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.     

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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms \(\alpha _{1}u_{1}(t)+\beta _{1}u_{1}(t-\tau )+{\int }_{-\tau }^{0}g_{1}(\eta )u_{1} (t+\eta )d\eta \) and \(\alpha _{2}u_{2}(t)+\beta _{2}u_{2}(t-\tau ) +{\int }_{-\tau }^{0}g_{2}(\eta )u_{2}(t+\eta )d\eta \) respectively, where u ₁(t) and u ₂(t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially.     

Stochastic Control of a Class of Dynamical Systems via Path Limits

Journal of Dynamical and Control Systems - Some limit theorems are derived for a class of controled Markov systems with small noises. The aim is to understand the effects of strategies of actions...     

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.      

A Multivariable Optimal Control System for a Generator

A new digital generator control system has been developed, which coordinates excitation and governing systems of a generator by using multivariable optimal control scheme on the basis of modern control theory. Control performance of the developed system was investigated with computer simulations and tested at a 200KVA pilot power plant with 3KV transmission lines. This paper first outlines the control system, next describes the control schemes including process models, optimal control algorithms, control sequence, and state estimation schemes, and finally shows its control performance obtained from the pilot plant tests and the computer simulations.     

Existence and Uniqueness of Optimal Control for a Distributed-Parameter Bilinear System

In this paper, we study an optimal control problem of bilinear type. The system is governed by a fourth-order parabolic operator. The performance index is of the form J(u) = ₀ ^T L(z(t), u(t)) dt. Under suitable hypotheses, it is shown that there exists an optimal control and it satisfies an appropriate optimality system. Further, for a small initial state is unique.     

Stabilization of Euler-Bernoulli Beam Equation with Boundary Moment Control and Disturbance by Active Disturbance Rejection Control and Sliding Mode Control Approaches

We consider boundary stabilization for a one-dimensional Euler-Bernoulli equation with boundary moment control and disturbance. The active disturbance rejection control (ADRC) and sliding mode control (SMC) approaches are adopted in investigation. By the ADRC approach, a state feedback disturbance estimator with time-varying gain is designed to estimate the disturbance. It is shown that the closed-loop system is asymptotically stable by canceling the disturbance in the feedback loop with its online estimation. In the second part, the SMC is applied to reject the disturbance. The well-posedness of the closed-loop system via SMC is proven, and the monotonicity of the “reaching condition” is presented without differentiating the sliding mode function which may not always exist for the weak solution. The numerical experiments are presented to illustrate the convergence and the peaking value reduction caused by the constant high gain. In addition, the control energies are compared numerically for two approaches.     

Optimal Control with State Constraints and the Space Shuttle Re-entry Problem

In this article, we initialize the analysis under generic assumptions of the small time optimal synthesis for single input systems with state constraints. We use geometric methods to evaluate the small time reachable set and necessary optimality conditions. Our work is motivated by the optimal control of the atmospheric arc for the re-entry of a space shuttle, where the vehicle is subject to constraints on the thermal flux and on the normal acceleration. A multiple shooting techniqueis finally applied to the computation of the optimal longitudinal arc.