A class of optimal state-delay control problems

We consider a general nonlinear time-delay system with state-delays as control variables. The problem of determining optimal values for the state-delays to minimize overall system cost is a non-standard optimal control problem–called an optimal state-delay control problem–that cannot be solved using existing optimal control techniques. We show that this optimal control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables. We then develop an efficient numerical method for determining the cost function’s gradient. This method, which involves integrating an auxiliary impulsive system backwards in time, can be combined with any standard gradient-based optimization method to solve the optimal state-delay control problem effectively. We conclude the paper by discussing applications of our approach to parameter identification and delayed feedback control.     

Control of substance solidification in a complex-geometry mold

The control of metal solidification in a mold of complex geometry is studied. The underlying mathematical model is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the optimal control problem for the solidification process is presented. This problem was solved numerically using gradient optimization methods. The gradient of the cost function was computed by applying the fast automatic differentiation technique, which yields the exact value of the cost function gradient for the chosen discrete version of the optimal control problem. The results of the study are described and analyzed. Some of the results are illustrated as plots.     

Investigation of the optimal control problem for metal solidification in a new formulation

New formulations of the optimal control problem for metal solidification in a furnace are proposed and studied. The underlying mathematical model of the process is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The formulated problems are solved numerically with the help of gradient optimization methods. The gradient of the cost function is computed by applying the fast automatic differentiation technique, which yields the exact value of the cost function gradient for a chosen discrete version of the optimal control problem. The research results are described and analyzed. Some of the results are illustrated.     

带阀点效应水火联合调度问题的一种半定凸松弛求解法

水火联合调度问题是电力系统中一类复杂的优化问题。合理安排调度周期内的水火电出力,确定一个最优发电计划,可以带来巨大的经济效益。在实际系统中,汽轮机调汽阀开启时出现的拔丝现象会使机组耗量特性产生阀点效应。忽略阀点效应,在一定程度上降低求解的精度。本文考虑带阀点效应的水火联合调度问题。该问题非凸非光滑,且带有非线性约束,直接使用确定性全局优化方法求解是相当困难的。本文使用高效的半定规划求解此问题。首先用耗量特性函数的初始周期代替其余有限的周期,并对其进行二次拉格朗日插值拟合。再通过引进0-1变量,得到整个耗量特性函数的近似,进而把问题松弛为半定规划模型。最后,采用凸规划应用软件包CVX求解一个仿真算例,得到一个近似全局最优解。     

Choosing Optimal Parameters in Iterative Processes

We consider the problem of choosing optimal parameters in certain iterative procedures. Specifically, we are interested in finite-step processes for which it is possible to estimate the computational cost and the error relaxation in terms of the process parameters. The problem of finding the optimal process that provides the required error relaxation with a minimal total computational cost is defined and studied. To solve the problem, it is generally necessary to solve a series of mathematical programming problems with rapidly increasing dimension. We suggest two ways to avoid that difficulty. The first is to find a process that is close to the optimal process, by solving only one mathematical programming problem. The second is to define optimal processes in some special cases when this problem can be simplified. We define conditions under which processes with geometrically decreasing error are optimal or asymptotically optimal. The methods of finding parameters of such processes are also provided. We illustrate our ideas with two examples: the bilevel gradient method for unconstrained function minimization and the iterative process for solving an optimal design problem.     

Optimal synchronization of two different in‐commensurate fractional‐order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional‐order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional‐order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. © 2016 Wiley Periodicals, Inc. Complexity 21: 401–416, 2016     

Über Normminimale Steuerungen von L-Prozessen und Lineare Optimierung bei Normbeschränkten Steuerungen

This paper is devoted to two problems in the theory of optimal control for linear processes. The first one is characterized by a cost of the form ess sup {p(u(t)):t∈[a, b]}, whereby p denotes the distance function of a compact convex set C ° ?^m containing the origin as an interior point and u:[a, b] → ?^m represents the control. In the second problem the cost depends linear on the controls, which are limited by a bound for ess sup {p(u(t)):t∈[a, b]}. There will be proved two duality theorems leading to a method for the construction of optimal controls in the case of a strict convex C. For linear processes defined by piecewise analytic functions these controls are piecewise continuous.     

Control of phase boundary evolution in metal solidification for new thermodynamic parameters of the metal

The problem of controlling the phase boundary evolution in the course of solidification of metals with different thermodynamic properties is studied. The underlying mathematical model of the process is based on a three-dimensional nonstationary two-phase initial–boundary value problem of the Stefan type. The control functions are determined by optimal control problems, which are solved numerically with the help of gradient optimization methods. The gradient of the cost function is exactly computed by applying the fast automatic differentiation technique. The research results are described and analyzed. Some of them are illustrated.     

Computational method for optimal control of switched systems with input and state constraints

In this paper, we consider an optimal control problem of switched systems with input and state constraints. Since the complexity of such constraint and switching laws, it is difficult to solve the problem using standard optimization techniques. In addition, although conjugate gradient algorithms are very useful for solving nonlinear optimization problem, in practical implementations, the existing Wolfe condition may never be satisfied due to the existence of numerical errors. And the mode insertion technique only leads to suboptimal solutions, due to only certain mode insertions being considered. Thus, based on an improved conjugate gradient algorithm and a discrete filled function method, an improved bi-level algorithm is proposed to solve this optimization problem. Convergence results indicate that the proposed algorithm is globally convergent. Three numerical examples are solved to illustrate the proposed algorithm converges faster and yields a better cost function value than existing bi-level algorithms.     

A computational method for free terminal time optimal control problem governed by nonlinear time delayed systems

This paper considers a free terminal time optimal control problem governed by nonlinear time delayed system, where both the terminal time and the control are required to be determined such that a cost function is minimized subject to continuous inequality state constraints. To solve this free terminal time optimal control problem, the control parameterization technique is applied to approximate the control function as a piecewise constant control function, where both the heights and the switching times are regarded as decision variables. In this way, the free terminal time optimal control problem is approximated as a sequence of optimal parameter selection problems governed by nonlinear time delayed systems, each of which can be viewed as a nonlinear optimization problem. Then, a fully informed particle swarm optimization method is adopted to solve the approximate problem. Finally, two free terminal time optimal control problems, including an optimal fishery control problem, are solved by using the proposed method so as to demonstrate its applicability.     

Evaluating Gradients in Optimal Control: Continuous Adjoints Versus Automatic Differentiation

This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim to make available to the optimization software not only the discrete objective functional, but also its gradient. The objective gradient can be computed either from forward (sensitivity) information or backward (adjoint) information. The purpose of this paper is to discuss various ways of adjoint computation. It will be shown both theoretically and numerically that methods based on the continuous adjoint equation require a careful choice of both the integrator and gradient assembly formulas in order to obtain a gradient consistent with the discretized control problem. Particular attention is given to automatic differentiation techniques which generate automatically a suitable integrator.     

Three-term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping

Many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain nonexpansive mapping, and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set. This paper presents a novel gradient method with a three-term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems. It is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions. Numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm.     

A projected Newton method in a Cartesian product of balls

We formulate a locally superlinearly convergent projected Newton method for constrained minimization in a Cartesian product of balls. For discrete-time,N-stage, input-constrained optimal control problems with Bolza objective functions, we then show how the required scaled tangential component of the objective function gradient can be approximated efficiently with a differential dynamic programming scheme; the computational cost and the storage requirements for the resulting modified projected Newton algorithm increase linearly with the number of stages. In calculations performed for a specific control problem with 10 stages, the modified projected Newton algorithm is shown to be one to two orders of magnitude more efficient than a standard unscaled projected gradient method.This work was supported by the National Science Foundation, Grant No. DMS-85-03746.     

A COMPUTATIONAL APPROACH FOR OPTIMAL CONTROL SYSTEMS GOVERNED BY PARABOLIC VARIATIONAL INEQUALITIES

Iterative techniques for solving optimal control systems governed by parabolic variational inequalities are presented. The techniques we use are based on linear finite elements method to approximate the state equations and nonlinear conjugate gradient methods to solve the discrete optimal control problem. Convergence results and numerical experiments are presented.     

Conjugate gradient techniques for the optimal control evolution dam problem

This paper considers the numerical simulation of optimal control evolution dam problem by using conjugate gradient method.The paper considers the free boundary value problem related to time dependent fluid flow in a homogeneous earth rectangular dam.The dam is taken to be sufficiently long that the flow is considered to be two dimensional.On the left and right walls of the dam there is a reservoir of fluid at a level dependent on time.This problem can be transformed into a variational inequality on a fixed domain.The numerical techniques we use are based on a linear finite element method to approximate the state equations and a conjugate gradient algorithm to solve the discrete optimal control problem.This algorithm is based on Armijo's rule in the unconstrained optimization theory.The convergence of the discrete optimal solutions to the continuous optimal solutions,and the convergence of the conjugate gradient algorithm are proved.A numerical example is given to determine the location of the minimum surface     

An exact penalty method for solving optimal control problems

This paper discusses an algorithm for solving optimal control problems. An optimal control problem is presented where the final time is unknown. The algorithm consists of an integrator and a minimizer; the latter is an exact penalty function used to solve constrained nonlinear programming problems. Essentially, the optimal control problem is converted to a mathematical programming problem such that a point satisfying the differential equations via the integrator is provided to the minimizer, a lower performance index is obtained, the integrator is reinitiated, etc., until a suitable stopping criterion is satisfied.     

Design Sensitivity for a Hyperelastic Rod in Large Displacement with Respect to Its Midcurve Shape

The paper is concerned with an optimal design problem for a hyperelastic rod. The function describing the position of a point at the line of rod cross-section centroids in its reference configuration is the variable subject to optimization. The necessary optimality condition is formulated. The continuation method combined with the gradient descent algorithm are employed to solve this problem numerically. Numerical results are provided.     

On the optimal control of the Schlögl-model

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation. The well-posedness of the optimal control problem and the regularity of its solution are proved. First-order necessary optimality conditions are established by standard adjoint calculus. The state equation is solved by the implicit Euler method in time and a finite element technique with respect to the spatial variable. Moreover, model reduction by Proper Orthogonal Decomposition is applied and compared with the numerical solution of the full problem. To solve the optimal control problems numerically, the performance of different versions of the nonlinear conjugate gradient method is studied. Various numerical examples demonstrate the capacities and limits of optimal control methods.     

动态投入产出灰色最优控制模型

根据灰色系统理论,建立了动态投入产出问题的灰色最优控制模型.利用灰集合理论,把灰色最优控制问题转化为以隶属度为目标函数的(非灰色的)非线性规划问题,从而可利用非线性规划的方法求解这个灰色最优控制问题.