Asymptotic stability in hybrid systems via nested Matrosov functions

Transcription

1 Asympoic sabiliy in hybrid sysems via nesed Marosov funcions Ricardo G. Sanfelice and Andrew R. Teel Absrac A heorem on nesed Marosov funcions is exended o ime-varying hybrid sysems. I provides sufficien condiions for uniform global asympoic sabiliy of a compac se. An applicaion o parameer idenificaion wih sae reses is made and illusraed on an example. I. INTRODUCTION Marosov s heorem provides sufficien condiions for uniform global asympoic sabiliy of he origin in ime-varying differenial equaions. The sufficien condiion repored by Marosov [3] is ha, given a coninuously differeniable (C ) funcion V ha esablishes uniform global sabiliy of he origin, here exiss an auxiliary C funcion wih derivaive ha is definiely nonzero in he se where he derivaive of V vanishes. Several alernaive versions of Marosov s heorem have appeared in he lieraure; see [8] and he references herein. Marosov s heorem has been applied o nonlinear conrol problems, including racking conrol [7], oupu feedback [6], and adapive conrol [2], among ohers. The mos recen versions of Marosov s heorem have provided exra flexibiliy by using muliple auxiliary funcions raher han only one as in he original work of Marosov. For coninuous-ime sysems see [8], where five auxiliary funcions are used in sabiliy analysis for nonholonomic vehicles, and [2], where 3n 2 auxiliary funcions are used for he inerconnecion of n subsysems; for discree-ime sysems see [5] and []. These auxiliary funcions need o saisfy nesed condiions specifying he poins where hey are negaive. A Marosov heorem wih one auxiliary funcion bu a weaken negaiviy condiion, expressed in erms of persisency of exciaion, has been proposed for a class of single-valued, ime-varying hybrid sysems in [] (see also []). In [], and also in [4], hese condiions have been shown o faciliae he consrucion of sricly decreasing Lyapunov funcions. To he bes of our knowledge, all insances of Marosov s heorem in he lieraure have focused on ime-varying sysems. In his noe, we emphasize ha i can also be used for ime-invarian sysems o assis in applying invarianceprinciple-based sabiliy analysis ools; see [9], [4], and [9]. In fac, o apply Marosov s heorem, neiher noions of invariance nor specific condiions guaraneeing sequenial compacness of soluions are needed for is applicaion. Building from he ideas in [8], we develop a nesed Marosov heorem for hybrid sysems allowing for se-valued dynamics, nonuniqueness of soluions, and Zeno soluions. R.G. Sanfelice: AME Dep., Universiy of Arizona, AZ 8572, Tel (52) , Fax (52) 62-89, sricardo@ .arizona.edu; A.R. Teel: ECE Dep., Universiy of California, Sana Barbara, CA , eel@ece.ucsb.edu. Research parially suppored by NSF grans CNS and ECS , and by AFOSR gran FA Afer a brief inroducion o hybrid sysems and sabiliy in Secion II, in Secion III we presen he resul for imeinvarian sysems and apply i o he bouncing ball sysem. In Secion IV, we give he main resul, which is for imevarying sysems, and apply i o parameer idenificaion wih sae reses. Is proof is given in Secion V. R n denoes n-dimensional Euclidean space, R he real numbers, R := [, ), Z he inegers, and Z k he inegers greaer han or equal o he ineger k. B is he open uni ball in an Euclidean space. Given a se S, S is is closure. Given S R n and a poin x R n, x S := inf y S x y, where is Euclidean norm. Given S R n and consans δ,, δ, Ω S (δ, ) := x R n δ x S }. A funcion α : R R is said o belong o class-k if i is coninuous, zero a zero, sricly increasing, and unbounded. II. HYBRID SYSTEMS A. Modeling framework and soluions We follow he presenaion in [6] and [7]. Cf. [2], [2], [9]. Hybrid sysems are dynamical sysems wih a sae x R n ha can change coninuously during flows, and disconinuously a jumps. The sae may include physical variables, like posiions and velociies, as well as logic variables aking values like on and off, which are ypically idenified wih inegers embedded in Euclidean space. A hybrid sysem H is defined by four objecs comprising is daa: Flow map: A se-valued mapping F : R n R n defining he flows (or coninuous evoluion). Flow se: A se C R n specifying he se of poins where flows are possible. Jump map: A se-valued mapping G : R n R n defining he jumps (or discree evoluion). Jump se: A se D R n specifying he se of poins where jumps are possible. A hybrid sysem H := (F, C, G, D) can be wrien in he form ẋ F(x) x C H : x + G(x) x D he soluions for which are now made precise. Definiion 2.: (hybrid ime domain) A se E R Z is a compac hybrid imej domain if E = ([ j, j+ ], j) j= for some finie sequence of imes = 2... J. A se E R Z is a hybrid ime domain if for all (T, J) E, E ([, T],,...J}) is a compac hybrid ime domain. Definiion 2.2: (hybrid arc) A funcion x : domx R n is a hybrid arc if domx is a hybrid ime domain and if for

2 2 each j Z, he funcion x(, j) is locally absoluely coninuous. A hybrid arc x is a soluion o he hybrid sysem H if x(, ) C D and: (S) For all j Z and almos all such ha (, j) dom x, x(, j) C and ẋ(, j) F(x(, j)). (S2) For all (, j) domx such ha (, j + ) domx, x(, j) D and x(, j + ) G(x(, j)). A soluion x is called maximal if here does no exis a soluion x such ha x is a runcaion of x o some proper subse of domx. The resuls in [7] give mild condiions on he daa (F, C, G, D) o guaranee cerain regulariy properies for he se of soluions o a hybrid sysem. These condiions are criical for sequenial compacness of soluions and inheren robusness of asympoic sabiliy [7], invariance principles [9], and converse Lyapunov heorems [3]. However, hese condiions are no criical in sufficien condiions for nominal asympoic sabiliy, like hose in his paper. B. Globally asympoically sable ses To esablish sufficien condiions for uniform global asympoic sabiliy for hybrid sysems, we consider closed, no necessarily compac, ses. Definiion 2.3: (UGS, UGA & UGAS) The closed se A R n for he sysem H is said o be uniformly globally sable (UGS) if here exiss α K such ha any soluion x saisfies x(, j) A α( x(, ) A ) for all (, j) domx; uniformly globally aracive (UGA) if for each ε > and r > here exiss T > such ha, for any soluion x, x(, ) A r, (, j) domx, and + j T imply x(, j) A ε; uniformly globally asympoically sable (UGAS) if i is boh UGS and UGA. UGAS does no imply ha soluions exis from every poin in R n nor ha maximal soluions have unbounded ime domains. A sufficien condiion for UGS of a closed se A is given nex. Theorem 2.4: The closed se A R n is UGS for he hybrid sysem H = (F, C, G, D) if here exiss a funcion V : R n R, C on an open se conaining C, and class-k funcions α, α 2 such ha α ( x A ) V (x) α 2 ( x A ) for all x C D G(D), sup f F(x) V (x), f for all x C, and sup g G(x) V (g) V (x) for all x D. C. Time-varying sysems Time-varying hybrid sysems wih sae x R n have he form ẋ F(x, τ, k) (x, τ, k) C, H v : x + () G(x, τ, k) (x, τ, k) D, where τ incremens wih ordinary ime and k incremens wih jumps, and C, D R n R Z. Time-varying sysems can be convered o ime-invarian sysems by considering he augmened sysem wih sae (x, τ, k) } ẋ F(x, τ, k) (x, τ, k) C, τ =, k = H aug : } x + G(x, τ, k) τ + = τ, k + (x, τ, k) D. = k + Then UGAS of a compac se A for he sae x in H v is idenified wih UGAS of he closed (no compac) se A R Z for he augmened sysem H aug. This requires giving special aenion o he possibly unbounded saes τ and k. See Secion IV. When H v is periodic, wih period T > wih respec o τ and period N Z wih respec o k, he augmened sysem can be wrien as ẋ F(x, z, k) ż = [ ] (x, z, k) C, z2 2πT z z S, k,...,n}, H v,p : k = x + } G(x, z, k) (x, z, k) D, z + = z z S, k + = (k mod N) + k,...,n}, where S denoes he uni circle and : S [, 2π) is such ha z denoes he angle, posiive in he counerclockwise direcion, beween z and he posiive horizonal axis. In his case, UGAS of a compac se A for he sae x in H v is idenified wih UGAS of he compac se A S,...,N} for H v,p. Thus, he ime-invarian resul in Secion III can be applied direcly o periodic ime-varying hybrid sysems. III. NESTED MATROSOV FUNCTIONS: THE TIME-INVARIANT CASE We firs sae a Marosov heorem for ime-invarian hybrid sysems. I provides sufficien condiions for UGAS of compac ses ha relax classical Lyapunov condiions. I is a convenien alernaive o LaSalle s invariance principle for esablishing araciviy of a sable compac se. In conras o invariance principles, no knowledge abou soluions is required. Like Lyapunov heorems, only bounds on derivaives and differences mus be esablished. Theorem 3.: (Time-invarian nesed Marosov) Le A R n be a compac, UGS se for he hybrid sysem H = (F, C, G, D). A is UGAS if here exis m Z and, for each < δ <, a number µ >, coninuous funcions u c,i : C Ω A (δ, ) R, u d,i : D Ω A (δ, ) R, i, 2,..., m}, funcions V i : R n R, i, 2,...,m}, C on an open se conaining C Ω A (δ, ), such ha, for each i, 2,..., m}, V i (x) µ x (C D) Ω A (δ, ), (2) sup V i (x), f u c,i (x) x C Ω A (δ, ), (3) f F(x) When UGAS holds for he soluion concep ha uses (S), i also holds for he more resricive soluion concep where x(, j) C for all excep possibly a he beginning and end of inervals of non-zero lengh. When C is closed, hese wo soluion conceps are equivalen. sup V i (g) V i (x) u d,i (x) g G(x) (C D) Ω A(δ, ) x D Ω A (δ, ), (4)

3 3 and, wih he consan funcions u c,, u d, : R n } and u c,m+, u d,m+ : R n }, for each j,,...,m}, ) if x C Ω A (δ, ) and u c,i (x) = for all i,,...,j} hen u c,j+ (x), 2) if x D Ω A (δ, ) and u d,i (x) = for all i,,...,j} hen u d,j+ (x). The heorem imposes a nesed negaive semi-definie condiion on he funcions u c,i and u d,i, which bound he change in V i along flows and jumps, respecively. Through he definiion of u c, and u d,, he nesed condiion requires ha u c, and u d, are never posiive. The funcion u c,2 (respecively, u d,2 ) can be posiive only where u c, (respecively, u d, ) is negaive, and so on. Finally, hrough he definiions of u c,m+ and u d,m+, here are no poins in Ω A (δ, ) where all of he u c,i (respecively, u d,i ) are zero. The exisence of µ saisfying (2) is guaraneed when V i is coninuous on Ω A (δ, ). However, coninuiy is no required in general. The heorem is saed for funcions V i ha are coninuously differeniable on an open se conaining C Ω A (δ, ), bu a similar resul holds for funcions locally Lipschiz on his se. Such a resul requires working wih a generalized noion of derivaive, like he Clarke generalized gradien [5]. When he firs funcion in Marosov s heorem saisfies he condiions in Theorem 2.4, i can be used o esablish he required UGS propery. This fac and Theorem 3. are illusraed nex. Example 3.2: (Bouncing ball) Consider a ball bouncing on he ground wih verical posiion x and verical velociy x 2. In beween bounces, he equaions of moion are given by ẋ = x 2, ẋ 2 = γ, where γ > is he graviaional consan and he sae x := (x, x 2 ) is in he se C := x R 2 x > }. Bounces occur when he sae x is in he se D := x R 2 x = and x 2 < } wih he updae rule x + =, x+ 2 = x 2, where [, ) is he resiuion coefficien. There are various ways o show ha A := (, ) (or even he sysem wih C and D replaced by heir closures) is UGAS. In [3] a sricly decreasing Lyapunov funcion was presened. Invariance principles, like hose in [9], can also be applied using he energy funcion V (x) := 2 x2 2 + γx. To use Marosov s heorem, we sar wih his same funcion V and find ha he condiions of Theorem 2.4 hold, so ha he origin is UGS, and condiions (3) and (4) in Theorem 3. hold for i = wih u c, (x) := x C, u d, (x) := 2 ( 2)x 2 2 x D. Since hese funcions are never posiive, iems ) and 2) in Theorem 3. hold for j =. In fac, since here are no poins ouside of A where u d, is zero (poins in D have x = ), iem 2 in Theorem 3. will hold for all j no maer wha u d,i is for i >. Nex, we pick V 2 : R 2 R o be given by V 2 (x) := γx 2. Condiions (3) and (4) in Theorem 3. hold for i = 2 wih u c,2 (x) := γ 2 x C, u d,2 (x) := γ( + )x 2 x D. Since u c,2 is always negaive, iem ) of Theorem 3. holds for all j. The origin is UGAS. To show UGAS of he origin of he sysem in Example 3.2 via invariance principles, i is required o define a noion of invariance, verify cerain regulariy condiions under which he invariance principle is applicable, as well as have rudimenary knowledge of he sysem soluions. The applicaion of Theorem 3. only requires he abiliy o consruc he Marosov funcions as Example 3.2 demonsraes. Anoher applicaion of Theorem 3. is given in [8, Example 4.2]. A. Main resul IV. NESTED MATROSOV FUNCTIONS: THE TIME-VARYING CASE Like in [5], [8], [], [], he ime-varying version is given in he spiri of he resul in [7], which is less general han Marosov s original resul bu is easier o sae and check. Given A R n, define Υ A (δ, ) := Ω A (δ, ) R Z. Given a se S R n R Z, define Π(S) := x R n (x, τ, k) S for some (τ, k) R Z }. Theorem 4.: (Time-varying nesed Marosov) Le A R n be a compac, UGS se for he hybrid sysem H v = (F, C, G, D). A is UGAS if here exis m, s Z and, for each < δ <, a number µ >, a funcion φ : R n R ( Z R s, ) coninuous funcions u c,i : Π(C) Ω A (δ, ) R s ( ) R, u d,i : Π(D) Ω A (δ, ) R s R, i, 2,...,m}, funcions V i : R n R Z R, i, 2,...,m}, C on an open se conaining C Υ A (δ, ), such ha, for each i, 2,..., m}, max V i (x, τ, k), φ(x, τ, k) } µ (x, τ, k) (C D) Υ A (δ, ), sup x V i (x, τ, k), f + τ V i (x, τ, k) u c,i (x, φ(x, τ, k)) f F(x,τ,k) (x, τ, k) C Υ A (δ, ), (6) sup V i (g, τ, k + ) V i (x, τ, k) g G(x,τ, k) Ω A(δ, ) (g,τ, k + ) (C D) u d,i (x, φ(x, τ, k)) (x, τ, k) D Υ A (δ, ), (7) and, wih he consan funcions u c,, u c,m+ : R n+s } and u d,, u d,m+ : R n+s }, for each j,,..., m}, ) if x Π(C) Ω A (δ, ), ψ µ, and u c,i (x, ψ) = for all i,,...,j}, hen u c,j+ (x, ψ), 2) if x Π(D) Ω A (δ, ), ψ µ, and u d,i (x, ψ) = for all i,,...,j}, hen u d,j+ (x, ψ). I can be verified ha he condiions for uniform asympoic sabiliy in [], [] can be cas as hose in Theorem 4.. The uiliy of Theorem 4. for coninuous-ime and discreeime sysems has been illusraed in [5], [8], [2]. Nex, we illusrae is usefulness for ime-varying hybrid sysems H v. (5)

4 4 B. Applicaion: Parameer idenificaion wih sae reseing Consider he class of sysems wih sae (ξ, ζ, ρ, τ) R n R n2 R R, flow and jump ses C := R n R n2 [, T 2 ] R, D := R n R n2 [T, T 2 ] R, respecively, where < T T 2 <, and dynamics ξ = Aξ + B(τ, ξ)ζ, ζ = exp( βρ)γb (τ, ξ)pξ, ρ = (8) when (ξ, ζ, ρ, τ) C, and ξ + ϕ(ξ), ζ + = ζ, ρ + =, (9) when (ξ, ζ, ρ, τ) D, where A, P R n n, B : R R n R n n2, β R, Γ R n2 n2, and ϕ : R n R n. In he case where β = and T = T 2 =, he sysem (8)-(9) corresponds o he dynamics of a classical parameer idenificaion algorihm. In paricular, for he sysem η = χ(, η) + ϑ(, η)θ where θ R n2 is an unknown consan vecor, consider he parameer idenificaion algorihm ˆη = A(ˆη η) + χ(, η) + ϑ(, η)ˆθ, ˆθ = Γϑ (, η)p(ˆη η). Defining ξ := ˆη η, ζ := ˆθ θ, and B(τ, ξ) := ϑ(τ, η(τ)) resuls in he sysem (8)-(9) wih β =, and T = T 2 =, so ha D is empy. See also he sysems considered in [8]. For he sysem (8)-(9), we make he following assumpions: Assumpion 4.2: The marices Γ and P are symmeric, posiive definie, v ϕ(ξ) = v Pv min exp( βt ), exp( βt 2 )} ξ Pξ, and here exis λ (, β) such ha A P + PA λp. Assumpion 4.3: The funcion τ B(τ, ) is coninuously differeniable. Moreover, here exis sricly posiive real numbers σ, ς, ε, and T, and α K, such ha, for all τ, ) B(τ, ) σ, 2) d B(τ, ) dτ ς, 3) B(τ, ξ) B(τ, ) α( ξ ), and 4) εi τ+t τ B (s, )B(s, )ds. The nex resul for he sysem (8)-(9) follows from Theorem 4.. Corollary 4.4: For he sysem (8)-(9) and under Assumpions 4.2 and 4.3, he compac se A := } } [, T 2 ] is UGAS. Proof. Le x := (ξ, ζ, ρ), F(x, τ) := (Aξ + B(τ, ξ)ζ, exp( βρ)γb (τ, ξ)pξ, ) for all (x, τ) C and G(x, τ) := (ϕ(ξ), ζ, ) for all (x, τ) D. We esablish UGS of A. Consider he Lyapunov funcion candidae V (x) := exp( βρ)ξ Pξ + ζ Γ ζ. There exis sricly posiive real numbers α and α such ha α (x, τ) 2 A V (x) α (x, τ) 2 A for all (x, τ) C D. Using Assumpion 4.2, for all (x, τ) C, V (x), F(x, τ) = β exp( βρ)ξ Pξ + exp( βρ)ξ (A P + PA)ξ + exp( βρ)2ξ PB(τ, ξ)ζ 2ζ exp( βρ)b (τ, ξ)pξ (β λ)exp( βρ)ξ Pξ, and, for all (x, τ) D and g G(x, τ), V (g) γξ Pξ + ζ Γ ζ V (x), where γ := min exp( βt ), exp( βt 2 )}. I follows from Theorem 2.4 ha A is UGS. Now we use Theorem 4. o esablish UGAS of A. Using Assumpion 4.2, le κ > be such ha, for all v ϕ(ξ), v v (exp(κ T ) )ξ ξ. Also, le c >. Then define V (x, τ) := V (x), V 2 (x, τ) := exp(κ ρ)ξ ξ, V 3 (x, τ) := ζ B (τ, )ξ, V 4 (x, τ) := τ exp(τ s) B(s, )ζ 2 ds, V 5 (x, τ) := exp(cρ) ζ 2, φ(x, τ) := B(τ, )ζ. Using Assumpion 4.3, i follows for each > here exiss µ > such ha (ξ, ζ) implies φ(x, τ) µ and V i (x, τ) µ for i, 2,..., 5}. Moreover, V 4 (x, τ) ε exp( T) ζ 2 for any T >. Define u c, (x, ψ) := (β λ)exp( βρ)ξ Pξ, u c,2 (x, ψ) := exp(κ T 2 )(κ ξ A ξ 2 +2(σ + α( ξ )) ξ ζ ), u c,3 (x, ψ) := ψ 2 + σ ζ 2 α( ξ ) + σ A ξ ζ + ς ξ ζ +σ (σ + α( ξ )) exp( βρ) ξ 2 P Γ, u c,4 (x, ψ) := ε exp( T) ζ 2 + ψ 2 +2σ 2 exp( βρ) Γ P (σ + α( ξ )) ξ ζ, u c,5 (x, ψ) := c exp(cρ) ζ 2 + exp((c β)ρ)2(σ + α( ξ )) Γ P ζ ξ, where by he norm of a marix we mean is marix 2-norm. Using Assumpion 4.3, rouine calculaions esablish he bound (6) for i, 2,...,5}. Using Assumpion 4.2, le κ 2 > be such ha, for all v ϕ(ξ), v κ 2 ξ. Then define u d, (x, ψ) :=, u d,2 (x, ψ) := ξ ξ, u d,3 (x, ψ) := (κ 2 + )σ ξ ζ, u d,4 (x, ψ) :=, u d,5 (x, ψ) := ( exp(ct )) ζ 2. Using Assumpion 4.3, rouine calculaions esablish he bound (7) for i, 2,...,5}. Finally, i is also sraighforward o verify condiions ) and 2) of Theorem 4.. Example 4.5: Consider he sysem (8)-(9) wih B(τ, ξ) = [ max, sin(τ)} min, sin(τ)} ], A =, P =, Γ = I, β =, T = T 2 = π, and ϕ(ξ) =. Assumpions 4.2 and 4.3 are saisfied excep for he condiion ha B is coninuously differeniable. In his case, he funcion V 3 in he proof of Corollary 4.4 would be only locally Lipschiz. However, he argumens for UGAS go hrough in he same way in his case bu using a more general noion of derivaive, like he Clarke generalized gradien [5]. Figure shows soluions, projeced ono he ordinary ime axis, for his sysem compared o soluions when using ϕ(ξ) = ξ, i.e., he sysem wihou reses. Because of he srucure of B in his example, sae reseing is able o decouple he idenificaion of he wo unknown parameers in he sysem. In paricular, when he iniial esimae of one parameer is correc, i remains correc hroughou he process of idenifying he second parameer.

5 5 ξ ξ ζ ζ ζ 2 ζ (a) (b) (c) (d) (e) (f) Fig.. Soluions o sysem (8)-(9) projeced ono he axis. (a)-(c) depic ξ, ζ, and ζ 2 for he case wihou reses while (d)-(f) show he case wih reses. In boh cases, he iniial condiions are ξ(, ) =, ζ (, ) =, and ζ 2 (, ) =. For he case wihou reses, ζ leaves zero a = 2π indicaing ha he esimae of θ becomes incorrec. On he oher hand, for he case wih reses ˆθ θ. V. PROOFS Theorem 3. follows from Theorem 4.. Our proof of Theorem 4. uses ideas in [5], [8]. A. Behavior of soluions UGS is assumed; we esablish UGA. Consider he hybrid sysem H v = (F, C, G, D), where G(x, τ, k) := g G(x, τ, k) (g, τ, k + ) C D } for each (x, τ, k) D. By assumpion, A is UGS for H v. Le ε > and r >. Le α K come from he UGS propery of H v and define := α(r) and δ = α (ε). UGA is esablished if here exiss T > such ha, for each soluion x o H v, he se Θ T (x) := (, j) domx T + j, x(s, i) Ω A (δ, ) (s, i) domx, s + i + j} is empy. To esablish his fac, we will use he following lemma which will be proved laer. Lemma 5.: Under he condiions of Theorem 4. for H v, here exis a funcion V :R n R Z R, C on an open se conaining C Υ A (δ, ), and numbers η, ρ > such ha max V (x, τ, k), φ(x, τ, k) } η (x, τ, k) (C D) Υ A (δ, ), sup x V (x, τ, k), f + τ V (x, τ, k) ρ f F(x,τ,k) (x, τ, k) C Υ A (δ, ), () () sup V (g, τ, k + ) V (x, τ, k) ρ g G(x,τ,k) Ω e A(δ, ) (2) (x, τ, k) D Υ A (δ, ). Using his lemma, we ake T > 2η/ρ. Now, suppose here exiss a soluion x o H v such ha Θ T (x) is nonempy, i.e., here exiss (, j) domx such ha + j T and x(s, i) Ω A (δ, ) Π(C D) (he laer se since all soluions x o H v ake values in his se) for all (s, i) domx wih s+i +j. I follows using () for x(, ) and x(, j), denoing he iniial value of (τ, k) by (τ, k ), inegraing and summing ()-(2), and using ha x(s, i) Π(C) for almos all imes s, ha η Ṽ (, j) Ṽ (, ) ( + j)ρ η Tρ < η, where Ṽ (, j) := V (x(, j), + τ, j + k ). This being impossible, we conclude ha Θ T (x) is empy for each soluion x and A is UGA for H v. Now, we show ha his implies ha A is UGA for H v. For arbirary ε, r >, le α K come from he UGS propery of H v and define δ = α (ε). Le T come from he UGA propery of Hv wih parameers δ and r (δ plays he role of ε in Definiion 2.3). For soluions x o H v wih x(, ) r ha are also soluions o H v here is nohing o check. Le x be a soluion o H v wih x(, ) r ha is no a soluion o H v. Then, here exiss (, j ) domx such ha x(, j ) C D. I follows ha he soluion resuling from runcaing x up o (, j ) is a soluion o H v. Then, using he UGA propery of H v, when +j T we have ha x(, j) A δ, +j +j, (, j) domx. Using he UGS propery of H v, UGA of A for H v follows wih parameers ε, δ, and T + (in he order inroduced in Definiion 2.3). B. Proof of Lemma 5. We use he following auxiliary lemmas, which are adapaions of Claim and Claim 2 in [8], culminaing in a heorem ha generalizes he nonlinear version of Finsler s lemma given in [, Theorem A.]. Lemma 5.2: Le Ψ R p be a nonempy compac se, Y i : Ψ R, i, 2,..., m}, be coninuous funcions and Y : Ψ }, Y m+ : Ψ } be consan funcions such ha for each j m, m}, if Y i (z) = for all i,,...,j} hen Y j+ (z). Then, here exiss ε > such ha: (3) Y i (z) = for all i, 2,..., m } imply Y m (z) ε. (4) Proof: By conradicion, suppose ha for each n Z here exis z n Ψ such ha Y i (z n ) = for all i,,..., m } and Y m (z n ) > n. By compacness of Ψ, he coninuiy of Y m, and he propery (3) wih j = m, he sequence z n } i= has an accumulaion poin z Ψ such ha Y i (z ) = for all i,,..., m}. Then, using he propery (3) wih j = m, his implies ha Y m+ (z ). This is a conradicion since, by definiion, Y m+ (z ) =. Lemma 5.3: Le Ψ R p be a nonempy compac se, Y i : Ψ R, i, 2,..., m}, m 2, be coninuous funcions and Y : Ψ }, Y m+ : Ψ } be consan funcions such ha for each j,,..., m}, if Y i (z) = for all i,,..., j} hen Y j+ (z). (5) Le l 2, 3,..., m}, ε >, and a coninuous funcion Ỹ l : Ψ R be given. Then, Propery implies Propery 2.

6 6 Propery : A) Y i (z) = for all i, 2,..., l } implies B) Ỹl(z) ε. Propery 2: here exiss K l > such ha: A) Y i(z) = for all i, 2,..., l 2} implies B) K l Y l (z)+ Ỹ l (z) ε/2 for all K l K l. Proof: By propery (5), Propery 2A implies Y l (z). Therefore, Propery 2A implies K l Y l (z) + Ỹl(z) Ỹl(z) K l. If Y l (z) = hen, due o Propery, Propery 2B holds for all K l whenever Propery 2A holds. We claim furher ha here exiss τ > such ha Propery 2B holds whenever Propery 2A holds and Y l (z) > τ. Suppose no, ha is, for each ineger n here exiss z n Ψ such ha Y l (z n ) > n and Ỹ l (z n ) > ε 2. (6) Then, by compacness of Ψ, coninuiy of Y l, and he nesed propery (5), he sequence z n } n= has an accumulaion poin z Ψ such ha Y l (z ) =. Then, here exiss a subsequence of z n } n=, which we will no relabel, converging o z. Then, Propery implies Ỹl(z ) ε. By coninuiy of Ỹ his conradics (6) for large enough n. I follows from he coninuiy of Ỹl and compacness of Ψ ha we can pick Kl > large enough o saisfy max z Ψ Ỹ l (z) τkl ε/2. Hence, Propery 2A implies Propery 2B. Theorem 5.4: Le Ψ c, Ψ d be compac subses of R p, Y c,i : Ψ c R, Y d,i : Ψ d R, i, 2,..., m}, be coninuous funcions and Y c,, Y d, : R p }, Y c,m+, Y d,m+ : R p } be consan funcions such ha, for each j,,..., m}, ) if z Ψ c and Y c,i (z) = for all i,,..., j} hen Y c,j+ (z), 2) if z Ψ d and Y d,i (z) = for all i,,..., j} hen Y d,j+ (z). Then here exis K i >, i, 2,...,m }, and ρ > such ha m m i= K iy c,i (z) + Y c,m (z) ρ z Ψ c, i= K (7) iy d,i (z) + Y d,m (z) ρ z Ψ d. Proof: If eiher Ψ c, Ψ d, or boh ses are empy, hen here is nohing o check for he corresponding inequaliies in (7). By assumpion, (3) in Lemma 5.2 holds for boh ses of funcions Y c,i and Y d,i. Apply Lemma 5.2 o each se of funcions o generae ε c >, ε d > and define ε := min ε c, ε d }. Lemma 5.2 implies ha Propery of Lemma 5.3 holds, for boh ses of funcions, wih l = m, ε = ε, and Ỹc,l = Y c,m and Ỹd,l = Y d,m. Since Propery A holds, Propery 2A holds. Then, from Propery 2 of Lemma 5.3, here exiss K m > such ha z Ψ c, Y c,i (z) =, i, 2,...,m 2} implies K m Y c,m (z) + Y c,m (z) ε 2 z Ψ d, Y d,i (z) =, i, 2,...,m 2} implies K m Y d,m (z) + Y d,m (z) ε 2 Then, Lemma 5.3 can be applied again wih l = m, ε = ε/2, and Ỹc,l = K m Y c,m + Y c,m and Ỹd,l = K m Y d,m +Y d,m. Proceeding in his way, he resul holds wih ρ := ε/2 m. ( Lemma 5. now follows ) from Theorem ( 5.4 by seing) Ψ c := Π(C) Ω A (δ, ) µb, Ψ d := Π(D) Ω A (δ, ) µb, Y c,i := u c,i, Y d,i := u d,i, i, 2,..., m}, and hen aking V := m i= K iv i + V m and η := µ + m i= K iµ. REFERENCES [] K. J. Arrow, F. J. Gould, and S. M. Howe. A general saddle poin resul for consrained opimizaion. Mahemaical Programming, 5: , 973. [2] M.S. Branicky, V. S. Borkar, and S. K. Mier. 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