Abstract
We deal with the simultaneous controllability properties of two one dimensional (strongly) coupled wave equations when the control acts on the boundary. Necessary and sufficient conditions for approximate and exact controllability are proved.
Highlights
This paper deals with the controllability properties of some systems of two coupled one dimensional hyperbolic equations
After we proved the results proposed in our article, some papers on this question appeared
We describe the spectrum of L∗d
Summary
This paper deals with the controllability properties of some systems of two coupled one dimensional hyperbolic equations. System (1) is said approximately controllable in L2 (0, π) × H−1 (0, π) at time T > 0 if for any y0, y1 , yT0 , yT1 ∈ L2 (0, π) × H−1 (0, π) and ε > 0, there exists a control function v ∈ L2 (0, T ) such that the associated solution to System (1) satisfies y (T ) − yT0 , yt (T ) − yT1 L2(0,π)×H−1(0,π) < ε. If λ ∈ σ (L∗d) , either λ ∈ (Λ1 ∪ Λ2) \Λ4 and in this case, as in the proof of Proposition 4, we get the two conditions in (37), or λ = ik ∈ Λ4 and from Proposition 3, this eigenvalue is algebraically double and its generalized eigenspace is spanned by: V1∗,λ = Φ∗1,λ, ikΦ∗1,λ , V2∗,λ =.
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