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

top
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

top

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

top
  1. [1] N.U. Ahmed, Systems governed by mean-field stochastic evolution equations on Hilbert spaces and their optimal control, Nonlinear Anal. (submitted). 
  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
  3. [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 
  4. [4] N.U. Ahmed and X. Ding, Controlled McKean-Vlasov equations, Commun. Appl. Anal. 5 (2001), 183-206. 
  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
  6. [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 
  7. [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. Zbl06494513
  8. [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). 
  9. [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 Zbl1332.49030
  10. [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 Zbl1293.93774
  11. [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions (Cambridge University Press, 1992). doi: 10.1017/CBO9780511666223 Zbl0761.60052
  12. [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 Zbl0849.60052
  13. [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 
  14. [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). Zbl0677.60106
  15. [15] N. Dunford and J.T. Schwartz, Linear Operators, Part 1 (Interscience Publishers, Inc., New York, 1958). Zbl0084.10402
  16. [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 Zbl0736.60051
  17. [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 Zbl1102.35044
  18. [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 Zbl0149.13501
  19. [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 Zbl1279.49015

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.