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.
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 ).
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