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Periodic stabilization for linear time-periodic ordinary differential equations

Gengsheng WangYashan Xu — 2014

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies the periodic feedback stabilization of the controlled linear time-periodic ordinary differential equation: () = ()() + ()(),  ≥ 0, where [(·)(·)] is a -periodic pair, , (·) ∈  (ℝ; ℝ) and (·) ∈  (ℝ; ℝ) satisfy respectively ( + ) = () for a.e.  ≥ 0 and ( + ) = () for a.e.  ≥ 0. Two periodic stablization criteria for a -period pair [(·)(·)] are established. One is an analytic criterion which is related to the transformation over time associated with...

An observability estimate for parabolic equations from a measurable set in time and its applications

Kim Dang PhungGengsheng Wang — 2013

Journal of the European Mathematical Society

This paper presents a new observability estimate for parabolic equations in Ω × ( 0 , T ) , where Ω is a convex domain. The observation region is restricted over a product set of an open nonempty subset of Ω 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.

Observability inequalities and measurable sets

Jone ApraizLuis EscauriazaGengsheng WangC. Zhang — 2014

Journal of the European Mathematical Society

This paper presents two observability inequalities for the heat equation over Ω × ( 0 , T ) . In the first one, the observation is from a subset of positive measure in Ω × ( 0 , T ) , while in the second, the observation is from a subset of positive surface measure on Ω × ( 0 , T ) . It also proves the Lebeau-Robbiano spectral inequality when Ω is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

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