A numerical method for determining a quasi solution of a backward time-fractional diffusion equation

In this study, we present existence and uniqueness theorems of a quasi solution to backward time-fractional diffusion equation. To do that, we consider a methodology, involving minimization of a least squares cost functional, to identify the unknown initial data. Firstly, we prove the continuous dependence on the initial data for the corresponding forward problem and then we obtain a stability estimate. Based on this, we give the existence theorem of a quasi solution in an appropriate class of admissible initial data. Secondly, it is shown that the cost functional is Fréchet-differentiable and its derivative can be formulated via the solution of an adjoint problem. These results help us to prove the convexity of cost functional and subsequently the uniqueness theorem of the quasi solution. In addition, in order to approximate the quasi solution, WEB-spline finite element method is used. Since the obtained system of linear equations is ill-posed, we apply the Levenberg-Marquardt regularization. Finally, a numerical example is given to show the validation of the introduced method.     

具分布观测的年龄相关的种群扩散系统生育率控制的非线性问题

本文讨论了具分布观测的年龄相关的种群扩散系统最优生育率控制的非线性问题,证明了系统解的存在唯一性和最优生育率控制的存在性,得到了控制为最优的一阶必要条件及其由偏微分方程组和变分不等式组成的最优性组,并进而建立了最优生育率控制的反馈表达式。这些结果可为种群扩散系统中的最优控制问题的实际研究提供理论基础。     

Optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow

This article considers a dynamic optimization problem arising in a one-dimensional magnetohydrodynamic flow, which can be modelled by coupled partial differential equations, where the external control input (external induction of magnetic field) takes a multiplicative effect on the system states (momentum and magnetic components). The aim is to drive the flow velocity to within close proximity of a desired target of flow velocity at the pre-indicated terminal time. First, the Galerkin method is utilized to reduce the original dynamic optimization problem to a lower finite-dimensional dynamic optimization problem governed by a set of ordinary differential equations. Then the control parameterization method is employed to parameterize the finite-dimensional optimization problem and thus obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques, such as sequential quadratic programming. The exact gradients of the cost functional with respect to the decision parameters, which are the key advantages of this approach, are computed using the analytical equations. Finally, numerical examples are illustrated to verify the effectiveness of the proposed method. The novel scheme of this article is to develop an effective computational optimal control method for realizing the optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow.     

Variational stability of infinite-dimensional optimal control problems

In this paper we examine the variational stability of infinite-dimensional optimal control problems governed by nonlinear evolution equations. Our tools are the Kuratowski—Mosco convergence of sets and the corresponding τ-convergence of functions. We prove the τ-convergence of cost functional and the convergence of the values of the problems, and we examine the variational stability of the solution and reachable sets. These results are then applied to a sequence of nonlinear parabolic distributed-parameter optimal control problems     

Plane interior boundary-value problems in microcontinuum fluid mechanics

This paper examines the fundamental question of existence and uniqueness of solutions of two plane interior boundary-value problems encountered in the field of microcontinuum fluid mechanics. These problems are analyzed using potential theoretic methods and with the aid of singular integral equations. The results obtained have their counterparts in the classical Navier-Stokes theory.     

Superconvergence of Fully Discrete Finite Elements for Parabolic Control Problems with Integral Constraints

A quadratic optimal control problem governed by parabolic equations with integral constraints is considered. A fully discrete finite element scheme is constructed for the optimal control problem, with finite elements for the spatial but the backward Euler method for the time discretisation. Some superconvergence results of the control, the state and the adjoint state are proved. Some numerical examples are performed to confirm theoretical results.     

一类非线性脉冲发展方程的最优反馈控制

本文考虑受控系统为Banach空间中一类G0紧半群的非线性脉冲积微分方程的最优反馈控制问题。首先证明容许控制对的存在性,进一步对Lagrange问题给出了一个新的存在性定理。     

A note on the existence and uniqueness of solutions of the micropolar fluid equations

We show that existence and uniqueness theorems, known in the theory of the Navier-Stokes equations, are valid for the incompressible micropolar equations too.     

Beam analogy for optimal control of linear dynamic systems

 Optimal control problems for linear dynamic systems with quadratic performance index are solved using the beam analogy. The governing equations for the optimal maneuver are derived in the form of coupled fourth order differential equations in the time domain. These equations are uncoupled using modal variables. Next, each independent equation is made analogous to the corresponding problem of a beam on an elastic foundation. The beam problem in the spatial domain is solved using standard FEM software. Finally the FEM results are transferred back to the time domain where they represent the optimal trajectories and controls for the dynamic system. Received 12 October 1999     

Further study of phantom solutions to the “TSD-Euler” equation

Summary It is known that the nonlinear Navier-Stokes equations will model most fluid flow of aeronautical interest. The existence and uniqueness of the solutions to the Navier-Stokes equations have not been proven although it is known that in certain cases only the most stable solution is obtained. This present work is concerned with identifying multiple solutions of the Navier-Stokes equations for transonic flow. The objective is to exploit the existence of these solutions rather than avoid them as has been the custom in the past. The present work has shown that the cause of multiple solutions in potential flow is a bifurcation of solutions at a specific Mach number distribution; airfoils could be designed to give such a distribution. It is also found that the presence of entropy and vorticity does not affect the occurrence of phantom solutions. A physical example of a phantom solution is explained by a study of the potential phantom solutions.     

基于二阶矩阵微分方程的机械振动系统线性二次型调节器设计

本文讨论机械振动系统线性二次型状态调节器（LQR）问题，直接针对系统二阶运动微分方程，性能指标为一个依赖于二阶导数的泛函。由欧拉-拉格朗日方程得出一个系统矩阵增广的二阶线性微分方程，指出该方程稳定的特征对就是最优控制振动系统闭环特征对，并给出求解最优控制状态反馈矩阵的方法，另外，由本文方法还可得出基于速度和加速度反馈的最优控制反馈矩阵。这里不涉及求解代数矩阵Riccati方程。     

非线性分数阶耦合泛函微分方程组边值问题的可解性

由于运动速度是有限的，因此在信号传输等过程中时滞现象往往是不可避免的。分数阶泛函微分方程是研究时滞系统运动规律的重要模型，当系统中具有两个或多个状态变量且这些状态变量相互作用时，常常运用耦合微分方程组来刻画。对一类具有 Riemann-Liouville 分数阶导数的非线性时滞耦合泛函微分方程组边值问题正解的存在唯一性进行了研究。首先，根据方程与边界条件的特点，建立了比较定理，构造了上解与下解的单调序列，并确定了上下解的关系。运用上下解的方法建立并证明了边值问题正解的存在性定理，同时得到了正解的取值范围。然后，利用迭代技术建立并证明了边值问题正解的存在唯一性定理。最后，给出了具体例子用于说明所得主要结论的适应性与广泛性。     

含控制时滞的海洋平台前馈反馈最优跟踪控制

通过引入状态变换,将含控制时滞的海洋平台系统转化为无时滞系统.然后,通过引入渐近稳定的期望系统,基于波浪力线性外系统近似模型,给出了海洋平台系统基于二次型性能指标泛函的前馈反馈最优跟踪控制器设计方法及其控制器的存在唯一性条件.控制器的前馈增益矩阵和反馈增益矩阵可分别经由求解代数Riccati方程和Sylvester方程得到.仿真算例表明了方法的有效性,且在减小平台的振动幅值和控制力方面,前馈反馈最优跟踪控制方法优于前馈反馈最优控制方法.     

Asymptotic behaviour of monotone multi-valued dynamical systems

We introduce the concept of a monotone multi-valued semi-flow as an order-preserving map. This definition is motivated by the applications in the theory of differential equations without uniqueness of solutions. For an order-preserving multi-valued semi-flow we prove several results on the structure of the global attractor. Some applications to models governed by ordinary differential equations and delay equations with continuous right-hand side are presented. In particular, the abstract results are applied to a biochemical control circuit.     

Brinkman-Forchheimer方程的加罚有限元方法

Brinkman-Forchheimer方程(BF方程)是具有强非线性项并满足无散度条件的流动控制方程,其中无散度条件的精确满足对控制方程的数值求解极其重要.为了放松无散度条件的限制,本文采用了加罚方法.为了得到加罚问题解的适定性,首先,利用加罚关系将压力项消去,证明了速度所满足的具有单调性的非线性椭圆变分问题等价于对应能量泛函的极小化问题,从而得到了速度的存在唯一性.进一步,利用LBB条件证明了BF方程加罚问题压力的存在唯一性.其次,证明了BF方程加罚问题的Galerkin变分问题的解关于加罚参数收敛到BF方程的Galerkin变分问题的解.最后,给出了BF方程加罚问题Galerkin变分问题的有限维逼近问题及其解的存在唯一性,并且得出了采用协调有限元离散的误差估计.数值算例表明加罚方法是有效的.     

费用结构一般化的"跳-停"奇异型随机控制

本文针对费用结构由一般性函数构成的情况,对一类带停时的奇异型折扣随机控制问题进行了研究.通过构造并对变分方程的深入分析,采用随机分析的方法证明了其最佳控制的存在性;在某些条件下,得到"跳-停"策略是其最优控制策略;给出了"跳-停"策略存在的条件、最优费用函数、最佳控制的解析式及控制方法.     

挠性结构主动减振中传感器和激振器的优化布置

本文提出了从有效衰减振动能量的角度，通过最小化Ｌｙａｐｕｎｏｖ第二方法中的优化性能指标，对挠性结构振动的主动控制进行优化的方法。研究了在以位移传感器或速度传感器作为检测元件进行部分状态变量反馈的约束条件下，传感器、激振器的最优位置以及最优反馈增益的确定，讨论了在包括截尾残余模态的情况下系统的稳定性。本文还对在两邻边固定、另两邻边自由的边界条件下矩形薄板振动的主动控制进行了优化计算和数字仿真。结果表明，优化的控制系统能够有效地衰减结构的振动。本方法具有明显的物理意义和较高的计算速度，并且通过选取不同的权系数，能够协调控制能量和振动的衰减效果，从而得到合理的位置和增益。     

A quadratic programming approach to image labelling

Image labelling tasks are usually formulated within the framework of discrete Markov random fields where the optimal labels are recovered by extremising a discrete energy function. The authors present an alternative continuous relaxation approach to image labelling, which makes use of a quadratic cost function over the class labels. The cost function to be minimised is convex and its discrete version is equivalent up to a constant additive factor to the target function used in discrete MRF approaches. Moreover, its corresponding Hessian matrix is given by the graph Laplacian of the adjacency matrix. Therefore the optimisation of the cost function is governed by the pairwise interactions between pixels in the local neighbourhood. This leads to a sparse Hessian matrix for which the global minimum of the continuous relaxation problem can be efficiently found by solving a system of linear equations using the Cholesky factorisation. The authors elaborate on the links between the method and other techniques elsewhere in the literature and provide results on synthetic and real-world imagery. The authors also provide a comparison with competing approaches.     

Existence and uniqueness of periodic orbits in the cubic autocatalator

Periodic orbits are sought in a mathematical model of a simple prototype chemical reaction involving essentially only two reacting species. Physically, these periodic orbits correspond to time-periodic oscillations in the concentrations of the two chemicals. Using the results for the existence and uniqueness of periodic orbits for Lienard systems, necessary and sufficient conditions are obtained for the existence of exactly one periodic orbit and of no periodic orbits. The results apply to a closed system where the quadratic autocatalytic reaction and decay step are present but the uncatalysed reaction is not and where there is only one physically relevant equilibrium solution     

刹车系统的摩擦自激振动和控制

研究刹车系统的摩擦自激振动和控制问题。采用LuGre 模型计算摩擦力,建立了两自由度盘式刹车系统的动力学模型。通过平衡点的稳定性分析,给出Hopf 分岔失稳的临界速度。应用基于微分几何法和线性二次型最优控制相结合的方法,设计单输入单输出的非线性系统控制器,以便通过推迟系统的分岔临界速度,减少减速型刹车过程中的摩擦颤振,避免刹车啸叫。最后分析了控制器和系统参数对控制效果的影响。仿真表明,该控制器能有效的抑制刹车系统中的摩擦自激振动。