离散双线性系统二次型最优控制的迭代算法

1引言众所周知,某些复杂的工业过程用线性近似动态模型是不适宜的,而且许多对象如工业生产,生态,生物和社会经济等过程可以用双线性模型自然描述,能够在稳态工作点的一个较大领域内描述一大类严重非线性系统的动态特性,描述精度优于传统的线性模型     

Optimal control problems with terminal functionals represented as the difference of two convex functions

Two control problems for a state-linear control system are considered: the minimization of a terminal functional representable as the difference of two convex functions (d.c. functions) and the minimization of a convex terminal functional with a d.c. terminal inequality contraint. Necessary and sufficient global optimality conditions are proved for problems in which the Pontryagin and Bellman maximum principles do not distinguish between locally and globally optimal processes. The efficiency of the approach is illustrated by examples.     

Regularized state-constrained boundary optimal control of the Navier-Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.     

Optimal control of the convective Cahn–Hilliard equation

This article studies the problem for optimal control of the convective Cahn–Hilliard equation in one-space dimension. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.     

Optimal control for systems described by hyperbolic equations with strong nonlinearity

An optimization control problem for a hyperbolic equation is considered. The system is nonlinear with respect to the state derivative. The regularization technique for the state equation is applied. The necessary conditions of optimality for the regularized control problem are proved. It uses the extended differentiability of the control-state mapping for the regularized equation. The convergence of the regularization method is proved. Thus the optimal control for the regularized problem with a small enough regularization parameter can be chosen as an approximate solution of the initial optimization problem.     

Control of Cauchy system for an elliptic operator

The control of a Cauchy system for an elliptic operator seems to be globally an open problem. In this paper, we analyze this problem using a regularization method which consists in viewing a singular problem as a limit of a family of well-posed problems. Following this analysis and assuming that the interior of considered convex is non-empty, we obtain a singular optimality system （S.O.S.） for the considered control problem.     

Optimal control problem of a generalized Ginzburg–Landau model equation in population problems

In this paper, we consider the problem for distributed optimal control of the generalized Ginzburg–Landu model equation in population. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved, and the optimality system is established. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal control of the obstacle problem: optimality conditions

^** E-mail: k.aithadi{at}ucam.ac.ma In this paper, we investigate optimal control problem governedby variational inequality of the obstacle type. Existence ofsolution for the problem is proved and we also show how to obtainoptimality conditions for a penalized problem issued from theoriginal one.     

Dynamics and control of a system of two non‐interacting preys with common predator

In this paper, we consider a Holling type model, which describes the interaction between two preys with a common predator. First, we give some sufficient conditions for the globally asymptotic stability and prove that local stability implies global stability. Then, we present a set of sufficient conditions for the existence of a positive periodic solution with strictly positive components. Finally, the optimal control strategy is developed to minimize the number of predator and maximize the number of preys. We also show the existence of an optimal control for the optimal control problem and derive the optimality system. The technical tool used to determine the optimal strategy is the Pontryagin Maximum Principle. Finally, the numerical simulations of global stability and the optimal problem are given as the conclusion of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite Difference Method for an Optimal Control Problem for a Nonlinear Time-dependent Schrödinger Equation

The finite difference method is applied to an optimal control problem for a system governed by a nonlinear Schrödinger equation with a complex coe?cient. The optimal control problem is discretized by the finite difference method, the error estimate for the finite difference scheme is established and the convergence of difference approximations of the optimal control according to the functional is proved.     

Optimal control problem for the modified Swift–Hohenberg equation in 3D case

In this paper, for 3D modified Swift–Hohenberg equation, the optimal control problem is considered, the existence of optimal solution is proved, and the optimality system is established. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal Control Problems of Phase Relaxation Models

This work is concerned with optimal control problems with convex cost criterion governed by the relaxed Stefan problem with or without memory. The existence of an optimal control is proved and necessary conditions for a given function to be an optimal control are found. Moreover, an asymptotic analysis is performed as the time relaxation parameter tends to zero.     

A convergent simplicial algorithm with ω-subdivision and ω-bisection strategies

The simplicial algorithm is a kind of branch-and-bound method for computing a globally optimal solution of a convex maximization problem. Its convergence under the ω-subdivision strategy was an open question for some decades until Locatelli and Raber proved it (J Optim Theory Appl 107:69–79, 2000). In this paper, we modify their linear programming relaxation and give a different and simpler proof of the convergence. We also develop a new convergent subdivision strategy, and report numerical results of comparing it with existing strategies.     

Quickest search over Brownian channels

In this paper we resolve an open problem proposed by Lai, Vincent Poor, Xin, and Georgiadis [Quickest search over multiple sequences. IEEE Trans. Inf. Theory 57(8) (2011), pp. 5375–5386]. Consider a sequence of Brownian motions with unknown drift equal to one or zero, which may be observed one at a time. We give a procedure for finding, as quickly as possible, a process which is a Brownian motion with non-zero drift. This original quickest search problem, in which the filtration itself is dependent on the observation strategy, is reduced to a single filtration impulse control and optimal stopping problem, which is in turn reduced to an optimal stopping problem for a reflected diffusion, which can be explicitly solved.     

Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid

A stationary convection-diffusion problem with a small parameter multiplying the highest derivative is considered. The problem is discretized on a uniform rectangular grid by the central-difference scheme. A new class of two-step iterative methods for solving this problem is proposed and investigated. The convergence of the methods is proved, optimal iterative methods are chosen, and the rate of convergence is estimated. Numerical results are presented that show the high efficiency of the methods.     

分数阶抛物方程控制系统最优控制的稳定性

本文研究了一类分数阶抛物方程的最优控制问题,主要讨论了其最优控制与最优值的稳定性.利用了凸方法获得了获得了这类问题最优控制的稳定性结果,并且推广了在参考文献[3]中的最优控制稳定性结论.     

一类线性乘积规划问题的分支定界缩减方法

提出了求解一类线性乘积规划问题的分支定界缩减方法, 并证明了算法的收敛性.在这个方法中, 利用两个变量乘积的凸包络技术, 给出了目标函数与约束函数中乘积的下界, 由此确定原问题的一个松弛凸规划, 从而找到原问题全局最优值的下界和可行解. 为了加快所提算法的收敛速度, 使用了超矩形的缩减策略. 数值结果表明所提出的算法是可行的.     

最优控制问题参数化研究—基于勒让德正交多项式逼近

使用勒让德正交多项式逼近方法,将Lagrange型最优控制问题转化为非线性规划问题.采用序列二次规划方法对此非线性规划问题的求解,并对多项式逼近和非线性规划求解后得到的解是否收敛给出了证明和实例分析.     

Simultaneous Optimal Controls for Non-Stationary Stokes Systems

This paper deal with optimal control problems for a non-stationary Stokes system. We study a simultaneous distributed-boundary optimal control problem with distributed observation. We prove the existence and uniqueness of a simultaneous optimal control and we give the first order optimality condition for this problem. We also consider a distributed optimal control problem and a boundary optimal control problem and we obtain estimations between the simultaneous optimal control and the optimal controls of these last ones. Finally, some regularity results are presented.     

A descent algorithm without line search for unconstrained optimization

In this paper, a new descent algorithm for solving unconstrained optimization problem is presented. Its search direction is descent and line search procedure can be avoided except for the first iteration. It is globally convergent under mild conditions. The search direction of the new algorithm is generalized and convergence of corresponding algorithm is also proved. Numerical results show that the algorithm is efficient for given test problems.