Gaussian estimates on networks with applications to optimal control

Luca Di Persio; Giacomo Ziglio; Luca Di Persio; Giacomo Ziglio

doi:10.3934/nhm.2011.6.279

Networks and Heterogeneous Media

2011, Volume 6, Issue 2: 279-296. doi: 10.3934/nhm.2011.6.279

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Gaussian estimates on networks with applications to optimal control

Luca Di Persio ^{1
,},
Giacomo Ziglio ^{1
,}

1.
Department of Mathematics, University of Trento, Povo (TN), 38123

Received: 01 April 2010 Revised: 01 April 2011
Primary: 35R02, 60H15, 93E20; Secondary: 90B15.

We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local, stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in [21]. In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in [13] to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.
- Stochastic partial differential equations on networks,
- Gaussian estimates,
- Optimal stochastic control
Citation: Luca Di Persio, Giacomo Ziglio. Gaussian estimates on networks with applications to optimal control[J]. Networks and Heterogeneous Media, 2011, 6(2): 279-296. doi: 10.3934/nhm.2011.6.279

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Abstract

We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local, stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in [21]. In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in [13] to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.

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