This paper presents a new observability estimate for parabolic equations in × (0, T ), where is a convex domain. The observation region is restricted over a product set of an open nonempty subset of and a subset of positive measure in (0, T ). This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

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In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation ∂ t u − u = g(u), with the homogeneous Dirichlet boundary condition, over Ω × (0, T * ). Ω is a bounded, convex open subset of R d , with a smooth boundary for the subset. The function g : R → R satisfies certain conditions. We establish some observation estimates for (u − v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω × {T }, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0, T ]. At least two results can be derived from these estimates: (i) if (u − v)(·, T ) L 2 (ω) = δ, then (u − v)(·, T ) L 2 (Ω) Cδ α where constants C > 0 and α ∈ (0, 1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω × {T }, then they coincide over Ω × [0, T m ). T m indicates the maximum number such that these two solutions exist on [0, T m ).

This work concerns with the existence of the time optimal controls for some linear evolution equations without the a priori assumption on the existence of admissible controls. Both global and local existence results are presented. Some necessary conditions, sufficient conditions, and necessary and sufficient conditions for the existence of time optimal controls are derived by establishing the relationship between controllability and time optimal control problems.

Using Fourier integral operators with special amplitude functions, we analyze the stabilization of the wave equation in a three-dimensional bounded domain on which exists a trapped ray bouncing up and down infinitely between two parallel parts of the boundary.

Abstract. We consider the exact controllability and stabilization of Maxwell equation by using results on the propagation of singularities of the electromagnetic field. We will assume geometrical control condition and use techniques of the work of Bardos et al. on the wave equation. The problem of internal stabilization will be treated with more attention because the condition divE = 0 is not preserved by the system of Maxwell with Ohm's law.Résumé. Nousétudions la contrôlabilité exacte et la stabilisation deséquations de Maxwell par le biais de la propagation des singularités du champélectromagnétique dans un domaine borné. Les résultats présentés s'inspirent des travaux de Bardos et al. sur le contrôle géométrique. Notre intéret se portera plus particulièrement sur la stabilisation interne où l'on doit considérer le système de Maxwell avec loi d'Ohm avec une densité de charge non nulle.

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We prove the approximate controllability for the heat equation with potential with a cost of order e c/ε when the target is in H 1 0 (Ω) with a precision in L 2 (Ω) norm. Also a quantification estimate of the unique continuation for initial data in L 2 (Ω) of the heat equation with potential is established.