Minimax control of ambiguous linear stochastic systems using the Wasserstein metric

In this paper, we propose a minimax linear-quadratic control method to address the issue of inaccurate distribution information in practical stochastic systems. To construct a control policy that is robust against errors in an empirical distribution of uncertainty, our method is to adopt an adversary, which selects the worst-case distribution. To systematically adjust the conservativeness of our method, the opponent receives a penalty proportional to the amount, measured with the Wasserstein metric, of deviation from the empirical distribution. In the finite-horizon case, using a Riccati equation, we derive a closed-form expression of the unique optimal policy and the opponent's policy that generates the worst-case distribution. This result is then extended to the infinite-horizon setting by identifying conditions under which the Riccati recursion converges to the unique positive semi-definite solution to an associated algebraic Riccati equation (ARE). The resulting optimal policy is shown to stabilize the expected value of the system state under the worst-case distribution. We also discuss that our method can be interpreted as a distributional generalization of the $H_\infty$-method.