Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays
Kai Liu
Discrete & Continuous Dynamical Systems - B 2013, 18(6): 1651-1661 doi: 10.3934/dcdsb.2013.18.1651
A class of stochastic optimal control problems of infinite dimensional Ornstein-Uhlenbeck processes of neutral type are considered. One special feature of the system under investigation is that time delays are present in the control. An equivalent formulation between an adjoint stochastic controlled delay differential equation and its lifted control system (without delays) is developed. As a consequence, the finite time quadratic regulator problem governed by this formulation is solved based on a direct solution of some associated Riccati equation.
keywords: Ornstein-Uhlenbeck process of neutral type quadratic stochastic optimal control Riccati equation.
Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives
Kai Liu
Discrete & Continuous Dynamical Systems - B 2018, 23(9): 3915-3934 doi: 10.3934/dcdsb.2018117

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially concerned about those delays appearing in both spatial and temporal derivative terms in which the coefficient operator under spatial variables may take the same form as the infinitesimal generator of the equation. We establish the stationary property of the neutral system under investigation by focusing on distributed delays. In the end, an illustrative example is analyzed to explain the theory in this work.

keywords: Stochastic functional differential equation of neutral type strongly continuous or \begin{document}$ c_0$\end{document} semigroup resolvent operator stationary solution

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