Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality

N.U. Ahmed

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2015)

  • Volume: 35, Issue: 2, page 165-195
  • ISSN: 1509-9407

Abstract

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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.

How to cite

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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 35.2 (2015): 165-195. <http://eudml.org/doc/276523>.

@article{N2015,
abstract = {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.},
author = {N.U. Ahmed},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {McKean-Vlasov stochastic differential equation; Hilbert spaces; relaxed controls; existence of optimal controls},
language = {eng},
number = {2},
pages = {165-195},
title = {Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality},
url = {http://eudml.org/doc/276523},
volume = {35},
year = {2015},
}

TY - JOUR
AU - N.U. Ahmed
TI - Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2015
VL - 35
IS - 2
SP - 165
EP - 195
AB - 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.
LA - eng
KW - McKean-Vlasov stochastic differential equation; Hilbert spaces; relaxed controls; existence of optimal controls
UR - http://eudml.org/doc/276523
ER -

References

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  2. [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 Zbl1141.93062
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  5. [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 Zbl1290.49034
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