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PreprintCC BY 4.0arXiv

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 ๐’ฎ=(Iโˆ’ฮ”)โˆ’1 on โ„d. The state is described by the law of a controlled diffusion, equivalently by a controlled Fokker-Planck equation with coefficients depending on ๐’ฎu. 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 ๐’ฎ. 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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Cite

BibTeXRIS
BibTeX
@misc{anita2026nonlocal,
  title = {{A Nonlocal Stochastic Optimal Control with Elliptic-Type Smoothing}},
  author = {Stefana-Lucia Anita and Luca Di Persio},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10029},
  eprint = {2610.10029},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10029},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Nonlocal Stochastic Optimal Control with Elliptic-Type Smoothing
AU  - Stefana-Lucia Anita
AU  - Luca Di Persio
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10029
UR  - https://arxiv.org/abs/2610.10029
AB  - We study a nonlocal stochastic optimal control problem in which the control acts through the elliptic smoothing operator ๐’ฎ=(Iโˆ’ฮ”)^โˆ’1 on โ„^d. The state is described by the law of a controlled diffusion, equivalently by a controlled Fokker-Planck equation with coefficients depending on ๐’ฎu. 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 ๐’ฎ. 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.
ER  -