Controllability of a Generalized Damped Wave Equation Controllability of a Generalized Damped Wave Equation

Hugo Leiva
2006 unpublished
In this paper we give a necessary and sufficient algebraic condition for the controllability of the following generalized damped wave equation on a Hilbert space X ¨ w + ηA α ˙ w + γA β w = d 1 u 1 + · · · + d m u m , if α > 0 u(t), if α = 0, where t ≥ 0, γ > 0, η > 0, β ≥ 0 and d i ∈ X; the scalar control functions u i ∈ L 2 (0, t 1 ; IR); the distributed control u ∈ L 2 (0, t 1 ; X) and A : D(A) ⊂ X → X is a positive defined self-adjoint unbounded linear operator in X with compact resolvent.
more » ... compact resolvent. The equation¨wequation¨ equation¨w +ηA α ˙ w +γA β w = 0 can be written as a first order system in the space D(A β/2) × X with corresponding linear operator A. Then, we prove the following statements: I) A generates a strongly continuous semigroup {T (t)} t≥0 such that for some positive constants M (η, γ) and µ we have T (t) ≤ M (η, γ)e −µt , t ≥ 0. II) If 2α ≥ β, then {T (t)} t≥0 is analytic in the space D(A α) × X. III) If 2α ≥ β > α or 2α ≤ β, the system is approximatelly controllable on [0, t 1 ]. IV) If 2α < β, then {T (t)} t≥0 is not analytic. V) If α = 0, the system is exactly controllable on [0, t 1 ]. VI) If α ≥ β > 0, the question about the controllability of this system is opened.
fatcat:wi46zo45u5fyjehnwtj74gprti