Iterative Solution Methods for a Class of State and Control Constrained Optimal Control Problems

2012 ◽  
Vol 03 (12) ◽  
pp. 1862-1867 ◽  
Author(s):  
Erkki Laitinen ◽  
Alexander Lapin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Iterative Solution ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
Solution Methods ◽  
Iterative Solution Methods ◽  
And Control

scholarly journals Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

2020 ◽  
Vol 146 (2) ◽  
pp. 335-368
Author(s):  
Owe Axelsson ◽  
János Karátson
Keyword(s):  
Optimal Control ◽  
Rate Of Convergence ◽  
Large Scale ◽  
Optimal Control Problems ◽  
Iterative Solution ◽  
Control Problems ◽  
Block Form ◽  
Solution Methods ◽  
Preconditioned Matrix ◽  
Iterative Solution Methods

Abstract Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear.

On the Iterative Solution Methods for Finite-Dimensional Inclusions with Applications to Optimal Control Problems

2010 ◽  
Vol 10 (3) ◽  
pp. 283-301 ◽  
Author(s):  
E. Laitinen ◽  
A. Lapin ◽  
S. Lapin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Function ◽  
Elliptic Boundary ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
Finite Dimensional ◽  
The Right

AbstractIterative methods for finite-dimensional inclusions which arise in applying a finite-element or a finite-difference method to approximate state-constrained optimal control problems have been investigated. Specifically, problems of control on the right- hand side of linear elliptic boundary value problems and observation in the entire domain have been considered. The convergence and the rate of convergence for the iterative algorithms based on the finding of the control function or Lagrange multipliers are proved.

Explicit algorithms to solve a class of state constrained parabolic optimal control problems

2015 ◽  
Vol 30 (6) ◽  
Author(s):  
Alexander Lapin ◽  
Erkki Laitinen ◽  
Sergey Lapin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Time Derivative ◽  
Iterative Solution ◽  
State Equation ◽  
The State ◽  
Linear Parabolic Equation ◽  
Control Problems ◽  
State Function ◽  
Solution Methods

AbstractWe consider an optimal control problem of a system governed by a linear parabolic equation with the following features: control is distributed, observation is either distributed or final, there are constraints on the state function and on its time derivative. Iterative solution methods are proposed and investigated for the finite difference approximations of these optimal control problems. Due to explicit in time approximation of the state equation and the appropriate choice of the preconditioners in the iterative methods, the implementation of all constructed methods is carried out by explicit formulae. Computational experiments confirm the theoretical results.

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