Displaying similar documents to “Controllability of analytic functions for a wave equation coupled with a beam.”

Exact Neumann boundary controllability for second order hyperbolic equations

Weijiu Liu, Graham Williams (1998)

Colloquium Mathematicae

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Using HUM, we study the problem of exact controllability with Neumann boundary conditions for second order hyperbolic equations. We prove that these systems are exactly controllable for all initial states in L 2 ( Ω ) × ( H 1 ( Ω ) ) ' and we derive estimates for the control time T.

Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off.

Cédric Villani (1999)

Revista Matemática Iberoamericana

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We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function f ∈ L(R) yields a control of √f in Sobolev norms as soon as f is locally bounded below. Under this additional assumption of lower bound, our result is an improvement of a recent estimate given by P.-L. Lions, and is optimal in a certain sense.