Abstract
In this paper, we investigate the approximate controllability of the coupled system with boundary degeneracy. The control functions act on the degenerate boundary. We prove the Carleman estimate and the unique continuation of the adjoint system. Then we get the approximate controllability by constructing the control functions.
Highlights
1 Introduction In this paper, we investigate the approximate controllability of the coupled degenerate system ut – xpux x + λ1u + λ2v = 0, (x, t) ∈ (0, 1) × (0, T), vt – xpvx x + λ3u + λ4v = 0, (x, t) ∈ (0, 1) × (0, T), u(0, t) = g1χ[T1,T2], u(1, t) = 0, t ∈ (0, T), v(0, t) = g2χ[T1,T2], v(1, t) = 0, t ∈ (0, T), u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ (0, 1), (1.1) (1.2) (1.3) (1.4) (1.5)
Note that it is not necessary that we propose the boundary condition on the degenerate boundary when 1 ≤ p < 2; see [13]
We prove the approximate controllability for the system (1.1)-(1.5)
Summary
The controllability of the following degenerate parabolic equation has been investigated; see references [1,2,3,4,5]: ut – xpux x + c(x, t)u = hχω, (x, t) ∈ (0, 1) × (0, T), (1.6) Theorem 2.1 There exists a unique solution (y, z) ∈ B × B to the problem (2.1)-(2.5) satisfying y B + z B + xpyx(0, t) L2(T1,T2) + xpyx(0, t) L2(T1,T2) + xpzx(0, t) L2(T1,T2) ≤ C1 f1 L2((0,1)×(0,T)) + f2 L2((0,1)×(0,T)) + yT L2(0,1) + zT L2(0,1) , where C1 is depending only on T , T1, λi L∞((0,1)×(0,T)), i = 1, 2, 3, 4.
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