# American Institute of Mathematical Sciences

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April  2017, 37(4): 2023-2043. doi: 10.3934/dcds.2017086

## Performance bounds for the mean-field limit of constrained dynamics

 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

Received  August 2016 Revised  October 2016 Published  December 2016

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.

Citation: Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086
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##### References:
Computation of $\alpha_N$ for different values of the regularization parameter $\nu$.
,Theorem 5.4]. For $N\leq 4$ no estimate of the type (10) could be established.">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.
Experimental results for the optimization problem with varying optimization horizon $N$ and regularization constant $\nu=10^2$.
Experimental results for the optimization problem with varying optimization horizon $N$ and regularization constant $\nu=10^3$.
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