Bulletin of the
Korean Mathematical Society BKMS

This paper investigates infinite horizon optimal control problems driven by a class of backward stochastic delay differential equations in Hilbert spaces. We first obtain a prior estimate for the solutions of state equations, by which the existence and uniqueness results are proved. Meanwhile, necessary and sufficient conditions for optimal control problems on an infinite horizon are derived by introducing time-advanced stochastic differential equations as adjoint equations. Finally, the theoretical results are applied to a linear-quadratic control problem.

Keywords: Infinite horizon, backward stochastic delay differential equations, stochastic maximum principle, optimal control

MSC numbers: 93E20, 60H30, 49K27, 49N10

Supported by: This work is partially supported by the Natural Science Foundation of Shaanxi Province (Grant No. 2017JM1016) and the Fundamental Research Funds for the Central Universities (Grant No. 2452019075).