Dynamic Programming Method for Constrained Discrete-Time Optimal Control

A dynamic programming method is presented for solving constrained, discrete-time, optimal control problems. The method is based on an efficient algorithm for solving the subproblems of sequential quadratic programming. By using an interior-point method to accommodate inequality constraints, a modification of an existing algorithm for equality constrained problems can be used iteratively to solve the subproblems. Two test problems and two application problems are presented. The application examples include a rest-to-rest maneuver of a flexible structure and a constrained brachistochrone problem.     

A Dual Dynamic Programming for Multidimensional Elliptic Optimal Control Problems

This paper deals with the optimal control problems with multiple integrals and an elliptic partial differential equation. The sufficient conditions for optimality in these problems are proved through a dual dynamic programming. The concept of an optimal dual feedback is introduced, and the theorem guaranteeing its existence is established. For the purposes of numerical methods, the ε-version of the verification theorem provided appears to be very useful.     

A Stability Theorem for Constrained Optimal Control Problems

This paper presents the stability of difference approximations of an optimal control problem for a quasilinear parabolic equation with controls in the coefficients, boundary conditions and additional restrictions. The optimal control problem has been convered to one of the optimization problem using a penalty function technique. The difference approximations problem for the considered problem is obtained. The estimations of stability of the solution of difference approximations problem are proved. The stability estimation of the solution of difference approximations problem by the controls is obtained.     

A Multigrid Scheme for Elliptic Constrained Optimal Control Problems

A multigrid scheme for the solution of constrained optimal control problems discretized by finite differences is presented. This scheme is based on a new relaxation procedure that satisfies the given constraints pointwise on the computational grid. In applications, the cases of distributed and boundary control problems with box constraints are considered. The efficient and robust computational performance of the present multigrid scheme allows to investigate bang-bang control problems.AMS Subject Classification: 49J20, 65N06, 65N12, 65N55Supported in part by the SFB 03 “Optimization and Control”     

Conditions and Algorithm for Optimal Solution of Probabilistic Constrained Programming

1.IntroductionAsweknow,theerroralwaysexistsinmeasurementanditisnormallydistributed,soweconsideraprobabilisticconstrainedprogrammingasbelow:WhereAeRTxnisastochaJsticmatrix,A1)A2,...,A.aretherowsofAandeachAinormallydistributedwithmeanmiandcovariancematrixDi,BER"x"isdeterministicmatrix,xisann-dimensionaldecisionvariable,b,c,daredeterministicvectorsofdimensionrtnfmrespectively,pisconstantreliabilitylevel,1相似文献   

一些约束规划问题的近似全局最优解

首先将一个具有多个约束的规划问题转化为一个只有一个约束的规划问题,然后通过利用这个单约束的规划问题,对原来的多约束规划问题提出了一些凸化、凹化的方法,这样这些多约束的规划问题可以被转化为一些凹规划、反凸规划问题．最后,还证明了得到的凹规划和反凸规划的全局最优解就是原问题的近似全局最优解．     

Duality for Set-Valued Multiobjective Optimization Problems. Part 2: Optimal Control

The present paper is a continuation of a paper by Azimov (J. Optim. Theory Appl. 2007, accepted), where we derived duality relations for some general multiobjective optimization problems which include convex programming and optimal control problems. As a consequence, we established duality results for multiobjective convex programming problems. In the present paper (Part 2), based on Theorem 3.2 of Azimov (J. Optim. Theory Appl. 2007, accepted), we establish duality results for several classes of multiobjective optimal control problems.     

Existence of Neighboring Feasible Trajectories: Applications to Dynamic Programming for State-Constrained Optimal Control Problems

In this paper, the value function for an optimal control problem with endpoint and state constraints is characterized as the unique lower semicontinuous generalized solution of the Hamilton-Jacobi equation. This is achieved under a constraint qualification (CQ) concerning the interaction of the state and dynamic constraints. The novelty of the results reported here is partly the nature of (CQ) and partly the proof techniques employed, which are based on new estimates of the distance of the set of state trajectories satisfying a state constraint from a given trajectory which violates the constraint.     

Computational Sensitivity Analysis for State Constrained Optimal Control Problems

A theoretical sensitivity analysis for parametric optimal control problems subject to pure state constraints has recently been elaborated in [7,8]. The articles consider both first and higher order state constraints and develop conditions for solution differentiability of optimal solutions with respect to parameters. In this paper, we treat the numerical aspects of computing sensitivity differentials via appropriate boundary value problems. In particular, numerical methods are proposed that allow to verify all assumptions underlying solution differentiability. Three numerical examples with state constraints of order one, two and four are discussed in detail.     

A Dynamic Programming Approach for Approximate Optimal Control for Cancer Therapy

In the last 15 years, tumor anti-angiogenesis became an active area of research in medicine and also in mathematical biology, and several models of dynamics and optimal controls of angiogenesis have been described. We use the Hamilton–Jacobi approach to study the numerical analysis of approximate optimal solutions to some of those models earlier analysed from the point of necessary optimality conditions in the series of papers by Ledzewicz and Schaettler.     

不等式约束优化全局收敛的滤子线性方程组算法

设计了求解不等式约束非线性规划问题的一种新的滤子序列线性方程组算法,该算法每步迭代由减小约束违反度和目标函数值两部分构成.利用约束函数在某个中介点线性化的方法产生搜索方向.每步迭代仅需求解两个线性方程组,计算量较小.在一般条件下,证明了算法产生的无穷迭代点列所有聚点都是可行点并且所有聚点都是所求解问题的KKT点.     

Optimal Control of a Constrained Bilinear Dynamic System

In this paper, an optimal feedback, for a free vibrating semi-active controlled plant, is derived. The problem is represented as a constrained optimal control problem of a single input, free vibrating bilinear system, and a quadratic performance index. It is solved by using Krotov’s method and to this end, a novel sequence of Krotov functions that suits the addressed problem, is derived. The solution is arranged as an algorithm, which requires solving the states equation and a differential Lyapunov equation in each iteration. An outline of the proof for the algorithm convergence is provided. Emphasis is given on semi-active control design for stable free vibrating plants with a single control input. It is shown that a control force, derived by the proposed technique, obeys the physical constraint related with semi-active actuator force without the need of any arbitrary signal clipping. The control efficiency is demonstrated with a numerical example.     

Simple Sequential Quadratically Constrained Quadratic Programming Feasible Algorithm with Active Identification Sets for Constrained Minimax Problems

In this paper, the nonlinear minimax problems with inequality constraints are discussed. Based on the idea of simple sequential quadratically constrained quadratic programming algorithm for smooth constrained optimization, an alternative algorithm for solving the discussed problems is proposed. Unlike the previous work, at each iteration, a feasible direction of descent called main search direction is obtained by solving only one subprogram which is composed of a convex quadratic objective function and simple quadratic inequality constraints without the second derivatives of the constrained functions. Then a high-order correction direction used to avoid the Maratos effect is computed by updating the main search direction with a system of linear equations. The proposed algorithm possesses global convergence under weak Mangasarian–Fromovitz constraint qualification and superlinear convergence under suitable conditions with the upper-level strict complementarity. At last, some preliminary numerical results are reported.     

A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems

We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.     

Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation, namely, a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second-order optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second-order sufficient condition holds, we give a formula for the second-order expansion of the value of the problem as well as the directional derivative of the optimal control, when the cost function is perturbed. Then we extend the theory of second-order optimality conditions to the case of vector-valued controls when the feasible set is defined by local and smooth convex constraints. When the space dimension n is greater than 3, the results are based on a two norms approach, involving spaces L ² and L ^s , with s>n/2 . Accepted 27 January 1997     

Optimal Battery Aging: An Adaptive Weights Dynamic Programming Algorithm

We present an algorithm to handle the optimization over a long horizon of an electric microgrid including a battery energy storage system. While the battery is an important and costly component of the microgrid, its aging process is often not taken into account by the energy management system, mostly because of modeling and computing challenges. We address the computing aspect by a new approach combining dynamic programming, decomposition and relaxation techniques. We illustrate this adaptive weight’ method with numerical simulations for a toy microgrid model. Compared to a straightforward resolution by dynamic programming, our algorithm decreases the computing time by more than one order of magnitude, can be parallelized, and allows for online implementations. We believe that this approach can be used for other applications presenting fast and slow variables.     

A Shooting Algorithm for Optimal Control Problems with Singular Arcs

In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss–Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method.     

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

In this work, we present an adaptive Newton-type method to solve nonlinear constrained optimization problems, in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first-order optimality conditions by the Newton-type method. This strategy allows one to balance the two errors and to derive effective stopping criteria for the Newton iterations. The algorithm proceeds with the search of the optimal point on coarse grids, which are refined only if the discretization error becomes dominant. Using computable error indicators, the mesh is refined locally leading to a highly efficient solution process. The performance of the algorithm is shown with several examples and in particular with an application in the neurosciences: the optimal electrode design for the study of neuronal networks.