On Pareto set in control and filtering problems under stochastic and deterministic disturbances

We consider two-criteria control or filtering problems for linear systems, where one criterion is the level of suppression for Gaussian white noise with unknown covariance, and another is the level of suppression for a deterministic signal of bounded power. We define a new criterion, the level of suppression for stochastic and deterministic disturbances that act jointly in the general case on different inputs. This criterion is characterized in terms of solutions of Riccati equations or linear matrix inequalities. We establish that for the choice of optimal controller or filter with respect to this criterion relative losses with respect to each of the original criteria compared to Pareto optimal solutions do not exceed the value $1-\sqrt2/2$. We extend these results to dual control and filtering problems for systems with one input and two outputs, generalize them to the case of $N$ criteria with loss estimate $1-\sqrt N/N$, and also apply them for systems with external and initial disturbances. We show a numerical example.