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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 H2 and H∞ norms of the system. Stochastic minimax filtering and control problems with unknown covariance matrices of random factors are considered. Equations of generalized H∞-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. [ABSTRACT FROM AUTHOR]

It is demonstrated how to design optimal digital controllers on a finite horizon for the linear time-varying objects with uncertain initial conditions and external disturbances. The square of an optimality criterion, called the maximum output deviation, represents the exact guaranteed maximum value of the squared Euclidean norm of the system output normalized by the sum of the squared Euclidean norms of disturbances and a quadratic form of the initial state of a system. The maximum output deviations and the worst-case disturbances and/or initial conditions causing them, as well as the minimax controllers (including the multiobjective ones that minimize the maximum deviations of several outputs) are calculated as solutions to a semidefinite programming problem. The necessary and sufficient conditions for the finite-time stability and boundedness of a system are established; they can be used for controller design. [ABSTRACT FROM AUTHOR]

We consider a linear time-varying system with an initial state and disturbance that are known imprecisely and satisfy a common constraint. The constraint is the sum of a quadratic form of the initial state and the time integral of a quadratic form of the disturbance, and these quadratic forms are allowed to be degenerate. We obtain a linear matrix differential Lyapunov equation describing the evolution of the ellipsoidal reachability set. In the problem of estimating the state based on output observations, this result is used to find the minimum-size ellipsoidal set of admissible system states, which is determined by the optimal observer and by the reachability set of the corresponding observation error equation. A method for control law synthesis ensuring that the system state reaches the target set or the system trajectory remains in a given ellipsoidal tube is proposed. Illustrative examples are given for the Mathieu equation, which describes parametric oscillations of a linear oscillator. [ABSTRACT FROM AUTHOR]

The maximum output deviation of a linear time-varying system is defined as the worst-case measure of the maximum value of the output Euclidean norm over a finite horizon provided that the sum of the squared energy of an external disturbance and a quadratic form of the initial state is 1. Maximum deviation is characterized in terms of solutions to differential matrix equations or inequalities. A modified concept of the boundedness of the system on a finite horizon under an external and initial disturbances is introduced and its connection with the concept of maximum deviation is established. Necessary and sufficient conditions for the boundedness of the system on a finite horizon are obtained. It is demonstrated that optimal controllers (including the multiobjective ones that minimize the maximum deviations of several outputs) as well as controllers ensuring the boundedness of the system can be designed using linear matrix inequalities. [ABSTRACT FROM AUTHOR]