Performance bounds for the mean-field limit of constrained dynamics

Michael Herty; Mattia Zanella

doi:10.3934/dcds.2017086

Article Contents

2017, Volume 37, Issue 4: 2023-2043. Doi: 10.3934/dcds.2017086

This issue Previous Article A unified approach to weighted Hardy type inequalities on Carnot groups Next Article Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics

Performance bounds for the mean-field limit of constrained dynamics

Michael Herty^1, and
Mattia Zanella^2, ,

1.
Department of Mathematics, IGPM, RWTH Aachen University, Templergraben 55, Aachen, 52062, Germany
2.
Department of Mathematics and Computer Science, University of Ferrara, Via N. Machiavelli 35, Ferrara, 44121, Italy

* Corresponding author: Mattia Zanella
* Corresponding author: Mattia Zanella

Received: August 2016

Revised: October 2016

Published: May 2017

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Abstract

In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.

Keywords:

Mathematics Subject Classification: Primary:35Q93, 49N35, 93B05, 93B52;Secondary:91A23, 92D25.

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Figure 1. Computation of $\alpha_N$ for different values of the regularization parameter $\nu$.

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Figure 2. Value of the cost functional $J_T^{u^{MPC}_N}(X_0)$ for controls obtained using a MPC strategy with control horizon $N$ (red) and presentation of the optimal costs $V_T^{*}(X_0)$ multiplied by $\frac{1}{\alpha_N}$ where $\alpha_N$ is computed as in [35,Theorem 5.4]. For $N\leq 4$ no estimate of the type (10) could be established.

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Figure 3. Experimental results for the optimization problem with varying optimization horizon $N$ and regularization constant $\nu=10^2$.

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Figure 4. Experimental results for the optimization problem with varying optimization horizon $N$ and regularization constant $\nu=10^3$.

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