Divergence of neighbouring optimal state trajectories of a dynamic system

The boundary of the set of possible states of a dynamic system which are attainable in a given time fro an initial state expands in the state space as the system evolves. A random disturbance to the system will perturb this boundary such that even weak, irregular disturbances can lead to the divergence of optimal trajectories which are initially parallel and arbitrarily close. It is shown how the statistics of the relative separation between such neighbouring optimal trajectories can be obtained for a second order dynamic system.