Optimal control with delays and mixed initial condition

This paper studies optimal control problems with unfixed initial instant for systems described by differential equations with variables delays and mixed initial condition. The mixed initial condition means that at the initial instant of time, some coordinates of the trajectory do not coincide with the corresponding coordinates of the initial function (a discontinuous part of the initial condition), whereas the others coincide (a continuous part of the initial condition). The authors prove necessary optimality conditions for the control and the initial function for the initial and final instants of time. From these conditions should be isolated the condition for the initial instant containing the effect of mixed initial condition. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 58, Optimal Control, 2008.     

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.     

Variation formulas of solution for a functional differential equation with delay function perturbation

Variation formulas of solution for a nonlinear functional differential equation with variable delay and continuous initial condition are proved. The effects of delay function perturbation and continuous initial condition are detected in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.     

论双曲型方程初—边值问题和初值问题解法的统一性

作者探索把双曲型方程初—边值问题中的初始函数在相应的本征函数族下展开成为Ｆｏｕｒｉｅｒ级数，把初始函数延拓到整个空间，把初—边值问题转化成为初值问题，从而使这两类不同的定解问题的解法统一起来．     

几种典型板考虑初始荷载效应的基频近似解

基于两组板考虑初始荷载效应的动力控制微分方程:一般形式的动力控制微分方程和极坐标形式的动力控制微分方程,运用Galerkin（伽辽金)法求解得到了简支矩形板、固支矩形板、简支等边三角形板、固支椭圆形板、简支圆形板和固支圆形板6种典型板考虑初始荷载效应的自由振动基频（第一阶频率）近似解．通过与相关文献提出的有限元法计算结果对比,验证了公式的正确性．基频近似解表达式简单明了,物理意义明确,清楚地说明了初始荷载及相关因素对板自由振动基频的影响,直观地说明了板的初始荷载效应这一概念．计算分析表明:初始荷载的存在增加了板的弯曲刚度,提高了板的自振频率．这种初始荷载效应对频率的影响主要受初始荷载大小、跨厚比及边界条件等因素的影响．在计算分析和设计中应考虑并重视这种初始荷载效应对板计算分析带来的影响.     

取油管振动方程解的整体不存在性

讨论推广的海底取油管振动方程的初边值问题和初值问题解的整体不存在性,对初边值问题推广了Gmira和Guedda得到的结果,对初值问题的结果是新的.     

Variation formulas of solutions and optimal control problems for differential equations with retarded argument

The authors prove a theorem on the continuous dependence of solutions of nonlinear systems of differential equations with variable delay on the perturbations of initial data (initial instant, initial function, and initial value of the trajectory) and the right-hand side in the case where these perturbations are small in the Euclidean and integral topology, respectively. The variation formulas of solutions of a differential equation with discontinuous and continuous initial condition are deduced; as compared with those known earlier, these formulas take into account the variation of the initial instant and the discontinuity and continuity of the initial data. A necessary condition for criticality of mappings defined on a finitely locally convex set is obtained. The quasiconvexity of filters in studying optimal problems with delays in controls is proved. Necessary optimality conditions and existence theorems are proved for optimal problems with variable delays in phase coordinates and controls having a nonfixed initial instant, a discontinuous and a continuous initial condition, and functional and boundary conditions of general form. Necessary optimality conditions are obtained for optimal problems with variable structure and delays. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 25, Optimal Control, 2005.     

Sharp critical thresholds for a class of nonlocal traffic flow models

We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.     

飞机初始备件品种优化的集对-粗糙集方法

飞机初始备件品种的确定是飞机初始使用和维修工作的必要条件.本文针对初始备件品种影响因素信息不完备的特点,依据集对分析方法在系统中的某一特性上具有同异反定量刻画的优势,结合经典粗糙集理论,提出初始备件品种优化的集对-粗糙集方法,最终得到初始备件品种优化的决策规则.实例计算表明,该方法能够为军方优化初始备件清单提供科学合理的决策.     

Vortex Perturbation Dynamics

An initial value approach is used to examine the dynamics of perturbations introduced into a vortex under strain. Both the basic vortex considered and the perturbations are taken as three-dimensional. An explicit solution for the time evolution of the vorticity perturbations is given for arbitrary initial vorticity. Analytical solutions for the resulting velocity components are found when the initial vorticity is assumed to be localized. For more general initial vorticity distributions, the velocity components are determined numerically. It is found that the variation in the radial direction of the initial vorticity disturbance is the most important factor influencing the qualitative behavior of the solutions. Transient growth in the magnitude of the velocity components is found to be directly attributable to the compactness of the initial vorticity.     

Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

Asymptotic Simplicity and Static Data

The present article considers time-symmetric initial data sets for the vacuum Einstein field equations, which are conformally related to static initial data sets in such a way that in a neighbourhood of infinity the two initial data sets have the same massless part. It is shown that for this class of data, the solutions to the regular finite initial value problem at spatial infinity for the conformal Einstein field equations extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data sets coincide with static data in a neighbourhood of infinity. This result highlights the special role played by static data among the class of initial data sets for the Einstein field equations whose development gives rise to a spacetime with a smooth conformal compactification at null infinity.     

Linearized maximum principle for neutral-type,variable-structure optimal problems with delays in controls

The authors state and study an optimal problem for variable-structure systems described by neutral-type quasilinear differential equations with discontinuous initial condition and incommensurable delays. The variation of the system structure means that in the process of motion, at a certain instant of time not known in advance, the object considered can pass from one law of motion to another, and, moreover, the initial condition for each of the subsequent states of the system depends on the state of the next to the last. The discontinuity of the initial condition means that at the initial instant of time, the value of the initial function and that of the trajectory do not coincide in general. The necessary optimality conditions are proved in the form of the linearized integral maximum principle for controls and initial functions and in the form of inequalities and equalities for the initial and final instants of structure change. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 42, Optimal Control, 2006.     

一类非线性演化方程初值问题的幂级数解

研究了一类非线性演化方程初值问题.通过不变子空间方法,这类初值问题被约化为常微分方程组的初值问题.这类初值问题是适定的.本文给出了这类初值问题关于时间变量t的幂级数解.     

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.     

具有初应力的1-3型压电复合材料的横向共振分析

1-3型压电复合材料在极化处理过程中会出现极化残余应力,基于线性压电理论,采用解析方法研究了极化残余应力对1-3型压电复合材料的横向共振模态的影响．首先建立含有初始应力的平面波波动方程,基于Bloch波函数理论构建了位移和电势函数,从而最终得到方程的解．数值结果表明初始残余应力的出现降低了复合材料的横向振动频率,而材料的压电性则使得横向振动频率提高．初始剪应力对对横向振动的频率影响较小,可以忽略．第一阶及第二阶振动频率将随着初始应力的大小呈线性变化．得出的结论对超声换能器的制作和研究将提供有意义的指导作用．     

Convergence from an electromagnetic fluid system to the full compressible MHD equations

We are concerned with the zero dielectric constant limit for the full electro-magneto-fluid dynamics in this article. This singular limit is justified rigorously for global smooth solution for both well-prepared and ill-prepared initial data. The explicit convergence rate is also obtained by a elaborate energy estimate. Moreover, we show that for the well-prepared initial data, there is no initial layer, and the electric field always converges strongly to the limit function. While for the ill-prepared data case, there will be an initial layer near t=0. The strong convergence results only hold outside the initial layer.     

基于初值修正的组合灰色Verhulst模型

对灰色Verhulst模型的拟合精度进行了分析,表明初值的选取对模型精度有重要影响.为提高灰色模型的拟合精度,利用最小二乘原理确定初值中的待定常数,给出了初值修正的灰色Verhulst模型.进一步利用修正模型和传统灰色Verhulst模型建立了组合灰色Verhulst模型,在平均相对误差最小的原则下,利用蚁群算法确定组合权系数.最后通过两个应用实例进行了计算和分析,结果表明,通过初值修正和组合模型能够提高灰色Verhulst模型的拟合精度,便于通过程序实现.     

On the application of genetic algorithms and gradient methods to the problem of the reconstruction of the initial stress field in a poroelastic inhomogeneous column

Initial stress fields play an important role in deformational processes in porous structures. Accurate modeling of dynamics behavior of the porous medium taking initial stresses into account requires an appropriate mathematical model. One of the common approaches for the construction of a mathematical model of the dynamic behavior of a medium taking the initial state into account is linearization. This paper presents equations of longitudinal vibrations of a poroelastic inhomogeneous body in the presence of the initial stress field. The influence of the initial stress rate and Biot’s modulus on the dynamic behavior of the environment is analyzed. The problem of reconstructing the initial stress field for the poroelastic inhomogeneous column based on the genetic algorithm and nonlinear optimization with the use of modern packages of numerical simulation is studied. A series of numerical experiments on the reconstruction of different initial stress field distribution laws using the described methods is carried out.     

Nonsymmetric bifurcations of solutions of the 2D Navier–Stokes system

We consider the 2D Navier–Stokes system written for the stream function with periodic boundary conditions and construct a set of initial data such that initial critical points bifurcate from 1 to 2 and then to 3 critical points in finite time. The bifurcation takes place in a small neighborhood of the origin. Our construction does not require any symmetry assumptions or the existence of special fixed points. For another set of initial data we show that 3 critical points merge into 1 critical point in finite time. We also construct a set of initial data so that bifurcation can be generated by the Navier–Stokes flow and do not require the existence of an initial critical point.