Non-uniform stabilization of control systems

Transcription

1 IMA Journal of Mathematical Control and Information 2002) 19, Non-uniform stabilization of control systems IASSON KARAFYLLIS Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece [Received on 10 March 2002; accepted on 6 June 2002] A version of non-uniform in time robust global asymptotic stability is proposed and enables us to derive: 1) sufficient conditions for the stabilization of uncertain nonlinear triangular time-varying control systems; 2) sufficient conditions for the solution of the partial-state global stabilization problem for autonomous systems. The results are obtained via the method of integrator backstepping and are generalizations of the existing corresponding results in the literature. Keywords: integrator backstepping; time-varying feedback; time-varying systems. 1. Introduction The notion of non-uniform in time Robust Global Asymptotic Stability RGAS) has been proved to be fruitful for the solution of several problems in Control Theory see Karafyllis & Tsinias, 2002a,b; Tsinias & Karafyllis, 1999; Tsinias, 2000) for applications to tracking problems and to the robust stabilization of uncertain systems that cannot be stabilized by continuous static time-invariant feedback and Karafyllis & Tsinias, 2001) for the extension of the notion of Input-to-State Stability ISS) to the time-varying case). It is shown in Karafyllis & Tsinias 2001, 2002b) that, even for autonomous systems for which uniform in time asymptotic stabilization by a continuous static feedback is not feasible, it is possible to exhibit non-uniform in time asymptotic stabilization by means of a smooth time-varying feedback. In this paper our interest is focused on uncertain nonlinear time-varying triangular systems. In order to find sufficient conditions for the robust stabilization of such systems, we first strengthen the notion of Robust Global Asymptotic Stability RGAS) given in Karafyllis & Tsinias 2001), by introducing the notion of φ-rgas in such a way that it allows the estimation of the rate of convergence to the equilibrium point. Roughly speaking, for the system ẋ = f t, x, d) x R n, t 0, d D 1.1) where D R m is a compact set and f t, 0, d) = 0 for all t, d) R + D, wesay that 0 R n is φ-rgas if it is in general non-uniformly in time RGAS and particularly, there exists a smooth function φ :R + [1, + ) such that every solution of 1.1) satisfies the following property: lim t + φ p t) xt) =0, p 0 1.2) c The Institute of Mathematics and its Applications 2002

2 420 I. KARAFYLLIS In Section 3 we develop the main tool for the integrator backstepping method that it is used in this paper, and in Section 4 this tool is used for the following triangular system: ẋ i = f i t,θ,x 1,...,x i ) + g i t,θ,x 1,...,x i )x i+1 i = 1,...,n u := x n+1 1.3) x = x 1,...,x n ) T R n, t 0, u R where the uncertainty θ = θt) is any measurable function taking values in a compact set Ω R l.weobtain a set of sufficient conditions Proposition 4.1) for the robust global asymptotic stabilization of 1.3), which is a direct generalization of the corresponding set of sufficient conditions given in the literature for the autonomous case Jiang et al., 1994; Tsinias, 1996). The problem of the stabilization by means of partial-state time-varying feedback is also addressed Proposition 4.2). Specifically, we study systems of the form ż = f 0 t,θ,z, x 1 ) ẋ i = f i t,θ,z, x 1,...,x i ) + g i t,θ,z, x 1,...,x i )x i+1 i = 1,...,n 1.4) u := x n+1 where z R m, x = x 1,...,x n ) T R n, u R, θ = θt) is any measurable function taking values in a compact set Ω R l, and we obtain sufficient conditions for the robust global asymptotic stabilization of 1.4) by means of a partial state smooth time-varying feedback of the form u = kt, x). Using these results, we next study the applications of time-varying feedback to autonomous control systems. In Section 5, the following two applications of time-varying feedback to autonomous control systems are studied: 1) We prove that for every function φ ) there exists a smooth time-varying feedback of the form u = kt, x),such that 0 R n is φ-rgas for the system ẋ i = f i θ, x 1,...,x i ) + g i θ, x 1,...,x i )x i+1 i = 1,...,n 1.5) u := x n+1 where x = x 1,...,x n ) T R n, u R, θ = θt) is any measurable function taking values in a compact set Ω R l Corollary 5.1). Roughly speaking, this means that we can design a smooth time-varying feedback so that the solutions of 1.5) converge to the equilibrium point as fast as desired. We emphasize that this feature cannot be accomplished by the use of locally Lipschitz time-invariant feedback. 2) The stabilization of autonomous systems by means of partial-state smooth timevarying feedback Theorem 5.4). Specifically, consider the system ż = f 0 z, x, u) 1.6a) ẋ i = f i θ, x 1,...,x i ) + g i θ, x 1,...,x i )x i+1 i = 1,...,n u := x n+1 1.6b) where z R m, x = x 1,...,x n ) T R n, u R, θ = θt) is any measurable function taking values in a compact set Ω R l, f 0, f i and g i are continuous with respect to θ Ω and locally Lipschitz with respect to z, x, u), uniformly in θ Ω, with f 0 0, 0, 0) = 0, f i θ, 0,...,0) = 0 for all θ Ω, for i = 1,...,n.

3 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 421 System 1.6) can be regarded as the cascade connection of two subsystems. Sufficient conditions for the global asymptotic stability of the equilibrium point for a cascade connection of two independent subsystems were given in Jiang et al. 1994) and recently in Panteley & Loria 1998). We provide sufficient conditions for the existence of a smooth partial-state time-varying feedback of the form u = kt, x), such that 0 R m R n is GAS. This is achieved in Theorem 5.4 and to this end we are using the Lyapunov characterization of forward completeness given in Angeli & Sontag 1999). We guarantee the existence of such a feedback, under the hypotheses: i) Subsystem 1.6a) is forward complete with x, u) as input. ii) 0 R m is GAS for the unforced subsystem ż = f 0 z, 0, 0) 0-GAS property). This is a generalization of the existing results since forward completeness and 0-GAS is weaker than ISS or even iiss as shown in Angeli et al., 2000). Notation * By C j A)C j A; Ω)), where j 0isanon-negative integer, we denote the class of functions taking values in Ω) that have continuous derivatives of order j on A. * By B r B r ), where r > 0, we denote the open closed) ball of radius r in R n, centred at 0 R n. * For definitions of classes K, K, KL see [8]. * By D + f t) we denote the upper right-hand side Dini derivative of the scalar function f, i.e. D + f t+h) f t) f t) = lim sup h 0 + h. * We denote by M D the class of measurable functions d :R + D. 2. Definitions and preliminary technical results In this section we give the notion of φ-rgas for time-varying systems and we present definitions and technical lemmas that play a key role in proving the main results of the paper. Their proofs can be found in the Appendix. DEFINITION 2.1 We denote by K + the class of non-decreasing C functions φ :R + R with φ0) 1, and we denote by K K +, the class of C functions that belong to K + φt) and satisfy lim t + φ r t) = 0, for some r 1. For example the functions φt) = 1, φt) = 1 + t, φt) = expt) all belong to the class K. The following lemma states some of the properties of these classes of functions. LEMMA 2.2 For every p K, q K and for all constants M 1 and a 0, it holds that the functions p )+q ), p )q ) and Mp a ) are of class K as well. Furthermore, for every function φ of class K +, there exists a function φ of class K,such that φt) φt) for all t 0. We next give the notion of φ-rgas, which directly extends the notion of RGAS presented in Karafyllis & Tsinias 2001). This notion is introduced in such a manner that we can have an estimate of the rate of convergence of the solution to the equilibrium point. Consider the system 1.1), where D is a compact subset of R m and the vector field f :R + R n D R n satisfies the following conditions:

4 422 I. KARAFYLLIS 1) The function f t, x, d) is measurable in t, for all x, d) R n D. 2) The function f t, x, d) is continuous in d, for all t, x) R + R n. 3) The function f t, x, d) is locally Lipschitz with respect to x, uniformly in d D, in the sense that for every bounded interval I R + and for every compact subset S of R n, there exists a constant L 0 such that f t, x, d) f t, y, d) L x y t I, x, y) S S, d D 2.1) with f t, 0, d) = 0, for all t, d) R + D. Let us denote by xt, t 0, x 0 ; d) = xt) the unique solution of 1.1) at time t that corresponds to input d M D with initial condition xt 0 ) = x 0 see Fillipov, 1988). DEFINITION 2.3 Let φ be a function of the class K +. We say that 0 R n is φ-robustly Globally Stable φ-rgs), if for every T 0, p 0 and ε>0, it holds that sup{φ p t) xt) :d M D, t t 0, x 0 ε, t 0 [0, T ]} < + and there exists a δ = δε, T, p) >0such that φ p t) xt) ε, for all x 0 δ, t t 0, t 0 [0, T ] and input d M D. 0 R n is called a φ-robust Global Attractor φ-rga), if for every R > 0, ε>0, p 0 and T 0, there is a τ := τε,t, R, p) 0 such that φ p t) xt) ε for all x 0 B R, t t 0 + τ, t 0 [0, T ] and input d M D. 0 R n is called φ-robustly Globally Asymptotically Stable φ-rgas), if it is φ-rgs and φ-rga. If 0 R n is φ-rgas for φt) := 1 then we simply write that 0 R n is RGAS. In Tsinias & Karafyllis 1999) we required t N xt) ε for every integer N 0in the definition of the L t Global Asymptotic Stability. It is clear that the present definition includes this case with φt) = 1+t K. The following lemma clarifies the consequences of the notion of φ-rgas and provides estimates of the solutions. LEMMA 2.4 Suppose that 0 R n is φ-rgas, for system 1.1) with d D as input. Then the following statements hold: i) For every function φ K + that satisfies φt) φt) for all t 0, 0 R n is φ-rgas for system 1.1). Particularly, 0 R n is RGAS. ii) For every pair of constants R 1 and p 0, 0 R n is φ-rgas, for system 1.1), where φt) = Rφ p t). iii) For every p 0, there exist functions σ ) KL and β ) K +, such that the following estimate holds for the solution of 1.1): xt) 1 φ p t) σ βt 0) x 0, t t 0 ), t t 0, d M D. 2.2) For the construction of a smooth time-varying feedback, we need to introduce the following class of convex functions. DEFINITION 2.5 We say that a :R + R + belongs to K con if:

5 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 423 1) a C 1 R + ) K. 2) The function da ds s) is non-decreasing. The following technical lemmas state some important properties of class K con that are used in the subsequent sections of this paper. LEMMA 2.6 The following statements hold: i) K con K. ii) If a,β belong to K con then a + β,aβ,aoβ also belong to K con. iii) For every a C 0 R + ) there exists a constant R > 0 and a function β K con such that as) R + βs), s 0. iv) If a ) K con the following properties hold: λas) aλs), s 0, λ 1 2.3a) as 1 ) + as 2 ) as 1 + s 2 ) a2s 1 ) + a2s 2 ), s 1, s 2 ) R +) b) LEMMA 2.7 Consider the vector field f C 0 Ω R n ;R m ), where Ω R l is a compact set, which is locally Lipschitz with respect to x R n, uniformly with respect to θ Ω and satisfies f θ, 0) = 0, for all θ Ω. Then there exists a K con such that: f θ, x) a x ), θ, x) Ω R n. 2.4) LEMMA 2.8 For every a K con, there exists an odd function β C R), functions γ K con, ã K con C R + ) and a constant R > 0, with the following properties: as) ãs), s 0 2.5) as) βs) γs), s 0 2.6) dβ ds s) R + γs), s ) The next lemma shows a fundamental property of forward complete time-varying systems. It shows that the reachable set contains a closed ball of positive radius at all times. This fact is going to be used in Section 3 of the paper. LEMMA 2.9 Consider system 1.1) and suppose that there exists a function ρ ) K con and a function φ ) K + such that f t, x, d) ρφt) x ), t, x, d) R + R n D. 2.8) Suppose, furthermore, that for all r 0, t 0 0 and t t 0 we have sup { xt) ; x 0 r, d M D } < ) Then it holds that B βt,t0,r) {xt); x 0 r, d M D } 2.10) where βt, t 0, r) is the unique solution of initial value problem: ẇ = ρφt)w) w R,wt 0 ) = r )

6 424 I. KARAFYLLIS 3. Adding a time-varying integrator The following technical lemma is the basic tool in the integrator backstepping method that we intend to use. Notice that in the time-varying case, there are many technical difficulties to obtain such a result, concerning the rate of convergence of the solution to the equilibrium point, as well as the issue of whether the dynamics converge to zero or not. Most of the technical assumptions introduced below are automatically satisfied in the autonomous case. LEMMA 3.1 Consider the system ẋ = Ft,θ,x, y) ẏ = f t,θ,x, y) + gt,θ,x, y)u x R n, y R, u R, t 0,θ Ω 3.1a) 3.1b) where Ω R l is a compact set, with Ft,θ,0, 0) = 0, f t,θ,0, 0) = 0, for all t,θ) R + Ω and F, f, g are measurable with respect to t, continuous with respect to θ, and locally Lipschitz with respect to x, y) uniformly in θ Ω. Suppose that there exists φ K such that the following hold: H1) There exists a function γ K, being locally Lipschitz on R +,ac j j 1) mapping k :R + R n Rwith k, 0) = 0, a constant µ 0, such that 0 R n is φ-rgas for ẋ = F t,θ,x, kt, x) + d γ x ) ) φ µ 3.2) t) with θ, d) D := Ω [ 1, 1] as input. H2) There exists a function p K con such that s pγ s)), s ) H3) There exists a K con such that the following inequalities hold for all t,θ,x, y) R + Ω R n R: Ft,θ,x, y) aφt) x, y) 3.4) kt, x) + k t, x) t aφt) x ) 3.5) k t, x) t φt) + aφt) x ). 3.6) H4) There exist constants K > 0andδ 0 such that the following inequalities hold for all t,θ,x, y) R + Ω R n R: 1 K φ δ gt,θ,x, y) φt) + aφt) x, y) ) 3.7) t) f t,θ,x, y) aφt) x, y) ). 3.8)

7 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 425 Then for every locally Lipschitz function Γ ) K, there exists a C mapping k : R + R Rwith k, 0) = 0 and constants M 1, q 1, such that 0 R n+1 is φ-rgas for d dt x = Ft,θ, x, kt, y kt, x)) + dγ x )) 3.9) with θ, d) D := Ω [ 1, ) 1] as input, where x := x, y), Ft, θ, x, u) := Ft,θ,x, y) and f t,θ,x, y) + gt,θ,x, y)u φt) := Mφ q t). 3.10) Furthermore, there exists a function ã ) K con,such that hypothesis H3) is satisfied with F ), x, kt, x) := kt, y kt, x)), ã ) and φ ) instead of F ), x, kt, x), a ) and φ ), respectively. When φ ) is bounded then the mapping k can be chosen to be independent of t. Proof. Let Φt, t 0, x 0 ; θ, d)) denote the solution of 3.2) initiated from x 0 R n at time t 0 0 and corresponding to input θ, d) M D and xt) = xt), yt)) denote the solution of 3.9) initiated from x 0 R n+1 at time t 0 0 and corresponding to input θ, d) M D. The proof is based on the following observations: i) By property ii) of Lemma 2.4 and definition 3.10), it suffices to show that 0 R n+1 is φ-rgas for 3.9) with θ, d) D := Ω [ 1, 1] as input. ii) In order to prove that 0 R n+1 is φ-rgas for 3.9) with θ, d) D := Ω [ 1, 1] as input, it suffices to show that there exists a function G ) K con and a C 0 function E :R + R + R + with Et, ) K for all t 0 and E, s) being non-decreasing, in such a way that the following inequality holds for all t t 0, s 0: sup{ xt) ; x 0 s,θ,d) M D } 3.11) Gφt) sup{ Φt, t 0, x 0 ; θ, d)) ; x 0 Et 0, s), θ, d) M D }). Indeed, notice that by virtue of 3.11), property iv) of Lemma 2.6 and the facts that φt) 1 and E, s) is non-decreasing, we have for all q 0, T 0, s 0 and h 0: sup {φ q t 0 + h) xt 0 + h) ; x 0 s, t 0 [0, T ],θ,d) M D } G φ q+1 t 0 + h) sup { Φt 0 + h, t 0, x 0 ; θ, d)) ; x 0 ET, s), t 0 [0, T ],θ,d) M D } ). 3.12) Furthermore, by virtue of iii) of Lemma 2.4 and the fact that 0 R n is φ-rgas for 3.2), it follows that there exist functions σ ) KL and β ) K +, such that the following estimate holds for the solution of 3.2): φ q+1 t) Φt, t 0, x 0 ; θ, d)) σ βt 0 ) x 0, t t 0 ), t t 0, θ, d) M D. 3.13) Combining 3.12) with 3.13) we obtain the following estimate, which holds for all h, s, T, q 0: sup { φ q t 0 + h) xt 0 + h) ; x 0 s, t 0 [0, T ],θ,d) M D } G σβt )ET, s), h)),

8 426 I. KARAFYLLIS which implies that 0 R n+1 is φ-rgas for 3.9) with θ, d) D := Ω [ 1, 1] as input. Since the proof is long and technical, we divide it into two parts. First part: Construction of Feedback. Given a locally Lipschitz function Γ ) K,we construct the feedback law kt, x) := kt, y kt, x)), where k ) C R + R) with k, 0) = 0, such that the analogue of H3) is satisfied. Second part: Stability Analysis. Exploiting the properties of the constructed feedback and Lemma 2.9, we prove that 3.11) holds for appropriate functions G ) K con and a E ) C 0 R + R + ). The methodology used is entirely different from the methodology used in Tsinias & Karafyllis 1999) for the case φt) = 1 + t. Without loss of generality, we may assume that the function p ), involved in 3.3), is of class K con C R + ). Indeed, this follows from Lemma 2.8, which guarantees the existence of a function p ) K con C R + ) that satisfies ps) ps), for all s 0. Consequently, if p ) is not of class K con C R + ),wecan replace it by p ). Similarly, without loss of generality, we may assume that γ K con C R + ), because if this is not the case then we can replacep ) K con C R + ) in 3.3) by ps) := ps) + s and γs) by γs) := p 1 s) γs), which belongs to K C R + ). First part: Construction of Feedback. In this part of proof we use repeatedly inequalities 2.3a), 2.3b) of Lemma 2.6 for functions of class K con,aswell as the fact that φt) 1 for all t 0. Notice that, by application of Lemma 2.7, to the even extensions of γ and Γ ) which are locally Lipschitz), there exist functions γ ) K con and Γ ) K con such that γs) γs), s a) Γ s) Γ s), s b) Define u := kt, z) + dγ x ) 3.15a) z := y kt, x) 3.15b) where k ) is yet to be selected. We get from 3.1) and 3.15a), 3.15b): ż = f t,θ,x, kt, x) + z) + dgt,θ,x, kt, x) + z)γ x ) k t, x) t k 3.15c) t, x)ft,θ,x, kt, x) + z) + gt,θ,x, kt, x) + z)kt, z). t

9 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 427 Moreover, by 3.3), 3.5), 3.14a) and 3.15b), there exist functions G i ) K con i = 1,...,3) such that ) x G 1 φ µ+1 t) z, when γ x ) 2φ µ t) z 3.16a) x G 2 φt) x ), when φ µ t) z γ x ) 3.16b) z G 3 φt) x ), t, x) R + R n c) where µ 0isthe constant involved in 3.2). Furthermore, property iii) of Lemma 2.6 implies the existence of a function G 4 ) K con and a constant R 1 > 0 such that { } dγ 0 max 0 ξ s ds ξ) R 1 + G 4 s), s d) It follows from 3.4), 3.5), 3.6), 3.7), 3.8), 3.14b), 3.15c), 3.16a) and 3.16d) that there exists a constant ν 2 and functions a i ) K con, i = 1, 2, that satisfy the following inequalities: d dt zt) a 1φ µ+ν t) z ) + sgnz)gt,θ,x, kt, x) + z)kt, z), for 2φ µ t) z γ x ) and z = ) d dt γ xt) ) a 2φ µ+ν t) z ), for 2φ µ t) z γ x ) >0 3.18) Since φ K, there exist constants K 0 and r 1 with 0 φt) K φ r t), t ) Inequalities 3.17), 3.18) in conjunction with 3.19) imply d φ µ t) zt) γ xt) ) ) K µφ σ t) z +a 1 φ σ t) z ) + a 2 φ σ t) z ) dt + φ µ t)sgnz)gt,θ,x, kt, x) + z)kt, z) for 2φ µ t) z γ x ) >0 3.20a) where d φ µ t) zt) ) K µφ σ t) z +a 1 φ σ t) z ) dt + φ µ t)sgnz)gt,θ,x, kt, x) + z)kt, z) for 2φ µ t) z γ x ) and z = b) We define the function a 3 ) K con : σ := 2µ + ν + r ) a 3 s) := a 1 s) + a 2 s) + K µs + R 1 + G 4 s))ρs) 3.22)

10 428 I. KARAFYLLIS ρs) := a s + as) + γs)) 3.23) where γ ) K con, R 1 > 0 and G 4 ) K con are defined in 3.14a) and 3.16d), respectively. Notice that, by virtue of 3.4), 3.5), 3.14a) and definition 3.23), we have F t,θ,x, kt, x) + d γ x ) ) φ µ ρφ 2 t) x ). 3.24) t) Furthermore, by virtue of Lemma 2.8, there exists an odd C function ψ ), afunction G 5 ) K con and a constant R 2 > 0, with the following properties: a 3 s) ψs) G 5 s), s 0 dψ ds s) R 2 + G 5 s), s a) 3.25b) We also define kt, z) := ψ 1 + K )φ σ +δ t)z ) 3.26) where K > 0 and δ 0 are the constants involved in 3.7). It is clear that the mapping kt, x, y) = kt, y kt, x)) is of class C j R + R n+1 ).Moreover, inequalities 3.5), 3.6), 3.19) in conjunction with 3.25a), 3.25b) imply that there exists a function ã ) K con and constants q 1, M 1, such that H3) is satisfied with x, k ), ã ) and φ ) instead of x, k ), a ) and φ ), respectively, where φ ) is defined in 3.10) and is of class K by virtue of iii) of Lemma 2.2. When φ ) is bounded we may select for R := sup t 0 φt): kt, z) := kz) = ψ ) 1 + K ) R σ +δ z ) The major property of the constructed feedback is the following inequality, which is a consequence of 3.7), 3.16d), 3.22), 3.25a), 3.26) and the fact that the function ψ ) is odd: sgnz)φ µ t)gt,θ,x, kt, x) + z)kt, z) K µφ σ t) z a 1 φ σ t) z ) a 2 φ σ t) z ) dγ φ µ t) z ) ) ρ φ 2+µ t) z. 3.27) ds Notice that by virtue of inequalities 3.20a), 3.20b) and 3.27), it follows that: d φ µ t) zt) γ xt) ) ) 0, when 2φ µ t) z γ x ) >0 3.28a) dt d dt φ µ t) zt) ) dγ ds φ µ t) z ) ρ φ 2+µ t) z ), when 2φ µ t) z γ x ) and z = b)

11 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 429 Second part: Stability Analysis. We define: L t := { x, y) R n R:φ µ t) y kt, x) γ x ) }. 3.29) Notice that by virtue of 3.28a) and definitions 3.15b), 3.29) of z, L t, respectively, it follows that the region L t is positively invariant the case x =0implies z = 0, which is the equilibrium position of 3.9)). As long as the trajectory of the solution of 3.9) remains outside L t we obtain using 3.15b), 3.28b) and 3.29) that D + φ µ t) zt) ) dγ ) ds φµ t) zt) )ρ φ 2+µ t) zt). 3.30) Let βt, t 0, r) denote the unique solution of the following initial value problem: ẇ = ρ φ 2 t)w ) w R,wt 0 ) = r 0. Indeed, by virtue of inequality 3.24) and Lemma 2.9, we guarantee that 3.31) B βt,t0,r) {Φt, t 0, x 0 ; θ, d)), x 0 r,θ,d) M D } 3.32) where Φt, t 0, x 0 ; θ, d)) denotes the solution of 3.2) initiated from x 0 R n at time t 0 0 and corresponding to input θ, d) M D. Furthermore, differential inequality 3.30) implies by virtue of the comparison principle in Khalil, 1996): φ µ t) zt) γβt, t 0,γ 1 φ µ t 0 ) zt 0 ) ))) as long as the trajectory of 3.9) remains outside L t. 3.33) Thus by 3.14a), 3.16a), 3.16c) and 3.33) we obtain the estimate for the solution of 3.9): xt) G 1 φt) γ βt, t0,γ 1 φ µ t 0 )G 3 φt 0 ) xt 0 ) ))) )) as long as the trajectory of 3.9) remains outside L t. 3.34) Moreover, by virtue of 3.32), 3.33) and the facts that G 3 ), γ ) K con, φt) 1 and using property iv) of Lemma 2.6, we obtain that the following estimate holds for G 6 := G 1 o γ : sup { xt) ; x 0 s,θ,d) M D } { G 6 φt) ) }) sup Φt, t 0, x 0 ; θ, d)) ; x 0 γ 1 G 3 φ µ+1 t 0 )s),θ,d) M D as long as the trajectory of 3.9) remains outside L t. 3.35) Let us denote by T + the first time that the solution is entering L T. Notice that for all t T,bypositive invariance of L t, the solution remains inside L t. Moreover, there exists an input θ, d) M D such that component xt) of the solution xt) of 3.9) satisfies xt) Φt, T, xt ); θ, d)), for all t T. Let t t 0 be arbitrary. We distinguish the cases:

12 430 I. KARAFYLLIS a) T = t 0.Inthis case we have xt) sup{ Φt, t 0, x 0 ; θ, d)) ; θ, d) M D } and consequently using 3.16b) we obtain the estimate: sup { xt) ; x 0 s,θ,d) M D } 3.36) G 2 φt) sup { Φt, t 0, x 0 ; θ, d)) ; x 0 s,θ,d) M D }). b) T / [t 0, t].inthis case estimate 3.35) holds. c) t 0 < T t.inthis case we have: xt) sup { Φt, T, xt ); θ, d)) ; T [t 0, t],θ,d) M D } By definition 3.29) and continuity of the solution we also have for the case c): γ xt ) ) = φ µ T ) zt ). Moreover, estimate 3.33) implies φ µ T ) zt ) γβt, t 0,γ 1 φ µ t 0 ) z 0 ))). These estimates in conjunction with 3.16b), 3.16c) give sup { xt) ; x 0 s,θ,d) M D } G 2 φt) sup I ) 3.37a) { I := Φt, T, xt ); θ, d)) ; T [t 0, t],θ,d) M D, xt ) β T, t 0,γ 1 φ µ t 0 )G 3 φt 0 )s) ))}. 3.37b) Inclusion 3.32) in conjunction with estimate 3.37) and the fact that G 3 ) K con and φt) 1, shows that the following estimate holds for case c): sup { xt) ; x 0 { s,θ,d) M D } G 2 φt) sup Φt, t0, x 0 ; θ, d)) ; x 0 γ 1 G 3 φ µ+1 t 0 )s) ) }),θ,d) M D. 3.38) Combining estimates 3.35), 3.36) and 3.38) for the cases a), b) and c), respectively, we obtain the desired inequality 3.11) for Gs) := 2G 2 s) + G 6 s) 3.39a) { } Et 0, s) := max s,γ 1 G 3 φ µ+1 t 0 )s)). 3.39b) Indeed, by virtue of property ii) of Lemma 2.6, we have G ) K con. The proof is complete. The following lemma provides a sufficient condition for hypothesis H2). In fact, Lemma 3.2 shows that hypothesis H2) is the analogue of the hypothesis of local exponential stability made in Jiang et al. 1994), Tsinias 1996) for the autonomous case. LEMMA 3.2 Suppose that γ K is a locally Lipschitz function that satisfies γs) Λs, s [0,η], for some constants η, Λ > 0. Then H2) is satisfied. Proof. Clearly, γ 1 s) Λ 1 s, for all s [0,γη)] and consequently the function γ gs) := sup 1 u) 0<u s u is continuous on 0, + ), positive and non-decreasing with gs) Λ 1 for all s 0,γη)]. Defining g0) := lim s 0 + gs), wehaveγ 1 s) gs)s, forall s 0. We also define hs) := s+1 0 gr)dr, which satisfies hs) gs), for all s 0and notice that the function ps) := hs)s is of class K con. Combining, we get γ 1 s) ps), for all s 0, which implies H2).

13 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS Applications to time-varying control systems We are now ready to apply induction using Lemma 3.1 and the method of integrator backstepping for the stabilization of system 1.3). PROPOSITION 4.1 Consider system 1.3), where x = x 1,...x n ) T R n, θ Ω R l, Ω being a compact set, f i and g i are measurable with respect to t 0, continuous with respect to θ Ω and locally Lipschitz with respect to x 1,...x i ), uniformly in θ Ω, for i = 1,...,n. Suppose that there exist functions φ K, a K con, constants δ 0, K > 0, such that the following hold for all t, x,θ) R + R n Ω and i = 1,...,n: f i t,θ,x 1,...,x i ) aφt) x 1,...,x i ) ) 4.1) 1 K φ δ t) g it,θ,x 1,...,x i ) φt) + aφt) x 1,...,x i ) ). 4.2) Then for every locally Lipschitz function Γ ) K there exists a C mapping k : R + R n Rwith k, 0) = 0, a function η K con and a constant p 0, with kt, x) η φ p t) x ), t, x) R + R n 4.3) such that 0 R n is φ-rgas for the following system: ẋ i = f i t,θ,x 1,...,x i ) + g i t,θ,x 1,...,x i )x i+1 i = 1,...,n x n+1 = kt, x) + dγ x ) 4.4) with θ, d) D := Ω [ 1, 1] as input. Moreover, when φ ) is bounded then the mapping k can be chosen to be independent of t. Proof. The proof of the general case is based on Lemma 3.1 and follows by using standard induction arguments like those given in Jiang et al. 1994). For reasons of simplicity we consider the case n = 2. The general case follows similarly by induction. First we consider the one-dimensional subsystem ẋ 1 = f 1 t,θ,x 1 ) + g 1 t,θ,x 1 )x 2 with x 2 as input. 4.5) Let M > 0. We define ãs) := 1 + s)as) M)s 4.6) which obviously by Lemma 2.6 is of class K con.bylemma 2.8 there exists an odd function ψ ) C R),afunction β ) K con and a constant R 1 > 0 such that ãs) ψs) βs), s 0 dψ ds s) R 1 + βs), s a) 4.7b)

14 432 I. KARAFYLLIS Furthermore, since φ ) K, there exists a constant r 1 such that lim t + φt) φ r t) = c) We set k 1 t, x 1 ) := ψ1 + K )φ 1+δ+r t)x 1 ) 4.8) or for the case of bounded φ ) K with R 2 := sup t 0 φt): k 1 t, x 1 ) := k 1 x 1 ) = ψ1 + K )R 1+r+δ 2 x 1 ) 4.8 ) where K,δ are the constants involved in 4.2). Obviously k 1 ) is a function of class C R + R) with k 1, 0) = 0. It follows from 4.2), 4.6), 4.7a) and definition 4.8), the fact that ψ ) is odd and φt) 1 for all t 0, as well as iv) of Lemma 2.6, that the following inequality holds for all t,θ,x 1 ) R + Ω R: ) sgnx 1 )g 1 t,θ,x 1 )k 1 t, x 1 ) ã φ r+1 t) x 1 Mφ r t) x x 1 )aφt) x 1 ) φt) x ) Moreover, inequalities 4.7a) 4.7c) and definition 4.8) imply the existence of constants R 3 1, σ 1 and a function ζ ) K con,such that the following inequalities hold for all t, x 1 ) R + R: k 1 t, x 1 ) + k 1 t t, x ) 1) ζ φt) x a) k 1 t, x 1 ) x φt) + ζ φt) x 1 ) b) φt) := R 3 φ σ t). 4.10c) We claim that 0 Ris φ-rgas for the closed-loop system 4.5) with x 2 = k 1 t, x 1 ) + d x 1, d [ 1, 1]. Inorder to prove this claim, notice that by 4.1), 4.2) and 4.9) we have for all x 1 = 0, t 0 and θ, d) D = Ω [ 1, 1]: d dt x 1 =sgnx 1 ) f t,θ,x 1 ) + sgnx 1 )gt,θ,x 1 )k 1 t, x 1 ) + sgnx 1 )gt,θ,x 1 )d x 1 Mφ r t) x 1. Since x 1 = 0isthe equilibrium point of the closed-loop system 4.5) with x 2 = k 1 t, x 1 )+ d x 1, d [ 1, 1] we get D + x 1 t) Mφ r t) x 1 t), for all t t )

15 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 433 The differential inequality 4.11) and the comparison lemma in Khalil 1996) give for all q 0, s 0: sup { φ q } t t) x 1 t) ; x 1 t 0 ) s,θ,d) M D φ q t 0 ) exp Mφ r τ) q φτ) ) ) dτ s. t 0 φτ) 4.12) By 4.7c) it follows that for every q 0there exists a finite time T := T q) 0, such that 1 φt) 2 Mφt) q φ r t) for all t T. This implies + 0 Mφ r τ) q φτ) ) φτ) dτ =+ for all q 0 and consequently by 4.12), we have that 0 Ris φ-rgas for the closedloop system 4.5) with x 2 = k 1 t, x 1 ) + d x 1, d [ 1, 1]. Byii) of Lemma 2.4 and definition 4.10c) it also follows that 0 Ris φ-rgas for the closed-loop system 4.5) with x 2 = k 1 t, x 1 ) + d x 1, d [ 1, 1]. Consider next the two-dimensional subsystem ẋ 1 = f 1 t,θ,x 1 ) + g 1 t,θ,x 1 )x 2 ẋ 2 = f 2 t,θ,x 1, x 2 ) + g 1 t,θ,x 1, x 2 )x 3 with x 3 as input. We apply Lemma 3.1 for this system. Clearly, by the previous analysis hypothesis H1) is satisfied for φ ) K as defined by 4.10c), γs) := s and µ = 0. Hypothesis H2) is trivially satisfied, while hypotheses H3) and H4) are consequences of inequalities 4.1), 4.2), 4.10a), 4.10b). The desired conclusion follows from the application of Lemma 3.1. The proof is complete. The following proposition is concerned with the problem of partial-state robust feedback stabilization of uncertain nonlinear time-varying systems. It is the analogue of Corollary 3.4 in [11] and Theorem 2.4 in [12], although here we consider uncertain systems. Its proof follows directly by induction and Lemmas 3.1 and 3.2. PROPOSITION 4.2 Consider the system 1.4), where z R m, x = x 1,...x n ) T R n, θ Ω R l, Ω is a compact set, f i i = 0,...,n) and g i i = 1,...,n) are measurable with respect to t 0, C 0 with respect to θ Ω and locally Lipschitz with respect to z, x), uniformly in θ Ω. Suppose that there exist functions φ K, a K con, constants δ 0 and K > 0, such that the following hold for all t, z, x,θ) R + R m R n Ω and i = 1,...,n: f 0 t,θ,z, x 1 ) aφt) z T, x 1 ) ) 4.13) f i t,θ,z, x 1,...,x i ) aφt) z T, x 1,...,x i ) ) 4.14) 1 K φ δ t) g it,θ,z, x 1,...,x i )) φt) + aφt) z T, x 1,...,x i ) ). 4.15) Suppose, furthermore, that 0 R m is φ-rgas for the following system ż = f 0 t,θ,z, d γ z ) ) φ µ 4.16) t)

16 434 I. KARAFYLLIS with θ, d) D := Ω [ 1, 1] as input, where µ 0 and γ K is a locally Lipschitz function that satisfies the following inequality for certain constants η, Λ > 0: Λs γs), s [0,η]. 4.17) Then for every locally Lipschitz function Γ ) K there exists a C mapping k : R + R n Rwith k, 0) = 0, such that 0 R m R n is φ-rgas for the following system: ż = f 0 t,θ,z, x 1 ) ẋ i = f i t,θ,z, x 1,...,x i ) + g i t,θ,z, x 1,...,x i )x i+1 i = 1,...,n 4.18) x n+1 = kt, x) + dγ z, x) ) with θ, d) D := Ω [ 1, 1] as input. Moreover, when φ ) is bounded then the mapping k can be chosen to be independent of t. EXAMPLE 4.3 Consider the system: ż = 2tz + θ 1 t) expµt)x ẋ = expat)θ 2 t)wz, x) + u 4.19) z, x) R 2, t 0, u R,θ = θ 1,θ 2 ) B0, r) where a, r, µ > 0 are known constants, wz, x) is a locally Lipschitz function satisfying w0, 0) = 0 and θ denotes the vector of the uncertain parameters of the system. It is immediate to verify that 0 Ris e t -RGAS for the subsystem ż = 2tz + θ 1 t)dt) z 4.20) where dt) 1. Indeed, for every θ ), d )) M B0,r) [ 1,1] the solution of 4.20) satisfies the estimate { )} zt) exp rt t 0 ) t 2 t0 2 z 0. Thus by Proposition 4.2 there exists a C mapping k :R + R Rwith k, 0) = 0 with such that the origin for 4.19) with u = kt, x) is e t -RGAS. 5. Applications to autonomous control systems In this section we study the applications of time-varying feedback laws to autonomous control systems. We first consider the case 1.5). We establish that the use of time-varying feedback can robustly accelerate the rate of convergence of the solution to the equilibrium point. This is shown by the following corollary, which is an immediate consequence of Proposition 4.1 and Lemma 2.7. COROLLARY 5.1 Consider system 1.5), where x = x 1,...x n ) T R n, θ Ω R l, Ω being a compact set, f i and g i are continuous with respect to θ Ω and locally Lipschitz with respect to x 1,...x i ), uniformly in θ Ω, with f i θ, 0,...,0) = 0, for all θ Ω,

17 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 435 for i = 1,...,n. Suppose that there exists a constant K > 0 such that the following hold for all x,θ) R n Ω and i = 1,...,n: 1 K g iθ, x 1,...,x i ). 5.1) Then for every φ ) K and Γ ) K being locally Lipschitz, there exists a C mapping k :R + R n Rwith k, 0) = 0, a function η ) K con and a constant p 0, with kt, x) η φ p t) x ), t, x) R + R n 5.2) such that 0 R n is φ-rgas for the following system with θ, d) D := Ω [ 1, 1] as input: ẋ i = f i θ, x 1,...,x i ) + g i θ, x 1,...,x i )x i+1 i = 1,...,n x n+1 = kt, x) + dγ x ). 5.3) Proof. Let φ ) K. Since each f i and g i is locally Lipschitz, uniformly in θ Ω, Lemma 2.7 implies that there exists a function a K con,such that the following inequalities hold for all t,θ,x) R + Ω R n and i = 1,...,n: f i θ, x 1,...,x i ) a x 1,...,x i ) ) aφt) x 1,...,x i ) ) 5.4a) g i θ, x 1,...,x i ) g i θ, 0,...,0) a x 1,...,x i ) ) aφt) x 1,...,x i ) ). 5.4b) Furthermore, inequality 5.1) gives for i = 1,...,n: 1 K φt) 1 K g iθ, x 1,...,x i ). 5.4c) Inequalities 5.4a) 5.4c) establish that all hypotheses of Proposition 4.1 are fulfilled for φt) = Rφt), where R := 1+sup θ Ω,i=1,...,n g i θ, 0,...,0). Notice that it holds: φt) φt) for all t 0. Therefore for every locally Lipschitz function Γ ) K, there exists a C mapping k : R + R n R,afunction η ) K con,aconstant p 0, with kt, x) ηφ p t) x ), such that 0 R n is φ-rgas for 3.63) with θ, d) Ω [ 1, 1] as input. Consequently, by Lemma 2.4 and inequality φt) φt), itfollows that 0 R n is φ-rgas for 5.3) with θ, d) Ω [ 1, 1] as input. Moreover, 5.2) is satisfied for ηs) := ηr p s). REMARK 5.2 Notice that the origin for system 1.5) cannot become φ-rgas for φt) = expt) with the application of locally Lipschitz static time-invariant feedback i.e. the rate of convergence to 0 R n of the solution of the closed-loop system 1.5) with a static locally Lipschitz time-invariant feedback cannot be faster than the exponential rate). EXAMPLE 5.3 Consider the two dimensional system ẋ 1 = θt)x x 2 ẋ 2 = u x 1, x 2 ) R 2, u R 5.5)

18 436 I. KARAFYLLIS where θ ) :R + [ 1, 1] is an unknown time-varying parameter. Let φ ) K be φt) a function that satisfies lim t + φ r t) = 0, for some constant r 1. Then 0 R 2 is φ-rgas for the following system with θ, d) [ 1, 1] 2 as input: ẋ 1 = θt)x x 2 ẋ 2 = kt, x 1, x 2 ) + dt) x 1 +dt) x 2 5.6) where kt, x 1, x 2 ) is the C time-varying feedback law, defined for some M > 0 sufficiently large, by the following relation: kt, x 1, x 2 ) := Mφ 2r t) z + z 3 + z 5) z := x 2 + 3φ r t) x 1 + x1 3 ) 5.7). Notice that for the selection φt) 1 K, the feedback law defined by 5.7) is actually time-invariant and guarantees uniform global asymptotic stability of the origin for system 5.6). Next we consider the problem of partial-state feedback stabilization of autonomous systems of the form 1.6). In the literature the usual assumption is that subsystem 1.6a) is ISS with x, u) as input. Here we intend to relax this hypothesis, by making use of time-varying feedback of the form u = kt, x). THEOREM 5.4 Consider the system 1.6) and suppose that there exists a constant K > 0, such that the following hold for all x,θ) R n Ω and i = 1,...,n: 1 K g iθ, x 1,...,x i ). 5.8) Furthermore suppose that the subsystem ż = f 0 z, x, u) is forward complete with x, u) as input and that 0 R m is GAS for the system ż = f 0 z, 0, 0). Then for every locally Lipschitz function Γ ) K there exists a C mapping k : R + R n R, with kt, 0) = 0, for all t 0, such that 0 R m R n is RGAS for the following system with θ, d) D := Ω [ 1, 1] as input: ż = f 0 z, x, u) ẋ i = f i θ, x 1,...,x i ) + g i θ, x 1,...,x i )x i+1 i = 1,...,n u = x n+1 = kt, x) + dγ x ). 5.9) Proof. The proof is divided into three parts. First part. We obtain estimates for the solution zt) of the subsystem 1.6a), with x, u) R n Ras input. Second part. We design a feedback law kt, x), such that 0 R n is φ-rgas for the subsystem 1.6b) with x n+1 = kt, x) + dγ x ), for an appropriate choice of φ K. Third part. We prove that 0 R m R n is RGAS for 5.9).

19 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 437 First part Since 0 R m is GAS for the subsystem ż = f 0 z, 0, 0), then in Angeli et al. 2000, Lemma IV.10) and Sontag 1998, Theorem 3), there exists a smooth function V :R n R +, K functions a 1, a 2,λ,δ, such that for all z, x, u) R m R n Rwe have a 1 z ) V z) a 2 z ) 5.10a) V z z) f 0z, x, u) V z) + λ z )δ x, u) ). 5.10b) Furthermore, in Angeli & Sontag 1999, Corollary 2.11), there exists a smooth and proper function W :R n R + and functions a 3, a 4,σ of class K and a constant R > 0, such that for all z, x, u) R m R n Rwe have a 3 z ) W z) a 4 z ) + R 5.10c) W z z) f 0z, x, u) W z) + σ x, u) ). 5.10d) Notice that by virtue of 5.10c), 5.10d) the solution zt) initiated at zt 0 ) = z 0 of the subsystem 1.6a) satisfies a 3 zt) ) expt t 0 )a 4 z 0 ) + R) + t t 0 expt τ)σ xτ), uτ)) )dτ, t t ) On the other hand, by 5.10a), 5.10b) and 5.11) we obtain the estimate for λs) := λ2a 1 3 2s)) and for all ξ [t 0, t]: a 1 zt) ) exp t ξ))a 2 zξ) ) t τ + λ expτ s)σ xs), us)) )ds + ξ t ξ t 0 ) δ xτ), uτ)) )dτ λexpτ t 0 )a 4 z 0 ) + R))δ xτ), uτ)) )dτ, t ξ. 5.12) Second part In Sontag 1998, Corollary 10), there exist functions q 1, q 2,µ K C 0, + )), such that σrs) q 1 r)q 1 s), r, s a) δrs) q 2 r)q 2 s), r, s b) λrs) µr)µs), r, s c) Without loss of generality, we may assume that q 1 1, q 1 2 K C 0, + )). Define φt) := exp t)) + 1 q 1 q 1 2 exp t) µexpt)) ). 5.14a)

20 438 I. KARAFYLLIS Notice that φ K +. Consequently, by Lemma 2.2, there exists φ K such that φt) φt), t b) Furthermore, by 5.8) and Corollary 5.1, we have that for every locally Lipschitz function Γ ) K there exists a C mapping k :R + R n R,afunction η Q, aconstant p 0, with kt, x) η φ p t) x ), t, x) R + R n 5.15) such that 0 R n is φ-rgas for the following system with θ, d) D := Ω [ 1, 1] as input: ẋ i = f i θ, x 1,...,x i ) + g i θ, x 1,...,x i )x i+1 i = 1,...,n x n+1 = kt, x) + dγ x ). 5.16) Third part Since is φ-rgas for 5.16) with input θ, d) M D,byLemma 2.4 it follows that there exist a KL function ζ and a K + function β, such that φ p+1 t) xt) ζ βt 0 ) x 0, t t 0 ), t t ) where p 0isthe constant involved in 5.15). Moreover, applying Lemma 2.7 for the even extension of Γ ) on R which is locally Lipschitz), we find that there exists a function Γ ) K con such that Γ s) Γ s), s ) Inequalities 5.15), 5.17), 5.18) and the fact that Γ, η K con,imply that the following estimate holds for all t t 0 : xt), ut)) xt) + ut) = xt) + kt, xt)) + dt)γ xt) ) xt) +η φ p t) xt) ) + Γ xt) ) 1 + dη φ p t) xt) ) + d Γ φ p t) xt) )) φ p t) xt) ds ds 1 φt) H βt 0) x 0 ) ζ βt 0 ) x 0, t t 0 ) 5.19) where Hs) := 1+ dη d Γ ds ζs, 0))+ ds ζs, 0)) is a continuous, positive and non-decreasing function. By 5.13a) and 5.19) it follows that τ τ ) 1 expτ s)σ xs), us)) )ds expτ)q 1 Z βt 0 ) x 0 )) q 1 ds, τ t 0 t 0 t 0 φs) ) where Zs) := Hs)ζs, 0), Z K.Onthe other hand 5.14) implies that q 1 1 φt) exp t), which in conjunction with the latter inequality gives τ expτ s)σ xs), us)) )ds expτ)q 1 Z βt 0 ) x 0 )), τ t ) t 0

21 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 439 The above inequality, combined with 5.12), 5.13b), 5.13c) and 5.19) gives for all ξ [t 0, t]: a 1 zt) ) exp t ξ))a 2 zξ) ) +q 2 Z βt 0 ) x 0 )) [µ q 1 Z βt 0 ) x 0 ))) + µ a 4 z 0 ) + R)] I t,ξ) 5.21a) t ) 1 I t,ξ):= µexpτ))q 2 dτ. ξ φτ) 5.21b) ) As previously, using 5.14) we may establish that µexpt))q 1 2 φt) exp t) and thus a 1 zt) ) exp t ξ))a 2 zξ) ) + + exp ξ)q 2 Z βt 0 ) x 0 )) [ µ q 1 Z βt 0 ) x 0 ))) µ a 4 z 0 ) + R) ] for all ξ [t 0, t] and t ξ. 5.22) The inequality above and 5.17) imply that 0 R m R n is RGAS for 5.9). In order to establish this fact, notice that, by virtue of 5.17) and 5.22), for all T 0 and r 0, it holds sup { zt), xt)) ; θ, d) M D, t t 0, z 0, x 0 ) r, t 0 [0, T ]} GβT )r) 5.23a) Gs) := ζs, 0) + a 1 1 a 2 s) + q 2 Zs)) [µ q 1 Zs))) + µ R + a 4 s))]) 5.23b) where G is a class K function. This establishes stability. In order to prove attractivity let ε > 0, r > 0 and T 0. There exists a τ := τε,t, r) T, such that exp ξ)q 2 ZβT )r))[µq 1 ZβT )r))) + µa 4 r) + R)] a 1 ε), for all ξ τ. It follows from 5.17), 5.22) and 5.23) that for all t τ it holds sup { zt), xt)) ; θ, d) M D, z 0, x 0 ) r, t 0 [0, T ]} ζβt )r, t T ) + a 1 1 exp t τ))a 2GβT )r)) + a 1 ε)). This proves that lim sup { zt), xt)) ; θ, d) M D, z 0, x 0 ) r, t 0 [0, T ]} ε. t + Since ε>0isarbitrary we have that lim sup{ zt), xt)) ; θ, d) M D, z 0, x 0 ) r, t + t 0 [0, T ]} = 0. The proof is complete. Acknowledgements The author thanks Professor John Tsinias for his comments and his suggestions. REFERENCES ANGELI, D.& SONTAG, E. D.1999) Forward completeness, unbounded observability and their Lyapunov characterizations. Systems and Control Letters, 38, ANGELI, D., SONTAG, E. D.& WANG, Y.2000) A characterization of integral input-to-state stability. IEEE Trans. Automat. Contr., 45,

22 440 I. KARAFYLLIS FILLIPOV, A. F.1988) Differential Equations with Discontinuous Right-Hand Sides. Dordrecht: Kluwer. JIANG, Z.P.,TEEL, A.&PRALY, L.1994) Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems, 7, KARAFYLLIS, I.& TSINIAS, J.2001) Converse Lyapunov Theorems for Non-Uniform In Time Global Asymptotic Stability and Stabilization By Means of Time-Varying Feedback. Proceedings of NOLCOS. KARAFYLLIS, I.& TSINIAS, J.2002a) Global stabilization and asymptotic tracking for a class of nonlinear systems by means of time-varying feedback. Int. Journal of Robust and Nonlinear Control, toappear. KARAFYLLIS, I.& TSINIAS, J.2002b) A converse Lyapunov theorem for non-uniform in time global asymptotic stability and its application to feedback stabilization. SIAM Journal Control and Optimization, submitted. KHALIL, H.K.1996) Nonlinear Systems, 2nd Edition. Englewood Cliffs, NJ: Prentice-Hall. PANTELEY, E.& LORIA, A.1998) On global uniform asymptotic stability of nonlinear timevarying systems in cascade. Systems and Control Letters, 33, SONTAG, E. D.1998) Comments on integral variants of ISS. Systems and Control Letters, 34, TSINIAS, J.1996) Versions of Sontag s input to state stability condition and output feedback stabilization. J. Math. Systems Estimation and Control, 6, TSINIAS, J.2000) Backstepping design for time-varying nonlinear systems with unknown parameters. Systems and Control Letters, 39, TSINIAS, J.& KARAFYLLIS, I.1999) ISS property for time-varying systems and application to partial-static feedback stabilization and asymptotic tracking. IEEE Trans. Automat. Contr., 44, Appendix Proof of Lemma 2.2. The implications i) iii) are obvious. We only prove the last statement of the lemma. Define the function { φlog s) + s if s 1 µs) := A.1) φ0) + 1)s if 0 s < 1. Clearly µ ) K and satisfies Define the function φt) µ e t), t 0. A.2) { 0 if s = 0 ρs) := 1 ) µ 1 1s if 0 < s. A.3) Again we have that ρ ) K and let ρ ) K C 0, + )) be a function with d ρ ds s) 1 for all s > 0 and lim s 0 ρs) + s =+, that satisfies ρs) ρs) for all s 0. Thus by A.3) we have 1 ) ρs) s > 0 µ e t) 1 µ 1 1s ρ 1 e t) t 0. A.4)

23 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 441 Define φt) := 1 ρ 1 e t). A.5) Since ρ ) K C 0, + )) with d ρ ds s) 1 for all s > 0, we easily establish that φ ) C R + ) and that φ ) is non-decreasing. Furthermore, since d ρ ds s) 1 for all s > 0 and lim ρs) s 0 + s =+,itfollows that 0 d ρ 1 ds s) 1, for all s 0. This fact and definition A.5) imply d φ dt t) = φ 2 t) d ρ 1 e t ) e t φ 2 t)e t. ds 1 d φ The latter inequality gives lim t + φ 2 t) dt t) = 0. Notice that we can easily establish using the implied inequality ρ 1 s) s for all s 0) that φt) e t 1, for all t 0. Consequently, we conclude that φ ) K. Moreover, by A.2), A.4) and definition A.5) we get that φt) φt), for all t 0. The proof is complete. Proof of Lemma 2.4. Implications i) and ii) are immediate consequences of the definition of φ-rgas and the fact that φt) 1, for all t 0. We focus on implication iii). Let p 0 and consider the time-varying transformation z := φ p t)x. A.6) Clearly, zt) satisfies the following system of differential equations: ż = p φt) ) φt) z + φ p z t) f t, φ p t), d z R n, t 0, d D. The fact that 0 R n is φ-rgas for 1.1) and definition A.6) imply that 0 R n is RGAS for the system above with d as input. Furthermore, Proposition 2.2 in Karafyllis & Tsinias 2002b) guarantees the existence of a KL function σ ) and a K + function β ) such that ) zt) σ βt 0 ) z 0, t t 0, t t 0. A.7) The desired 2.2) is a consequence of inequality A.7), definition A.6) and the selection βt) := βt)φ p t). Proof of Lemma 2.6. The statements i) ii) are obvious. We prove statements iii) iv). iii) Define the function γs) := s + sup 0 τ s aτ). A.8) Clearly γ C 0 R + ) and is strictly increasing with as) γs), s 0. A.9)

24 442 I. KARAFYLLIS Let h : R R + be a C function with hs) = 0ifs / 0, 1) and R hs)ds = 1 0 hs)ds = 1. Define the function 1 γs) := γw)hw s)dw = γs + w)hw)dw. A.10) Clearly γ C R + ),isnon-decreasing and satisfies R 0 γs) γs) γs + 1), s 0. A.11) Let r > 0beanarbitrary constant and define the following functions: d γ δs) := ds r) + 1 if 0 s r sup r τ s d γ A.12) ds τ) + 1 if s > r βs) = s 0 δw)dw. A.13) Since by definition A.12), δ is continuous, non-decreasing and satisfies δs) 1, s 0, we have that β K con. Notice that for s > r we get by A.12) and A.13) βs) s r δw)dw γs) γr) and the latter inequality in conjunction with A.9) and A.11) implies that as) R+βs), s 0, for R = γr). iv) Inequality 2.3a) is a consequence of the inequality ȧs) ȧλs), which holds for all λ 1 and s 0. The right-hand side of inequality 2.3b) is a well-known property of the functions of class K. The left-hand side of inequality 2.3b) is a consequence of the inequality ȧs 1 ) ȧs 1 + s 2 ), which holds for all s 1 0 and s 2 0. The proof is complete. Proof of Lemma 2.7. Since f is locally Lipschitz in x, uniformly in θ, there exist constants L, r > 0, such that Consider the function f θ, x) L x, θ Ω, for x r. A.14) βs) := sup f θ, x). A.15) θ Ω x s Clearly, the mapping β ) is continuous and non-decreasing with β0) = 0. Furthermore by A.14) we have βs) Ls, for s r. A.16)

25 NON-UNIFORM STABILIZATION OF CONTROL SYSTEMS 443 2s Clearly the function ζs) = 1 s s βw)dw belongs to the class K C 1 0, + )) and satisfies ζs) βs) for all s 0. Define the functions: dζ r for 0 s < ds 2) r 2 δs) := dζ ds τ) for s r A.17) 2 sup r 2 τ s s as) := λs + δw)dw 0 A.18) { λ := max L + 1, 2 r r 2) } ζ. A.19) Since δ is a continuous, non-decreasing function we have that a K con.weclaim that 2.4) is satisfied. Notice that as) λs L + 1)s and consequently for all x r and θ Ω,weobtain using A.14) f θ, x) L x a x ). On the other hand, for s r 2,byA.17) and A.19) we have that r 2 s as) = λs + δw)dw + δw)dw λ r s r r 2 The proof is complete. dζ w)dw ζs) βs) ds Proof of Lemma 2.8. Let h :R + R + be a C function that satisfies hs) = 0, s / 0, 1) and R + hs)ds = 1 0 hs)ds = 1. Define ãs) := au)hu s)du au)hu)du. A.20) R + R + Notice that ã0) = 0 and ã C R + ). Furthermore, we have ãs) = 1 0 as + w) aw))hw)dw. A.21) Clearly by virtue of 2.3b) of Lemma 2.6 and A.21), it follows that 2.5) holds. Moreover by A.21) we get dã 1 ds s) = da s + w)hw)dw 0 ds A.22) which implies that dã ds s) is non-decreasing and positive. Therefore ã K con C R + ). Define R := da ds 2r) for some r > 0 and notice that we also have as) Rs, s [0, 2r]. A.23)

26 444 I. KARAFYLLIS Finally, define βs) := Rs Rs + exp for 0 s r 1 r 1 ) ãs) for s > r. s r A.24) Clearly β C R + ) and its odd extension β ) is also a smooth function on R. Moreover by 2.5), A.23) and definition A.24) we have that as) βs), s 0. By virtue of iii) of Lemma 2.6, there exist M > 0 and δ K con such that dã ds s) M + δs) and since ) 1 sup s>0 exp 1 s 2 s 4, we can easily verify that 2.6) and 2.7) are satisfied for ) 1 R := R + M exp r γs) := 5exp ) 1 ãs) + δs)) + Rs. r The proof is complete. Proof of Lemma 2.9. In order to prove 2.10), let r 0, t 0 0, t t 0 and notice that for the solution βt, t 0, r) of 2.11) we have τ βt τ,t 0, r) = ρφt τ)βt τ,t 0, r)), τ [0, t t 0 ]. A.25) Let a vector x B βt,t0,r) be the initial condition for the problem dξ τ) = f t τ,ξ,d) dτ ξ0) = x, d D,τ 0. A.26) This is the time-reversed system 1.1). It is clear by virtue of 2.8) that the following differential inequality is satisfied: D + ξτ) ρφt τ) ξτ) ), τ [0, t]. A.27) It follows by A.25), A.27) and the comparison Lemma in Khalil 1996), that ξτ) βt τ,t 0, r), τ [0, t t 0 ]. A.28) Since βt 0, t 0, r) = r, inequality A.28) shows that there exists x 0 B r and d M D such that xt, t 0, x 0 ; d) = x and it is given by x 0 = ξt t 0 ) for the particular d M D. Since x B βt,t0,r) is arbitrary, it follows that 2.10) holds. The proof is complete.