Asymptotic stability of time-varying distributed parameter semi-linear systems Ilyasse Aksikas ∗ ∗ Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar (Tel: 974 3305 34615; e-mail: aksikas@qu.edu.qa). Abstract: The asymptotic behaviour is studied for a class of non-linear distributed parameter time- varying dissipative systems. This is achieved by using time-varying infinite-dimensional Banach state space description. Stability criteria are established, which are based on the dissipativity of the system in addition to another technical condition. The general development is applied to semi-linear systems with time varying nonlinearity. Stability criteria are extracted from the previous conditions. These theoretical results are applied to a class of transport-reaction processes. Different types of nonlinearity are studied by adapting the criteria given in the early portions of the paper. 1. INTRODUCTION Stability is one of the most important aspects of system theory. The fundamental theory of stability is extensively developed for finite-dimensional systems. Many results on the asymptotic behavior of nonlinear infinite-dimensional systems are known, for which the dissipativity property plays an important role, see e.g. [7], [9], [8], [12] , [16], [17], [18], [19]. In [2] and [3], asymptotic stability was studied for a class of semi-linear infinite-dimensional systems. With respect to the domain of definition of the nonlinearity in the system, two scenarios were treated. The first one deals with semi-linear systems with a nonlinear term defined everywhere on the state space. In this case, some stability criteria were established on the basis of the m-dissipativity concept (see [2, Theorem 12 and Corollary 13]). The second case is when the nonlinear term is not necessarily defined everywhere, but only defined on a closed convex subset of the state space. This case is more important from application point of view due to the fact that some physical limitations are imposed. [2, Theorem 16] proves the asymptotic stability under some technical conditions, which will not be easy to check for practicing control engineers. Motivated by this fact, [4] deals with more investigation on the case when the nonlinearity is defined only on a convex subset by adapting some conditions. The objective of this paper is to extend the results developed in [2] and [4] to the time-varying case. This case has many applications, for example catalytic reactors are most often modelled as plug flow reactors and play an important role in many industrial processes (e.g. methanol, ammonia, sulphuric acid, nitric acid and other petrochemicals). Commonly, the conversion within a catalytic reactor decreases with time, as catalytic deactivation or catalyst decay occurs. As this catalytic deactivation occurs, the chemical kinetics change with time. The paper is organized as follows. Section 2 contains some basic results on nonlinear evolution system theory. In Section 3, an asymptotic stability criterion in Banach space is proved, which is based on a technical and weaker condition than the m- dissipativity concept. In Section 3, asymptotic stability criteria are established for a class of semi-linear infinite-dimensional systems by applying the result stated in the previous section. Section 4 deals with the asymptotic stability of an infinite- dimensional description of a transport-reaction process model. 2. PRELIMINARIES Let X be a Banach space and ‖·‖ is the norm on X. Let D be a closed subset of X and let {A(t)} t∈IR be a family of non-linear operators from D to X. Let us consider the following abstract Cauchy problem:  ˙ x(t)= A(t)x(t) x(s)= x s (1) Let us consider the following conditions: (C1) For all (t, x) ∈ IR × D lim inf h→0 + d(x + hA(t)x; D)/h =0, (2) (C2) For all x, y ∈ D, there exists a bounded linear functional f on X such that f (x − y)= ‖x − y‖ 2 = ‖f ‖ 2 and there exists a continuous function µ : IR → IR such that for all t ≥ 0 f (A(t)x − A(t)y) ≤ µ(t)‖x − y‖. (3) The following theorem is an immediate consequence of [18, Theorem 5.1, p. 238] since all its conditions are satisfied by using (C1) and (C2): Theorem 1. Assume that D is a closed subset of X and A(·)· : IR × D → X is continuous. Suppose that (C1) and (C2) are satisfied. Then, there is a unique solution x(t, x s ) to the system (1) on the interval [s, ∞). Moreover, if x ′ s ∈ D, then for all t ≥ s: ‖x(t, x s ) − x(t, x ′ s )‖≤‖x s − x ′ s ‖ exp  t s µ(r)dr  (4) Let us consider the family of operators Γ(t, s): D → D defined as follows Γ(t, s)x s = x(t, x s ) for all t ≥ s ≥ 0 and x ∈ D, (5) Preprints of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 2014 Copyright © 2014 IFAC 677