JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 118, 42-62 (1986) Control of Nonlinear Variable Structure Systems G. BARTOLINI D.I.S.T., Universitri di Genova. Via all’opera Pia, 16145 Genoa, Italy T. ZOLEZZI * Istituto di Matematica. Universitri di Genova. Via L. B. Alberti 4. 16132 Genoa, Italy Submitted by George Leitmann 0. INTRODUCTION We consider control systems described by state equations f =f(t, -5 u), (1) where the state variable x = (x1 ,..., x,)’ E R”, the control variable u = (u, ,..., u,)’ E R” and a prime denotes transpose; sliding manifold s(x) = (Sl(X),..., s,(x))’ = 0 (2) and control constraints given by u(t, X)E% (3) for some given subset 42 of R”. We wish to control the system by using feedback control laws u = u(t, x) which are discontinuous along the surfaces given by Sj(X) = 0, j = l,..., m. Often the control law takes the form zg t, x) = I q (t, xl if sj(x) > 0 u,: (6 x) if sj(x) < 0, (4) * Supported in part by M.P.I. 0022-241X/86 $3.00 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved. 42