Control problems for semi-linear retarded integro-differential equations by the Fredholm theory

ABSTRACT

In this paper, we deal with the approximate controllability for semi-linear retarded functional integro-differential equations by using the Fredholm theory in Hilbert spaces. We no longer require the compactness of structural operators to obtain the approximate controllability for the nonlinear differential system, but instead we use the theory of interpolation spaces and the regularity of solutions of semi-linear given equations with unbounded principal operators. Finally, based on the properties of general degree theory in infinite dimensional spaces, we investigate the relation between the reachable set of trajectories of the semi-linear retarded functional integro-differential system and that of its corresponding linear system excluded by the nonlinear term.     

Controllability for semilinear functional differential equations with unbounded delays

In this paper, the approximate controllability of the semilinear functional differential equations with unbounded delays is first investigated. Then the regularity of solutions of the given system is addressed. Finally, a simple example to which our main result can be applied is given.     

Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille–Yosida operators

In this paper, we mainly focused the approximate controllability results for a class of non-densely defined fractional neutral differential control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of fractional differential inclusions with infinite delay where the linear part is non-densely defined and satisfies the Hille–Yosida condition. The main techniques rely on Bohnenblust– Karlin's fixed point theorem, operator semigroups and fractional calculus. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. Finally, an example is also given for the illustration of the obtained theoretical results.     

高维拟线性抛物系统能控性的近期进展(英文)

本文综述高维拟线性抛物型方程、拟线性复Ginzburg-Landau方程以及只含一个控制变量的高维耦合拟线性抛物型方程组的能控性方面的一些近期的结果.通过使用不动点技术,采用主部具有C1系数的线性抛物型方程或方程组一些新的精细的Carleman估计.这一方法的要点是在古典解的框架下考虑能控性问题,并且当给定的数据具有一定的正则性时,线性抛物型方程或方程组在H¨older空间中来选取控制函数.利用类似的方法,还建立了拟线性抛物型方程不灵敏控制的存在性,其关键是将不灵敏问题转化为由拟线性抛物型方程和线性抛物型方程构成的耦合方程组在单个控制下一个非标准的能控性问题.     

有界区域上Petrowsky系统的能控性: 极小泛函方法

利用极小泛函方法研究Petrowsky系统的控制问题.通过构造下半连续,严格凸的强制的泛函,把能控性问题,近似能控性问题转化为求泛函极小值存在的问题,从而得到了一些能控制性理论的充分必要条件,这些结果是采用了极小泛函方法得到的,改进了利用乘子方法得到的结果.     

Accuracy of Approximation of a Solution to an Abstract Cauchy Problem

A constructive image of the solution to a homogeneous Cauchy problem in a Banach space with a densely specified linear closed logarithmically-sector operator is investigated. The uniform accuracy of estimation of an approximate solution in t ≥ 0 is proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 145–152, May–June 2005.     

Local and global controllability for nonlinear systems

A technique to study local and global controllability properties for a wide class of nonlinear systems is presented. In addition to an extensive study of systems on ² we propose — as an application of our method — a new criterion for global controllability of systems with polynomial drift term, defined on ⁿ. In the case of linear systems, the criterion reduces to the classical Kalman condition.A study of local controllability at the equilibrium point of the drift term of systems defined on ² enables us not only to rederive a local controllability result derived by Hermes [4,5], but also to extend this result when no assumptions are made on the boundedness of the controls.     

Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions

This paper develops an iterative algorithm for the solution to a variable-coefficient semilinear heat equation with nonlocal boundary conditions in the reproducing space. It is proved that the approximate sequence u _n (x, t) converges to the exact solution u(x, t). Moreover, the partial derivatives of u _n (x, t) are also convergent to the partial derivatives of u(x, t). And the approximate sequence u _n (x, t) is the best approximation under a complete normal orthogonal system.     

Numerical approach for solving fractional Fredholm integro-differential equation

In this article, we present a new method which is based on the Taylor Matrix Method to give approximate solution of the linear fractional Fredholm integro-differential equations. This method is based on first taking the truncated Taylor expansions of the functions in the linear fractional differential part and Fredholm integral part then, substituting their matrix forms into the equation. We solve this matrix equation with the assistance of Maple 13. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method.     

Existence and uniqueness of mild solutions for abstract delay fractional differential equations

In this paper, by means of solution operator approach and contraction mapping theorem, the existence and uniqueness of mild solutions for a class of abstract delay fractional differential equations are obtained.     

Converse Lyapunov–Krasovskii theorems for systems described by neutral functional differential equations in Hale's form

In this article, we show that the existence of a Lyapunov–Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability (GES) of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map on the right-hand side of the equation is Lipschitz on bounded sets. A link between GES and input-to-state stability is also provided.     

广义动态系统的微分算子矩阵方法

本文用微分算子矩阵方法讨论了一类广义动态系统的结构性质,并且用微分算子多项式矩阵的系数阵,给出了各类动态系统能控(观)性的直接判据.它是用微分算子矩阵方法研究广义动态系统的开端.     

二阶分布参数系统的稳定性和能控性

讨论Hilbert空间上两个二阶线性系统的稳定性和能控性,在较一般的假设下,得到了这两个系统的指数稳定性和精确能控性,渐近稳定性和近似能控性之间的关系.最后,给出线性系统渐近稳定的一个充分必要条件.     

非线性动力学方程的李级数解法及其应用

分别从推广的微分方程幂级数解的理论和线性算子半群理论等不同的角度研究了非线性动力学方程的求解问题,得到了所谓的李级数解法．并进一步讨论了算法的具体实施过程,它可以用于构造非线性动力学方程任意高阶的显式积分格式．最后,把李级数解法应用于求解广义Hamilton系统,它能保持广义Hamilton系统真解的典则性．数值算例显示该方法是有效的。     

Well posedness and numerical solution for a non-local pseudohyperbolic initial boundary value problem

ABSTRACT

In this paper, we study a non-local initial boundary value problem for a one-dimensional pseudohyperbolic equation. We first establish the existence and uniqueness of strong solution, then a numerical solutions for the system will be derived by using the finite-difference method.     

也谈二分法求方程的近似解

用二分法求方程的近似解是《普通高中数学课程标准》的必修内容之一。利用平分区间及无限逼近的数学思想,是对解析性较好的一元函数避开其复杂运算、近似地计算、有效的解题,是一种行之有效的算法。从另一个角度考虑逼近,会收到良好的效果,结果比教材上的结果更加精确。     

Solutions of delay differential equations by using differential transform method

Differential transform method (DTM) is extended for delay differential equations. By using DTM, we manage to obtain the numerical, analytical, and exact solutions of both linear and nonlinear equations. In comparison with the existing techniques, the DTM is a reliable method that needs less work and does not require strong assumptions and linearization.