Abstract
We first study the existence results and properties of the solution set of a control system described by fractional differential equations with nonconvex control constraint. Then a problem of minimizing an integral functional over the solution set of the control system is considered. Along with the original minimizing problem, we also consider the problem of minimizing the integral functional whose integrand is the bipolar (with respect to the control variable) of the original integrand over the solution set of the same system but with the convexified control constraint. We prove that the relaxed problem has an optimal solution and obtain some relationships between these two minimizing problems. Finally, an example is given to illustrate the results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Z.B. Bai., On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, No 2 (2010), 916–924. DOI: 10.1016/j.na.2009.07.033.
K. Balachandran, J. Kokila, J.J. Trujillo., Relative controllability of fractional dynamical systems with multiple delays in control. Comput. Math. Appl. 64, No 10 (2012), 3037–3045. DOI: 10.1016/j.camwa.2012.01.071.
K. Balachandran, J.Y. Park., J.J. Trujillo., Controllability of nonlinear fractional dynamical systems. Nonlinear Anal. 75 (2012), 1919–1926. DOI: 10.1016/j.na.2011.09.042.
D. Băleanu, J.A.T. Machado., A.C.J. Luo., Fractional Dynamics and Control. Springer, New York (2012).
M. Benchohra, S. Hamani, S.K. Ntouyas., Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3 (2008), 1–12.
N.N. Bogolyubov., Sur quelques méthodes nouvelles dans le calcul des variations. Ann. Mat. Pura Appl. 7 (1930), 249–271.
A. Cernea, On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. J. Appl. Math. Comput. 38 (2012), 133–143. DOI: 10.1007/s12190-010-0468-6.
A. Debbouchea, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62 (2011), 1442–1450. DOI: 10.1016/j.camwa.2011.03.075.
F.S. De. Blasi, G. Pianigiani, A.A. Tolstonogov., A bogolyubov-type theorem with a nonconvex constraint in banach spaces. SIAM J. Control Optim. 43, No 2 (2004), 466–476. DOI: 10.1137/S0363012903423156.
J. Dixon, S. McKee, Weakly Singular Discrete Gronwall Inequalities. ZAMM·Z. Angew. Math. Mech. 68, No 11 (1986), 535–544. DOI: 10.1002/zamm.19860661107.
A. Dzieliński, W. Malesza, Point to point control of fractional differential linear control systems. Adv. Differ. Equ. 2011 (2011), 13; DOI: 10.1186/1687-1847-2011-13.
I. Ekeland, R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).
S.C. Hu., N.S. Papageorgiou., Handbook of Multivalued Analysis: Volume I: Theory. Kluwer Academic Publishers, Dordrecht-Boston-London (1997).
A.D. Ioffe., On lower semicontinuity of integral functionals, I. SIAM J. Control Optim. 15, No 4 (1977), 521–538. DOI: 10.1137/0315035.
A.A. Kilbas., S.A. Marzan., Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differential Equations 41 (2005), 84–89. DOI: 10.1007/s10625-005-0137-y.
A.A. Kilbas., H.M. Srivastava., J.J. Trujillo., Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V, Amsterdam (2006).
X.P. Li., F.L. Chen., X.Z. Li., Generalized anti-periodic boundary value problems of impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 28–41. DOI: 10.1016/j.cnsns.2012.06.014.
X.Y. Liu., X. Fu, Control systems described by a class of fractional semilinear evolution equations and their relaxation property. Abstr. Appl. Anal. 2012 (2012), Article ID 850529, 20 p.; DOI: 10.1155/2012/850529.
X.Y. Liu., Y.L. Liu., Fractional differential equations with fractional non-separated boundary conditions. Electron. J. Diff. Equ. 2013 (2013), # 25, 13 p.
Z.H. Liu., X.W. Li., On the controllability of impulsive fractional evolution inclusions in Banach spaces. J. Optim. Theory Appl. 156 (2013), 167–182. DOI: 10.1007/s10957-012-0236-x.
Z.H. Liu., J.H. Sun., Nonlinear boundary value problems of fractional differential systems. Comput. Math. Appl. 64, No 4 (2012), 463–475. DOI: 10.1016/j.camwa.2011.12.020.
E.J. MacShane., Existence theorems for Bolza problems in the calculus of variations. Duke Math. J. 7, No 1 (1940), 28–61.
S. Migórski, Existence and relaxation results for nonlinear evolution inclusions revisited. J. Appl. Math. Stoch. Anal. 8, No 2 (1995), 143–149.
S. Migórski, Existence and relaxation results for nonlinear second order evolution inclusions. Discuss. Math. Differ. Incl. Control Optim. 15 (1995), 129–148.
K.S. Miller., B. Ross, An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York (1993).
J. Sabatier, O.P. Agrawal., J.A.T. Machado. (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007).
S.I. Suslov., The Bogolyubov theorem with a differential inclusion as constraint. Siberian Math. J. 35, No 4 (1994), 802–814. DOI: 10.1007/BF02106624.
Z.X. Tai., X.C. Wang., Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces. Appl. Math. Lett. 22 (2009), 1760–1765. DOI: 10.1016/j.aml.2009.06.017.
S.A. Timoshin., A.A. Tolstonogov., Bogolyubov-type theorem with constraints induced by a control system with hysteresis effect. Nonlinear Anal. 75 (2012), 5884–5893. DOI: 10.1016/j.na.2012.05.028.
A.A. Tolstonogov., Bogolyubov’s theorem under constraints generated by a controlled second-order evolution system. Izv. Math. 67, No 5 (2003), 1031–1060. DOI: 10.1070/IM2003v067n05ABEH000456.
A.A. Tolstonogov., Bogolyubov’s theorem under constraints generated by a lower semicontinuous differential inclusion. Sb. Math. 196, No 2 (2005), 263–285. DOI: 10.1070/SM2005v196n02ABEH000880.
A.A. Tolstonogov., Relaxation in non-convex control problems described by first-order evolution equations. Mat. Sb. 190, No 11 (1999), 135–160. Engl. transl., Sb. Math. 190 (1999), 1689–1714. DOI: 10.1070/SM1999v190n11ABEH000441.
A.A. Tolstonogov., D.A. Tolstonogov., Lp-continuous extreme selectors of multifunctions with decomposable values: Existence theorems. Set-Valued Anal. 4 (1996), 173–203. DOI: 10.1007/BF00425964.
A.A. Tolstonogov., D.A. Tolstonogov., Lp-continuous extreme selectors of multifunctions with decomposable values: Relaxation theorems. Set-Valued Anal. 4 (1996), 237–269. DOI: 10.1007/BF00419367.
G.T. Wang., B. Ahmad, L.H. Zhang., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 74, No 3 (2011), 792–804. DOI: 10.1016/j.na.2010.09.030.
J.R. Wang., M. Fečkan, Y. Zhou, Relaxed controls for nonlinear fractional impulsive evolution equations. J. Optim. Theory Appl. 156, No 1 (2013), 13–32. DOI: 10.1007/s10957-012-0170-y.
J.R. Wang., L.L. Lv., Y. Zhou, Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces. J. Appl. Math. Comput. 38 (2012), 209–224. DOI: 10.1007/s12190-011-0474-3.
J.R. Wang., Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal.-Real World Appl. 12, No 6 (2011), 3642–3653. DOI: 10.1016/j.nonrwa.2011.06.021.
J. Wei, The controllability of fractional control systems with control delay. Comput. Math. Appl. 64, No 10 (2012), 3153–3159. DOI: 10.1016/j.camwa.2012.02.065.
Q.J. Zhu., On the solution set of differential inclusions in Banach space. J. Differ. Equat. 93, No 2 (1991), 213–237. DOI: 10.1016/0022-0396(91)90011-W.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Xiaoyou, L., Youjun, X. Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System. FCAA 19, 94–115 (2016). https://doi.org/10.1515/fca-2016-000
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2016-000