Abstract
This paper is concerned with optimal control of neutral stochastic functional differential equations (NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type (VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well.
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References
Ahmed N U. Dynamic Systems and Control with Applications. New Jersey-London-Beijing-Singapore: World Scientific, 2006
Ahmed N U. Deterministic and stochastic neutral systems on Banach spaces and their optimal feedback controls. J Nonlinear Syst Appl, 2013, 4: 1–10
Banks H, Kent G. Control of functional differential equations of retarded and neutral type to target sets in function space. SIAM J Control Optim, 1972, 10: 567–593
Bismut J. Théorie Probabiliste du Contrôle des Diffusions. Providence, RI: Amer Math Soc, 1976
Bismut J. Linear quadratic optimal stochastic control with random coefficients. SIAM J Control Optim, 1976, 14: 419–444
Bismut J. An introductory approach to duality in optimal stochastic control. SIAM Rev, 1978, 20: 62–78
Chen L, Wu Z. Maximum principle for stochastic optimal control problem of forward-backward system with delay. In: Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. New York: IEEE, 2009, 2899–2904
Chen L, Wu Z. Maximum principle for the stochastic optimal control problem with delay and application. Automatica, 2010, 46: 1074–1080
Cushing C M. Integro-Differential Equations and Delay Models in Population Dynamics. Berlin: Springer-Verlag, 1977
El Karoui N, Peng S, Quenez M C. Backward stochastic differential equations in finance. Math Finance, 1997, 7: 1–71
Elsanosi I, Øksendal B, Sulem A. Some solvable stochastic control problems with delay. Stoch Stoch Rep, 2000, 71: 69–89
Gut A. An introduction to the theory of asymptotic martingales. In: Amarts and Set Function Processes. Lecture Notes in Mathematics, vol. 1042. Berlin-New York: Springer, 1983, 1–49
Hale J. Oscillations in neutral functional differential equations. In: Non-linear Mechanics. Rome: Edizioni Cremonese, 1973, 97–111
Hale J. Theory of Functional Differential Equations. New York: Springer-Verlag, 1977
Hale J K. Introduction to Functional Differential Equations. Berlin-New York: Springer-Verlag, 1993
Hu Y, Peng S. Maximum principle for optimal control of stochastic system of functional type. Stoch Anal Appl, 1996, 14: 283–301
Huang L, Mao X. Delay-dependent exponential stability of neutral stochastic delay systems. IEEE Trans Automat Control, 2009, 54: 147–152
Kent G. A maximum principle for optimal control problems with neutral functional differential systems. Bull Amer Math Soc, 1971, 77: 565–570
Kolmanovskii V, Khvilon E. Necessary conditions for optimal control of systems with deviating argument of neutral type. Autom Remote Control, 1969, 30: 327–339
Kolmanovskii V, Myshkis A. Introduction to the Theory and Applications of Functional Differential Equations. Boston, MA: Kluwer Academic Publisher, 1999
Kolmanovskii V B, Nosov V R. Stability and Periodic Modes of Control Systems with Aftereffect. Moscow: Nauka, 1981
Kolmanovskii V B, Nosov V R. Stability of Functional Differential Equations. London: Academic Press, 1986
Kolmanovskii V B, Shaikhet L. Construction of Lyapunov functionals for stochastic hereditary systems: A survey of some recent results. Math Comput Modelling, 2002, 36: 691–716
Lin J. Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch Anal Appl, 2002, 20: 165–183
Liu K. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete Contin Dyn Syst Ser B, 2013, 18: 1651–1661
Mao X. Exponential stability in mean square of neutral stochastic differential functional equations. Systems Control Lett, 1995, 26: 245–251
Øksendal B, Sulem A. A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In: Optimal Control and Partial Differential Equation. Amsterdam: IOS Press, 2001, 64–79
Pardoux E, Peng S. Adapted solution of a backward stochastic differential equation. Systems Control Lett, 1990, 14: 55–61
Protter P. Stochastic Integration and Differential Equations: A New Approach. Berlin-New York-London-Paris-Tokyo-Hong Kong: Springer-Verlag, 1990
Ren Y, Xia N. Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay. Appl Math Comput, 2009, 210: 72–79
Salamon D. Control and Observation of Neutral Systems. London: Pitman Advanced Pub Program, 1984
Shaikhet L. Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations. Appl Math Lett, 1997, 10: 111–115
Shaikhe L. Some new aspects of Lyapunov-type theorems for stochastic differential equations of neutral type. SIAM J Control Optim, 2010, 48: 4481–4499
Wei W. Neutral backward stochastic functional differential equations and their application. ArXiv:1301.3081, 2013
Yong J. Well-posedness and regularity of backward stochastic Volterra integral equations. Probab Theory Related Fields, 2008, 142: 21–77
Yu Z. The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls. Automatica, 2012, 48: 2420–2432
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Wei, W. Maximum principle for optimal control of neutral stochastic functional differential systems. Sci. China Math. 58, 1265–1284 (2015). https://doi.org/10.1007/s11425-015-4972-x
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DOI: https://doi.org/10.1007/s11425-015-4972-x
Keywords
- neutral stochastic functional differential equation
- neutral backward stochastic functional equation of Volterra type
- stochastic optimal control
- Pontryagin maximum principle