Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.This work was supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.The author is thankful to Professor L. Cesari for many helpful suggestions and also for calling his attention to the recent papers of Professor K. Narukawa.     

Numerical approximation of nonconvex optimal control problems defined by parabolic equations

In this paper, we consider an optimal control problem for distributed systems governed by parabolic equations. The state equations are nonlinear in the control variable; the constraints and the cost functional are generally nonconvex. Relaxed controls are used to prove existence and derive necessary conditions for optimality. To compute optimal controls, a descent method is applied to the resulting relaxed problem. A numerical method is also given for approximating a special class of relaxed controls, notably those obtained by the descent method. Convergence proofs are given for both methods, and a numerical example is provided.     

A class of stochastic games with ordered field property

It is shown that discounted general-sum stochastic games with two players, two states, and one player controlling the rewards have the ordered field property. For the zero-sum case, this result implies that, when starting with rational data, also the value is rational and that the extreme optimal stationary strategies are composed of rational components.     

A vindication of centralized controls in inventory management

We study here a set of quasi-variational inequalities related to inventory/production stochastic problems. We mainly focus our attention on two subjects: (i) From a theoretical point of view, we compare the advantages of global controls versus a decentralized approach via a model of an inventory serial system with Gaussian demand. (ii) We consider discretized systems, we solve the simple model of (i), and we apply a similar technique for solving a more complex system with Poissonian demand. The centralized approach naturally leads to large-scale problems; we solve them using a fast algorithm of resolution with very good performances. We conclude with some numerical results.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 1: Problems with No State Constraints

This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints, we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive Pontryagin's principles for problems with pointwise state constraints. Accepted 12 July 2001. Online publication 21 December 2001.     

Penalty technique for a min-max control problem

Using relaxed controls, optimality conditions are derived for a control problem where the cost ess supG(x,u,t) is minimized. In reformulating the problem, we impose a unilateral constraint, which we deal with by a penalty method. Associated with the unilateral constraint, we obtain a multiplier which is related to a measuredp. This measure is a positive Radon measure ifG is continuous in all its arguments and is a finitely additive positive Radon measure ifG is only measurable in thet-variable. We have indicated how the results of Girsanov follow from our results and how our results are related to Holmaker's.     

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Sorin Micu ^{This paper studies the numerical approximation of the boundarycontrol for the wave equation in a square domain. It is knownthat the discrete and semi-discrete models obtained by discretizingthe wave equation with the usual finite-difference or finite-elementmethods do not provide convergent sequences of approximationsto the boundary control of the continuous wave equation as themesh size goes to zero. Here, we introduce and analyse a newsemi-discrete model based on the space discretization of thewave equation using a mixed finite-element method with two differentbasis functions for the position and velocity. The main theoreticalresult is a uniform observability inequality which allows usto construct a sequence of approximations converging to theminimal L2-norm control of the continuous wave equation. Wealso introduce a fully discrete system, obtained from our semi-discretescheme, for which we conjecture that it provides a convergentsequence of discrete approximations as both h and t, the timediscretization parameter, go to zero. We illustrate this factwith several numerical experiments.}     

基于无线网络的塔吊群防碰撞系统的研究

根据无线网络的特点与塔吊群协同作业的安全要求,研究了工业环境下无线网络的实时性和可靠性,成功研制了塔吊群防碰撞系统,预防塔吊间的碰撞问题.     

Computational water purification controls using second‐order adjoint equations with stability confirmation

This paper presents a computational method for water purification using second‐order adjoint equations. In Japan, the waters of polluted rivers are purified by conveying the waters from other rivers into the main rivers or by using outflows from sewage plants. The shallow water flow equation based on the water velocity and elevation and the advection diffusion equation of COD concentration are governing equations. The control problem involves finding a flow velocity into the main river that can reduce the COD concentration as close to the target value as possible. In other words, the problem is to find a water velocity to minimize the performance function, which is the square sum of the discrepancy between the computed and the observed COD concentrations. The present research was motivated by the need to apply water purification controls to practical projects. We have found that the controls occasionally tend to be unstable, and the stability of control must be ensured. By expanding the extended performance function into the Taylor series, the necessary condition for the stationary state is derived. Based on this condition, the first‐and second‐order adjoint equations can be obtained. The backward solution of the adjoint equation leads to the gradient and the Hessian product; these serve as the basis of the quasi‐Newton method. From the condition that the performance function must be minimum, the stability confirmation index can be determined. Using this index, we have derived the trust region method, the computation of which confirms the stability of control. Verification was carried out using a simple channel model. By varying the peak value of the inflow velocity, the outlet velocity has been determined such that the water elevation at the target point is zero. Depending on the peak value of the inflow, unstable control arises; this is determined by the stability confirmation index presented in this paper. The trust region method with the stability confirmation index is shown to be adaptable to judge the stability of control. The present method was applied to the water purification of Teganuma river in Japan. The steady fundamental state was computed with the inflow, outflow, and COD concentration at the inlet being specified. The control velocity at the control point can be determined for a fixed control duration with and without the stability confirmation index. The inflow, outflow, and COD concentration are specified as functions of time. It is shown that this method is suitable for practical use because control stability can be ensured. Moreover, it is also noted that the maximum flow velocity for stable control depending on the given control duration can be obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence of almost periodic solution in a ratio-dependent Leslie system with feedback controls

A ratio-dependent Leslie system with feedback controls is studied. By using a comparison theorem and constructing a suitable Lyapunov function, some sufficient conditions for the existence of a unique almost periodic solution (periodic solution) and the global attractivity of the solutions are obtained. Examples show that the obtained criteria are new, general, and easily verifiable.