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 that nonlinear minimax controller problems might possess, namely the certainty equivalence and generalized certainty equivalence. The certainty equivalence is a generalization of the one introduced earlier by the authors (1990, 1991), 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