Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems
Abstract
:1. Introduction
2. Preliminary Results
- ()
- The control operator is dense and exists;
- ()
- exists and .
3. Regional Fractional Controllability
- (1)
- (2)
- ;
- (3)
- .
- (1)
- (2)
- ;
- (3)
- .
4. Optimal Control with a Regional Target
5. Examples
5.1. Example 1: Zonal Actuator
5.2. Example 2: Pointwise Actuator
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Miller, K.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; John Wiley & Sons, Inc.: New York, NY, USA, 1993. [Google Scholar]
- Machado, J.A.T.; Kiryakova, V.; Mainardi, F. A poster about the old history of fractional calculus. Fract. Calc. Appl. Anal. 2010, 13, 447–454. [Google Scholar]
- Machado, J.A.T.; Kiryakova, V.; Mainardi, F. A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 2010, 13, 329–334. [Google Scholar]
- Sene, N. Fractional input stability for electrical circuits described by the Riemann–Liouville and the Caputo fractional derivatives. AIMS Math. 2019, 1, 147–165. [Google Scholar] [CrossRef]
- Abuaisha, T.; Kertzscher, J. Fractional-order modelling and parameter identification of electrical coils. Fract. Calc. Appl. Anal. 2019, 22, 193–216. [Google Scholar] [CrossRef]
- Podlubny, I. Fractional Differential Equations; Academic Press, Inc.: San Diego, CA, USA, 1999. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies, 204; Elsevier Science B.V.: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Engel, K.J.; Nagel, R. A Short Course on Operator Semigroups; Springer: New York, NY, USA, 2006. [Google Scholar]
- Renardy, M.; Rogers, R.C. An Introduction to Partial Differential Equations; Springer: New York, NY, USA, 2004. [Google Scholar]
- Jai, A.E.; Pritchard, A.J.; Simon, M.C.; Zerrik, E. Regional controllability of distributed systems. Int. J. Control 1995, 62, 1351–1365. [Google Scholar] [CrossRef]
- Aadi, A.A.; Zerrik, E.H. Constrained regional control problem of a bilinear plate equation. Int. J. Control 2022, 95, 996–1002. [Google Scholar] [CrossRef]
- Jai, A.E.; Pritchard, A.J. Sensors and Controls in the Analysis of Distributed Systems; Halsted Press: New York, NY, USA, 1988. [Google Scholar]
- Ge, F.; Chen, Y.Q.; Kou, C. Regional gradient controllability of sub-diffusion processes. J. Math. Anal. Appl. 2016, 440, 865–884. [Google Scholar] [CrossRef]
- Ge, F.; Chen, Y.Q.; Kou, C. Regional boundary controllability of time fractional diffusion processes. IMA J. Math. Control Inf. 2016, 34, 871–888. [Google Scholar] [CrossRef] [Green Version]
- Kaczorek, T. Selected Problems of Fractional Systems Theory; Lecture Notes in Control and Information Sciences; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Kaczorek, T.; Rogowski, K. Fractional Linear Systems and Electrical Circuits; Studies in Systems, Decision and Control; Springer: Cham, Switzerland, 2015. [Google Scholar]
- Ghasemi, M.; Nassiri, K. Controllability of linear fractional systems with delay in control. J. Funct. Spaces 2022, 2022, 5539770. [Google Scholar] [CrossRef]
- Jerzy, K. Controllability of fractional linear systems with delays. In Proceedings of the 25th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland, 23–26 August 2021; pp. 331–336. [Google Scholar]
- Singh, B.K.; Agrawal, S. Study of time fractional proportional delayed multi-pantograph system and integro-differential equations. Math. Methods Appl. Sci. 2022, 45, 8305–8328. [Google Scholar] [CrossRef]
- Sakulrang, S.; Moore, E.J.; Sungnul, S.; Gaetano, A.D. A fractional differential equation model for continuous glucose monitoring data. Adv. Differ. Equ. 2017, 2017, 150. [Google Scholar] [CrossRef]
- Gaetano, A.D.; Sakulrang, S.; Borri, A.; Pitocco, D.; Sungnul, S.; Moore, E.J. Modeling continuous glucose monitoring with fractional differential equations subject to shocks. J. Theor. Biol. 2021, 526, 110776. [Google Scholar] [CrossRef] [PubMed]
- Kheyrinataj, F.; Nazemi, A. On delay optimal control problems with a combination of conformable and Caputo-Fabrizio fractional derivatives via a fractional power series neural network. Netw. Comput. Neural Syst. 2022, 33, 62–94. [Google Scholar] [CrossRef] [PubMed]
- Agrawal, O.P. A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 2004, 38, 323–337. [Google Scholar] [CrossRef]
- Bahaa, G.M. Fractional optimal control problem for differential system with control constraints. Filomat 2016, 30, 2177–2189. [Google Scholar] [CrossRef]
- Mophou, G.M. Optimal control of fractional diffusion equation. Comput. Math. Appl. 2011, 61, 68–78. [Google Scholar] [CrossRef] [Green Version]
- Tomovski, Z.; Metzler, R.; Gerhold, S. Fractional characteristic functions, and a fractional calculus approach for moments of random variables. Fract. Calc. Appl. Anal. 2022, 25, 1307–1323. [Google Scholar] [CrossRef]
- Mainardi, F.; Paradisi, P.; Gorenflo, R. Probability distributions generated by fractional diffusion equations. arXiv 2007, arXiv:0704.0320v1. [Google Scholar]
- Wei, J. The controllability of fractional control systems with control delay. Comput. Math. Appl. 2012, 64, 3153–3159. [Google Scholar] [CrossRef] [Green Version]
- Zhou, Y.; Jiao, F. Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 2010, 59, 1063–1077. [Google Scholar] [CrossRef] [Green Version]
- Ge, F.; Chen, Y.-Q.; Kou, C.; Podlubny, I. On the regional controllability of the sub-diffusion process with Caputo fractional derivative. Fract. Calc. Appl. Anal. 2016, 19, 1262–1281. [Google Scholar] [CrossRef]
- Małgorzata, K. On Solutions of Linear Fractional Differential Equations of a Variational Type; Czestochowa University of Technology: Czestochowa, Poland, 2009. [Google Scholar]
- Pudlubny, I.; Chen, Y.Q. Adjoint fractional differential expressions and operators. In Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007, Las Vegas, NV, USA, 4–7 September 2007. [Google Scholar]
- Bahaa, G.M.; Tang, Q. Optimality conditions of fractional diffusion equations with weak Caputo derivatives and variational formulation. J. Fract. Calc. Appl. 2018, 9, 100–119. [Google Scholar]
- Sakamoto, K.; Yamamoto, M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 2011, 382, 426–447. [Google Scholar] [CrossRef] [Green Version]
- Sakamoto, K.; Yamamoto, M. Approximate controllability for fractional diffusion equations by interior control. Appl Anal. 2014, 93, 1793–1810. [Google Scholar]
- Mozyrska, D.; Torres, D.F.M. Minimal modified energy control for fractional linear control systems with the Caputo derivative. Carpathian J. Math. 2010, 26, 210–221. [Google Scholar]
- Mozyrska, D.; Torres, D.F.M. Modified optimal energy and initial memory of fractional continuous-time linear systems. Signal Process. 2011, 91, 379–385. [Google Scholar] [CrossRef]
- Eroğlu, B.B.; şkan, D.Y. Comparative analysis on fractional optimal control of an SLBS model. J. Comput. Appl. Math. 2023, 421, 114840. [Google Scholar] [CrossRef]
- Mamehrashi, K. Ritz approximate method for solving delay fractional optimal control problems. J. Comput. Appl. Math. 2023, 417, 114606. [Google Scholar] [CrossRef]
- Jai, A.E.; Pritchard, A.J. Sensors and Actuators in Distributed Systems Analysis; Wiley: New York, NY, USA, 1988. [Google Scholar]
- Zerrik, E. Regional Analysis of Distributed Parameter Systems. PhD Thesis, University of Rabat, Rabat, Morocco, 1993. [Google Scholar]
- Karite, T.; Boutoulout, A. Regional enlarged controllability for parabolic semilinear systems. Int. J. Appl. Pure Math. 2017, 113, 113–129. [Google Scholar] [CrossRef] [Green Version]
- Karite, T.; Boutoulout, A. Regional boundary controllability of semilinear parabolic systems with state constraints. Int. J. Dyn. Syst. Differ. Equ. 2018, 8, 150–159. [Google Scholar]
- Karite, T.; Boutoulout, A.; Torres, D.F.M. Enlarged controllability of Riemann–Liouville fractional differential equations. J. Comput. Nonlinear Dyn. 2018, 13, 6. [Google Scholar] [CrossRef] [Green Version]
- Lions, J.L. Optimal Control of Systems Governed by Partial Differential Equations; Springer: New York, NY, USA, 1971. [Google Scholar]
- Lions, J.-L. Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1; Recherches en Mathématiques Appliquées: Paris, France, 1988. [Google Scholar]
- Huang, H.; Fu, X. Optimal control problems for a neutral integro-differential system with infinite delay. Evol. Equ. Control Theory 2022, 11, 177–197. [Google Scholar] [CrossRef]
- Xi, X.X.; Hou, M.; Zhou, X.F.; Wen, Y. Approximate controllability of fractional neutral evolution systems of hyperbolic type. Evol. Equ. Control Theory 2022, 11, 1037–1069. [Google Scholar] [CrossRef]
- Wen, Y.; Xi, X.X. Complete controllability of nonlinear fractional neutral functional differential equations. Adv. Contin. Discret. Models 2022, 2022, 1–11. [Google Scholar] [CrossRef]
- Elshenhab, A.M.; Wang, X.; Cesarano, C.; Almarri, B.; Moaaz, O. Finite-time stability analysis of fractional delay system. Mathematics 2022, 10, 1883. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Karite, T.; Khazari, A.; Torres, D.F.M. Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems. Mathematics 2022, 10, 4813. https://doi.org/10.3390/math10244813
Karite T, Khazari A, Torres DFM. Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems. Mathematics. 2022; 10(24):4813. https://doi.org/10.3390/math10244813
Chicago/Turabian StyleKarite, Touria, Adil Khazari, and Delfim F. M. Torres. 2022. "Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems" Mathematics 10, no. 24: 4813. https://doi.org/10.3390/math10244813
APA StyleKarite, T., Khazari, A., & Torres, D. F. M. (2022). Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems. Mathematics, 10(24), 4813. https://doi.org/10.3390/math10244813