scholarly journals Binary optimal control by trust-region steepest descent

2022 ◽  
Author(s):  
Mirko Hahn ◽  
Sven Leyffer ◽  
Sebastian Sager
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Function ◽  
Trust Region ◽  
Descent Method ◽  
Steepest Descent ◽  
Topological Optimization ◽  
Steepest Descent Method ◽  
Control Problems ◽  
Theoretical Argument

AbstractWe present a trust-region steepest descent method for dynamic optimal control problems with binary-valued integrable control functions. Our method interprets the control function as an indicator function of a measurable set and makes set-valued adjustments derived from the sublevel sets of a topological gradient function. By combining this type of update with a trust-region framework, we are able to show by theoretical argument that our method achieves asymptotic stationarity despite possible discretization errors and truncation errors during step determination. To demonstrate the practical applicability of our method, we solve two optimal control problems constrained by ordinary and partial differential equations, respectively, and one topological optimization problem.

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.

Regularization and discretization error estimates for optimal control of ODEs with group sparsity

2018 ◽  
Vol 24 (2) ◽  
pp. 811-834 ◽  
Author(s):  
Christopher Schneider ◽  
Gerd Wachsmuth
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Function ◽  
Semismooth Newton Method ◽  
Control Problems ◽  
Time Interval ◽  
Stability Estimates ◽  
Optimal Controls ◽  
Group Sparsity ◽  
Linear Quadratic Optimal Control

It is well known that optimal control problems with L1-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L2-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method.

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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