Abstract
The present paper is devoted to the problem of stabilization of the one-dimensional semilinear heat equation with nonlocal initial conditions. The control is with boundary actuation. It is linear, of finite-dimensional structure, given in an explicit form. It allows to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel, then to apply a fixed point argument in a convenient space.
Highlights
We address the problem of asymptotic exponential stabilization of (1)
We look for a control uw given in a feedback form, i.e., uw(t) = uw(w(t)), such that once inserted into equation (1) it yields that the corresponding solution of the closed-loop equation (1) satisfies the exponential decay w2(t, x) dx Ce−ρt ∀t 0 for a constant C > 0 and arbitrarily large ρ
We discussed about the semilinear heat equation on the rod with polynomial nonlinearity and with nonlocal initial conditions
Summary
We are interested in the following equation:. ∂tw(t, x) = ∂xxw(t, x) + a(x) + b(t) w(t, x) + c(t, x)wn(t, x), t ∈ (0, ∞), x ∈ (0, 1), w(t, 0) = uw(t), w(t, 1) = 0, t > 0,. The problem of exponential stability associated to (1), i.e., whether the solution of the uncontrolled equation (1) (uw ≡ 0) satisfies an exponential decay as above, has been addressed in many works, see, for example, [2,4,9]. Its simple form allows us to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel similar to the heat kernel. 2.4.1], considered for the particular case: definition interval (0, 1), function p = 1, r(x) = a(x)+α, constants α1 = β1 = 1 and α2 = β2 = 0, we have that A is self-adjoint and has a countable set of simple real eigenvalues {λk}k∈N∗ with the corresponding eigenfunctions {φk}k∈N∗.
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