Automation and Remote Control, Vol. 63, No. 7, 2002, pp. 1199–1203. Translated from Avtomatika i Telemekhanika, No. 7, 2002, pp. 179–183. Original Russian Text Copyright c  2002 by Venets. NOTES Ideal Sliding Mode in the Problems of Convex Optimization V. I. Venets Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia Received February 14, 2002 Abstract—The characteristics of the sliding mode that appears with using the continuous convex-programming algorithms based on the exact penalty functions were discussed. For the case under study, the ideal sliding mode was shown to occur in the absence of infinite number of switchings. 1. INTRODUCTION The sliding modes are widely used to study various applications of control. They enable one to design systems featuring the desired characteristics where application of the ordinary smooth controls is difficult or impossible at all. By the ideal sliding mode is usually [1] meant the motion along the discontinuity surface that occurs if the phase trajectories in the neighborhood of this surface are directed “toward” each other. The authors mostly believe (see [2–5], for example) that an infinite number of switchings corresponds to the ideal sliding modes, but sometimes they assert [6] that these (ideal) modes exist only as theoretical constructions. It turns out, however, that the ideal sliding modes do not necessarily have an infinite number of switchings. This fact may be illustrated by the optimization problems that are often used as examples in the study of the sliding modes [1]. 2. CONTINUOUS ALGORITHMS OF CONVEX OPTIMIZATION In what follows, we employ the terminology and notation of nonlinear programming, but one could pass with no trouble to the notation used by the experts in the field of sliding modes (for example, in [1]). Let us consider a rather general formulation of the problem of convex programming formulated as follows: f (x) → min, x ∈ G = {x ∈ R n : g i (x) ≤ 0,i =1,...,m}, (1) where all functions f, g i , i =1,...,n, are the proper convex functions that are definite and continuously differentiable everywhere on R n . The requirement on continuous differentiability is not a must and was introduced here just to simplify the presentation. We assume that a solution of problem (1) exists on a nonempty set X * ⊂ G. To solve the original problem (1), we use the method of penalty functions or, to be more correct, exact penalty functions. In doing so, the original problem is replaced by that of determining the unconditional minimum of the nonsmooth convex function F (x, q)= f (x)+ q m  i=1 g + i (x), (2) 0005-1179/02/6307-1199$27.00 c  2002 MAIK “Nauka/Interperiodica”