I deeply appreciate the honor of delivering this fifth SIAM von Neumann lecture and above all I feel that it is not an easy task to measure up to it. I have known von Neumann rather well and we were associated as editors of the Annals of Mathematics for twenty years-perhaps his most active scientific period. Like everyone else who knew him I was lost in admiration before his great originality and his operating angle of vision which must have been little short of 1800: from symbolic logic, to quantum theory, Lie groups, continuous geometry, theory of computing nmachines! At all events John von Neumann was one of those mathematicians who believe that to apply mathematics is to enrich mathematics. I think, therefore, that for this reason if for no other he would have approved of my choice of topic. 1. The problem of Lurie. Since we shall be dealing entirely with differential equations in many variables, vectors and matrices are practically indispensable. The general notations are standard: A, B, -*. (X, U, V excepted) represent constant square matrices of order n; transposes: A', B', * ; in particular, En or E denotes the unit rnatrix of order n. The letters a, b, c, d, * *, x, y stand for column n-vectors (n X 1 matrices). The same written as row vectors (1 X n matrices) are designated by a', b', * . The components of x, for example, are denoted by xi, x, , Xn . Greek letters represent scalars. It was the great merit of Lurie of Leningrad to extract from the general nonlinear control problem a tractable stylized mathematical formulation. With a slight but convenient modification by V. M. Popov, Lurie's system is