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## Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2016 | 36 | 2 | 181-206
Tytuł artykułu

### A general class of McKean-Vlasov stochastic evolution equations driven by Brownian motion and Lèvy process and controlled by Lèvy measure

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EN
In this paper we consider McKean-Vlasov stochastic evolution equations on Hilbert spaces driven by Brownian motion and Levy process and controlled by Levy measures. We prove existence and uniqueness of solutions and regularity properties thereof. We consider weak topology on the space of bounded Le´vy measures on infinite dimensional Hilbert space and prove continuous dependence of solutions with respect to the Le´vy measure. Then considering a certain class of Le´vy measures on infinite as well as finite dimensional Hilbert spaces, as relaxed controls, we prove existence of optimal controls for Bolza problem and some simple mass transport problems
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EN
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Tom
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181-206
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wydano
2016
otrzymano
2016-11-30
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Bibliografia
• [1] \bibitem {1} N.U. Ahmed, Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality, Discuss. Math. Diff. Incl. Control and Optim. 35 (2015), 165-195. doi: 10.7151/dmdico.1171
• [2] N.U. Ahmed, Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control, SIAM J. Cotrol and Optim. 46 (2007), 356-378. doi: 10.1137/050645944
• [3] N.U. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stochastic Process. Appl. 60 (1995), 65-85. doi: 10.1016/0304-4149(95)00050-X
• [4] N.U. Ahmed and X. Ding, Controlled McKean-Vlasov equations, Commun. Appl. Anal. 5 (2001), 183-206.
• [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 (2013), 3235-3257. doi: 10.1137/120885656
• [6] N.U. Ahmed, Systems governed by mean-field stochastic evolution equations on Hilbert spaces and their optimal control, Dynamic Systems and Applications (DSA) 25 (2016), 61-88.
• [7] 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-Wiely & Sons, Inc. New York, 1991).
• [8] 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
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• [15] 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
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