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### A new method to obtain decay rate estimates for dissipative systems

ESAIM: Control, Optimisation and Calculus of Variations

### A new method to obtain decay rate estimates for dissipative systems with localized damping.

Revista Matemática Complutense

We consider the wave equation damped with a locally distributed nonlinear dissipation. We improve several earlier results of E. Zuazua and of M. Nakao in two directions: first, using the piecewise multiplier method introduced by K. Liu, we weaken the usual geometrical conditions on the localization of the damping. Then thanks to some new nonlinear integral inequalities, we eliminate the usual assumption on the polynomial growth of the feedback in zero and we show that the energy of the system decays...

### A new method to obtain decay rate estimates for dissipative systems

ESAIM: Control, Optimisation and Calculus of Variations

We consider the wave equation damped with a boundary nonlinear velocity feedback . Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction...

### Stabilization of the wave equation by on-off and positive-negative feedbacks

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback $a\left(t\right){u}_{t}$. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions $a$: typically $a$ is equal to $1$ on $\left(0,T\right)$, equal to $0$ on $\left(T,qT\right)$ and is $qT$-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases,...

### Stabilization of the wave equation by on-off and positive-negative feedbacks

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback $a\left(t\right){u}_{t}$. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions : typically is equal to on , equal to on and is -periodic. We study the boundary case and next the locally distributed case, and we give . In both cases, we prove that there are of...

### Null controllability of the heat equation in unbounded domains by a finite measure control region

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically ${ℝ}_{+}$ or ${ℝ}^{N}$. Considering an unbounded and disconnected control region of the form $\omega :={\cup }_{n}{\omega }_{n}$, we prove two null controllability results: under some technical assumption on the control parts ${\omega }_{n}$, we prove that every initial datum in some weighted ${L}^{2}$ space can be controlled to zero by usual control functions, and every initial datum in ${L}^{2}\left(\Omega \right)$ can...

### Null controllability of the heat equation in unbounded domains by a finite measure control region

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically ${ℝ}_{+}$ or ${ℝ}^{N}$. Considering an unbounded and disconnected control region of the form $\omega :={\cup }_{n}{\omega }_{n}$, we prove two null controllability results: under some technical assumption on the control parts ${\omega }_{n}$, we prove that every initial datum in some weighted space can be controlled to zero by usual control functions, and every initial...

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