Regional averaged controllability for hyperbolic parameter dependent systems
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Control Theory and Technology http://link.springer.com/journal/11768
Regional averaged controllability for hyperbolic parameter dependent systems Mouna ABDELLI† , Abdelhak HAFDALLAH, Fay¸cal MERGHADI, Meriem LOUAFI Laboratory of Mathematics Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria Received 9 January 2020; revised 22 May 2020; accepted 22 May 2020
Abstract The purpose of this paper is to extend the notion of regional controllability for hyperbolic parameter dependent systems. The key idea is the characterization of the averaged regional control with minimal energy. This control steers the state average (with respect to such a parameter) towards the desired state only on a given part of the system evolution domain. In this paper, we give the precis definition and the properties of this new concept. Then, we use an approach based on an extension of the Hilbert uniqueness method devoted to the calculation of the control in two different cases: zone control and pointwise control. Keywords: Parameter dependent system, averaged regional controllability, averaged control, Hilbert uniqueness method, minimum energy DOI https://doi.org/10.1007/s11768-020-0006-5
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Introduction
Controllability is one of the most important notions of system theory. It has attracted attention in different fields and has experienced wide developments [1,2]. Recently, the regional controllability concept is commonly applied in practical applications. For a distributed parameter system evolving on a spatial domain Ω ⊂ Rn , the term regional controllability consists of steering the initial state to a prescribed state defined only in a given subregion ω ⊂ Ω. This concept has been introduced and extended in [3–6].
However, in many physical processes, modeling by parameter dependent system appeared to be a challenging problem. In such a situation where the parameter has an unknown value, we cannot control every realization of the system by a single control (using a control independent of the parameter). In this context, Zuazua introduced the notion of averaged controllability. Its goal is to control the averaged state of a parameterized system instead of the state with respect to the unknown parameter. This new notion was first introduced in [7]. Also, the problem of averaged controllability was previously introduced both in the context of PDE (partial
† Corresponding author. E-mail: [email protected]. This work was supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT).
© 2020 South China University of Technology, Academy of Mathematics and Systems Science, CAS and Springer-Verlag GmbH Germany, part of Springer Nature
M. Abdelli et al. / Control Theory Tech, Vol.
differential equations) and finite-dimensional linear systems (See [8, 9]). In this paper, we present an extension of the regional controllability results to hyperbolic parameter dependent system. Thus, we are interested only in the system averaged evolution on a subregi
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