Asymptotic upper bound for the peak-effect in linear control systems

The large deviations of linear control system trajectories from the equilibrium position during the stabilization process, known as peak-effect, represent a serious obstacle to the design of cascade control systems and to guidance stabilization. Evaluation of such deviations is an important problem not only for control theory but also for other branches of mathematics where similar phenomenon appears as, for example, in numerical analysis, in the study of hydrodynamic stability and transition to turbulence, in the dynamics of vehicular platoons, and many others. The aim of this paper is to improve one of the main results concerning upper bounds for the peak-effect in linear control systems in the work by B.T. Polyak and myself in 2016. As the lower bound for overshoot proved by R.N. Izmailov in 1987 shows, the asymptotic estimate for the upper bound obtained here cannot be improved.