Viscosity solutions for a system of PDEs and optimal switching

Abstract

In this paper, we study the existence and uniqueness of viscosity solutions for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case of this system is the deterministic version of the Verification Theorem of the Markovian optimal m-states optimal switching problem in finite horizon. The switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central valuation in cases where such organizations or state give grants or financial assistance to power plants that promotes green energy in their production activity or that uses less polluting modes in their production. Our main tools is an approximation scheme and the notion of systems of reflected backward stochastic differential equations.

Management strategies for run-of-river hydropower plants: an optimal switching approach

“…Optimal switching is a relatively new and fast growing field of mathematics combining optimization, SDEs and partial differential equations (PDEs) (Barkhudaryan et al 2020;Biswas et al 2010;Djehiche et al 2010;El-Asri and Fakhouri 2017;El-Asri and Hamadéne 2009;Hamadéne and Morlais 2013;Hu and Tang 2010;Kharroubi 2016;Lundström et al 2014aLundström et al , b, 2019Martyr 2016a, b;Perninge 2018, Lundström and Olofsson 2021. However, a literature survey shows that, although the mathematical theory is well developed, applications of optimal switching to real life problems is a far less explored area.…”

System of variational inequalities with interconnected obstacles

“…For example, the authors of [18] establish Harnack-type inequalities for viscosity solutions of fully nonlinear cooperative elliptic systems, and, in [19], the authors consider Liouville-type theorems and a priori estimates for solutions of boundary value problems for systems of elliptic PDEs. More results connected with optimal switching problems with signed switching costs can be found in [20][21][22][23][24][25].…”

Management strategies for run-of-river hydropower plants -an optimal switching approach

“…Optimal switching is a relatively new and fast growing field of mathematics combining optimization, SDEs and partial differential equations (PDEs) [5,6,15,17,18,23,25,27,28,29,30,31,32,38,39,40]. However, a literature survey shows that, although the mathematical theory is well developed, applications of optimal switching to real life problems is a far less explored area.…”