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The ellipsoidal estimation of reachable sets is an efficient technique for the set-membership modelling of uncertain dynamical systems. In this paper, the optimal outer ellipsoidal approximation of reachable sets is considered, and attention is paid to the new criterion associated with the projection of the approximating ellipsoid onto a given direction. Nonlinear differential equations governing the evolution of ellipsoids are analysed and simplified. The asymptotic behaviour of ellipsoids near the initial point and at infinity is studied. It is shown that the optimal ellipsoids under consideration touch the corresponding reachable sets at all time instants. A control problem for a system subjected to uncertain perturbations is investigated in the framework of the optimal ellipsoidal estimation of reachable sets.

Linear elastic systems controlled by bounded distributed forces are considered. We propose a method of control based on decomposing the original system into an infinite number of subsystems with one degree of freedom each. After that, we apply the time-optimal control to each mode of oscillations. As a result, we present explicit formulae for the distributed control and upper estimates on the time of the control process. Sufficient controllability conditions are also obtained. Examples are presented.