Regional Boundary Observability of Parabolic Linear Systems with Constraints on the Gradient
- PDF / 789,683 Bytes
- 9 Pages / 595.276 x 790.866 pts Page_size
- 96 Downloads / 208 Views
Regional Boundary Observability of Parabolic Linear Systems with Constraints on the Gradient Hayat Zouiten1 · Ali Boutoulout1 · Fatima-Zahrae EL Alaoui1 Received: 23 August 2017 / Revised: 15 November 2017 / Accepted: 12 February 2018 © Brazilian Society for Automatics–SBA 2018
Abstract We investigate regional boundary observability of parabolic linear systems with constraints on the gradient. The Hilbert uniqueness method is used to reconstruct the initial gradient state between two prescribed functions f 1 and f 2 on a part of the boundary ∂ of the whole domain without the knowledge of the state. The obtained results are illustrated by numerical simulations performed through an example which led to successful results. Keywords Distributed systems · Parabolic systems · Observability with constraints · Regional boundary gradient reconstruction · HUM approach
1 Introduction The theory of optimal control is certainly, at the present, one of the most interdisciplinary areas of research which arises in most modern applications modeled using partial differential equations (PDEs). In control theory, various notions have been developed, particularly the concept of observability which was introduced at the beginning of the sixties by Kalman (1960), and has been generalized to the infinitedimensional context by Delfour and Mitter (1972), Dolecki and Russell (1977), Russell and Weiss (1994). Roughly speaking, observability generally means the possibility to reconstruct the initial state of a distributed system based on partial measurements takes on the system by the means of tools called sensors. In recent years, remarkable new results have been derived on regional analysis of distributed parameter systems, which target of interest is not fully the geometrical evolution domain , but just in an internal subregion ω of (see Amouroux et al. 1994; EL Jai et al. 1993) or on a part of the boundary subregion ∂ of (see Zerrik et al. 1999, 2002). Recently, important progress has been made in this domain and a several of new concepts were introduced to study, one of the most interesting notions is the observability with constraints (also
B 1
Hayat Zouiten [email protected]
called enlarged observability) (see Boutoulout et al. 2011a; Zouiten et al. 2017a). This concept was also developed when the constraints on the regional gradient case where we are interested just in an internal subregion ω of the whole domain (see Boutoulout et al. 2011b; Zouiten et al. 2017b). This problem is of great interest in the theory of optimal control since the existing systems that cannot be observable but gradient observable, and they provide a means to deal with some problem from the real world (see Reed and Simon 1972), for instance as a real application, we can try to observe the flux of the substrate concentration at the bottom boundary subregion of a biological reactor between two prescribed levels by giving an application of a parabolic system which is related to constrained observability. The aim of this paper is to give an a
Data Loading...
