On the Lagrange duality of stochastic and deterministic minimax control and filtering problems

As shown below, the linear operator norms in the deterministic and stochastic cases are optimal values of the Lagrange-dual problems. For linear time-varying systems on a finite horizon, the duality principle leads to stochastic interpretations of the generalized $H_2$ and $H_\infty$ norms of the system. Stochastic minimax filtering and control problems with unknown covariance matrices of random factors are considered. Equations of generalized $H_\infty$-suboptimal controllers, filters, and identifiers are derived to achieve a trade-off between the error variance at the end of the observation interval and the sum of the error variances on the entire interval.