Abstract
The aim of this paper is to investigate the stability and the stabilization of the fractional spatial derivative of order \(\alpha \in ]0, 1[\) for a class of infinite dimensional linear systems. Firstly, we explore some properties of the fractional derivative stability. Then, we establish the characterization of the fractional spatial derivative stabilization with an approach based on the steady state Riccati equation. Moreover, we discuss a minimization fractional problem. Finally, we illustrate the developed results by numerical simulations.
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References
Balakrishnan, A.V.: Strong stabilizability and the steady state Riccati equation. Appl. Math. Optim. 7(1), 335–345 (1981)
Curtain, R.F., Pritchard, A.J.: Infinite Dimensional Linear Systems Theory. Springer, Berlin (1978)
Curtain, R.F., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Limited (2006)
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic, New York and London (1974)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Podlubny, I., Chen, Y.: Adjoint fractional differential expressions and operators. In: ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society Of Mechanical Engineers, pp. 1385–1390 (2007)
Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)
Pritchard, A.J., Zabczyk, J.: Stability and stabilizability of infinite dimensional systems. SIAM Rev. 23(1), 25–51 (1981)
Ross. B.: Fractional Calculus and Its Applications, Springer Verlag, in Lecture Notes in Mathematics, Proc. int. conf. New Haven, 457 (1974)
Triggiani, R.: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52(3), 383–403 (1975)
Zerrik, E., Benslimane, Y.: An output gradient stabilization of distributed linear systems approaches and simulations. Intell. Control Autom. 3(2), 159–167 (2012)
Zerrik, E., Ouzahra, M.: Regional stabilization for infinite dimensional systems. Int. J. Control, 76(1), 73–81 (2003)
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Zitane, H., Larhrissi, R., Boutoulout, A. (2020). On the Fractional Output Stabilization for a Class of Infinite Dimensional Linear Systems. In: Zerrik, E., Melliani, S., Castillo, O. (eds) Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Studies in Systems, Decision and Control, vol 243. Springer, Cham. https://doi.org/10.1007/978-3-030-26149-8_18
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DOI: https://doi.org/10.1007/978-3-030-26149-8_18
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