Vanishing viscosity for linear-quadratic mean-field control problems

TL;DR

This paper demonstrates that adding a small heat diffusion term to linear-quadratic mean-field control problems and then letting it vanish yields the original problem's optimal control, confirming the method's validity.

Contribution

It establishes the validity of the vanishing viscosity method for linear-quadratic mean-field control problems, showing the limit of regularized solutions matches the original optimal control.

Findings

Vanishing viscosity approach converges to the original control solution.

Regularized problems with heat diffusion approximate the original problem.

The method confirms the robustness of the control solution under regularization.

Abstract

We consider a mean-field control problem with linear dynamics and quadratic control. We apply the vanishing viscosity method: we add a (regularizing) heat diffusion with a small viscosity coefficient and let such coefficient go to zero. The main result is that, in this case, the limit optimal control is exactly the optimal control of the original problem.