Optimal Control of Elliptic Differential Inclusions with Dirichlet and Neumann Boundary Conditions

In this paper, we consider optimization Dirichlet and Neumann problems for differential inclusions in which the right-hand sides are governed by multivalued function (mapping), which depends not only of the unknown functions, but also on the first partial derivatives of these functions. This generalization is very important, and the results obtained cannot be deduced from the results of the first author considered earlier. Formulations of sufficient conditions are based on the discretization idea of the continuous problem and equivalence theorems. Thus in the form of the Euler–Lagrange inclusion, sufficient optimality conditions are derived; for this, locally adjoint mappings are used. In general, we establish necessary and sufficient conditions for the so-called discrete approximation problem on a uniform grid. These conditions take an intermediate place between discrete and continuous problems. The results are generalized to the multidimensional case with a second-order elliptic operator.     

基于时滞微分方程的过水系统描述及其水击计算方法

针对解析法和数值分析法均难以精确建立过水系统连续模型的问题,在水击基本方程组的基础上进行变形求解,分析并推导出基于时滞微分方程的过水系统描述理论。基于该理论建立复杂过水系统的时滞微分方程组,研究了其离散求解方法并证明了解的存在性和唯一性。结合实际电站进行仿真计算并与特征线法的计算结果进行比较,验证了所提方法的适用性与优越性,为精确建立水力发电机组连续模型及研究控制策略奠定了基础。     

考虑局部阴影的光伏阵列MPPT优化控制策略

摘要： 针对局部阴影下光伏阵列出现多峰值输出的情况,提出一种优化粒子群算法与模糊控制扰动观察法相结合的最大功率点追踪方法。利用优化粒子群算法在全局范围内寻找极值点及最大功率点;通过设置语言变量、定义模糊子集及模糊控制推理,进行跟踪调节和稳定控制,实现光伏系统最大功率输出。提高了搜寻精度,缩短了搜寻时间,避免了在最大功率点处的震荡。仿真结果验证了所提算法的有效性和合理性。     

Controllability of the Impulsive Semilinear Heat Equation with Memory and Delay

For many control systems in real life, impulses and delays are intrinsic properties that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not modify certain properties such as controllability. In other words, the controllability is robust by looking the impulses and delays as perturbations. In this regard, in this paper, we prove the interior approximate controllability of the Impulsive Semilinear Heat Equation with Memory and Delay. This is done by employing A.E. Bashirov and at. technique which avoids fixed point theorems. In this case, the delay helps us to prove the approximate controllability by pulling back the control solution to a fixed curve in a short time interval.     

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.     

Infinite Horizon Optimal Control Problem for Stochastic Evolution Equations in Hilbert Spaces

The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated infinite horizon backward stochastic differential equations with quadratic growth. As an application, an optimal control of stochastic heat equation is also given.     

Nonoscillation and Exponential Stability of Delay Differential Equations with Oscillating Coefficients

We prove that under some additional conditions, the nonoscillation of the scalar delay differential equation

随机时滞电力系统稳定性分析

以励磁系统输入信号延时作为时滞环节,并通过引入负荷扰动项,建立单机无穷大随机时滞系统的数学模型;仿真获得随机时滞电力系统的受扰轨迹,分析不同时滞、不同扰动强度下随机时滞电力系统的稳定性;负荷模型分别采用恒阻抗、恒功率、恒电流模型,分析比较采用不同负荷模型对系统稳定性的影响,分析表明3种负荷模型所能承受的时滞上限能力从大到小依次为:恒阻抗模型、恒电流模型、恒功率模型。     

Null Controllability with Constraints on the State for Stochastic Heat Equation

This paper is concerned with a null controllability problem for stochastic heat equation with finite number of constraints on the state. First, we transform the problem into a null controllability problem with constraint on the control. Then, we prove an adapted Carleman estimate. Applying this estimate, we establish a bilinear form and obtain the main result.     

Observability of Singularly Perturbed Linear Time-Dependent Differential Systems with Small Delay

A singularly perturbed linear time-dependent differential system with multiple point and distributed time-delays in state variables is considered. Values of the delays are of order of a small positive parameter multiplying a part of derivatives in the system. Two much simpler systems (the reduced-order and the boundary-layer ones), associated with the original system, are introduced. These systems are in separate time scales, and they do not contain the small parameter any more. Connections between properties of observability of these systems and such a property of the original system itself, valid for all sufficiently small values of the parameter of singular perturbations, are established. These results are obtained for both, standard and nonstandard, types of the original system, and for two, Euclidean space and state-space, types of the observability.     

A Differential Quadrature Solution of Natural Convection in an Enclosure with a Partial Partition

Natural convection in a partially divided enclosure has been examined numerically using a differential quadrature method. Governing equations for the flow and heat transfer have been constructed by the vorticity-stream function formulation and computational results have been obtained for the Rayleigh numbers, 10⁴, 10⁵, 10⁶, the locations of the partition, 0.2, 0.4, 0.6, 0.8, and the partition ratios, 0.25, 0.50, 0.75, 1.0. The results show that circulation strength, and therefore heat transfer, decreases considerably with increasing partition height especially for the higher values of the Rayleigh number. As the distance of the partition from the hot wall increases, first a decrease and then an increase is observed in the average Nusselt number for the low Rayleigh numbers and first an increase and then a decrease turns out for the high Rayleigh numbers. As for the average Nusselt number on the partition, first it shows a decreasing trend and then an increasing trend as the partition is distanced from the hot wall towards the cold wall. The increase in the partition height brings about significant increase in the average Nusselt number on the partition.     

Hypergeometric Functions and Parabolic Partial Differential Equations

We investigate the summability of formal power series solutions of parabolic linear partial differential equations with exponential coefficients.2000 Mathematics Subject Classification. 35C10, 35C20.This research was supported by a Marie Curie Fellowship of the European Community program Improving Human Research Potential under contract number HPMF-CT-2002-01818.     

Index Criteria for Differential Algebraic Equations Arising from Linear-Quadratic Optimal Control Problems

The index of DAE systems arising from linear quadratic optimal control problems is considered. Necessary and sufficient conditions ensuring regularity with tractability index one are proved. Then, it is shown that if the control problem DAE is regular with index one and if the leading term of the DAE to be controlled is given by one full-column-rank and one full-row-rank matrix, then it has a Hamiltonian inherent explicit ODE. For problems with regular index zero or index one DAEs to be controlled, the DAE of the control problem is shown to be regular with tractability index one or three, depending on whether the control coefficient R is singular. 2000 Mathematics Subject Classification. 49N10, 49J15, 34A09. Supported by the DFG research center MATHEON in Berlin. This work was partially supported by the Russian Fundamental Research Foundation Project Nos. 02-01-00351 and 06-01-00296.     

Hierarchical Control for the One-dimensional Plate Equation with a Moving Boundary

In this paper, we investigate the controllability for the one-dimensional plate equation in intervals with a moving boundary. This equation models the vertical displacement of a point x at time t in a bar with uniform cross section. We assume the ends of the bar with small and uniform variations. More precisely, we have introduced functions α(t) and β(t) modeling the motion of these ends. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.     

On Expansion of a Solution of General Nonautonomous Polynomial Differential Equation

We give a recursive formula for an expansion of a solution of a general nonautonomous polynomial differential equation. The formula is given on the algebraic level with the use of a shuffle product. This approach minimizes the number of integrations on each order of expansion. Using combinatorics of trees, we estimate the radius of convergence of the expansion.     

Feedback-Invariant Optimal Control Theory and Differential Geometry, II. Jacobi Curves for Singular Extremals

This is the second article in the series that began in [4]. Jacobi curves were defined, computed, and studied in that paper for regular extremals of smooth control systems. Here we do the same for singular extremals. The last section contains a feedback classification and normal forms of generic single-input affine in control systems on a 3-dimensional manifold.     

On the Accuracy of a Finite-Difference Method for Parabolic Partial Differential Equations with Discontinuous Boundary Conditions

Although the numerical solution of parabolic partial differential equations (PDEs) is widely documented, the effect of discontinuous boundary conditions on numerical accuracy is not. This article employs the Keller box finite-difference method to study the effect of such discontinuities when solving the linear one-demensional transient heat equation. We demonstrate that this formally second-order-accurate scheme can lose accuracy, but that an analytical understanding of the behavior of the solution helps in providing an accuracy-restoring formulation. Benchmark computations are presented that will provide guidance in the numerical solution of nonlinear parabolic PDEs for which there are no closed-form analytical solutions.     

Regularity of an Euler-Bernoulli Equation with Neumann Control and Collocated Observation

This paper studies the regularity of an Euler-Bernoulli plate equation on a bounded domain of ℝⁿ, n ≥ 2, with partial Neumann control and collocated observation. It is shown that the system is not only well posed in the sense of D. Salamon but also regular in the sense of G. Weiss. It is also shown that the corresponding feedthrough operator is zero. 2000 Mathematics Subject Classification. 93C20, 34G10, 93D25, 35L20. This work was supported by the National Natural Science Foundation of China and the National Research Foundation.     

兴利随机优化调度模型及实例应用

与确定型优化调度将入流作为已知系列相比,随机动态规划将入流描述为随机变量,则更具合理性,因此该法是目前求解多库优化调度的最好方法。由于入流的随机描述,极大地增加了计算量,从而经常导致“维数灾”问题,故而又限制了该法在实际中的应用,本文以黄河上游５个梯级水电站群为例,应用随机优化调度方法研究其兴利优化调度问题,通过合理的入流描述,有效地避免了“维数灾”问题。     

On the Summability of Formal Solutions of Nonlinear Partial Differential Equations with Shrinkings

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.