On external estimates of reachable sets of control systems with integral constraints

In this paper, we consider the problem of constructing external estimates of reachable sets as a level set of a certain differentiable Lyapunov-Bellman function (depending only on the state vector) for a control system with an integral control constraint. In particular, with its suitable choice, one can obtain ellipsoidal and rectangular estimates. The proposed constructions are based on integral estimates, the maximum solution, and the comparison principle for systems of differential inequalities. By using time in the arguments of the Lyapunov-Bellman function, it is possible to obtain more accurate estimates. In the linear nonstationary case, the latter can coincide with the set of reachability. A number of illustrative examples for nonlinear systems are given.