More formally, we consider a system whose dynamics is
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Both players try to achieve their objective by minimizing their respective cost functions:
The squared Wasserstein distance between Gaussian distributions is given by:
In this project, we consider two approaches to find the Nash Equilibrium solution.
First, we consider affine disturbance history feedback control policy which is a common control policy for linear control problems. This way, we turn the stochastic optimal control problem into a finite dimensional optimization problem for each player then we use iterative best response algorithm to find Nash Equilibrium.
Secondly, we use the linearity of the underlying system dynamics. Since the control cost that we consider is a convex quadratic function of control input for both players, we approximate the terminal state distribution cost as a linear-quadratic function of the terminal state.
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