Topological Methods in Nonlinear Analysis

Topological characteristics of solution sets for fractional evolution equations and applications to control systems

Shouguo Zhu, Zhenbin Fan, and Gang Li

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper explores an abstract Riemann-Liouville fractional evolution model with a weighted delay initial condition. We develop the resolvent technique, a generalization of semigroup method, to formulate an appropriate notion of mild solutions to this abstract system and present the topological characteristics of the corresponding solution set in a weighted space. Furthermore, in view of the topological characteristics, we analyze the approximate controllability of the abstract system without Lipschitz assumption. We end up addressing an infinite dimensional fractional delay diffusion control system and a finite dimensional fractional ordinary differential control system by utilizing our theoretical findings.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 54, Number 1 (2019), 177-202.

Dates
First available in Project Euclid: 8 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1562551225

Digital Object Identifier
doi:10.12775/TMNA.2019.033

Citation

Zhu, Shouguo; Fan, Zhenbin; Li, Gang. Topological characteristics of solution sets for fractional evolution equations and applications to control systems. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 177--202. doi:10.12775/TMNA.2019.033. https://projecteuclid.org/euclid.tmna/1562551225


Export citation

References

  • R.P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative, Adv. Differential Equations (2009), DOI:10.1155/2009/981728.
  • J. Andres and M. Pavlačková, Topological structure of solution sets to asymptotic boundary value problems, J. Differential Equations 248 (2010), 127–150.
  • R. Bader and W. Kryszewski, On the solution sets of differential inclusions and the periodic problem in Banach spaces, Nonlinear Anal. 54 (2003), 707–754.
  • K. Borsuk, Theory of Retracts, Monogr. Mat., vol. 44, PWN, Warsaw, 1967.
  • F.E. Browder and C.P. Gupta, Topological degree and nonlinear mappings of analytic type in Banach spaces, J. Math. Anal. Appl. 26 (1969), 390–402.
  • D.H. Chen, R.N. Wang and Y. Zhou, Nonlinear evolution inclusions: topological characterizations of solution sets and applications, J. Funct. Anal. 265 (2013), 2039–2073.
  • S. Djebali, L. Górniewicz and A. Ouahab, Solutions Sets for Differential Equations and Inclusions, De Gruyter Ser. Nonlinear Anal. Appl., vol. 18, Walter de Gruyter, Berlin, 2013.
  • M.L. Du and Z.H. Wang, Initialized fractional differential equations with Riemann–Liouville fractional-order derivative, Eur. Phys. J. Special Topics 193 (2011), 49–60.
  • M.M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals 14 (2002), 433–440.
  • Z. Fan, Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives, Indag. Math. 25 (2014), 516–524.
  • Z. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput. 232 (2014), 60–67.
  • L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, 2nd edition, Springer, Dordrecht, 2006.
  • L. Górniewicz and M. Lassonde, Approximation and fixed points for compositions of $R_{\delta}$-maps, Topology Appl. 55 (1994), 239–250.
  • R. Hilfer, Fractional diffusion based on Riemann–Liouville fractional derivatives, J. Phys. Chem. B 104 (2000), 3914–3917.
  • L.H. Hoa, N.N. Trong and L.X. Truong, Topological structure of solution set for a class of fractional neutral evolution equations on the half-line, Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 235–255.
  • S. Hu and N.S. Papageorgious, Handbook of Multivalued Analysis \rom(Theory), Kluwer Academic Publishers, Dordrecht Boston, London, 1997.
  • D.M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1969), 91–97.
  • E. Iturriaga and H. Leiva, A necessary and sufficient condition for the controllability of linear systems in Hilbert spaces and applications, IMA J. Math. Control Inform. 25 (2009), no. 3, 269–280.
  • M. Kamenskiĭ, V. Obukhovskiĭ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., vol. 7, Walter de Gruyter, Berlin, New York, 2001.
  • T.D. Ke, V. Obukhovskiĭ, N.C. Wong and J.C. Yao, Approximate controllability for systems governed by nonlinear Volterra type equations, Differ. Equ. Dyn. Syst. 20 (2012), no. 1, 35–52.
  • S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differential Equations 252 (2012), 6163–6174.
  • K. Li and J. Peng, Fractional resolvents and fractional evolution equations, Appl. Math. Lett. 25 (2012), 808–812.
  • K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives, J. Funct. Anal. 263 (2012), 476–510.
  • Z. Liu and X. Li, Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives, SIAM J. Control Optim. 53 (2015), 1920–1933.
  • N.I. Mahmudov and N. Semi, Approximate controllability of semilinear control systems in Hilbert spaces, TWMS J. App. Eng. Math. 2 (2012), no. 1, 67–74.
  • D. Matignon and B. d'Andréa-Novel, Some results on controllability and observability of finite-dimensional fractional differential systems, in: Proceedings of the Computational Engineering in Systems and Application Multiconference, vol. 2, Lille, France, July 1996, IMACS, IEEE-SMC, pp. 952–956.
  • K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim. 25 (1987), 715–722.
  • Z. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993.
  • J. Sabatier, C. Farges and J.C. Trigeassou, Fractional systems state space description: some wrong ideas and proposed solutions, J. Vib. Control 20 (2014), no. 7, 1076–1084.
  • T.I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim. 25 (1987), 1173–1191.
  • R.N. Wang, D.H. Chen and T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations 252 (2012), 202–235.
  • R.N. Wang, Q.M. Xiang and Y. Zhou, Fractional delay control problems: topological structure of solution sets and its applications, Optimization 63 (2014), 1249–1266.
  • J.R. Wang and Y. Zhou, Existence of mild solutions for fractional delay evolution systems, Appl. Math. Comput. 218 (2011), 357–367.
  • J.R. Wang, Y. Zhou and W. Wei, Impulsive problems for fractional evolution equations and optimal controls in infinite dimensional spaces, Topol. Methods Nonlinear Anal. 38 (2011), 17–43.
  • H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075–1081.
  • Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), 1063–1077.
  • S. Zhu, Z. Fan and G. Li, Optimal controls for Riemann–Liouville fractional evolution systems without Lipschitz assumption, J. Optim. Theory Appl. 174 (2017), 47–64.
  • S. Zhu, Z. Fan and G. Li, Approximate controllability of Riemann–Liouville fractional evolution equations with integral contractor assumption, J. Appl. Anal. Comput. 8 (2018), 532–548.