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Performance bounds for the mean-field limit of constrained dynamics

  • * Corresponding author: Mattia Zanella

    * Corresponding author: Mattia Zanella
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  • 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.

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

    Citation:

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

    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.

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

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

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