# POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation

@article{Strazzullo2020PODGalerkinMO, title={POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation}, author={Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza}, journal={Journal of Scientific Computing}, year={2020}, volume={83}, pages={1-35} }

In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent…

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## References

SHOWING 1-10 OF 84 REFERENCES

Reduced Basis Method for Parametrized Elliptic Optimal Control Problems

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2013

The methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable is developed and recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems.

Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering

- Environmental Science, MathematicsSIAM J. Sci. Comput.
- 2018

This work proposes reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering and proposes a POD-Galerkin reduction of the optimality system.

Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2018

This paper introduces reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable and proposes two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional.

Reduced basis methods for optimal control of advection-diffusion problems ∗

- Mathematics
- 2007

The reduced basis (RB) method is proposed for the approximation of multi-parametrized equations governing an optimal control problem. The idea behind the RB method is to project the so- lution onto a…

Optimal flow control for Navier–Stokes equations: drag minimization

- Mathematics
- 2007

Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag…

A certified reduced basis method for parametrized elliptic optimal control problems

- Mathematics
- 2014

In this paper, we employ the reduced basis method as a surrogate model for the solu- tion of linear-quadratic optimal control problems governed by parametrized elliptic partial dierential equations.…

Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations

- Computer Science, MathematicsComput. Math. Appl.
- 2015

The reduced basis method is extended to the case of noncoercive (elliptic) equations, such as the Stokes equations, and applied to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

On the stability of the reduced basis method for Stokes equations in parametrized domains

- Mathematics
- 2007

We present an application of reduced basis method for Stokes equations in domains with affine parametric dependence. The essential components of the method are (i) the rapid convergence of global…

A Space-Time Petrov-Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2014

We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but…

Certified Reduced Basis Methods for Parametrized Distributed Elliptic Optimal Control Problems with Control Constraints

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2016

This paper employs the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations and proposes two different reduced basis approximations and associated error estimation procedures.