External estimates of reachable sets for control systems with integral constraints

In this paper, we consider the problem of constructing external estimates for reachable sets as level sets of some differentiable Lyapunov–Bellman function for a control system with an integral control constraint from $\mathbb{L}_p$, $p>1$. For an appropriate choice of this function, ellipsoidal and rectangular estimates can be obtained. Constructions proposed are based on integral estimates, maximal solutions, and the comparison principle for systems of differential inequalities. Using time as an argument of the Lyapunov–Bellman function, we obtain more accurate estimates. In the linear nonstationary case, such estimates may coincide with the reachable set. Also we present illustrative examples for nonlinear systems.