Optimal Control Problems with Nonlocal Conservation Laws: Existence of Optimizers and Singular Limits in Approximations of Local Conservation Laws

This contribution considers optimal control problems subject to nonlocal conservation laws – those in which the velocity depends nonlocally (i.e., via a convolution) on the solution – and the so-called singular limit. First, the existence of minimizers is demonstrated for a broad class of optimal control problems, involving optimization over the initial datum, velocity, and nonlocal kernel for classical tracking-type $L^2$ cost functionals. Then, it is proven that the obtained minimizers converge to minimizers of the corresponding local optimal control problem when the kernel function of the convolution is of exponential type and approaches a Dirac distribution. Finally, some numerical results are presented.

💡 Research Summary

This paper investigates optimal control problems constrained by nonlocal scalar conservation laws, where the flux velocity depends on the solution through a convolution kernel. The authors consider three control variables simultaneously: the initial datum (q_{0}), the velocity function (V), and the nonlocal kernel (\gamma). Admissible sets are defined with natural physical bounds: the initial data belong to a bounded subset of (L^{\infty}(\mathbb{R})), the velocity functions are non‑increasing and uniformly bounded in (W^{1,\infty}) (or (W^{2,\infty}) for stronger results), and the kernels are non‑increasing, normalized in (L^{1}), and uniformly bounded in (L^{\infty}).

The first major contribution is a well‑posedness theorem (Theorem 2.4) for the nonlocal Cauchy problem. Under the stated assumptions the equation admits a unique weak solution (q) that stays within the a priori bounds imposed by the initial data (a maximum‑minimum principle). Moreover, the solution can be expressed explicitly via characteristics (\xi_{W}) as
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