Optimality conditions for stochastic boundary control problems governed by semilinear parabolic equations. (English) Zbl 1245.93146

Yu, Huaiqiang; Liu, Bin

J. Math. Anal. Appl. 395, No. 2, 654-672 (2012).

Summary: We study the boundary control problems for stochastic parabolic equations with Neumann boundary conditions. Imposing super-parabolic conditions, we establish the existence and uniqueness of the solution of state and adjoint equations with non-homogeneous boundary conditions by the Galerkin approximations method. We also find that, in this case, the adjoint equation (BSPDE) has two boundary conditions (one is non-homogeneous, the other is homogeneous). Using these results, we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method.

MSC:

93E20	Optimal stochastic control
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
49K45	Optimality conditions for problems involving randomness
35K99	Parabolic equations and parabolic systems

Keywords:

stochastic partial differential equations; boundary control; necessary conditions; convex state constraints; BSPDE with non-homogeneous double boundary conditions

References:

[1]	Casas, E., Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35, 1297-1327 (1997) · Zbl 0893.49017
[2]	Casas, E.; Raymond, J. P.; Zidani, H., Pontryagin’s principle for local solutions of control problems with mixed control-state constraints, SIAM J. Control Optim., 39, 1182-1203 (2000) · Zbl 0984.49011
[3]	Kushner, H. J., Necessary conditions for continuous parameter stochstic optimization problems, SIAM J. Control Optim., 10, 550-565 (1973) · Zbl 0242.93063
[4]	Haussmann, U. G., General necessary conditions for optimal control of stochastic systems, Math. Program. Stud., 6, 30-48 (1976) · Zbl 0369.93048
[5]	Haussmann, U. G., (A Stochstic Maximum Principle for Optimal Control of Diffusions. A Stochstic Maximum Principle for Optimal Control of Diffusions, Pitman Research Notes in Math Series, vol. 151 (1986), John Wiley & Sons: John Wiley & Sons New York) · Zbl 0616.93076
[6]	Arkin, V. I.; Saksonov, M. T., Necessary optimality conditions for stochastic differential equations, Sov. Math. Dokl., 20, 1-5 (1979) · Zbl 0414.49017
[7]	Bensoussan, A., Lecture on stochastic control Part I, Lecture Notes in Math., 972, 1-39 (1983)
[8]	Bismut, J. M., An introductory approach to duality in optimal stochastic control, SIAM Rev., 20, 61-78 (1978) · Zbl 0378.93049
[9]	Cadenillas, A.; Karatzas, I., The stochastic maximum principle for linear convex optimal control with random coefficients, SIAM J. Control Optim., 33, 590-624 (1995) · Zbl 0826.93069
[10]	Elliott, R. J., The optimal control of diffusions, Appl. Math. Optim., 22, 229-240 (1990) · Zbl 0718.49013
[11]	Elliott, R. J.; Kohlmann, M., The second order minimum principle and adjoint process, Stoch. Stoch. Rep., 46, 15-39 (1994) · Zbl 0824.60051
[12]	Peng, S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 966-979 (1990) · Zbl 0712.93067
[13]	Yong, J.; Zhou, X., (Stochastic Control, Hamilton Systems and HJB Equations. Stochastic Control, Hamilton Systems and HJB Equations, Appl. Math., vol. 43 (1999), Springer-Verlag: Springer-Verlag New York)
[14]	Bahlali, S., Necessary and sufficient optimality conditions for relaxed and strict control problems, SIAM J. Control Optim., 47, 2078-2095 (2008) · Zbl 1167.49024
[15]	Bensoussan, A., Stochastic maximum principle for distributed parameter system, J. Franklin Inst., 315, 387-406 (1983) · Zbl 0519.93042
[16]	Hu, Y.; Peng, S., Maximum principle for semilinear stochastic evolution controls systems, Stochastics, 33, 159-180 (1990) · Zbl 0722.93080
[17]	Zhou, X. Y., On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31, 1462-1478 (1993) · Zbl 0795.93104
[18]	Hu, Y.; Peng, S., Adapted solution of a backward semilinear stochastic evolution equation, Stoch. Anal. Appl., 9, 445-459 (1991) · Zbl 0736.60051