energy identity $${\parallel x\left(t\right)\parallel }_X^2-{\parallel x\left(0\right)\parallel }_X^2={\int }_0^T{\parallel u\left(t\right)\parallel }_U^2\mathrm{d}t-{\int }_0^T{\parallel y\left(t\right)\parallel }_U^2\mathrm{d}t.$$ We show that the system described above is isomorphic to its dual, so that a similar...

We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.

This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check...

In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem....

We survey the literature on well-posed linear systems, which has been an area of rapid development in recent years. We examine the particular subclass of conservative systems and its connections to scattering theory. We study some transformations of well-posed systems, namely duality and time-flow inversion, and their effect on the transfer function and the generating operators. We describe a simple way to generate conservative systems via a second-order differential equation in a Hilbert space....

We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are...