Structural properties of minimax policies for a class of differential games arising in nonlinear H^∞-control and filtering

This paper introduces, in precise mathematical terms, two properties (named, certainty equivalence and generalized certainty equivalence) that nonlinear minimax controller problems might possess. The certainty equivalence is a generalization of the one introduced earlier in [2] and [3], which applies to problems where the `worst-case disturbance' may not be unique (but the worst-case state trajectory is). The generalized certainty equivalence, on the other hand, extends this to accommodate nonunique worst-case state trajectories, and leads to the construction of controllers that guarantee a bounded upper value for the underlying game. The paper also shows that for a large class of games (and under certain conditions) certainty-equivalent (as well as generalized certainty-equivalent) controllers admit (infinite-dimensional) estimator (Kalman-filter) structures, where the estimator gain depends on the state of the estimator. These results are then applied to the nonlinear minimax filtering problem, which is treated here as a special case of the general control problem.