Control of Conservation Laws in the Nonlocal-to-Local Limit

TL;DR

The paper addresses controlling entropy solutions of scalar conservation laws by treating the initial data as the control and regularizing the problem with a nonlocal velocity. By establishing Gamma-convergence of the nonlocal control functionals to the local ones, it shows that minimizers of the nonlocal problems converge to minimizers of the local problem in the $L^1_{loc}$ sense, both in the continuous and discrete (Eulerian-Lagrangian) frameworks. A key innovation is the improved nonlocal-to-local limit that handles simultaneous kernel and initial data convergence, plus a rigorous discrete Gamma-convergence analysis. Numerical experiments corroborate the theory, including a diagonal double-limit where $\Delta x$ and $H$ vanish together, suggesting the approach reliably recovers the local optima from nonlocal approximations.

Abstract

We analyze a class of control problems where the initial datum acts as a control and the state is given by the entropy solution of (local) conservation laws by a nonlocal-to-local limiting strategy. In particular we characterize the limit up to subsequence of minimizers to nonlocal control problems as minimizer of the corresponding local ones. Moreover, we also prove an analogous result at a discrete level by means of a Eulerian-Lagrangian scheme.

Figures (9)

• Figure 1.1: The $\Gamma$-convergence of functionals here proved: the solid lines represents the convergence in the nonlocal-to-local limit in the continuum and discrete framework (\ref{['teo:4']} and \ref{['teo:8']}); the dotted lines are consequences of the Eulerian-Lagrangian scheme in the approximation of solution to \ref{['eq:23']} and \ref{['eq:24']}; at last, the diagonal dashed line is verified here numerically for some sequences $(H, \Delta x) \to 0$ (see \ref{['sec:diagonal convergence']}).
• Figure 4.1: Scheme of the moving regions $D_j^m$ "impermeable" to the flux defined in \ref{['eq:49']}. In cyan, we plotted the no-flux curves described by \ref{['eq:13']}; in magenta we add the linear approximation of the curves (see \ref{['eq:50']} and \ref{['eq:39']}).
• Figure 4.2: Representation of the Eulerian-Lagrangian scheme \ref{['eq:21']}. In black we plot the solution $U$ at times $t^m$ and $t^{m+1}$; in magenta dashed we report the approximation of no-flux curves and the values $U^{m+1}_{j+\frac{1}{2}}$ defined in \ref{['eq:26']}.
• Figure 4.3: The kernel function $\eta_H$ and its piecewise constant projection onto the grid.
• Figure 5.1: On the left, we present the comparison between the real minimizer to $\mathcal{G}_\Delta$ (see \ref{['eq:46']}) $U_{o}^d$ and approximation $U_{o}^{min}$ in the case $\Delta x= 0.01$ and 'StepTolerance' equal to $\Delta x^3$. The discrepancy at the right boundary may be justified by the fact that there the initial datum is rapidly leaving the domain, thus has no contribution in the objective functional, while on the same time the algorithm is initialized by the value 0.45 on that boundary. On the right, we show the evolution in time of $U^d$ and $U^{min}$, the last being the solution to \ref{['eq:24']} correspondent to the approximate optimal initial datum $U_o^{min}$.

Theorems & Definitions (28)

• Example 1.1
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Theorem 3.2: Improved result on the nonlocal-to-local limit
• proof