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

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

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

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topN.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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