Synthesis of restricted inertial controls for systems with multivariate control

The local positional synthesis problem for the control system ˙x = f (x,u), x ∈ Rn, u ∈ Rr , under preassigned restrictions u(k)(x) dk on derivatives of control with respect to the closed system up to given order, 0 k l, is considered. The controllability function method is the basis of the investigation. Linear systems are considered and the family of controllability functions Θα(x) is constructed, for which the corresponding controls uα(x) solve the cited synthesis problem if α 2l + 1, so that the time of motion from x0 to 0 equals αΘ 1/α α (x0). Nonlinear systems are considered with respect to the first approximation in a partial case l = 1; we give the upper estimate for the time of motion. In the limiting case Θ∞(x) is the Lyapunov function and u∞(x) solves the stabilization problem and satisfies the preassigned restrictions.  2002 Elsevier Science (USA). All rights reserved.