On the maximal parameter range of global stability for a nonlocal thermostat model
- PDF / 769,055 Bytes
- 37 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 163 Views
Journal of Evolution Equations
On the maximal parameter range of global stability for a nonlocal thermostat model Patrick Guidotti
and Sandro Merino
Abstract. The global asymptotic stability of the unique steady state of a nonlinear scalar parabolic equation with a nonlocal boundary condition is studied. The equation describes the evolution of the temperature profile that is subject to a feedback control loop. It can be viewed as a model of a rudimentary thermostat, where a parameter controls the intensity of the heat flow in response to the magnitude of the deviation from the reference temperature at a boundary point. The system is known to undergo a Hopf bifurcation when the parameter exceeds a critical value. Results on the characterization of the maximal parameter range where the reference steady state is globally asymptotically stable are obtained by analyzing a closely related nonlinear Volterra integral equation. Its kernel is derived from the trace of a fundamental solution of a linear heat equation. A version of the Popov criterion is adapted and applied to the Volterra integral equation to obtain a sufficient condition for the asymptotic decay of its solutions.
1. Introduction The nonlocal nonlinear problem ⎧ ⎪ u − uxx = 0 ⎪ ⎪ t ⎪ ⎨u (t, 0) = tanhβu(t, π ) x ⎪u x (t, π ) = 0 ⎪ ⎪ ⎪ ⎩ u(0, ·) = u 0
in (0, ∞) × (0, π ), for t ∈ (0, ∞), for t ∈ (0, ∞),
(1.1)
in (0, π ),
with parameter β ∈ R was first introduced in [12] as a simple model for a thermostat where the sensor is not placed in the same position as the heating/cooling actuator that receives temperature feedback. In that paper, it was shown that the local exponential stability of the trivial solution u(t, ·) ≡ 0 is lost when β ∈ (0, ∞) exceeds the critical value β0 ≈ 5.6655. In fact, in [12] it is shown that a Hopf bifurcation occurs at β0 which produces a local branch of periodic solutions for β ∈ (β0 , β0 + ε) and some ε > 0. In this paper, we show that the trivial solution is globally attractive for β ∈ (0, β0 ). As a matter of fact, a more precise statement can be proven (see Theorem 6.1) which requires the terminology used in [18] and distinguishes between the concept of attractor Aˆ β and B-attractor Aβ . While the precise definitions are given Keywords: Nonlinear reaction diffusion systems, Nonlocal boundary conditions, Nonlinear feedback control systems, Popov criterion, Volterra integral equation, Global attractor.
P. Guidotti and S. Merino
J. Evol. Equ.
in Sect. 6, we point out that, in general, the attractor Aˆ β is a proper subset of the Battractor Aβ . We prove that the continuous semiflow β induced on H1 (0, π ) by the system (1.1) has a global attractor Aˆ β = {0} for β ∈ (0, β0 ) and that for β ∈ (0, π4 ) the B-attractor and the attractor coincide, i.e. Aβ = Aˆ β = {0}. In [7], the authors prove that the global B-attractor Aβ exists for β ∈ (0, ∞) and that Aβ = {0} for β ∈ (0, π1 ). We thus extend previous results by determining larger parameter ranges where the global stability of the trivial solution holds and where the
Data Loading...
