Xiuxiang Zhou

• Lingnan Normal University

This paper is addressed to a study of the stability of heat and wave equations with memory The necessary and sufficient conditions of the exponential stability are investigated by the theory of Laplace transform. The results show that the stability depends on the decay rate and the coefficient of the kernel functions of the memory. Besides, the fee...

This paper studies a minimal time control problem for a linear heat equation with memory. The purpose of such a problem is to find a control (among certain control constraint set), which steers the solution of the heat equation with memory from a given initial state to a given target as soon as possible. In this paper, we study the existence of opt...

This paper is devoted to a studying of the controllability properties for the Coleman–Gurtin-type equation, which is a class of multidimensional integral–differential equations. The goal is to prove the existence of a control function which steers the state variable and the integral term to the neighborhood of two given final configurations at the...

This paper is devoted to analyzing the controllability to rest of the Gurtin-Pipkin model, which is a class of differential equations with memory terms. The goal is not only to derive the state to vanish at some time but also to require the memory term to vanish at the same time, ensuring that the controlled system is controllable to rest. In order...

In this work, we mainly study the exact controllability of stochastic differential equations with memory. For time invariant systems, by applying forward backward stochastic differential equations, some rank criteria ensuring systems’ exact controllability are provided. For time variant systems, equivalent conditions guaranteeing systems’ exact con...

In this paper, we study the null controllability and approximate controllability for a class of weakly degenerate parabolic equations with memory by means of boundary controls. Unlike the known result for the degenerate parabolic equation, the degenerate parabolic equation with memory in general is not null controllable. This is based on the observ...

In this paper, we investigate the stability of heat equation with a constant memory kernel. We prove that the system does not decay for the positive kernel, and is polynomially stable but not exponentially for the negative kernel. In particular, a more detailed description of the decay property for the negative kernel is obtained by dividing initia...

This paper addresses the study of the integral-type approximate controllability of linear parabolic integro-differential equations. This new controllability is defined by imposing some additional integral-type constraints on the usual approximate controllability and therefore, can be used to keep the state close to the target. The paper is concerne...

This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whe...

This paper is addressed to the disturbance decoupling and almost disturbance decoupling problems in infinite dimensions. We introduce a class of approximate finite dimensional systems, and show that if the systems are disturbance decoupled, so does the original infinite dimensional system. It is also shown that this approach can be employed to solv...

In this paper, we study the approximate and null controllability of the heat equation with memory. We mainly consider the distributed control case, with an arbitrarily small support of the control function. For each positive final time TT, we deduce that the approximate controllability property holds and the null controllability property fails.