Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (Stochastic Partial Differential Equations and Applications, Series of Lecture Notes in Pure and Appl. Math., vol. 245, pp. 253–260, Chapman and Hall, London, 2006) and Goreac (Applied Analysis and Differential Equations, pp. 153–164, World Scientific, Singapore, 2007) from the finite dimensional to the infinite dimensional case.     

Approximate Controllability of Nonlinear Delay Integro Differential Evolution Equations with Random Impulses

The basic concept of this research is to analyse the approximate controllability (AC) of a nonlinear delay integrodifferential evolution system (NDIDES) with random impulse of the type \begin{align*}&z''(\zeta)=\mathfrak{A}(\zeta)z(\zeta)+(\mathfrak{B}x)(\zeta)+\int_{0}^{\zeta}\mathcal{H}(\zeta, s,z(\beta(s))), \ \sigma_{q} <\zeta < \sigma_{q+1}, \ \zeta\in [\zeta_{0}, \mathcal{T}], \\ &z(\sigma_{q})=a_{q}(\tau_{q})z(\sigma^{-}_{q}), ~~q = 1,2,\ldots,\\ &z_{\zeta_{0}}=\upsilon,\end{align*} by assuming that the linear system is approximately controllable. The existence and uniqueness of the mild solution to above system have been determined by using the Banach contraction principle and trajectory accessible sets. We generalize the results for NDIDES with and without fixed-type impulsive moments.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.     

Controllability for a class of second-order evolution differential inclusions without compactness

In this paper, we discuss the existence and controllability for a class of second-order evolution differential inclusions without compactness in Banach spaces. By applying the technique of weak topology and Glicksberg–Ky Fan fixed point theorem, we prove our main results without the hypotheses of compactness on the operator generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Further, we extend our study to existence and controllability of second-order evolution differential inclusions with nonlocal conditions and impulses. Finally, an example is given for the illustration of the obtained theoretical results.     

Exact Controllability for a Class of Nonlinear Evolution Control Systems

In this paper, we study the exact controllability of the nonlinear control systems. The controllability results by using the monotone operator theory are es-tablished. No compactness assumptions are im...     

Existence and Approximate Controllability of Solutions for an Impulsive Evolution Equation with Nonlocal Conditions in Banach Space

In this article, we study the existence of mild solutions and approximate controllability for non-autonomous impulsive evolution equations with nonlocal conditions in Banach space. The existence of mild solutions and some conditions for approximate controllability of these non-autonomous impulsive evolution equations are given by using the Krasnoselskii''s fixed point theorem, the theory of evolution family and the resolvent operator. In particular,the impulsive functions are supposed to be continuous and the nonlocal item is divided into Lipschitz continuous and completely bounded. An example is given as an application of the results.     

On the Approximate Controllability for Hilfer Fractional Evolution Hemivariational Inequalities

The paper is concerned with the approximate controllability of some Hilfer fractional evolution hemivariational inequalities. Using two classes of operators and their fundamental properties, we derive sufficient conditions for approximate controllability of linear and semilinear controlled systems via a fixed point theorem for multivalued maps. Finally, an example is given to illustrate our theory.     

Local Controllability of Functional Integrodifferential Equations in Banach Space

In this paper, we establish a set of sufficient conditions for the local controllability of functional integrodifferential equations in Banach space. The results are obtained by using the methods of analytic semigroups, fractional powers of operators, and a fixed-point argument. These results generalize previous results on bounded linear operators to unbounded linear operators in which the equation involves a nonlinear delay term. An application to a partial integrodifferential equation is given.     

Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms

This paper deals with the approximate controllability of the semilinear heat equation, when the nonlinear term depends on both the state y and its spatial gradient y and the control acts on any nonempty open subset of the domain. Our proof relies on the fact that the nonlinearity is globally Lipschitz with respect to (y, y). The approximate controllability is viewed as the limit of a sequence of optimal control problems. Another key ingredient is a unique continuation property proved by Fabre (Ref. 1) in the context of linear heat equations. Finally, we prove that approximate controllability can be obtained simultaneously with exact controllability over finite-dimensional subspaces.     

Approximate controllability of backward stochastic evolution equations in Hilbert spaces

In this paper, we examine the approximate controllability of a semilinear backward stochastic evolution equations in Hilbert spaces with non-Lipschitz coefficient.     

Approximate Controllability of Neutral Functional Differential Systems with State-Dependent Delay

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.     

Approximate Controllability of Neutral Stochastic Functional Differential Systems with Infinite Delay

In this article, we consider a class of control systems governed by the neutral stochastic functional differential equations with unbounded delay and study the approximate controllability of the system. An example is given to illustrate the result.     

一类二阶混合型泛函微分方程多重周期解

本文利用变分原理和Z2不变群指标研究了二阶混合型泛函微分方程x"(t-τ) f(t,x(t),x(t-τ),x(t-2τ))=0和x"(t-r) λf1(t,x(t),x(t-τ),x(t-2τ))=x(t-τ)多重周期解．得出了有关新结果．     

Approximate Controllability of a Stochastic Wave Equation

In this paper we discuss the controllability of a wave equation with random noise. Our main tools are the Ito representation theorem and an adaptation of the Hilbert uniqueness method for the exact controllability of deterministic equations.     

Approximate Controllability of a Stochastic Wave Equation

In this paper we discuss the controllability of a wave equation with random noise. Our main tools are the Ito representation theorem and an adaptation of the Hilbert uniqueness method for the exact controllability of deterministic equations.     

二阶非线性时滞差分方程解的振动性

研究了二阶非线性时滞差分方程△(α_n(△(x_n+p_nx_(n-r)))~γ)+bn(△(x_n+p_nx_(n-T)))~γ+f(n,x_(n-σ))=0给出方程振动的充分条件,推广和改进了中立时滞差分方程的许多结果.     

一类非线性发展方程的动力学行为

本文通过引入一类Frechet空间,证明了一类与流体力学有关的非线性发展方程的弱解存在着整体吸引子．