Abstract
We study the duality theory for fractional resolvents, extending and improving some corresponding theorems on semigroups. As applications, we develop the variational technique to analyze the finite-approximate controllability of a backward fractional control system with a right-sided Riemann-Liouville fractional derivative. Moreover, validity of our theoretical findings is given by a fractional diffusion model.
Acknowledgements
The authors are grateful to the editor and the referees for their constructive comments and suggestions for the improvement of the paper. Furthermore, the work was supported by the NSF of China (11871064, 11771378) and the NSF of the JiangSu Higher Education Institutions (18KJB110019).
References
[1] P. Balasubramaniam, P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi’s function. Appl. Math. Comput. 256 (2015), 232–246.10.1016/j.amc.2015.01.035Search in Google Scholar
[2] J. Baumeister, Stable Solution of Inverse Problems. Vieweg, Braunschweig (1987).10.1007/978-3-322-83967-1Search in Google Scholar
[3] J.B. Conway, A Course in Functional Analysis (2nd Edition). Springer-Verlag, New York (1990).Search in Google Scholar
[4] G. Da Prato, M. Iannelli, Linear abstract integrodifferential equations of hyperbolic type in Hilbert spaces. Rend. Sem. Mat. Padova 62 (1980), 191–206.Search in Google Scholar
[5] G. Da Prato, M. Iannelli, Linear integrodifferential equations in Banach spaces. Rend. Sem. Mat. Padova 62 (1980), 207–219.Search in Google Scholar
[6] J.P. Dauer, N.I. Mahmudov, M.M. Matar, Approximate controllability of backward stochastic evolution equations in Hilbert spaces. J. Math. Anal. Appl. 323 (2006), 42–56.10.1016/j.jmaa.2005.09.089Search in Google Scholar
[7] Z. Fan, Characterization of compactness for resolvents and its applications. Appl. Math. Comput. 232 (2014), 60–67.10.1016/j.amc.2014.01.051Search in Google Scholar
[8] M. Jung, Duality theory for solutions to Volterra integral equation. J. Math. Anal. Appl. 230 (1999), 112–134.10.1006/jmaa.1998.6174Search in Google Scholar
[9] S. Kumar, N. Sukavanam, Controllability of fractional order system with nonlinear term having integral contractor. Fract. Calc. Appl. Anal. 16, No 4 (2013), 791–801; 10.2478/s13540-013-0049-0; https://www.degruyter.com/journal/key/FCA/16/4/html.Search in Google Scholar
[10] S. Kumar, N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay. J. Diff. Equ. 252 (2012), 6163–6174.10.1016/j.jde.2012.02.014Search in Google Scholar
[11] K. Li, J. Peng, Fractional resolvents and fractional evolution equations. Appl. Math. Lett. 25 (2012), 808–812.10.1016/j.aml.2011.10.023Search in Google Scholar
[12] X. Li, J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Basel (1994).10.1007/978-1-4612-4260-4Search in Google Scholar
[13] Z. Liu, X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives. SIAM J. Control Optim. 53 (2015), 1920–1933.10.1137/120903853Search in Google Scholar
[14] C. Lizama, Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243 (2000), 278–292.10.1006/jmaa.1999.6668Search in Google Scholar
[15] C. Lizama, H. Prado, On duality and spectral properties of (a, k)-regularized resolvents. Proc. Roy. Soc. Edinburgh 139A (2009), 505–517.10.1017/S0308210507000364Search in Google Scholar
[16] N.I. Mahmudov, Finite-approximate controllability of fractional evolution equations: variational approach. Fract. Calc. Appl. Anal. 21, No 4 (2018), 919–936; 10.1515/fca-2018-0050; https://www.degruyter.com/journal/key/FCA/21/4/html.Search in Google Scholar
[17] N. I. Mahmudov, Variational approach to finite-approximate controllability of sobolev-type fractional systems. J. Optim. Theory Appl. (2018); 10.1007/s10957-018-1255-z.Search in Google Scholar
[18] N.I. Mahmudov, S. Zorlu, On the approximate controllability of fractional evolution equations with compact analytic semigroups. J. Comput. Appl. Math. 259 (2014), 194–204.10.1016/j.cam.2013.06.015Search in Google Scholar
[19] G. Mophou, Controllability of a backward fractional semilinear differential quation. Appl. Math. Comput. 242 (2014), 168–178.Search in Google Scholar
[20] Z. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).10.1007/978-1-4612-5561-1Search in Google Scholar
[21] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar
[22] J. Prüss, Evolutionary Integral Equations and Applications. Birkhäuser, Basel (1993).10.1007/978-3-0348-8570-6Search in Google Scholar
[23] J. Zhang, Y. Li, Duality theory of regularized resolvents operator family. J. Appl. Anal. Comput. 1 (2011), 279–290.Search in Google Scholar
[24] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59 (2010), 1063–1077.10.1016/j.camwa.2009.06.026Search in Google Scholar
[25] S. Zhu, Z. Fan, G. Li, Optimal controls for Riemann-Liouville fractional evolution systems without Lipschitz assumption. J. Optim. Theory Appl. 174 (2017), 47–64.10.1007/s10957-017-1119-ySearch in Google Scholar
[26] S. Zhu, Z. Fan, G. Li, Approximate controllability of Riemann-Liouville fractional evolution equations with integral contractor assumption. J. Appl. Anal. Comput. 8 (2018), 532–548.Search in Google Scholar
[27] S. Zhu, Z. Fan, G. Li, Topological characteristics of solution sets for fractional evolution equations and applications to control systems. Topol. Methods Nonlinear Anal. (2019); doi:10.12775/TMNA. 2019.033.10.12775/TMNA. 2019.033Search in Google Scholar
© 2021 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial Note
- Anniversary of Prof. S.G. Samko, FC Events (FCAA–Volume 24–2–2021)
- Research Paper
- Operational calculus for the general fractional derivative and its applications
- Riesz potentials and orthogonal radon transforms on affine Grassmannians
- Characterizations of variable martingale Hardy spaces via maximal functions
- Survey Paper
- Contributions on artificial potential field method for effective obstacle avoidance
- Research Paper
- Self-similar cauchy problems and generalized Mittag-Leffler functions
- Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives
- A boundary value problem for a partial differential equation with fractional derivative
- Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications
- Duality theory of fractional resolvents and applications to backward fractional control systems
- Kinetic solutions for nonlocal stochastic conservation laws
- A fractional analysis in higher dimensions for the Sturm-Liouville problem
- Averaging theory for fractional differential equations
Articles in the same Issue
- Frontmatter
- Editorial Note
- Anniversary of Prof. S.G. Samko, FC Events (FCAA–Volume 24–2–2021)
- Research Paper
- Operational calculus for the general fractional derivative and its applications
- Riesz potentials and orthogonal radon transforms on affine Grassmannians
- Characterizations of variable martingale Hardy spaces via maximal functions
- Survey Paper
- Contributions on artificial potential field method for effective obstacle avoidance
- Research Paper
- Self-similar cauchy problems and generalized Mittag-Leffler functions
- Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives
- A boundary value problem for a partial differential equation with fractional derivative
- Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications
- Duality theory of fractional resolvents and applications to backward fractional control systems
- Kinetic solutions for nonlocal stochastic conservation laws
- A fractional analysis in higher dimensions for the Sturm-Liouville problem
- Averaging theory for fractional differential equations