# Feedback control of parametrized PDEs via model order reduction and dynamic programming principle

@article{Alla2020FeedbackCO, title={Feedback control of parametrized PDEs via model order reduction and dynamic programming principle}, author={Alessandro Alla and Bernard Haasdonk and Andr{\'e} Schmidt}, journal={Advances in Computational Mathematics}, year={2020}, volume={46}, pages={1-28} }

In this paper, we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of…

#### Figures, Tables, and Topics from this paper

#### 3 Citations

A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation

- Computer ScienceElectronic Research Archive
- 2021

A feedback design for numerical solution to optimal control problems, which is based on solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which can avoid solving the HJB equation repeatedly and efficaciously promote the computation efficiency and save memory.

Optimal bounds for numerical approximations of infinite horizon problems based on dynamic programming approach

- Computer Science, MathematicsArXiv
- 2021

The error bound of the fully discrete method is revised and it is proved, under similar assumptions to those of the time discrete case, that the error of the partially discrete case is in fact O(h+ k) which gives first order in time and space for the method.

Model Order Reduction via Moment-Matching: A State of the Art Review

- Computer ScienceArchives of Computational Methods in Engineering
- 2021

This manuscript presents a state-of-the-art review of the moment-matching based order reduction methods for linear and nonlinear dynamical systems, and tracks the progress of moment- matching methods from their inception to how they have emerged as the most commonly adopted platform for reducing systems in large-scale settings.

#### References

SHOWING 1-10 OF 50 REFERENCES

Data-driven surrogates of value functions and applications to feedback control for dynamical systems

- Computer Science
- 2018

A novel approximation technique for the value function of an infinite horizon optimal control based on solving optimal open loop control problems on a finite horizon with a sampling of the global value function along the generated trajectories is introduced.

Model Order Reduction Approaches for Infinite Horizon Optimal Control Problems via the HJB Equation

- Mathematics
- 2017

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations…

A POSTERIORI ERROR ESTIMATION FOR REDUCED ORDER SOLUTIONS OF PARAMETRIZED PARABOLIC OPTIMAL CONTROL PROBLEMS

- Mathematics
- 2014

We consider the ecient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial dierential equations. To this end, we employ the reduced basis…

Reduced-order minimum time control of advection-reaction-diffusion systems via dynamic programming.

- Mathematics
- 2014

A numerical approach for a time-optimal feedback control problem for an advection-reaction-diffusion model is considered. Our approach is composed by three main building blocks: approximation of the…

Reduced basis approximation of large scale parametric algebraic Riccati equations

- Mathematics
- 2018

The algebraic Riccati equation (ARE) is a matrix valued quadratic equation with many important applications in the field of control theory, such as feedback control, state estimation or ℋ ∞ -robust…

Error Analysis for POD Approximations of Infinite Horizon Problems via the Dynamic Programming Approach

- Mathematics, Computer ScienceSIAM J. Control. Optim.
- 2017

In this paper infinite horizon optimal control problems for nonlinear high-dimensional dynamical systems are studied and a reduced-order model is derived for the dynamical system, using the method of proper orthogonal decomposition (POD).

Suboptimal Feedback Control of PDEs by Solving HJB Equations on Adaptive Sparse Grids

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2017

An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A…

Polynomial Approximation of High-Dimensional Hamilton-Jacobi-Bellman Equations and Applications to Feedback Control of Semilinear Parabolic PDEs

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2018

A procedure for the numerical approximation of high-dimensional Hamilton--Jacobi--Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is…

Reduced Basis Method and A Posteriori Error Estimation for Parametrized Linear-Quadratic Optimal Control Problems

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2010

The reduced basis method is proposed for the solution of parametrized optimal control problems described by parabolic partial differential equations in the unconstrained case and allows at the on-line step large computational cost savings with respect to the “truth” approximation used for defining the reduced basis.

Nonlinear Model Predictive Control

- Mathematics
- 2011

In this chapter, we introduce the nonlinear model predictive control algorithm in a rigorous way. We start by defining a basic NMPC algorithm for constant reference and continue by formalizing state…