An efficient preconditioning method for state box-constrained optimal control problems

2018 ◽  
Vol 26 (4) ◽  
pp. 185-207 ◽  
Author(s):  
Owe Axelsson ◽  
Maya Neytcheva ◽  
Anders Ström
Keyword(s):  
Optimal Control ◽  
Control Variable ◽  
Penalization Method ◽  
Constrained Optimal Control ◽  
State Variable ◽  
Fine Mesh ◽  
Wide Range ◽  
Matrix Blocks ◽  
Preconditioning Method ◽  
Standard Regularization

Abstract An efficient preconditioning technique used earlier for two-by-two block matrix systems with square matrix blocks is shown to be applicable also for a state variable box-constrained optimal control problem. The problem is penalized by a standard regularization term for the control variable and for the box-constraint, using a Moreau–Yosida penalization method. It is shown that there occur very few nonlinear iteration steps and also few iterations to solve the arising linearized equations on the fine mesh. This holds for a wide range of the penalization and discretization parameters. The arising nonlinearity can be handled with a hybrid nonlinear-linear procedure that raises the computational efficiency of the overall solution method.

A Constraining Hyperplane Technique for State Variable Constrained Optimal Control Problems

1973 ◽  
Vol 95 (4) ◽  
pp. 380-389 ◽  
Author(s):  
K. Martensson
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Variable ◽  
Dynamic Programming Algorithm ◽  
Inequality Constraints ◽  
Programming Algorithm ◽  
Control Problems ◽  
State Control ◽  
Constrained Optimal Control ◽  
State Variable

A new approach to the numerical solution of optimal control problems with state-variable inequality constraints is presented. It is shown that the concept of constraining hyperplanes may be used to approximate the original problem with a problem where the constraints are of a mixed state-control variable type. The efficiency and the accuracy of the combination of constraining hyperplanes and a second-order differential dynamic programming algorithm are investigated on problems of different complexity, and comparisons are made with the slack-variable and the penalty-function techniques.

Multi-item Optimal control problem with fuzzy costs and constraints using Fuzzy variational principle

2019 ◽  
Vol 53 (3) ◽  
pp. 1061-1082
Author(s):  
Jotindra Nath Roul ◽  
Kalipada Maity ◽  
Samarjit Kar ◽  
Manoranjan Maiti
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Variational Principles ◽  
Control Variable ◽  
Equality Constraint ◽  
Raw Material ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
State Variable

An imperfect multi-item production system is considered against time dependent demands for a finite time horizon. Here production is defective. Following [Khouja and Mehrez J. Oper. Res. Soc. 45 (1994) 1405–1417], unit production cost depends on production, raw-material and maintenance costs. Produced items have same fixed life-time. Warehouse capacity is limited and used as a constraint. Available space, production, stock and different costs are assumed as crisp or imprecise. With the above considerations, crisp and fuzzy constrained optimal control problems are formulated for the minimization of total cost consisting of raw-material, production and holding costs. These models are solved using conventional and fuzzy variational principles with equality constraint condition and no-stock as end conditions. For the first time, the inequality space constraint is converted into an equality constraint introducing a pseudo state variable following Bang Bang control. [Roul et al., J. Intell. Fuzzy Syst. 32 (2017) 565–577], as stock is mainly controlled by production, for the control problems production is taken as the control variable and stock as state variable. The reduced optimal control problem is solved by generalised reduced gradient method using Lingo-11.0. The models are illustrated numerically. For the fuzzy model, optimum results are obtained as fuzzy numbers expressed by their membership functions. From fuzzy results, crisp results are derived using α-cuts.

scholarly journals Application of optimal control theory on optimal advertising expenditure in monopoly

2020 ◽  
Vol 83 ◽  
pp. 01017
Author(s):  
Nora Grisáková ◽  
Peter Štetka
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Market Share ◽  
Optimal Control Theory ◽  
Control Variable ◽  
New Product ◽  
Advertising Expenditure ◽  
Advertising Rate ◽  
State Variable ◽  
Economic Application

Presented paper is being focused on Optimal control theory, Variation Calculus and its economic application. Aim of this research paper is to shortly describe Optimal control and Variation Calculus and to present how can we deal with these type of issues. The last part of this paper is presenting possible economic application of Optimal control, based on the maximization of profit in monopoly while introducing new product on the market. Our control variable is the advertising rate, which affects the profit of monopoly through advertising expenditures and as a state variable was the market share defined.

scholarly journals Constants within Error Estimates for Legendre-Galerkin Spectral Approximations of Control-Constrained Optimal Control Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Jianwei Zhou
Keyword(s):  
Optimal Control ◽  
Error Estimate ◽  
Optimal Control Problems ◽  
Control Variable ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
One Dimensional ◽  
Error Indicator ◽  
Spectral Approximations ◽  
Explicit Formulae

Explicit formulae of constants within the aposteriorierror estimate for optimal control problems are investigated with Legendre-Galerkin spectral methods. The constrained set is put on the control variable. For simpleness, one-dimensional bounded domain is taken. Meanwhile, the corresponding aposteriorierror indicator is established with explicit constants.

scholarly journals The Constants in A Posteriori Error Indicator for State-Constrained Optimal Control Problems with Spectral Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jianwei Zhou
Keyword(s):  
Optimal Control ◽  
Spectral Methods ◽  
Optimal Control Problems ◽  
Posteriori Error ◽  
Control Problems ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Constrained Optimal Control ◽  
Error Indicator ◽  
State Variable

We employ Legendre-Galerkin spectral methods to solve state-constrained optimal control problems. The constraint on the state variable is an integration form. We choose one-dimensional case to illustrate the techniques. Meanwhile, we investigate the explicit formulae of constants within a posteriori error indicator.

scholarly journals Problème de contrôle optimal frontière pour l'équation de la chaleur : Approche variationnelle et pénalisation

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
H. Metoui
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Control Problem ◽  
Boundary Control ◽  
Control Variable ◽  
Nous Proposons ◽  
Penalization Method ◽  
Numerical Result ◽  
Boundary Control Problem ◽  
International Audience

International audience Our goal is to give a detailed analysis of an optimal control problem where the control variable is a rather boundary condition of Dirichlet type in L². We focus on establishing an appropriate variationnel approach to the optimal problem. We use the penalization method for the boundary control problem and we study the convergence between the penalized and the non-penalized boundary control problem. A numerical result is reported on to validate the convergence. Notre objectif est d'analyser un problème de contrôle frontière de l'équation de la chaleur définie avec une condition de Dirichlet de carré intégrable. Nous proposons une approche variationnelle adéquate au problème de contrôle frontière. Afin de prendre en compte la condition de Dirichlet, nous adoptons une procédure de pénalisation et nous étudions la convergence de la solution optimale pénalisée vers celle du problème initial. Un test numérique est discuté pour valider la convergence.

Analysis of Constrained Optimal Control with Related Constraints of Input Variables

2014 ◽  
Vol 39 (5) ◽  
pp. 679-689 ◽  
Author(s):  
Xiong-Lin LUO ◽  
Xiao-Long ZHOU ◽  
Shu-Bin WANG
Keyword(s):  
Optimal Control ◽  
Constrained Optimal Control ◽  
Input Variables
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