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arXiv · 2610.10029

A Nonlocal Stochastic Optimal Control with Elliptic-Type Smoothing

Stefana-Lucia Anita·Luca Di Persio
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Abstract

We study a nonlocal stochastic optimal control problem in which the control acts through the elliptic smoothing operator $\mathcal{S}=(I-Δ)^{-1}$ on $\mathbb{R}^d$. The state is described by the law of a controlled diffusion, equivalently by a controlled Fokker-Planck equation with coefficients depending on $\mathcal{S}u$. We prove the existence of an optimal control by the direct method in the calculus of variations. We then derive a Pontryagin-type maximum principle by spike variations, relying on sharp properties of the operator $\mathcal{S}$. The result yields a pointwise minimisation rule for the optimal control. We discuss the meaning of the optimal control problem in the case of population dynamics.

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Stefana-Lucia Anita, Luca Di Persio. 2026-10-07. A Nonlocal Stochastic Optimal Control with Elliptic-Type Smoothing. https://arxiv.org/abs/2610.10029

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