Proof of Theorem 2 Ceragioli EXTERNAL STABILIZATION OF DISCONTINUOS SYSTEMS

External Stabilization 119 G2 for each i ∈ {1, . . . , m} only one of the following mutually exclusive conditions holds: – for all p ∈ ∂V t, x p · 1, g i t, x 0, – for all p ∈ ∂V t, x p · 1, g i t, x 0, – for all p ∈ ∂V t, x p · 1, g i t, x = 0; G3 there exists ı ∈ {1, . . . , m} such that for each i ∈ {1, . . . , m}\{ı} only one of the following mutually exclusive conditions holds: – for all p ∈ ∂V t, x p · 1, g i t, x 0, – for all p ∈ ∂V t, x p · 1, g i t, x 0, – for all p ∈ ∂V t, x p · 1, g i t, x = 0; Let us remark that f3 ⇒ f2 ⇒ f1 and G1 ⇒ G2 ⇒ G3. T HEOREM 2. Let V : n+1 → be such that there exists L 0 such that V0 and V1 hold. If for all x ∈ n with kxk L one of the following couples of conditions holds for a.e. t ≥ 0: i f1 and G1, ii f2 and G2, iii f3 and G3, then system 1 is UBIBS stabilizable. Let us make some remarks. If for all x ∈ n with kxk L assumption f1 or f2 or f3 holds for a.e. t ≥ 0, then, by Theorem 1 in Section 2, system 2 is uniformly Lagrange stable. Actually in [4] the authors introduce the concept of robust uniform Lagrange stability and prove that it is equivalent to the existence of a locally Lipschitz continuous Lyapunov-like function. Then assumption f1 or f2 or f3 implies more than uniform Lagrange stability of system 2. In [9], the author has also proved that, under mild additional assumptions on f , robust Lagrange stability implies the existence of a C ∞ Lyapunov-like function, but the proof of this result is not actually constructive. Then we could still have to deal with nonsmooth Lyapunov-like functions even if we know that there exist smooth ones. Moreover Theorem 2 can be restated for autonomous systems with the function V not de- pending on time. In this case the feedback law is autonomous and it is possible to deal with a situation in which the results in [9] don’t help. Finally let us remark that if f is locally Lipschitz continuous, then, by [14] page 105, the Lagrange stability of system 2 implies the existence of a time-dependent Lyapunov-like function of class C ∞ . In this case, in order to get UBIBS stabilizability of system 1, the regularity assumption on G can be weakened to G ∈ L ∞ loc n+1 ; m as in [2].

4. Proof of Theorem 2

We first state and prove a lemma. L EMMA 2. Let V : n+1 → be such that there exists L 0 such that V0 and V1 hold. If t , x , with kxk L, is such that, for all p ∈ ∂V t, x p · 1, g i t, x 0, then there exists δ x 0 such that, for all x ∈ Bx, δ x , for all p ∈ ∂V t, x, p · 1, g i t, x 0. 120 F. Ceragioli Analagously if t , x , with kxk L, is such that for all p ∈ ∂V t, x, p · 1, g i t, x 0, then there exists δ x 0 such that, for all x ∈ Bx, δ x , for all p ∈ ∂V t, x, p·1, g i t, x 0. Proof. Let γ 0 be such that kxk L + γ , and let L x 0 be the Lipschitz constant of V in the set {t} × Bx, γ . For all t, x ∈ {t} × Bx, γ and for all p ∈ ∂V t, x kpk ≤ L x see [5], page 27. Since g i is continuous there exist η and M such that k1, g i t, x k ≤ M in {t} × Bx, η. Let d = min{ p · 1, g i t, x , p ∈ ∂V t, x}. By assumption d 0. Let us consider ǫ d 2L x +M . By the continuity of g i , there exists δ i such that, if kx − xk δ i , then k1, g i t, x − 1, g i t, x k ǫ. By the upper semi-continuity of ∂ V see [5], page 29, there exists δ V 0 such that, if kx − xk δ V , then ∂ V t , x ⊆ ∂V t, x + ǫ B0, 1, i.e. for all p ∈ ∂V t, x there exists p ∈ ∂V t, x such that kp − pk ǫ. Let δ x = min{γ, η, δ i , δ V }, x be such that kx − xk δ x and p ∈ ∂V t, x, p ∈ ∂V t, x be such that kp − pk ǫ. It is easy to see that |p · 1, g i t, x − p · 1, g i t, x | d 2 , hence p · 1, g i t, x p · 1, g i t , x − d 2 = d 2 0. The second part of the lemma can be proved in a perfectly analogous way. Proof of Theorem 2. For each x ∈ n , let N x be the zero-measure subset of + in which no one of the couples of conditions i , ii and iii holds. Let k : n+1 → m , kt, x = k 1 t, x, . . . , k m t, x, be defined by k i t, x =        −kxk if ∀ p ∈ ∂V t, x p · 1, g i t, x 0 if ∀ p ∈ ∂V t, x p · g i t, x = 0 , or f3 and G3 hold and i = ı, or t ∈ N x kxk if ∀ p ∈ ∂V t, x p · 1, g i t, x 0 . It is clear that k ∈ L ∞ loc n+1 , m . By Theorem 1 it is sufficient to prove that for all R 0 there exists ρ L , R such that for all x ∈ n and v ∈ m the following holds: kxk ρ, kvk R ⇒ max ˙V 3 t, x ≤ 0 for all t ∈ + \N x where ˙ V 3 t, x = {a ∈ : ∃w ∈ K f t, x + Gt, xkt, x + Gt, xv such that ∀ p ∈ ∂ V t, x p · 1, w = a}. Let x be fixed and t ∈ + \N x . Let a ∈ ˙V 3 t, x, w ∈ K f t, x + Gt, xkt, x + Gt, xv be such that for all p ∈ ∂V t, x p · w = a. By Theorem 1 in [8] we have that K f t, x + Gt, xkt, x + vx ⊆ K f t, x + P m i=1 g i t, xK k i t, x + v i , then there exists z ∈ K f t, x, z i ∈ K k i t, x + v i , i ∈ {1, . . . , m}, such that w = z + P m i=1 g i t, xz i . Let us show that a ≤ 0. We distinguish the three cases i, ii, iii. i b = p · 1, z = a − P m i=1 c i t,x z i does not depend on p, then b ∈ ˙V 2 t, x and, by f1, b ≤ 0. External Stabilization 121 Let us now show that for each i ∈ {1, . . . , m} c i t,x z i ≤ 0. If i is such that c i t,x = 0, obviously c i t,x z i ≤ 0. If i is such that c i t,x 0 then, by Lemma 1, there exists δ x such that k i t, y = −kyk in {t} × Bx, δ x , then k i is continuous at x with respect to y. This implies that K k i t, x + v i = −kxk + v i , i.e. z i = −kxk + v i and c i t,x z i ≤ 0, provided that kvk ρ ≥ max{L, R}. The case in which i is such that c i t,x 0 can be treated analogously. We finally get that a = b + P m i=1 c i t,x z i ≤ 0. ii By f2 there exists p ∈ ∂V t, x such that p · 1, z ≤ 0. a = p · 1, z + P m i=1 p · 1, g i t, xz i . The fact that for each i ∈ {1, . . . , m} we have p · 1, g i t, xz i ≤ 0 can be proved as in i we have proved that for each i ∈ {1, . . . , m} c i t,x z i ≤ 0. We finally get that a ≤ 0. iii Let us remark that if G2 is not verified, i.e. we are not in the case ii , there exists p ∈ ∂V t, x corresponding to ı such that p · 1, g ı t, x = 0. Indeed, because of the convexity of ∂ V t, x, for all v ∈ n , if there exist p 1 , p 2 ∈ ∂V t, x such that p 1 ·v 0 and p 2 · v 0, then there also exists p 3 ∈ ∂V t, x such that p 3 · v = 0. Let p ∈ ∂V t, x be such that p·1, g ı t, x = 0. For all p ∈ ∂V t, x a = p ·1, w. In particular we have a = p·1, w = p·1, z+ P i6=ı p ·1, g i t, xz i + p·1, g ı t, xz ı . By f3, p · 1, z ≤ 0. If i 6= ı the proof that p · 1, g i t, xz i ≤ 0 is the same as in ii. If i = ı, because of the choice of p, p · 1, g ı t, x = 0. Also in this case we can then conclude that a ≤ 0. References [1] B ACCIOTTI A., External Stabilizability of Nonlinear Systems with some applications, In- ternational Journal of Robust and Nonlinear Control 8 1998, 1–10. [2] B ACCIOTTI A., B ECCARI G., External Stabilizabilty by Discontinuous Feedback, Pro- ceedings of the second Portuguese Conference on Automatic Control 1996. [3] B ACCIOTTI A., C ERAGIOLI F., Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions, Preprint Dipartimento di Matematica del Politecnico di Torino. [4] B ACCIOTTI A., R OSIER L., Liapunov and Lagrange Stability: Inverse Theorems for Dis- continuous Systems, Mathematics of Control, Signals and Systems 11 1998, 101–128. [5] C LARKE F. H., Optimization and Nonsmooth Analysis, Wiley and Sons 1983. [6] F ILIPPOV A. F., Differential Equations with Discontinuous Right Handsides, Kluwer Aca- demic Publishers 1988. [7] M C S HANE E. J., Integration, Princeton University Press 1947. [8] P ADEN B., S ASTRY S., A Calculus for Computing Filippov’s Differential Inclusion with Application to the Variable Structure Control of Robot Manipulators, IEEE Transaction on Circuits and Systems, Vol. Cas-34, No. 1, January 1997, 73–81. [9] R OSIER L., Smooth Lyapunov Functions for Lagrange Stable Systems, Preprint. [10] S HEVITZ D., P ADEN B., Lyapunov Stability Theory of Nonsmooth Systems, IEEE Trans- action on Automatic Control 39 9, September 1994, 1910–1914. 122 F. Ceragioli [11] S ONTAG E. D., A Lyapunov-like Characterization of Asymptotic Controllability, SIAM J. Control and Opt. 21 1983, 462–471. [12] S ONTAG E. D., S USSMANN H., Nonsmooth Control Lyapunov Functions, Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications 1995, 2799– 2805. [13] V ARAIYA P. P., L IU R., Bounded-input Bounded-output Stability of Nonlinear Time- varying Differential Systems, SIAM Journal Control 4 1966, 698–704. [14] Y OSHIZAWA T., Stability Theory by Liapunov’s Second Method, The Mathematical Soci- ety of Japan 1966. AMS Subject Classification: ???. Francesca CERAGIOLI Dipartimento di Matematica “U. Dini” Universit`a di Firenze viale Morgagni 67A 50134 Firenze, ITALY e-mail: ✄ ✂ ✆ ✁ ✞ ✟ ✠ ✆ ✝ ✆ ✡ ☛ ✄ ☞ ✌ ✡ ✟ ✝ ✆ ✍ ✆ ✡ ✆ ☞ Rend. Sem. Mat. Univ. Pol. Torino Vol. 56, 4 1998

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