Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

Bilinear control of a degenerate hyperbolic equation

“…Furthermore, x α µ ′ (x) = (2 − α)x ∈ H 1 (0, 1) and so µ ∈ H 2 α (0, 1). Finally, we observe that x α/2 µ ′ (x) = (2 − α)x 1− α 2 ∈ L ∞ (0, 1) that implies that µ ∈ V (2,∞) α (0, 1); • α ∈ [1,2): in this case it easy to check that µ ∈ L 2 (0, 1). Moreover,…”

“…Proposition 2.2. Let α ∈ [1,2). Then, a) H 1 α (0, 1) is a Hilbert space, b) A : D(A) ⊂ L 2 (0, 1) → L 2 (0, 1) is a self-adjoint negative operator with dense domain.…”

“…• the spectral analysis: as often happens concerning degenerate elliptic operator, the eigenvalues and eigenfunctions of the associated spectral problem are given in terms of Bessel functions, see Propositions 3.1 when α ∈ [0, 1) and Proposition 3.2 when α ∈ [1,2). The spectral analysis here is entirely new; • the unboundness of the eigenfunctions: in the work by Beauchard [6], a key point is that the eigenfunctions (cosine functions) are bounded.…”

“…On the other hand, in many applications, one is naturally led to use bilinear controls, as such controls are more realistic than additive ones to govern the evolution of certain systems (see [2,3,9,14,17,18,19] for parabolic equations.). For example, [26] mentions in particular…”