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, 22(11): 1-20 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
Dynamics of an SIS reaction-diffusion epidemic model for disease transmission
Wenzhang Huang Maoan Han Kaiyu Liu
Mathematical Biosciences & Engineering 2010, 7(1): 51-66 doi: 10.3934/mbe.2010.7.51
Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition has been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many important and interesting properties have been obtained: such as the role of diffusion coefficients in defining the reproductive number; the global stability of disease-free equilibrium; the existence and uniqueness of a positive endemic steady; global stability of endemic steady for some particular cases; and the asymptotical profiles of the endemic steady states as the diffusion coefficient for susceptible individuals is sufficiently small. In this research we will study two modified SIS diffusion models with the Dirichlet boundary condition that reflects a hostile environment in the boundary. The reproductive number is defined which plays an essential role in determining whether the disease will extinct or persist. We have showed that the disease will die out when the reproductive number is less than one and that the endemic equilibrium occurs when the reproductive number is exceeds one. Partial result on the global stability of the endemic equilibrium is also obtained.
keywords: stability. epidemic model reaction-diffusion equations
Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes
Kai Liu Zhi Li
Discrete & Continuous Dynamical Systems - B 2016, 21(10): 3551-3573 doi: 10.3934/dcdsb.2016110
In this paper, we are concerned with a class of neutral stochastic partial differential equations driven by $\alpha$-stable processes. By combining some stochastic analysis techniques, tools from semigroup theory and delay integral inequalities, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the $p$-th moment to the stochastic systems are obtained. Subsequently, by employing a weak convergence approach, we try to establish some stability conditions in distribution of the segment processes of mild solutions to the stochastic systems under consideration. Last, an example is presented to illustrate our theory in the work.
keywords: Global attracting set exponential decay in the $p$-th moment $\alpha$-stable process. stability in distribution
Minimization of the lowest eigenvalue for a vibrating beam
Quanyi Liang Kairong Liu Gang Meng Zhikun She
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 2079-2092 doi: 10.3934/dcds.2018085

In this paper we solve the minimization problem of the lowest eigenvalue for a vibrating beam. Firstly, based on the variational method, we establish the basic theory of the lowest eigenvalue for the fourth order measure differential equation (MDE). Secondly, we build the relationship between the minimization problem of the lowest eigenvalue for the ODE and the one for the MDE. Finally, with the help of this built relationship, we find the explicit optimal bound of the lowest eigenvalue for a vibrating beam.

keywords: Eigenvalue minimization problem the fourth order equation

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