Systems with Time Delay in the Calculus of Variations: The Method of Steps

In a recent article (see [8]), we derived necessary and sufficientconditions for minima for the fixed-endpoint problem in thecalculus of variations involving a constant delay in the phasecoordinates. These conditions are expressed, explicitly, interms of the first and second variations. The vanishing of thefirst variation is characterized in terms of an extended Euler'sequation, just as for delay-free problems, but the characterizationof the conditions on the second variation remained unsolved.In this paper we convert, through the ‘method of steps’,the delay problem into one without delay. Although this problemwill not have fixed-endpoint constraints, it allows us to introduce,in a natural way, the concept of ‘conjugate sequence’;this solves the main difficulty for delay problems, namely,to give initial conditions for existence and uniqueness of solutionsof the Hamiltonian system (which is a difference–differentialsystem with both advanced and retarded arguments). The conditionson the second variation are then characterized by an extra conditionwhich is based exclusively on a solution of a given matrix Riccatiequation.     

A Time Finite Element Method Based on the Generalized Variational Calculus

In this contribution, a new finite element method in the temporal domain is presented, in which the time step size is introduced as an additional variable. Thus, the variation of the time integral of the Lagrangean resulting from Hamilton's principle has to be carried out with respect to the rules of the generalized variational calculus. Apart from the usual time integral of the Euler‐Lagrange differential equations, the so‐called transversality condition is obtained as an additional result representing a time‐boundary term, which is used to obtain an optimal step size in the time domain.     

A Bliss-Type Multiplier Rule for Constrained Variational Problems with Time Delay

This paper gives a Bliss-type multiplier rule for general constrained variational problems with time delay. Our formulation allows the time delay terms and their derivatives to be present in both the objective functional and the equality and inequality constraint equations. Our major results include a Lagrange Multiplier Rule involving Euler–Lagrange equations for delay differential equations and transversality conditions. Of special note is that these results will be used in later work by the authors to obtain new analytical techniques to solve general constrained problems involving delay differential equations for the Calculus of Variations and for Optimal Control Theory. They will also lead to new general, accurate, and efficient numerical methods to solve these important problems where no such methods exist at this time. Thus, for the first time, we will be able to solve this important class of problems obtaining analytic solutions for simpler problems and accurate numerical results for more complex problems when such a solution exists.     

The Inverse Problem of the Calculus of Variations: Separable Systems

This paper deals with the inverse problem of the calculus of variations for systems of second-order ordinary differential equations. The case of the problem which Douglas, in his classification of pairs of such equations, called the separated case is generalized to arbitrary dimension. After identifying the conditions which should specify such a case for n equations in a coordinate-free way, two proofs of its variationality are presented. The first one follows the line of approach introduced by some of the authors in previous work, and is close in spirit, though being coordinate independent, to the Riquier analysis applied by Douglas for n = 2. The second proof is more direct and leads to the discovery that belonging to the separated case has an intrinsic meaning for the given second-order differential equations: the system is separable in the sense that it can be decoupled into n pairs of first-order equations.     

一类具有阶段结构和时滞的非自治捕食系统

本文研究一类具有阶段结构和时滞的非自治捕食系统解的渐近性质 ,在适当条件下 ,得到系统是持续生存的且全局渐近稳定的     

Variational Approach to a Class of Second Order Hamiltonian Systems on Time Scales

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the second order Hamiltonian system on time scale \mathbbT\mathbb{T}

时滞系统在博弈中的应用研究

演化博弈理论解决策略均衡问题是目前比较认可的好方法.对于博弈方的经验行为,还未曾研究.本文以非对称2×2的博弈问题为研究对象,建立起带有时滞的博弈系统,通过动力学相关知识,分别研究了两类系统下博弈参与者策略的稳定状态.数值模拟分析了延时给决策带来的影响.实验表明,时滞作用不改变策略的稳定状态,改变的仅是达到稳态的速率.     

Exponential Stability of Singularly Perturbed Systems with Time Delay

In this article, we study the exponential stability of singularly perturbed systems with time delay. By using vector delay inequalities and Lyapunov functions, exponential stability criteria are derived for both linear and some classes of nonlinear singularly perturbed systems with time delay. Examples are given to verify the stability criteria.     

A Variational Approach to Dynamical Systems and Its Numerical Simulation

We propose and describe an alternative perspective to the study and numerical approximation of dynamical systems. It is based on a variational approach that seeks to minimize the quadratic error understood as a deviation of paths from being a solution of the corresponding system. Although this philosophy has been examined recently from the point of view of the direct method, we exploit optimality conditions and steepest descent strategies to establish precise and easy-to-implement numerical schemes for the approximation. We show the practical performance in a number of selected examples and indicate how this strategy, with minor changes, may also be used to deal with boundary value problems. Our emphasis is placed more so on relevant results that justify the numerical implementation and less on abstract theoretical results under optimal sets of assumptions.     

Fractional Variational Approach for Dissipative Mechanical Systems

More recently, a variational approach has been proposed by Lin and Wang for damping motion with a Lagrangian holding the energy term dissipated by a friction force. However, the modified Euler-Lagrange equation obtained within their for- malism leads to an incorrect Newtonian equation of motion due to the nonlocality of the Lagrangian. In this communication, we generalize this approach based on the fractional actionlike variational approach and we show that under some simple restric- tions connected to the fractional parameters introduced in the fractional formalism, this problem may be solved.     

随机时滞控制系统的均方BIBO稳定性

本文研究了具有时滞和非线性扰动的随机控制系统的均方有界输入-有界输出(BIBO)稳定.首先,探讨了具有离散时滞和非线性扰动的随机系统的均方BIBO稳定性问题,在此基础上,进一步研究带有离散时滞和分布时滞以及非线性扰动的随机系统的均方BIBO稳定性.通过设计合理的控制器,建立合适的Lyapunov泛函,结合Riccati矩阵方程,得到时滞依赖的均方BIBO稳定性条件.     

不确定随机时滞系统的时滞相关鲁棒镇定

研究一类不确定随机时滞系统的时滞相关鲁棒镇定问题.通过引入参数化的中立型模型变换,构造Lyapunov-krasovskii泛函,运用线性矩阵不等式方法,得到了使得闭环系统为均方指数稳定的保守性较小的时滞相关鲁棒镇定条件.     

Variational Problems with Symmetries: A Pfaffian System Approach

Reduction methods for completely integrable Pfaffian Systems with symmetry are applied to variational problems, providing analogues of the Arnold?CLiouville Theorem and Marsden?CWeinstein reduction in the Lagrangian setting. A?generalization of the Mishchenko?CFomenko Theorem for non Abelian integrability is given in terms of solvable structures.     

On the Variational Approach to Systems of Quasilinear Conservation Laws

The paper contains results concerning the development of a new approach to the proof of existence theorems for generalized solutions to systems of quasilinear conservation laws. This approach is based on reducing the search for a generalized solution to analyzing extremal properties of a certain set of functionals and is referred to as a variational approach. The definition of a generalized solution can be naturally reformulated in terms of the existence of critical points for a set of functionals, which is convenient within the approach proposed. The variational representation of generalized solutions, which was earlier known for Hopf-type equations, is generalized to systems of quasilinear conservation laws. The extremal properties of the functionals corresponding to systems of conservation laws are described within the variational approach, and a strategy for proving the existence theorem is outlined. In conclusion, it is shown that the variational approach can be generalized to the two-dimensional case.     

Optimization Approach to the Robustness of Linear Delay Systems

By using the Lyapunov equation approach and an improved Razumikhin-type theorem, this paper presents a new robust stability criterion for a linear system subject to delayed time-varying nonlinear perturbations. Then, by using a parameter optimization technique, an efficient algorithm is derived for determining a desirable matrix for the Lyapunov equation. As a consequence, less conservative robust stability bounds for the perturbed system are achieved. Numerical examples are included to demonstrate the effectiveness of the proposed approach.     

Borderline problems in the Calculus of Variations

The problem of compactness in the theory of Calculus of Variations is as old as the theory itself. In recent years important progress was made in understanding problems in the border region where compactness is “lost”. After discussing some problems with “lack of compactness”, and describing the ground-breaking result of Brezis-Nirenberg for elliptic equations with polynomial nonlinearities inR ^N, N≥3, some new results for elliptic equations on bounded domains inR ² are described. In particular, it is shown that (as in the case of Brezis-Nirenberg) the existence or non-existence of solutions depends on the terms with lower order (i.e. subcritical) growth.     

变分泛函的半连续性问题

研究了非多项式增长的变分泛函,利用Orlicz空间理论,得到了其在Orlicz-Sobolev空间中弱序列下半连续的充要条件,推广了关于多项式增长的变分泛函的相应结论。     

网络化非周期采样控制系统的主动时间滞后控制:随机脉冲切换系统方法

研究了一类带有随机丢包的非周期采样网络化控制系统的镇定问题.不同于传统观点往往将时滞看作系统稳定性的消极因素,考虑时间滞后对系统稳定性的积极影响,并提出一个新颖的主动时间滞后控制方法来镇定该系统.为了分析时间滞后控制的积极作用并获得较低保守性的结论,首先把带随机丢包的非周期采样系统建模为带固定切换率的随机脉冲切换系统,...     

一类带有不确定性的时滞系统的控制器设计

针对一类带有不确定性的单输入单输出的时滞非线性系统,提出了一种鲁棒非线性控制算法.利用反步设计的迭代设计思想,在每一步构造李亚普诺夫-克拉索夫斯基函数,用放大不等式的方法获得控制器,保证闭环系统的稳定性.以连续搅拌化学反应器为例的仿真结果也验证了控制器具有良好的控制特性.