A CHARACTERIZATION OF CLARKE'S STRICT TANGENT CONE VIA NONLINEAR SEMIGROUPS

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1 proceedings of the american mathematical society Volume 93, Number 1, January 1985 A CHARACTERIZATION OF CLARKE'S STRICT TANGENT CONE VIA NONLINEAR SEMIGROUPS JEAN - PAUL PENOT Abstract. Clarke's strict tangent cone Tf (a) at a e X to a closed subset of a Banach space E is shown to contain the limit inferior of tangent cones Tx(x) to X at iasi->u,iel Several characterizations of Tf (a) are presented. As a consequence various tangential and subtangential conditions for continuous vector fields on X are shown to be equivalent. It is well known that invariance results for dynamical systems on closed subsets of Banach spaces are linked with tangency conditons (see, for instance, [4-7, 10, 12, 16, 17] and their references). On the other hand, some results on tangent cones can be deduced from the study of dynamical systems (see, for instance, [12, 16]). In this note we follow this second streamline in order to give characterizations of the strict tangent cone (or Clarke's tangent cone) to a closed subset in a Banach space. These characterizations were given in [13] under restrictive assumptions, valid for instance in the finite dimensional case. Other studies containing some of these imphcations can be found in [1, 4, 7-9] for instance. 1. Characterizations of the strict tangent cone. Given a subset X of a Banach space E we denote by dx{e) the distance of e g E to X: dx(e) = inf{<7(e, x): x g A"}. Weset7J(a, r) = {x g X: d(a, x) < r). Let us recall that the classical tangent cone (also called contingent cone) at a G A" to A" is the set Tx(a) (also denoted elsewhere T(X, a) or 7^ A") of vectors v g E such that d'x(a, v) < 0, where d'x{a,v)=: liminf rl(dx{a + tw) - dx{a)) (/,h0-»(0+,o) = liminf t'1(dx(a + tv) - dx(a)). t-0+ The strict tangent cone (or Clarke's tangent cone) is the set Tff (a) of vectors v g E such that dx(a, v) < 0, where dx(a,v)= limsup t~1(dx(e + tv) - dx(e)). (f,e)-*(0+,a) Received by the editors December 19, Mathematics Subject Classification. Primary 58F35; Secondary 49A50, 49B27, 58C06. Key words and phrases. Distance function, flow invariance, limit inferior, semigroups, tangent cones, vector fields American Mathematical Society /85 $ $.25 per page

2 CLARKE'S STRICT TANGENT CONE 129 Both cones have been extensively used in optimization and nonlinear analysis. The first one gives a closer approximation to the set X around a than the second one; but it is not necessarily convex. The second one does not necessarily increase with X. Some algebraic links between the two cones have been delineated in [11 and 14]. Here we focus our attention on the relationships between the two cones obtained by taking limits asx->û,xel The following result was shown in [2] to be a consequence of a general ordering principle (see also [6] for related material). Proposition 1. Let F be a closed subset of a Banach space E, let c g R+, to g ]0, + oo [ and let S be a continuous semigroup on E such that (a) d(s(t)x, S(t)y) < e"'d(x, y) for each (t, x, y) g R+x E X E, (b) liminfí^0+ r^ísío*, F) < c for each x G F. Thend(S(t)z, F) < ea'd(z, F) + c( -\eat - 1) for each t G R+ and each z g F. In the following theorem the restrictive assumptions made in [13] are dropped. Theorem 1. Let a be a point of a closed subset X of a Banach space E. For any v g E the following assertions are equivalent: (a)» e 27(a); (b) limsup^fllimsup,^0+ r\dx(e + tv) - dx(e)) < 0; OO limx^a ;tea-hmsup,^0+ rldx{x + tv) = 0; (c) limsupe^aliminf,^0 t~l(dx{e + tv) - dx(e)) < 0; (cohmjt_a>j(.a.limiiif,_0+/-1rf^jc + tv) = 0. Proof. The implications (b) => (b'), (c) => (c'), (b) =» (c) and (b') => (c') are obvious; the implication (a) => (b) follows from the following inequality in which q(t,e) = t-\dx(e + tv)-dx(e)): inf sup inf sup #(r,e)< inf inf sup sup q(t,e). «>0 eej(o,«) 0>O /e]0,/8[ a>0 ß>0 e^b(a.a) /e]0,/3[ Thus it suffices to show that (c') imphes (a). There is no loss of generality in supposing y ^ 1. Let e > 0 be given; we will show that limsup t~1(dx(e + tv) dx(e)) < 3e. (t,e)^(0+,a) Using (c') we can find 5 > 0 such that liinmit_0t~1dx(x + tv) ^ e for each x G X n B, where B = B(a, 38) is the closed ball with center a and radius 3<S. Let F=A"nJ5, let X(e)= fi-1min(8, d(e, Bc)) with Bc = E\B, and let S be the semigroup generated by the lipschitzian vector field V: E -» E given by V(e) = X(e)v. As is well known, the flow of V is defined on R X F as V is globally lipschitzian (with Lipschitz constant «= S"1) so that S is well defined and satisfies condition (a) of Proposition 1 (cf. [3] for instance). Let us check condition (b): for each x g F t-ld(s(t)x, F) < rxd{s{t)x, x + tv(x)) + rld(x + tv{x), F).

3 130 JEAN-PAUL PENOT As limr^0 t~ï(s(t)x - x) = V(x), the first term of the right-hand side has limit 0; the second one has limit inferior X(x)liminfJ_0+5"1J(x + sv, X) < e when d(a, x) < 38, x g X and limit 0 when d(a, x) 38, x g X. Thus condition (b) is satisfied with c = e. For each z g B(a, 8) we have dx(z) < d{z, a) < 8, and any x g A such that d(z, x) < 25 is in B(a, 38), so that dx(z) = d(z, F). On the other hand, as F c A we have ^(z + if) < d(z + tv, F) for any / G R+. Let a g ]0, 8 ] be so small that 2coa < e, r\ewt - 1) < 2u for / g]0, a ]. Then for t g]0, a ], z g 7J(a, a) we have S(t)z = z + tv and r\dx{z + tv) - dx(z)) < ra(d(z + tv, F) - d{z, F)) < rl(«"'~ 1)(í/(z,F) +w^e) < 2co(a + co_1e) < 3e. D 2. Some consequences. Given x G X and u e E we define the contingency coefficient (or tangency coefficient) of y at x with respect to X as Let us set ^.(x, v) = lim inf t~1dx(x + in). r^0+ Tx(x)= {v^e:kx(x,v)^e\\v\\}, so that Tx(x) is a cone and Tx(x) = C\e>0Tx(x). Corollary 1. For any closed subset X of a Banach space and any a G A" one has liminf (x,e)-»(a,0) x(ex, e>0 Tx(x)czTxî(a). Proof. Suppose u belongs to the left-hand side of this inclusion. Then for any a > 0 there exists ß > 0 such that for any x g X n 7?(a, /J) and any e g ]0, /}[ there exists v' g T^x) n 7J(t;, a). Thus /^(x, v) < ^(x, f/) + d(v', v) < e( u + a) + a. As eg]0, ß[ is arbitrary, we get kx(x, v) < a for xelni?(a,j3) whence limx^a. xexkx(x, v) = 0 and u g Tff (a) by Theorem 1. D Corollary 2. Fot* a/jy closed subset X of a Banach space and any a g A" one has lim inf Tx(x) c Tf (a). x»a x«ea" This follows from the preceding corollary and the fact that Tx(x) c Tx(x) for any x G X and any e > 0. Corollary 3. Let X be a closed subset of a Banach space E and let a g E. Suppose v g F is i//c// that for any subset A of X with a in its closure one has v g limsupx^a_xeatx(x). Thenv^ Tf (a).

4 CLARKE'S STRICT TANGENT CONE 131 This follows from the fact that for any relation F: X -» E one has liminff(x) = f] limsupf(x), t"" /(erf x a where j^ is the family of subsets A of X whose closure contains a. Corollary 4. Let V: X -» E be a continuous vector field on a closed subset X of a Banach space E. Then the following assertions are equivalent: (a) for each x G X, V(x) g Tf (x); {h) for each x g A", V(x) g Tx(x); (c) for each x G A", hm,_0 t~ldx(x + tv(x)) = 0; (à) for each x g A", liminf,^0 t~ldx(x + tv(x)) = 0. Proof. The implications (a) => (b), (a) => (c), (c) =» (d), (d) <=> (b) are obvious. Suppose (b) is satisfied. Then for each x g X we have V(x) = limv^x yexv(y), hence V( x ) g lim inf^ _xyex Tx( y ) c Tf ( x ), hence (a) holds true. D Added in proof. Since the present paper has been submitted for publication, the inclusion of Corollary 2 has been proved by different methods and in an independent way in references [18 and 19] below. Both references contain counterexamples showing that the inclusion may be strict. References 1. J. P. Aubin, Gradients généralisés de Clarke, Microcours, Centre de Recherche Mathématique, Université de Montréal CRM 703, H. Brézis and F. E. Browder, A general ordering principle in nonlinear functional analysis. Adv. in Math. 21 (1976), H. Cartan, Calcul différentiel, Hermann, Paris MR 36 #6243; English transi., Houghton Mifflin, Boston, Mass., F. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), K. Deimling, Ordinary differential equations in Banach spaces. Lecture Notes in Math., vol. 546, Springer-Verlag, Berlin and New York, I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), B. Cornet, Contributions à la théorie mathématique des mécanismes dynamiques d'allocation des ressources, Thèse, Univ. Paris-Dauphine (Dec. 1981). 8. J.-B. Hiriart-Urruty, New concepts in nondifferentiable programming. Analyse Non Convexe, Bull. Soc. Math. France, Mémoire n 60,1979, pp _, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res. 4 (1979), V. Lakshmikantham, 77îe current status of abstract Cauchy problem: Nonlinear Systems and Applications (Proc. Internat. Conf. Univ. Texas, Arlington, Tex. 1976), Academic Près, New York, 1977, pp MR 56 # D. H. Martin and G. G. Watkins, Cores of tangent cones and Clarke's tangent cone (preprint) R. H. Martin, Nonlinear operators and differential equations in Banach spaces, Wiley, New York, J.-P. Penot, A characterization of tangential regularity, Nonlinear Analysis, Theory, Methods and Appl. 5 (1981), J.-P. Penot and P. Terpolilli, Cones tangents et singularités. C. R. Acad. Sei. Paris 296 (1983), R. R. Phelps, Support cones in Banach spaces and their applications. Adv. in Math. 13 (1974), MR 49 # P. Volkmann, New proof of a density theorem for the boundary of a closed set, Proc. Amer. Math. Soc. 60 (1976), MR 55 #8761.

5 132 JEAN-PAUL PENOT 17. J. A. Yorke, Invariance for ordinary differential equations, Math. Systems Th. 1 (1967), MR 37 #1695; correction, Math. Systems Th. 2 (1968), 381. MR 38 # S. Dolecki and J.-P. Penot, The Clarke's tangent cone and limits of tangent cones, Publ. Math. Pau (1983), J. S. Treiman, Characterization of Clarke's tangent and normal cones in finite and infinite dimensions, Nonlinear Analysis, Theory, Methods and Appl. 7 (1983), département de mathématiques, université de pau, avenue de l'université 64000, pau, France