Fractional stochastic evolution hemivariational inequalities and optimal controls

Yirong Jiang, Qiongfen Zhang, Nanjing Huang



This paper investigates the existence of mild solutions for fractional stochastic evolution hemivariational inequalities and optimal controls. An existence theorem concerned with the mild solution for the presented system is proved by means of the fractional calculation, stochastic analysis theory, Bohnenblust-Karlin fixed point theorem and some properties of the Clarke subdifferential. Moreover, an existence result of optimal control pair that governed by a fractional stochastic evolution hemivariational inequality is also obtained. Finally, an example is given for demonstration.


Fractional stochastic evolution inclusion; hemivariational inequality; Clarke subdifferential; mild solution; optimal control

Full Text:



P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl. 174 (2016), 1–17.

E.J. Balder, Necessary and sufficient conditions for L1 -strong weak lower semicontinuity of integral functionals, Nonlinear Anal. 11 (1987), 1399–1404.

H.F. Bohnenblust and S. Karlin, On a Theorem of Ville, Contributions to the Theory of Games, vol. I, Princeton University Press: Princeton, 1950, pp. 155–160.

F.H. Clarke, Optimization and nonsmooth analysis, Wiley, New York, 1983.

A. Debbouche and D.F.M. Torres, Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions, Fract. Calc. Appl. Anal. 18 (2015), 95–121.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.

K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992.

Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Springer, New York, 2003.

K. Diethelm, The Analysis of Fractional Differential Equations: An ApplicationOriented Exposition Using Differential Operators of Caputo Type, Springer, Berlin, 2010.

M.M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Soliton Fract. 14 (2002), 433–440.

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Springer, Berlin, 2006.

T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math. 20 (1919), 292–296.

T. Guendouzi, Existence and controllability results for fractional stochastic semilinear differential inclusions, Differ. Equ. Dyn. Syst. 23 (2015), 225–240.

A. Harrat, J.J. Nieto and A. Debbouche, Solvability and optimal controls of impulsive hilfer fractional delay evolution inclusions with clarke subdifferential, J. Comput. Appl. Math. (2018), DOI: 10.1016/

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I, Kluwer Academic Publishers, Dordrecht Boston, London, 1997.

Y.R. Jiang, Optimal feedback control problems driven by fractional evolution hemivariational inequalities, Math. Meth. Appl. Sci. 41 (2018), 4305–4326.

Y.R. Jiang and N.J. Huang, Solvability and optimal control of fractional delay evolution inclusions with Clarke subdifferential, Math. Meth. Appl. Sci. 40 (2017), 3026–3039.

Y.R. Jiang, N.J. Huang and J.C. Yao, Solvability and optimal control of semilinear nonlocal fractional evolution inclusion with Clarke subdifferential, Appl. Anal. 96 (2017), 2349–2366.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

S. Kumar, Mild solution and fractional optimal control of semilinear system with fixed delay, J. Optim. Theory Appl. 174 (2017), 108–121.

A. Lasota and Z. Opial, Application of the Kakutani–Ky–Fan theorem in the theory of ordinary differential equations or noncompact acyclic-valued map, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781–786.

J.H. Lightbourne and S.M. Rankin, A partial functional differential equation of Sobolev type, J. Math. Anal. Appl. 93 (1983), 328–337.

Z.H. Liu and X.W. Li, Approximate controllability for a class of hemivariational inequalities, Nonlinear Anal. Real World Appl. 22 (2015), 581–591.

L. Lu and Z.H. Liu, Existence and controllability results for stochastic fractional evolution hemivariational inequalities, Appl. Math. Comput. 268 (2015), 1164–1176.

L. Lu, Z.H. Liu, W. Jiang and J.L. Luo, Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities, Math. Methods Appl. Sci. 39 (2016), 5452–5464.

N.I. Mahmudov, Variational approach to finite-approximate controllability of Sobolevtype fractional systems, J. Optim. Theory Appl. (2018), DOI: 10.1007/s10957-018-1255-z.

N.I. Mahmudov and A. Denker, On controllability of linear stochastic systems, Internat. J. Control. 73 (2000), 144–151.

F. Mainardi, P. Paradisi and R. Gorenflo, Probability Distributions Generated by Fractional Diffusion Equations, Kluwer Academic Publisher, Dordrecht Boston, London, 2000.

S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim. 17 (2000), 285–300.

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal. 41 (2009), 1415–1435.

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Springer, Berlin, 2013.

P. Muthukumar, N. Durga, F.A. Rihan and C. Rajivganthi, Optimal control of second order stochastic evolution hemivariational inequalities with Poisson jumps, Taiwanese J. Math. 21 (2017), 1455–1475.

P.D. Panagiotopoulos, Nonconvex superpotentials in sense of F.H. Clarke and applications, Mech. Res. Comm. 8 (1981), 335–340.

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer, Berlin, 1993.

J.Y. Park and T.G. Ha, Existence of antiperiodic solutions for hemivariational inequalities, Nonlinear Anal. 68 (2008), 747–767.

J.Y. Park and S.H. Park, Optimal control problems for anti-periodic quasilinear hemivariational inequalities, Optimal Control Appl. Meth. 28 (2007), 275–287.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.

G.D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.

C. Rajivganthi and P. Muthukumar, Almost automorphic solutions for fractional stochastic differential equations and its optimal control, Optimal Control Appl. Meth. 37 (2016), 663–681.

Y. Ren, L.Y. Hu and R. Sakthivel, Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay, J. Comput. Appl. Math. 235 (2011), 2603–2614.

R. Sakthivela, Y. Ren, A. Debbouchec and N.I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal. 95 (2016), 2361–2382.

R. Sakthivel, S. Suganya and S.M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl. 63 (2012), 660–668.

Z.M. Yan and F.X. Lu, Solvability and optimal controls of a fractional impulsive stochastic partial integro-differential equation with state-dependent delay, Acta. Appl. Math. 155 (2018), 57–84.

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), 1063–1077.


  • There are currently no refbacks.

Partnerzy platformy czasopism