Stabilité Lp exponentielle d’un système d’échangeurs thermiques avec diffusion et sans diffusion

Abstract

In this paper we study exponential stability of a heat exchanger system with diffusion and without diffusion in the context of Banach spaces. The heat exchanger system is governed by hyperbolic partial differential equations (PDE) and parabolic PDEs, respectively, according to the diffusion impact ignored or not in the heat exchange. The exponential stability of the model with diffusion in the Banach space (C[0, 1])4 is deduced by establishing the exponential Lp stability of the considered system, and using the sectorial operator theory. The exponential decay rate of stability is also computed for the model with diffusion. Using the perturbation theory, we establish the exponential stability of the model without diffusion in the Banach space (C[0, 1])4 with the uniform topology. However the exponential decay rate of stability without diffusion is not exactly computed, since its associated semigroup is non analytic. Indeed the purpose of our paper is to investigate the exponential stability of a heat exchanger system with diffusion and without diffusion in the real Banach space X1 = (C[0, 1])4 with the uniform norm. The exponential stability of these two models in the Hilbert space X2 = (L2(0, 1))4 has been proved in [31] by using Lyapunov’s direct method. The first step consists to study the stability problem in the real Banach space Xp = (Lp(0, 1))4 equipped with the usual Lp norm, p > 1. By passing to the limit (p ! 1) we can extend some results of exponential stability from Xp = (Lp(0, 1))4 to the space X1 = (C[0, 1])4. In particular the dissipativity of the system in all the Xp spaces implies its dissipativity in X1 (see Lemma 3). The section 1 is dedicated to recall the heat exchanger models. The process with diffusion is governed by a system of parabolic PDEs, and the process without diffusion is described by degenerate hyperbolic PDEs of first order. The section 2 deals with exponential stability of the parabolic system in the Lebesgue spaces Lp(0, 1) , 1 < p < 1. Certain results can be extended to the X1 space. Unfortunately this study doesn’t allow us to deduce the expected stability of the system in X1. In the section 3, the sectorial operator theory is made use of to get exponential stability results on the model with diffusion in Xp. Specifically the theory enables us to determine the exponential decay rate in (C[0, 1])4 by computing the spectrum bound. In the section 4, using a perturbation technique we show the exponential stability for the model without diffusion in all Xp spaces, 1 < p < 1. We then take the limit, as p goes to 1, to deduce the exponential stability of the system in the Banach space X1. We call the diffusion model the heat exchanger model with diffusion taken into account and the convection model the heat exchanger without diffusion, respectively. We use the analyticity property of the semigroup associated to the diffusion model in order to determine its exponential decay rate. However the semigroup associated to the convection model is not analytic. In the latter case we have not yet found an efficient method to compute exactly the exponential decay rate. The main tools we use for our investigations are the notion of dissipativity in the Banach spaces, specifically in the Lp spaces, and the sectorial operator theory. As the reader will see our work presents some extensions of the Lyapunov’s direct method to a context of Banach spaces. We will denote the system operator associated to the diffusion model by Ad,p, and that of the convection model by Ac,p, respectively. The index p indicates the Lp( ) space in which the system evolves and the operator Ad,p or Ac,p is considered. Thus Ad,p (resp. Ac,p) indicates the diffusive (resp. convective) operator in the Xp space.