Fractional output stabilization for a class of bilinear distributed systems
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Fractional output stabilization for a class of bilinear distributed systems Hanaa Zitane1 · Rachid Larhrissi1 · Ali Boutoulout1 Received: 13 December 2018 / Accepted: 13 June 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, we consider the question of the state fractional spatial derivative stabilization, using Riemann–Liouville derivative of order α ∈]0, 1[, for a class of bilinear distributed systems. Firstly, we characterize the feedback control that ensure the strong and the weak stabilization of the fractional output. Then, we solve a minimization fractional problem. Finally, we provide an example with numerical simulations to illustrate the effectiveness of the given stabilization theorems. Keywords Bilinear distributed systems · Fractional spatial derivative · Strong stabilization · Weak stabilization · Riemann–Liouville Mathematics Subject Classification 93D15 · 93C10
1 Introduction Bilinear systems form an interesting subclass of nonlinear systems due to their suitability for modeling a wide array of real world problems. For instance, they model biological, physical and chemical processes [1,2]. The most significant non-linearity in mathematical models of these class of systems appears in the multiplication of the state and the control in the dynamical process. The problem of stabilization arises in almost every control problem. Consequently, the stabilization issue is one of the most important and extensively studied concept in systems theory [3,4]. The question of the state stabilization for bilinear systems has been considered and widely treated in many works [5–7]. In Ouzahra [5], sufficient condition that ensure the weak and
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Hanaa Zitane [email protected] Rachid Larhrissi [email protected] Ali Boutoulout [email protected]
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MACS Laboratory, Department of Mathematics, University of Moulay Ismail, 11201 Meknes, Morocco
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H. Zitane et al.
the exponential stabilization of the state for such systems are given. Moreover, in Ouzahra [7], the author presented a sufficient condition for a suitable feedback control to ensure the state strong stabilization with an explicit decay estimate is given. Later, the state gradient stabilization question for distributed bilinear systems has been explored. In Zerrik et al. [8], sufficient condition to obtain the state gradient weak stabilization were developed. Also, the state gradient exponential stabilization was characterized using the decomposition approach. Hence, the authors provided the strongly stabilizing control which minimizes a given performance cost. In the last few decades, the fractional calculus has attracted increasing attention of many researchers due to its demonstrated applications in enormous fields such as physics, chemistry, mechanics, electricity, engineering, economics, signal and image processing and biophysics [9–12]. Likewise, in control theory, fractional calculus was introduced to the main important concepts in systems theory which are stability and controllability of differen
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