Abstract and Applied Analysis

Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations

Abstract

We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 216919, 11 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412276913

Digital Object Identifier
doi:10.1155/2014/216919

Mathematical Reviews number (MathSciNet)
MR3198165

Zentralblatt MATH identifier
07021949

Citation

Pan, Xue; Li, Xiuwen; Zhao, Jing. Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations. Abstr. Appl. Anal. 2014 (2014), Article ID 216919, 11 pages. doi:10.1155/2014/216919. https://projecteuclid.org/euclid.aaa/1412276913

References

• D. Baleanu, A. K. Golmankhaneh, and A. K. Golmankhaneh, “On electromagnetic field in fractional space,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 288–292, 2010.
• A. Debbouche and M. M. El-Borai, “Weak almost periodic and optimal mild solutions of fractional evolution equations,” Electronic Journal of Differential Equations, vol. 46, 8 pages, 2009.
• A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elservier Science B.V., Amsterdam, The Netherlands, 2006.
• V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, pp. 2677–2682, 2008.
• Z. H. Liu and J. F. Han, “Integral boundary value problems for fractional order integrodifferential equations,” Dynamic Systems & Applications, vol. 21, pp. 535–548, 2012.
• Z. H. Liu and X. W. Li, “Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 1362–1373, 2013.
• X. Y. Liu, Z. H. Liu, and X. Fu, “Relaxation in nonconvex optimal control problems described by fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 409, no. 1, pp. 446–458, 2014.
• Z. H. Liu and X. W. Li, “On the controllability of impulsive fractional evolution inclusions in banach spaces,” Journal of Optimization Theory and Applications, vol. 156, pp. 167–182, 2013.
• Z. H. Liu, J. H. Sun, and I. Szántó, “Monotone iterative technique for Riemann-Liouville fractional integro-differential equations with advanced arguments,” Results in Mathematics, vol. 63, pp. 1277–1287, 2013.
• Z. H. Liu and J. H. Sun, “Nonlinear boundary value problems of fractional functional integro-differential equations,” Computers and Mathematics with Applications, vol. 64, pp. 3228–3234, 2012.
• Z. H. Liu, X. W. Li, and J. H. Sun, “Controllability of nonlinear fractional impulsive evolution systems,” Journal of Integral Equations and Applications, vol. 25, no. 3, pp. 395–405, 2013.
• K. X. Li and J. Peng, “Fractional resolvents and fractional evolution equations,” Applied Mathematics Letters, vol. 25, no. 5, pp. 808–812, 2012.
• K. X. Li, J. G. Peng, and J. X. Jia, “Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives,” Journal of Functional Analysis, vol. 263, pp. 476–510, 2012.
• M. P. Lazarevi'c and A. M. Spasi'c, “Finite-time stability analysis of fractional order timedelay systems: gronwall's approach,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 475–481, 2009.
• G. M. Mophou, “Optimal control of fractional diffusion equation,” Computers and Mathematics with Applications, vol. 61, no. 1, pp. 68–78, 2011.
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
• S. Zhang, “Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 5-6, pp. 2087–2093, 2009.
• N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765–771, 2006.
• M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433–440, 2002.
• M. M. El-Borai, “The fundamental solutions for fractional evolution equations of parabolic type,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2004, no. 3, pp. 197–211, 2004.
• J. R. Wang, Y. Zhou, and M. Medved, “On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 1, pp. 31–50, 2012.
• J. R. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 262–272, 2011.
• Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010.
• H. P. Ye, J. M. Gao, and Y. S. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.
• S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis (Theory), Kluwer Academic, Dordrecht, The Netherlands, 1997.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983.
• E. J. Balder, “Necessary and sufficient conditions for ${L}^{1}$-strong- weak lower semicontinuity of integral functionals,” Nonlinear Analysis, vol. 11, no. 12, pp. 1399–1404, 1987. \endinput