Optimal control of maximal output deviations over a finite horizon

A notion of the maximal output deviation for a linear time-varying system over a finite horizon is introduced. This is the worst-case peak value of the output in response to external and initial disturbances provided that sum of the squared disturbance energy and a quadratic form of the initial state is equal to unit. The maximal output deviations under either external disturbances or initial disturbances are defined respectively. The peak value of the multiple vector signal is considered as its generalized $L_{\infty}$ norm. The variational approach is applied to characterize the maximal output deviations in terms of solutions to matrix differential equations. By using discretization, semi-definite programs are derived to compute the optimal control and Pareto optimal controls minimizing the maximal deviation of a single output and several outputs, respectively. The efficiency of the approach proposed is demonstrated on two problems of optimal protection from shock and vibration.