Deterministic Optimality of the Steady-State Behavior of the Kalman-Bucy Filter

In this letter, we provide a deterministic characterization of optimality of the steady-state behavior of the Kalman-Bucy filter, via an inverse optimal control argument. The result is achieved in two steps, both interesting per se. First, a singular linear-quadratic (LQ) optimal control problem is formulated and solved with respect to the innovation term of a classic Luenberger observer, hence yielding a LQ optimal observer. Then, such a construction is employed to interpret the optimality of the steady-state behavior of the celebrated Kalman-Bucy filter in a purely deterministic sense.