A problem of attainability with constraints of asymptotic nature

We construct and study properties of attainability sets for a linear control system with discontinuous coefficients of the control action. This problem is tightly connected with the issue of an attainability “on average” (i.e. the type of an attainability in the class of mathematical expectations of random vectors). Focusing on the case of constraints of an asymptotic character, we tackle the mentioned problems by means of the universal approach. These constraints correspond to a control regime in the class of “narrow” pulses. We consider a control problem in the class of pulse-like controls complying with the requirement of full consumption of energy resources.