A general strategy for solving the motion planning problem for real analytic, controllable systems without drift is proposed. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added. Using formal calculations with a product expansion relative to P. Hall basis, another control is produced that achieves the desired result on the formal level. This provides an exact solution of the original problem if the given system is nilpotent. For a general system, an iterative algorithm is derived that converges very quickly to a solution. For nonnilpotent systems which are feedback nilpotentizable, the algorithm, in cascade with a precompensator, produces an exact solution. Results of simulations which illustrate the effectiveness of the procedure are presented.>

Let (w/sub ij/) be the weights of the connections of a neural network with n nodes, calculated from m data vectors v/sup 1/, ..., v/sup m/ in (1,-1)/sup n/, according to the Hebb rule. The author proves that if m is not too large relative to n and the v/sup k/ are random, then the w/sub ij/ constitute, with high probability, a perfect representation of the v/sup k/ in the sense that the v/sup k/ are completely determined by the w/sub ij/ up to their sign. The conditions under which this is established turn out to be less restrictive than those under which it has been shown that the v/sup k/ can actually be recovered by letting the network evolve until equilibrium is attained. In the specific case where the entries of the v/sup k/ are independent and equal to 1 or -1 with probability 1/2, the condition on m is that m should not exceed n/0.7 log n.>

This paper studies time-optimal control questions for a certain class of nonlinear systems. This class includes a large number of mechanical systems, in particular, rigid robotic manipulators with torque constraints. As nonlinear systems, these systems have many properties that are false for generic systems of the same dimensions.