In this paper we consider controlled McKean-Vlasov stochastic evolution equations on Hilbert spaces. We prove existence and uniqueness of solutions and regularity properties thereof. We use relaxed controls, adapted to a current of sub-sigma algebras generated by observable processes, and taking values from a Polish space. We introduce an appropriate topology based on weak star convergence. We prove continuous dependence of solutions on controls with respect to appropriate topologies. Theses results are then used to prove existence of optimal controls for Bolza problems. Then we develop the necessary conditions of optimality based on semi-martingale representation theory on Hilbert spaces. Next we show that the adjoint processes arising from the necessary conditions optimality can be constructed from the solution of certain BSDE.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1171,
author = {N.U. Ahmed},
title = {Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {35},
year = {2015},
pages = {165-195},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1171}
}
N.U. Ahmed. Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 35 (2015) pp. 165-195. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1171/
[000] [1] N.U. Ahmed, Systems governed by mean-field stochastic evolution equations on Hilbert spaces and their optimal control, Nonlinear Anal. (submitted).
[001] [2] N.U. Ahmed, Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control, SIAM J. Control and Optim. 46 (1) (2007), 356-378. doi: 10.1137/050645944 | Zbl 1141.93062
[002] [3] N.U. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stoch. Proc. Appl. 60 (1995), 65-85. doi: 10.1016/0304-4149(95)00050-X
[003] [4] N.U. Ahmed and X. Ding, Controlled McKean-Vlasov equations, Commun. Appl. Anal. 5 (2001), 183-206.
[004] [5] N.U. Ahmed, C.D. Charalambous, Stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and drviven by relaxed controls, SIAM J. Control and Optim. 51 (4) (2013), 3235-3257. doi: 10.1137/120885656 | Zbl 1290.49034
[005] [6] N.U. Ahmed, Stochastic neutral evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality, Nonlinear Anal. (A) Theory, Methods & Applications 101 (2014), 66-79. doi: 10.1016/j.na.2014.01.019
[006] [7] N.U. Ahmed, Stochastic neutral evolution equations on Hilbert spaces and their partially observed optimal relaxed control, J. Abstract Diff. Equ. Appl. 5 (1) (2014), 1-20. | Zbl 06494513
[007] [8] N.U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series, Vol. 246, Longman Scientific and Technical, U.K. (Co-published with John-Wiley & Sons, Inc. New York, 1991).
[008] [9] N.U. Ahmed, Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality, Discuss. Math. Diff. Incl., Control and Optim. 34 (2014), 105-129. doi: 10.7151/dmdico.1153 | Zbl 1332.49030
[009] [10] N.U. Ahmed, Stochastic initial boundary value problems subject to distributed and boundary noise and their optimal control, JMAA 421 (2015), 157-179. doi: 10.1016/j.jmaa.2014.06.078 | Zbl 1293.93774
[010] [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions (Cambridge University Press, 1992). doi: 10.1017/CBO9780511666223 | Zbl 0761.60052
[011] [12] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Math. Soc. Lecture Note Ser. 229 (Cambridge University Press, London, 1996). doi: 10.1017/CBO9780511662829 | Zbl 0849.60052
[012] [13] D.A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Stat. Phys. 31 (1983), 29-85. doi: 10.1007/BF01010922
[013] [14] D.A. Dawson and J. Gartner, Large Deviations, Free Energy Functional and Quasi-Potential for a Mean Field Model of Interacting Diffusions, Mem. Amer. Math. Soc. 398 (Providence, RI, 1989). | Zbl 0677.60106
[014] [15] N. Dunford and J.T. Schwartz, Linear Operators, Part 1 (Interscience Publishers, Inc., New York, 1958). | Zbl 0084.10402
[015] [16] Y. Hu and S. Peng, Adaptive solution of a backward semilinear stochastic evolution equation, Stoch. Anal. Appl. 9 (4) (1991), 445-459. doi: 10.1080/07362999108809250 | Zbl 0736.60051
[016] [17] N.I. Mahmudov and M.A. McKibben, Abstract second order damped McKean-Vlasov stochastic evolution equations, Stoch. Anal. Appl. 24 (2006), 303-328. doi: 10.1080/07362990500522247 | Zbl 1102.35044
[017] [18] H.P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA 56 (1966), 1907-1911. doi: 10.1073/pnas.56.6.1907 | Zbl 0149.13501
[018] [19] Yang Shen and Tak Kuen Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlin. Anal. 86 (2013), 58-73. doi: 10.1016/j.na.2013.02.029 | Zbl 1279.49015