Control Lyapunov Functions and Stabilization by Means of Continuous Time-Varying Feedback
1 CONTROL LYAPUNOV FUNCTIONS AND STABILIZATION BY MEANS OF CONTINUOUS TIME-VARYING FEEDBACK
Iasson Karafyllis * and John Tsinias ** * Department of Environmental Engineering, Technical University of Crete, 73100, Chania, Greece email: [email protected] ** Department of Mathematics, National Technical University of Athens, Zografou Campus 15780, Athens, Greece Email: [email protected]
Abstract
For a general time-varying system, we prove that existence of an “Output Robust Control Lyapunov Function” implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result towards feedback stabilization due to J. M. Coron and L. Rosier concerning stabilization of autonomous systems by means of time-varying periodic feedback.
Keywords:
Control Lyapunov Function, feedback stabilization, time-varying systems.
1. Introduction
Control Lyapunov functions play a central role to the solvability of the feedback stabilization problem and several important works are found in the literature, where sufficient conditions are provided in terms of Lyapunov functions for characterizations of various types of stability, as well for existence of feedback stabilizers (see for instance [1,2,4,5,6,7,9,11,12,13,15,18,19,20,21,25]). In the present work we consider time-varying uncertain systems of the general form: mkn
UuYtDdx xtHY uxdtfx
ℜ⊆∈ℜ∈≥∈ℜ∈== ,,0,, ),( ),,,( & (1.1) where )( ⋅ d is a time-varying disturbance which takes values on the set l D ℜ⊂ and Yu , play the role of input and output, respectively, of (1.1). We assume that n ℜ∈ is an equilibrium for (1.1), i.e. it holds = dtf and = tH for all Ddt ×ℜ∈ + ),( . We prove that existence of an “Output Robust Control Lyapunov Function” (ORCLF) implies existence of continuous time-varying feedback stabilizer ),( xtKu = (1.2) that guarantees global output asymptotic stability of the output ),( xtHY = with respect to the resulting closed-loop system (1.1) with (1.2), being uniform with respect to disturbances )( ⋅ d . Our main results constitute generalizations of an important result towards feedback stabilization obtained in [7] by J. M. Coron and L. Rosier concerning autonomous systems: ),( uxfx = & , mn ux ℜ×ℜ∈ ),( with n f ℜ∈= (1.3) Particularly, among other things in [7], it is established that existence of a time-independent control Lyapunov function, which satisfies the “small-control property” guarantees existence of a continuous time-varying periodic feedback (1.2) in such a way that n ℜ∈ is globally asymptotically stable for the resulting time-varying closed-loop system (1.1) with (1.2). In the present work we present generalizations of the result above for general time-varying systems (1.1). Particularly, in Theorem 2.8 of present paper we establish that existence of an ORCLF, which satisfies a time-varying version of the small control property, implies existence of a continuous feedback stabilizer UK n →ℜ×ℜ + : , which is continuously differentiable with respect to n x ℜ∈ on the set })0{\( n ℜ×ℜ + exhibiting Robust Global Asymptotic Output Stability (RGAOS) of the resulting closed-loop system, being uniform with respect to initial values of time . In Theorem 2.9 of present work it is shown that, under lack of the small control property, existence of an ORCLF, implies existence of a continuous feedback stabilizer UK n →ℜ×ℜ + : , being continuously differentiable with respect to n x ℜ∈ on the set n ℜ×ℜ + exhibiting RGAOS of the resulting closed-loop system, being in general non-uniform with respect to initial values of time . We note here, that various concepts of asymptotic stability being in general non-uniform with respect to initial values of time, their Lyapunov characterizations, as well applications to feedback stabilization and related problems are found in several recent works (see for instance [11,12,13] and references therein). As a consequence of Theorem 2.9 and the main result in [12], it is shown in Corollary 2.10 of the present work that that the converse claim of Theorem 2.9 is true; to be more precise the following three statements are equivalent: • existence of an ORCLF (under lack of the small control property), • existence of a continuous mapping UK n →ℜ×ℜ + : being continuously differentiable with respect to n x ℜ∈ on n ℜ×ℜ + , such that the closed-loop system (1.1) with (1.2) is (non-uniformly in time) RGAOS, • existence of an ORCLF satisfying the small control property. It should be emphasized here that, when the result of Theorem 2.9 is restricted to autonomous systems (1.3), we get the following result which generalizes both Artstein’s theorem on stabilization in [2,20] and Rosier-Coron main result in [7]: Assume that (1.3) possess a (time-independent) Control Lyapunov Function, namely, suppose that there exists a map );( + ℜℜ∈ n CV , functions ∞ ∈ Kaa , , and );( ++ ℜℜ∈ C ρ being positive definite such that )()()( xaxVxa ≤≤ , ( ) )(),()(min xVuxfxxV Uu ρ −≤∂∂ ∈ , n x ℜ∈ . Then there exists a time-varying continuous mapping UK n →ℜ×ℜ + : , being continuously differentiable with respect to n x ℜ∈ on n ℜ×ℜ + , such that the closed-loop time-varying system (1.3) with (1.2) is RGAOS in general non-uniformly with respect to initial values of time. Comparing with the results obtained in [7] the result above presents the important advantage that feedback stabilization is exhibited under lack of the small control property and the corresponding feedback is an ordinary map, being in general time-varying but non-periodic. On the other hand, our approach leads in general to non-uniform in time asymptotic stability for the resulting closed-loop system. Finally, it should be pointed out that the main results in the present work (Theorem 2.8 and Theorem 2.9) generalize the main result in [7] in the following additional directions: • The dynamics of systems we consider are in general time-varying, including disturbances, and the control set U is in general a positive cone of m ℜ . • The general problem of robust output stabilization is considered and feedback stabilization is exhibited under the presence of time-varying Control Lyapunov Functions. The proofs of the main results in the present work are inspired by the proof of the main result in [7], but are essentially different in many points. The paper is organized as follows. In Section 2, several stability notions and the concept of the Output Robust Control Lyapunov Function are presented, as well precise statements of our main results are given. Section 3 contains the proofs of the main results.
Notations
Throughout this paper we adopt the following notations: ∗ Let n A ℜ⊆ . By );( Ω AC , we denote the class of continuous functions on A , which take values in Ω . Likewise, );( Ω AC denotes the class of functions on A with continuous derivatives, which take values in Ω . ∗ For a vector n x ℜ∈ we denote by x its usual Euclidean norm and by x ′ its transpose. ∗ Z denotes the set of integers, + Z denotes the set of non-negative integers and + ℜ denotes the set of non-negative real numbers. ∗ We denote by ][ r the integer part of the real number r , i.e., the greatest integer, which is less than or equal to r . ∗ A continuous mapping
Uxtkxt n ∈→∋ℜ×ℜ + ),(),( , is continuously differentiable with respect to n x ℜ∈ on the open set n A ℜ×ℜ⊆ + (with respect to the n ℜ×ℜ + topology), if the mapping n xtxkxtA ℜ∈∂∂→∋ ),(),( is continuous and is called locally Lipschitz with respect to n x ℜ∈ on the open set n A ℜ×ℜ⊆ + , if for every compact set AS ⊆ it holds that +∞<⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ ≠∈∈−− yxSytSxtyx ytkxtk ,),(,),(:),(),(sup ∗ We denote by + K the class of positive C functions defined on + ℜ . We say that a function ++ ℜ→ℜ : ρ is positive definite if = ρ and > s ρ for all > s . By K we denote the set of positive definite, increasing and continuous functions. We say that a positive definite, increasing and continuous function ++ ℜ→ℜ : ρ is of class ∞ K if +∞= +∞→ )(lim s s ρ . ∗ Let l D ℜ⊆ be a non-empty set. By D M we denote the class of all Lebesgue measurable and locally essentially bounded mappings Dd →ℜ + : .
2. Basic Notions and Main Results
In this work, we consider systems of the form (1.1) under the following hypotheses: (H1)
The mappings nn UDf
ℜ→×ℜ××ℜ + : , kn H ℜ→ℜ×ℜ + : are continuous and for every bounded interval + ℜ⊂ I and every compact set US n ×ℜ⊂ there exists ≥ L such that vuLyxLvydtfuxdtf −+−≤− ),,,(),,,( for all DIdt ×∈ ),( , Sux ∈ ),( , Svy ∈ ),( . (H2) The set l D ℜ⊂ is compact and U is a closed positive cone, i.e., m U ℜ⊆ is a closed set and, if Uu ∈ , then Uu ∈ )( λ for all ]1,0[ ∈λ . (H3 ) Zero n ℜ∈ is an equilibrium; particularly, assume that = dtf , = tH for all Ddt ×ℜ∈ + ),( . Definition 2.1
We say that a function );( ++ ℜℜ×ℜ∈ n CV is an Output Robust Control Lyapunov Function (ORCLF) for (1.1) , if there exist functions ∞ ∈ Kaa , , + ∈ K βμ , , );( ++ ℜℜ∈ C ρ , being locally Lipschitz and positive definite and ( ) );}0{\( ++ ℜℜ×ℜ∈ n Cb , such that i) for every n xt ℜ×ℜ∈ + ),( it holds: ( ) ( ) xtaxtVxtxtHa )(),()(),( βμ ≤≤+ (2.1) ii) for every })0{\(),( n xt ℜ×ℜ∈ + it holds: ( ) ),(),,,(),(),(maxmin ),( xtVuxdtfxtxVxttV DdUu xtbu ρ−≤⎟⎟⎠⎞⎜⎜⎝⎛ ∂∂+∂∂ ∈∈≤ (2.2)
For the case, where (2.2) holds and, instead of (2.1), it holds: ( ) ( ) xtaxtVxa )(),( β≤≤ , n xt ℜ×ℜ∈∀ + ),( (2.1’) the corresponding );( ++ ℜℜ×ℜ∈ n CV is called State Robust Control Lyapunov Function (SRCLF).
We say that the ORCLF (SRCLF) satisfies the small-control property with respect to (1.1), if in addition to (2.1), (2.2), ((2.1’), (2.2)), there exist functions ∞ ∈ Ka , + ∈ K γ such that the following inequality holds for all })0{\(),( n xt ℜ×ℜ∈ + : ( ) xtaxtb )(),( γ≤ (2.3) where ( ) );}0{\( ++ ℜℜ×ℜ∈ n Cb is the function involved in (2.2). Remark 2.2:
If the ORCLF );( ++ ℜℜ×ℜ∈ n CV is − T periodic (namely, ),(),( xtVxTtV =+ for certain > T and for all n xt ℜ×ℜ∈ + ),( ), then (2.1) implies ( ) ),( xtVxMa ≤ for all n xt ℜ×ℜ∈ + ),( , where ],0[ >= ∈ tM Tt μ . Consequently, existence of a − T periodic OCLF implies existence of a − T periodic SRCLF. Moreover, the existence of a time-invariant ORCLF implies the existence of a time-invariant SRCLF. Remark 2.3:
The small-control property in Definition 2.1 constitutes a time-varying version of the small-control property for the autonomous case [2,9,20].
Remark 2.4:
Time -Varying ORCLFs have to be considered even for autonomous systems. It should be noticed that, in general, it is possible for an autonomous system (1.1) to possess a time-varying
ORCLF satisfying the small-control property, although a time-independent
ORCLF does not exists. Indeed, consider the elementary linear system xx = & , ux = & , with ℜ=∈ Uu and output xY = . Obviously, this system is not feedback stabilizable to zero ℜ∈ and therefore, according to [7], a time-invariant SRCLF does not exist. Neither a time-independent ORCLF exists, according to Remark 2.2 above. On the other hand it can be easily verified that the function xxtxtV +−= is an ORCLF, which in addition satisfies the small-control property. We next present certain stability concepts used in the present work. Consider the system ),,( xdtfx = & (2.4a) kn YDdx xtHY
ℜ∈∈ℜ∈= ,, ),( (2.4b) where the mappings nn Df ℜ→ℜ××ℜ + : , kn H ℜ→ℜ×ℜ + : are continuous with = dtf , = tH for all Ddt ×ℜ∈ + ),( and l D ℜ⊂ is compact. We assume that for every Dn Mdxt ×ℜ×ℜ∈ + ),,( there exists ],0( +∞∈ h and a unique solution n httdxtxx ℜ→+⋅=⋅ ),[:);,,()( of (2.4a) with )( xtx = . Definition 2.5:
We say that (2.4) is
Robustly Forward Complete (RFC), if for every ≥ T , ≥ r it holds that: { } +∞<∈⋅∈∈≤+ D MdThTtrxdxthtx )(,],0[,],0[,;);,,(sup (2.5) Clearly, the notion of robust forward completeness implies the standard notion of forward completeness, which simply requires that for every initial condition the solution of the system exists for all times greater than the initial time, or equivalently, the solutions of the system do not present finite escape time. Conversely, an extension of Proposition 5.1 in [16] to the time-varying case shows that every forward complete system (2.4) whose dynamics are locally Lipschitz with respect to ),( xt , uniformly in Dd ∈ , is RFC. All output stability notions used in the present work will assume RFC. We next provide the notion of (non-uniform in time) Robust Global Asymptotic Output Stability (RGAOS) (see [12,13]), which is a generalization of the notion of Robust Output Stability (see [22,23,26]). Let us denote by ));,,(,()( dxtxHY ⋅⋅=⋅ the output of (2.4) corresponding to input D Md ∈ and initial condition )( xtx = . Definition 2.6:
Consider system (2.4) and suppose that is RFC. We say that (2.4) is (non-uniformly in time) Robustly Globally Asymptotically Output Stable (RGAOS) if it satisfies the following properties:
P1(Output Stability)
For every > ε , ≥ T , it holds: { } +∞<∈⋅∈≤≥ D MdTtxtttY )(,],0[,,;)(sup ε (Robust Lagrange Output Stability) and there exists a ( ) >= T εδδ such that ,)(],0[, tttYTtx ≥∀≤⇒∈≤ εδ , D Md ∈⋅∀ )( (Robust Lyapunov Output Stability) P2(Uniform Output Attractivity on compact sets of initial data)
For every > ε , ≥ T and ≥ R , there exists a ( ) ≥= RT εττ such that τε +≥∀≤⇒∈≤ ,)(],0[, tttYTtRx , D Md ∈⋅∀ )( The notion of Uniform Robust Global Asymptotic Output Stability was originally given in [22,23] and is a special case of non-uniform in time RGAOS.
Definition 2.7:
Consider system (2.4) and suppose that is RFC. We say that (2.4) is
Uniformly
Robustly Globally Asymptotically Output Stable (URGAOS), if it satisfies the following properties:
P1(Uniform Output Stability)
For every > ε , it holds that { } +∞<∈⋅≥≤≥ D MdtxtttY )(,0,,;)(sup ε (Uniform Robust Lagrange Output Stability) and there exists a ( ) >= εδδ such that ,)(0, tttYtx ≥∀≤⇒≥≤ εδ , D Md ∈⋅∀ )( (Uniform Robust Lyapunov Output Stability) P2(Uniform Output Attractivity on compact sets of initial states)
For every > ε and ≥ R , there exists a ( ) ≥= R εττ such that τε +≥∀≤⇒≥≤ ,)(0, tttYtRx , D Md ∈⋅∀ )( Obviously, for the case xxtH = ),( the notions of RGAOS, URGAOS coincide with the notions of non-uniform in time Robust Global Asymptotic Stability (RGAS) as given in [11] and
Uniform Robust Global Asymptotic Stability (URGAS) as given in [16], respectively. Also note that, if there exists ∞ ∈ Ka with )),(( xtHax ≤ for all n xt ℜ×ℜ∈ + ),( , then (U)RGAOS implies (U)RGAS . We are now in a position to state our main results. Theorem 2.8:
Consider system (1.1) under hypotheses (H1-3) and assume that (1.1) admits an ORCLF which satisfies (2.1), (2.2) and the small-control property (2.3). Moreover, suppose that ≡ t β , where + ∈ K β is the function involved in (2.1). Then there exists a continuous mapping UK n →ℜ×ℜ + : with = tK for all ≥ t , being continuously differentiable with respect to n x ℜ∈ on the set })0{\( n ℜ×ℜ + , such that for all Dn Mdxt ×ℜ×ℜ∈ + ),,( the solution )( ⋅ x of the closed-loop system (1.1) with ),( xtKu = : )),(,,,( xtKxdtfx = & (2.6) with initial condition n xtx ℜ∈= )( , corresponding to input D Md ∈ is unique and system (2.6) is URGAOS. Theorem 2.9:
Consider system (1.1) under hypotheses (H1-3) and assume that (1.1) admits an ORCLF which satisfies (2.1), (2.2). Then there exists a continuous mapping UK n →ℜ×ℜ + : , with = tK for all ≥ t , which is continuously differentiable with respect to n x ℜ∈ on n ℜ×ℜ + , such that the closed-loop system (2.6) is RGAOS. It should be emphasized that the small-control property is not required for the validity of the result of Theorem 2.9. On the other hand, Theorem 2.9 cannot in general guarantee uniformity of solutions of the resulting closed-loop system (2.6) with respect to the initial time. Another advantage of Theorem 2.9 above is that the proposed feedback ),( xtK is locally Lipschitz with respect to n x ℜ∈ . The latter in conjunction with the converse Lyapunov theorem in [12] leads to the following result: Corollary 2.10:
Consider system (1.1) under hypotheses (H1-3). The following statements are equivalent: (i)
System (1.1) admits an ORCLF which satisfies (2.1), (2.2). (ii)
There exists a continuous mapping UK n →ℜ×ℜ + : , with = tK for all ≥ t , which is continuously differentiable with respect to n x ℜ∈ on n ℜ×ℜ + , such that the closed-loop system (1.1) with ),( xtKu = is RGAOS. (iii) System (1.1) admits an ORCLF which satisfies (2.1), (2.2) and the small-control property (2.3).
The following example illustrates the nature of Theorem 2.9.
Example 2.11:
Consider the following system kn Nj kj
YuDdx xtHY uxtgxdtfx j ℜ∈ℜ∈∈ℜ∈= += ∑ = ,,, ),( ),(),,( & (2.7) where l D ℜ⊂ is a compact set, nn Df ℜ→ℜ××ℜ + : , nnj g ℜ→ℜ×ℜ + : ( Nj ,...,0 = ) are locally Lipschitz mappings with = dtf for all Ddt ×ℜ∈ + ),( and kn H ℜ→ℜ×ℜ + : is a continuous mapping with = tH for all ≥ t . Assume that Njk j ,...,1, = are odd positive integers (2.8) and there exist functions );( ++ ℜℜ×ℜ∈ n CV , ∞ ∈ Kaa , , + ∈ K βμ , , );( ++ ℜℜ∈ C ρ being locally Lipschitz and positive definite, such that ( ) ( ) xtaxtVxtxtHa )(),()(),( βμ ≤≤+ , n xt ℜ×ℜ∈∀ + ),( (2.9) and in such a way that the following implication holds: ∑ = ⇒=⎟⎟⎠⎞⎜⎜⎝⎛ ∂∂ Nj j xtgxtxV ( ) ),(2),,(),(),(max xtVxdtfxtxVxttV Dd ρ −≤⎟⎟⎠⎞⎜⎜⎝⎛ ∂∂+∂∂ ∈ (2.10) We claim that );( ++ ℜℜ×ℜ∈ n CV is an ORCLF for system (2.7). Indeed, by exploiting (2.8) and implication (2.10) it follows that for every n xt ℜ×ℜ∈ + ),( there exists ℜ∈ u such that ( ) ),(2),(),(),,(),(),(max xtVuxtgxtxVxdtfxtxVxttV Nj kjDd j ρ −≤⎟⎟⎠⎞⎜⎜⎝⎛ ∂∂+∂∂+∂∂ ∑ =∈ (2.11) From (2.11), compactness of l D ℜ⊂ and continuity of f , j g ( Nj ,...,0 = ), it follows by applying standard partition of unity arguments, that there exists a function ( ) );}0{\( ++ ℜℜ×ℜ∈ n Cb such that ( ) ),(),(),(),,(),(),(maxmin xtVuxtgxtxVxdtfxtxVxttV Nj kjDdu xtbu j ρ−≤⎟⎟⎠⎞⎜⎜⎝⎛ ∂∂+∂∂+∂∂ ∑ =∈ℜ∈≤ (2.12) Hence, by (2.9) and (2.12) we may conclude that );( ++ ℜℜ×ℜ∈ n CV is an ORCLF for system (2.7). Consequently, according to statement of Theorem 2.9, there exists a continuous mapping UK n →ℜ×ℜ + : , with = tK for all ≥ t , which is continuously differentiable with respect to n x ℜ∈ on n ℜ×ℜ + , such that the closed-loop system (2.7) with ),( xtKu = is RGAOS. <
3. Proofs of the Main Results
The proof of the main results of the present work is based on three lemmas below. Particularly, Lemma 3.1 is a preparatory result for the construction of the desired feedback stabilizer. It constitutes a time-varying extension of Lemma 2.7 in [7], but its constructive proof differs from the corresponding proof of the previously mentioned result.
Lemma 3.1:
Consider system (1.1) under hypotheses (H1-3) and assume that (1.1) admits an ORCLF which satisfies (2.1), (2.2). Then there exists a C function ( ) Uk n →ℜ×ℜ× + }0{\]1,0[: with == xtkxtk (3.1a) =∂∂=∂∂ xttkxtsk ; =∂∂ xtxk (3.1b) =∂∂=∂∂ xttkxtsk ; =∂∂ xtxk (3.1c) for all ≥ t , }0{\ n x ℜ∈ , and in such a way that: ( ) ),(21)),,(,),(,(),(),( xtVdsxtskxsdtfxtxVxttV ρ −≤∂∂+∂∂ ∫ (3.2) for all })0{\(),( n xt ℜ×ℜ∈ + , D Md ∈ . Moreover, the following inequality holds for all })0{\(),( n xt ℜ×ℜ∈ + : ),(~),,(max ]1,0[ xtbxtsk s ≤ ∈ (3.3) where ⎭⎬⎫⎩⎨⎧ +≤≤≤≤= txyxybxtb ττ (3.4) Proof of Lemma 3.1:
Let ++ ℜ→ℜ×ℜ n b :~ as given by (3.4) that obviously is of class );( ++ ℜℜ×ℜ n C and let ),1[: +∞→ℜ×ℜ + n ϕ be any smooth ( ∞ C ) function satisfying ),(),,,(),(),(maxmax ),(~ xtuxdtfxtxVxttV DdUu xtbu ϕ ≤⎟⎟⎠⎞⎜⎜⎝⎛ ∂∂+∂∂ ∈∈≤ , })0{\(),( n xt ℜ×ℜ∈∀ + (3.5) Moreover, let )1,0(})0{\(: →ℜ×ℜ + n ε be a smooth function such that ( ) ),(),((4 )),((),(0 xtxtV xtVxt ϕρ ρε +≤< , })0{\(),( n xt ℜ×ℜ∈∀ + (3.6) and define ( ) ⎟⎟⎠⎞⎜⎜⎝⎛ +∂∂+∂∂=Ψ ∈ ),(43),,,(),(),(max:),,( xtVuxdtfxtxVxttVuxt Dd ρ , Uuxt n ×ℜ×ℜ∈ + ),,( (3.7a) ),,0(:),,( uxuxt Ψ=Ψ , Uuxt n ×ℜ×−∈ )0,1(),,( (3.7b) By virtue of (2.2), continuity of Ψ and compactness of l D ℜ⊂ , it follows that for each })0{\(),1(),( n xt ℜ×+∞−∈ there exist
Uxtuu ∈= ),( with ),( xtbu ≤ and ]1,0(),( ∈= xt δδ with xxt ≤δ such that ≤Ψ xtuy τ , { } δτττ <−+−ℜ×+∞−∈∈∀ xytyy n :),1(),(),( (3.8) Using (3.8) and standard partition of unity arguments, we can determine sequences ∞= ℜ×+∞−∈ })}0{\(),1(),{( inii xt , ∞= ∈ }{ ii Uu , ∞= ∈ )}1,0({ ii δ with )),,0(max( iii xtbu ≤ and iiii xxt ≤= δδ associated with a sequence of open sets ∞= Ω }{ ii with { } iiini xyty δττ <−+−ℜ×+∞−∈⊆Ω :),1(),( (3.9a) forming a locally finite open covering of })0{\(),1( n ℜ×+∞− and in such a way that: ≤Ψ i uy τ , i y Ω∈∀ ),( τ (3.9b) Also, a family of smooth functions ∞= }{ ii θ with ≥ xt i θ for all ( ) }0{\),1(),( n xt ℜ×+∞−∈ can be determined with ii upps Ω⊆ θ (3.9c) = ∑ ∞= i i xt θ , ( ) }0{\),1(),( n xt ℜ×+∞−∈∀ (3.9d) Next define recursively the following mappings for each })0{\(),( n xt ℜ×ℜ∈ + : ),(),(),( xtxtTxtT iii θ += − , ≥ i ; = xtT ; })0{\(),( n xt ℜ×ℜ∈ + (3.10) Notice that definition (3.10) implies ∑ = = ni in xtxT ),()( θ for all ≥ n . Since the open sets ∞= Ω }{ ii form a locally finite open covering of })0{\( n ℜ×ℜ + , it follows from (3.9c) and (3.10) that for every })0{\(),( n xt ℜ×ℜ∈ + there exists ,...}3,2,1{),( ∈= xtmm such that = xtT i for mi ≥ (3.11) We define the index set { } >∈= xtjxtJ j θ (3.12) which by virtue of (3.11) is a non-empty finite set. It follows from definitions (3.10) and (3.12) that [ ) )1,0[),(),,( =∪ −∈ xtTxtT jjxtJj , })0{\(]1,0[),( n xt ℜ×ℜ×∈∀ + (3.13) Let ]1,0[: →ℜ h be any smooth non-decreasing function with = sh for ≤ s and = sh for ≥ s (3.14a) and let ( ) ⎟⎟⎠⎞⎜⎜⎝⎛ −−+= −− −−−− j jjjjjj xt xtxthxtxtxtxtg ε εθθεθ , ,...}3,2,1{ ∈ j (3.14b) where ),( ⋅⋅ ε is the function defined by (3.6). Notice that according to (3.14a,b) it holds: { } ),(,2),(min),(),(21,2),(min xtxtxtgxtxt jjjjj θεθε −−−− ≤≤⎭⎬⎫⎩⎨⎧ (3.15a) −− = jj xtxtg ε for −− ≥ jj xtxt εθ (3.15b) We define the following map mn xtskxts ℜ∈→∋ℜ×ℜ× + ),,(),,(})0{\(]1,0[ : ⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛ −−⎟⎟⎠⎞⎜⎜⎝⎛= −−− ),(51 ),(51),(2),( ),(),,( xtg xtgxtTshxt xtguxtsk j jjjjj ε , for ⎟⎠⎞⎢⎣⎡ +∈ −− ),(21),(),,( xtxtTxtTs jjj θ , ),( xtJj ∈ (3.16a) ⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛ −−⎟⎟⎠⎞⎜⎜⎝⎛= −− ),(51 ),(51),(2),( ),(),,( xtg sxtgxtThxt xtguxtsk j jjjjj ε , for ⎟⎠⎞⎢⎣⎡ −∈ ),(),,(21),( xtTxtxtTs jjj θ , ),( xtJj ∈ (3.16b) = xtk (3.16c) Notice that because of (3.13), ),,( ⋅⋅⋅ k is well defined for all })0{\(]1,0[),,( n xts ℜ×ℜ×∈ + . Furthermore, according to definition (3.16), hypothesis (H2) guarantees that ),,( ⋅⋅⋅ k takes values in m U ℜ⊆ and is continuously differentiable on the region ( ) ( ) }0{\),(),,( njjxtJj xtTxtT ℜ×ℜ×⎟⎠⎞⎜⎝⎛ ∪ +−∈ . Furthermore, it holds that →∂∂ xtssk , →∂∂ xtssk , →∂∂ xtsxk as ),( xtTs j → for all ,...}3,2,1,0{ ∈ j (3.17) Next, we show that ),,( ⋅⋅⋅ k is continuously differentiable on the whole region })0{\(]1,0[ n ℜ×ℜ× + and simultaneously that (3.1b,c,d) are fulfilled. We distinguish the following cases: Case 1: Let )1,0( ∈ s , })0{\(),( n xt ℜ×ℜ∈ + and suppose that there exists a positive integer p with ),( xtTs p = . Then, there exist positive integers lm , with mpl ≤≤ in such a way that > + xt m θ , > xt l θ (3.18a) >=== xtTxtTs lm (3.18b) Equality (3.18b) in conjunction with definition (3.10) means === + xtxt lm θθ , if +≥ lm (3.19) Notice that definition (3.16a) and (3.18a) imply that in our case it holds = xtsk (3.20) By taking into account continuity of the mappings mlml TTgg ,,, + and (3.15a), it follows that there exists > δ such that ⎟⎠⎞⎜⎝⎛ +−∈′ + ),(51),(),,(51),( ygyTygyTs mmll ττττ })0{\(]1,0[),,( n ys ℜ×ℜ×∈′∀ + τ with δτ <−+−+−′ xytss (3.21) By virtue of definition (3.16a,b), (3.20) and (3.21) it follows that for every })0{\(]1,0[),,( n ys ℜ×ℜ×∈′ + τ with δτ <−+−+−′ xytss it holds: ⎟⎟⎠⎞⎜⎜⎝⎛≤−′ −−+= vvvmlv y yguxtskysk τε ττ , if +≥ lm (3.22a) ),,(),,( xtskysk =′ τ , if lm = (3.22b) If +≥ lm , then by (3.15a) and (3.19) we also get = xtg v for mlv ,...,1 += , hence, since the mappings v g are continuously differentiable, there exists a constant > L such that xyLtLy yg vmlv −+−≤ += ττε τ ),( ),(max ,...,1 , }0{\),( n y ℜ×ℜ∈∀ + τ with δτ <−+− xyt (3.23) It turns out from (3.22a,b) and (3.23) that ( ) ),,(),,( xytLxtskysk −+−′≤−′ ττ (3.24) for certain constant >′ L and for δτ <−+−+−′ xytss . We conclude from (3.24) that the derivatives of ),,( ⋅⋅⋅ k exist for ),( xtTs p = and it holds that =∂∂=∂∂ xtstkxtssk and =∂∂ xtsxk for ),( xtTs p = . The latter in conjunction with (3.17) implies that ),,( ⋅⋅⋅ k is continuously differentiable in a neighborhood of ),,( xts with )1,0(),( ∈= xtTs p . Case 2: Let = s , })0{\(),( n xt ℜ×ℜ∈ + and suppose that there exists an integer ≥ p with == xtTs p . Clearly, there exists an integer pm ≥ such that > + xt m θ (3.25a) ==== xtTxtTs m (3.25b) (note again that equality (3.25b) means that === xtxt m θθ for the case > m ). By virtue of definition (3.16a) it holds that = xtsk and continuity of the mappings m T and + m g implies that there exists > δ such that ⎟⎠⎞⎢⎣⎡ +∈′ + ),(51),(,0 ygyTs mm ττ for all })0{\(]1,0[),,( n ys ℜ×ℜ×∈′ + τ with δτ <−+−+−′ xytss . Then as in Case 1, it follows by (3.16) that ⎟⎟⎠⎞⎜⎜⎝⎛≤−′ −−= vvvmv y yguxtskysk τε ττ , if > m (3.26a) ),,(),,( xtskysk =′ τ , if = m (3.26b) for every δτ <−+−+−′ xytss , from which we get the desired conclusion, namely, that that ),,( ⋅⋅⋅ k is continuously differentiable in a neighborhood of ),,0( xt and further (3.1b) holds. Case 3: Let = s , })0{\(),( n xt ℜ×ℜ∈ + and let p a positive integer with ),( xtTs p = . Let ∞= Ω }{ ii be the locally finite open covering of })0{\(),1( n ℜ×+∞− and the associated sequence of functions { } i i θ ∞= in such a way that (3.9a,b,c,d) hold. Let })0{\(),1( n N ℜ×+∞−⊂ be a neighborhood containing ),( xt which intersects only a finite number of the open sets ∞= Ω }{ ii (see [10]). Consequently, by (3.9d) there exists an integer > m such that ∅=Ω∩ i N for all mi > and = y i τθ for all mi > , Ny ∈ ),( τ . Clearly, there exists },...,1{ ml ∈ with > xt l θ (3.27a) ==== xtTxtTs ml (3.27b) Without loss of generality we may assume that lm > . By virtue of definition (3.16c) we have = xtk and continuity of the mappings l T , l g , asserts existence of a constant > δ such that ⎥⎦⎤⎜⎝⎛ −∈′ ygyTs ll ττ and Ny ∈ ),( τ ; })0{\(]1,0[),,( n ys ℜ×ℜ×∈′∀ + τ with δτ <−+−+−′ xyts (3.28) Using (3.16) and (3.28) we get ⎟⎟⎠⎞⎜⎜⎝⎛≤−′ −−+= vvvmlv y yguxtkysk τε ττ from which it follows that (3.24) holds for all δτ <−+−+−′ xyts and for certain constant >′ L . This implies that the derivatives of ),,( ⋅⋅⋅ k exist for = s and particularly, (3.1c) holds. The latter in conjunction with (3.17) implies that ),,( ⋅⋅⋅ k is continuously differentiable in a neighborhood of ),,1( xt . We next establish (3.3). By virtue of (3.14a), (3.15a) and definition (3.16) we have jxtJjs uxtsk ),(]1,0[ max),,(max ∈∈ ≤ , ),( xtJ being the index set defined by (3.12). For every ),( xtJj ∈ there exist })0{\(),1(),( njj xt ℜ×+∞−∈ with )),,0(max( jjj xtbu ≤ (3.29) for which j xt Ω∈ ),( and in such a way that (3.9a) holds with ji = . Since ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧≤= jjjj xxt δδ , it follows that, when jjj xxtt δ <−+− , it holds that xxx j ≤≤ and +≤≤− ttt j . The latter in conjunction with (3.25) and definition (3.4) of ),(~ ⋅⋅ b implies (3.3). Finally, we establish (3.2). Notice that by (3.12), (3.14a), (3.15b), (3.16a,b), for any }0{\),( n xt ℜ×ℜ∈ + and integer ),( xtJj ∈ it holds: j uxtsk = ),,( , ⎥⎦⎤⎢⎣⎡ −+∈∀ − ),(52),(),,(52),( xtgxtTxtgxtTs jjjj when −− ≥ jj xtxt εθ (3.30) hence, the set }),(,),,(:]1,0[{: ),( xtJjuxtsksI jxt ∈≠∈= has Lebesgue measure, say ),( xt I , satisfying : ),(2),( ),( 1),( xtxtI xtJj jxt εε ≤≤ ∑ ∈ −− (3.31) Then for any D Md ∈ it follows by virtue of (3.7a), (3.9b) and (3.31) that ( ) ( ) ),,,(),(),(maxmax),(),(),(143 )),,(,),(,(),(),( ),(~10 uxdtfxtxVxttVxtxtVxt dsxtskxsdtfxtxVxttV DdUu xtbu ∂∂+∂∂+−−≤ ≤∂∂+∂∂ ∈∈≤ ∫ ερε Inequalities (3.5), (3.6) in conjunction with the above inequality imply (3.2) and the proof is complete. < The next lemmas (Lemma 3.2 and 3.3) constitute key results of the rest analysis and generalize Lemmas 2.8, 2.9 in [7]. Their proofs are based on certain appropriate generalizations of the technique employed in [7].
Lemma 3.2:
Consider system (1.1) under the same hypotheses with those imposed in Lemma 3.1. For every pair of sets { }
Zirr i ∈= : , { } Ziaa i ∈= : with > i r , > i a , ++ −<+ iiii arar for all Zi ∈ (3.32) +∞= +∞→ ii r lim , = −∞→ ii r (3.33) there exists a continuous mapping Uk nar →ℜ×ℜ + })0{\(: , , being continuously differentiable with respect to }0{\ n x ℜ∈ with , = xjk ar and 0),( , =∂∂ xjxk ar for all ( ) + ×ℜ∈ Zjx n }0{\),( (3.34) ),(~),( , xtbxtk ar ≤ , })0{\(),( n xt ℜ×ℜ∈∀ + (3.35) where ),(~ ⋅⋅ b is defined by (3.4), and in such a way that the following property holds for all ( ) ZMidxt Dn ××ℜ×ℜ∈ + }0{\),,,( : iiii ardxttxtVrrxtV +≤⇒∈ − , for all [ ] ( ) [ ) max00 ,1min, tttt +∈ (3.36) where );,,( dxtx ⋅ denotes the unique solution of )),(,,,( , xtkxdtfx ar = & , })0{\(),( n xt ℜ×ℜ∈ + (3.37) with initial condition }0{\)( n xtx ℜ∈= , corresponding to D Md ∈ , ),,(: tdxttt >= denotes its maximal existence time. Moreover, for each ( ) ZZijx n ××ℜ∈ + }0{\),,( , there exists a positive integer ≥ N such that ⎟⎠⎞⎜⎝⎛ −+≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎠⎞⎜⎝⎛ ++⇒−≤ −− iiiii NsxjVardxjNsjxNsjVarxjV μ ),(,2max;,,,2),( , for all D Md ∈ , { } Ns ,...,1,0 ∈ with max tNsj <+ (3.38) where { } ],[:)(min41: iii rrss − ∈= ρμ (3.39) Proof of Lemma 3.2:
Let ( ) Uk n →ℜ×ℜ× + }0{\]1,0[: be a C function which satisfies (3.1), (3.2), (3.3) and whose existence is guaranteed by Lemma 3.1. Let Zi ∈ , + ∈ Zj , define ]},[),(:]1,[),{( iinji rrxtVjjxt − ∈ℜ×+∈=Ω (3.40) ),,,min(: +−− = iiiii aaaa ρ (3.41) and let 0 , > ji δ satisfying: ( ) { } ( ) ( )( ) { } )),((41,]1,0[:,,,,,),,(,,,max),( ),(~,,:,,,max),(),(),(),( xtVDdsxtskxdtfxtskxdtfxtxV xtbuUuDdudxtfxtxVxtxVxttVxttV ρ≤∈∈−∂∂+ ≤∈∈∂∂−∂∂+∂∂−∂∂ ji xt ,00 ),( Ω∈∀ , })0{\(),( n xt ℜ×ℜ∈∀ + with ],[ ,00 ji ttt δ +∈ , ji xx ,0 δ≤− (3.42) Also, let + ∈ ZN ji , with , ≥ ji N be a family of integers which satisfies the following inequalities: ( ) jiiii NxtbuUuDdrrxtVjjtuxdtfxtxVxttV ,23 ),(~,,,],[),(,]2,[:,,,),(),(max4 ρ≤⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ ≤∈∈∈+∈∂∂+∂∂ +− (3.43) ( ) { } jijiii NxtbuUuDdrrxtVjjtuxdtf ,,23 ),(~,,,],[),(,]2,[:,,,max22 δ≤≤∈∈∈+∈+ +− (3.44) Consider next a smooth non-decreasing function ]1,0[: →ℜ h with = sh for ≤ s and = sh for ≥ s and define the desired Uk nar →ℜ×ℜ + })0{\(: , as follows: ⎪⎪⎩⎪⎪⎨⎧ ⎟⎟⎠⎞⎢⎣⎡ +∈⎟⎟⎠⎞⎜⎜⎝⎛ +−−⎟⎟⎠⎞⎜⎜⎝⎛ − ⎟⎟⎠⎞⎢⎣⎡ +∈⎟⎟⎠⎞⎜⎜⎝⎛ +−−⎟⎟⎠⎞⎜⎜⎝⎛ −= −− −−− − iiijijiiii iiijijiii iar rrrxtVxNljljtNkaa xtVrh rrrxtVxNljljtNkaa rxtVhxtk ,2),(,,,)(),min( ),(2 2,),(,,,)(),min( ),(2:),( , ( , ) i j t x ∈ Ω , ⎟⎟⎠⎞⎢⎢⎣⎡ +++∈ jiji NljNljt ,, for some }1,...,1,0{ , −∈ ji Nl (3.45) Obviously, (3.35) is a consequence of (3.3), (3.4) and (3.45). Moreover, by taking into account (3.1), (3.32), it follows that ),( , ⋅⋅ ar k above is continuous, continuously differentiable with respect to }0{\ n x ℜ∈ and satisfies 0),( , = xjk ar , , =∂∂ xjxk ar , ( ) + ×ℜ∈∀ Zjx n }0{\),( (3.46) Let Dn Mdx ×ℜ∈ })0{\(),( and ⎟⎟⎠⎞⎢⎢⎣⎡ +++∈ jiji NljNljt ,,0 for some }1,...,1,0{ , −∈ ji Nl with ],[),( +− ∈ ii rrxtV (3.47) Then by (3.43), (3.44) and (3.47) it can be easily established that for all ⎥⎥⎦⎤⎢⎢⎣⎡ ++∈ ji Nljtt ,0 it holds: ji xdxttxtt ,0000 );,,( δ≤−+− (3.48a) ),min(21),());,,(,(),min(21),( iiii aaxtVdxttxtVaaxtV −− +≤≤− (3.48b) Indeed, suppose on the contrary that there exist Dn Mdx ×ℜ∈ })0{\(),( , ⎟⎟⎠⎞⎢⎢⎣⎡ +++∈ jiji NljNljt ,,0 for some }1,...,1,0{ , −∈ ji Nl satisfying (3.47) and ⎥⎥⎦⎤⎢⎢⎣⎡ ++∈ ji Nljtt ,0 such that either (3.48a) or (3.48b) does not hold and consider the closed set ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ ≥⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ −+−−⎥⎥⎦⎤⎢⎢⎣⎡ ++∈= +−− , 0000112 0000,0 jiiiiiji xdxtxtaaaa xtVdxtxVNljtA δτττττ Notice that, since At ∈ , the set A is non-empty. Let At min: = . Clearly, since At ∉ , it holds that tt > . Definition of the set A above, (3.32) and (3.47) imply that ],[));,,(,( +− ∈ ii rrdxtxV ττ for every ),[ tt ∈τ . It follows from (3.35), (3.43), (3.44) that jii NdxtxVdd ,00 ρτττ ≤ and jiji Nx ,, )(22 δτ ≤+ & , a.e. for ),[ tt ∈τ which in conjunction with definition (3.41) and the fact that ji Nt ,0 ≤−τ imply that for all ],[ tt ∈τ we would have: ),,,min(41));,,(,(),());,,(,( +−− ≤≤− ∫ iiiit aaaadsdxtsxsVdsdxtVdxtxV τ ττ ; jit dssxtxdxtxt ,00000 δτττ τ ≤+−≤−+− ∫ & The previous inequalities for t =τ are in contradiction with the fact that At ∈ . In order to establish properties (3.36) and (3.38), we first need the following properties: Property P1:
Let D Md ∈ and let ji Nljt ,0 += , }1,...,1,0{ , −∈ ji Nl (3.49a) ]2,[),( iiii ararxtV −+∈ −− (3.49b) Then the following inequality is fulfilled: ⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ +≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++< ,4 1,;,,1,10 xNljVNxNljVdxNljNljxNljV jijijijijiji ρ (3.50) Proof of P1:
Using (3.48b) and definition (3.45) it follows: ⎟⎟⎠⎞⎜⎜⎝⎛ +−−= );,,(,,)());,,(,( dxttxNljljtNkdxttxtk jijiar , ⎥⎥⎦⎤⎢⎢⎣⎡ +++∈∀ jiji NljNljt ,, (3.51) For convenience let us denote here ji Nh , = , );,,()( dxtxx ⋅=⋅ and )(:)(~ httdtd += (notice that D Md ∈ ~ ). From (3.51) we have: ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ∫∫∫∫∫ ∫ −+++∂∂+ ++++⎥⎦⎤⎢⎣⎡ ∂∂−++∂∂+ +⎥⎦⎤⎢⎣⎡ ∂∂−++∂∂+ +⎥⎦⎤⎢⎣⎡ ∂∂+∂∂= ⎥⎦⎤⎢⎣⎡ ++++++∂∂+++∂∂= ⎥⎥⎦⎤⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ −∂∂+∂∂=−++ +
10 000000000010 0000000010 000010 0000000010 000000000 000000 ,,,),(~,)(,,),(),(~,),( )(,,),(),(~,),())(,( ),())(,( ,,,),(~,),(),( )(,,),(),(,))(,())(,( )(,,),(),(,))(,())(,(),())(,( dsxtskxsdtfhstxtskhstxsdhstfxtxVh dshstxtskhstxsdhstfxtxVhstxhstxVh dsxttVhstxhsttVh dsxtskxsdtfxtxVxttVh dshstxtskhstxhstdhstfhstxhstxVhstxhsttVh dxthtkxdfxxVxtVxtVhtxhtV ht t ττττττττττ (3.52) Using (3.2), (3.3), (3.42), (3.48), (3.49) and (3.52) we get the desired (3.50) and the proof of P1 is complete. The next property is a consequence of P1:
Property P2 : Suppose that iiji arxNljV −≤⎟⎟⎠⎞⎜⎜⎝⎛ +< for some }1,...,1,0{ , −∈ ji Nl (3.53) and assume that the solution of (3.37) with initial condition }0{\ nji xNljx ℜ∈=⎟⎟⎠⎞⎜⎜⎝⎛ + , corresponding to some D Md ∈ exists for ⎥⎥⎦⎤⎢⎢⎣⎡ +++∈ jiji NljNljt ,, . Then ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ −⎟⎟⎠⎞⎜⎜⎝⎛ ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++< −− ijijiiijijiji NxNljVardxNljNljxNljV μ ,0,110,,, (3.54) where > i μ is defined by (3.39). Proof of P2:
Obviously, the desired (3.54) is a consequence of (3.50), provided that (3.49b) is fulfilled. Consider the remaining case ,0 −− +≤⎟⎟⎠⎞⎜⎜⎝⎛ +< iiji arxNljV (3.55) We show by contradiction that, when (3.55) holds, then −− +≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++< iijijiji ardxNljNljxNljV . Indeed, suppose on the contrary that −− +>⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++ iijijiji ardxNljNljxNljV (3.56) Then, there would exist ⎟⎟⎠⎞⎜⎜⎝⎛ +++∈ jiji NljNljt ,,1 in such a way that 23;,,, −− +=⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ + iiji ardxNljtxtV . Using (3.48b) the latter implies −− +≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++< iijijiji ardxNljNljNljV φ which contradicts (3.56), and the proof of the P2 is complete. The following property is a direct consequence of property P2 and (3.32): Property P3:
Suppose that (3.53) holds and assume again that the solution of (3.37) with initial condition }0{\ nji xNljx ℜ∈=⎟⎟⎠⎞⎜⎜⎝⎛ + corresponding to some D Md ∈ exists for ⎥⎥⎦⎤⎢⎢⎣⎡ +++∈ jiji N sljNljt ,, , , for certain },...,2,1,0{ , lNs ji −∈ . Then ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ −⎟⎟⎠⎞⎜⎜⎝⎛ ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++< −− ijijiiijijiji NsxNljVardxNljN sljxN sljV μ ,0,110,,, ,,2max;,,,0 (3.57) The desired (3.38) follows from property P3 with = l , ji NN , = . We next proceed with the proof of (3.36). Combining property P3 with (3.48b) we obtain: Property P4:
If (3.53) is fulfilled then ( ) iijiiiji aaxNljVardxNljtxtV ,min21,,2max;,,,0 −−− +⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ ⎟⎟⎠⎞⎜⎜⎝⎛ ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +< , ( ) ⎟⎟⎠⎞⎢⎢⎣⎡ ++∈∀ max, jtNljt ji (3.58) Proof of P4:
Let }1,...,2,1,0{ , −−∈ lNs ji with max, tN slj ji <++ . By virtue of (3.57), we distinguish the following two cases: Case1: Suppose that ;,,, − ≥⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++ ijijiji rdxNljN sljxN sljV (3.59) Then by invoking (3.48b) we get from (3.57), (3.59) ( ) iijiiiji aaxNljVardxNljtxtV ,min21,,2max;,,,0 −−− +⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ ⎟⎟⎠⎞⎜⎜⎝⎛ ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +< , ⎥⎥⎦⎤⎢⎢⎣⎡ +++++∈∀ jiji NsljN sljt ,, (3.60) The desired (3.58) is a consequence of (3.60). Case 2: Suppose that ;,,,0 − <⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++< ijijiji rdxNljN sljxN sljV (3.61) We show that, when (3.61) holds, then ( ) iiiiji aaardxNljtxtV ,min212;,,, −−− ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ + , ⎟⎟⎠⎞⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++∈∀ jiji NsljtN sljt ,max,
Assume on the contrary that there would exist ⎟⎟⎠⎞⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++∈ jiji
NsljtN sljt ,max, such that ( ) iiiiji aaardxNljtxtV ,min212;,,, −−− ++>⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ + (3.63)
By (3.61), (3.63), there would exist ⎟⎟⎠⎞⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++∈ jiji
NsljtN sljt ,max,1 such that −− +=⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ + iiji ardxNljtxtV ; −− +≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ + iiji ardxNljxV ξξ , ⎥⎥⎦⎤⎢⎢⎣⎡ ++∈∀ , tN slj ji ξ (3.64) By (3.64) and (3.48b) we get ( ) iiiiji aaardxNljxV ,min212;,,,0 −−− ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +< ξξ , ⎟⎟⎠⎞⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛ +++∈∀ ji Nsljtt ,max1 ξ (3.65) Combining (3.64), (3.65) we obtain ( ) iiiiji aaardxNljxV ,min212;,,,0 −−− ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +< ξξ for all ⎟⎟⎠⎞⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++∈ jiji NsljtN slj ,max, ξ , which contradicts hypothesis (3.63). We conclude from (3.60) and (3.62) that in both cases above we have ( ) iijiiiji aaxNljVardxNljtxtV ,min21,,2max;,,,0 −−− +⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ ⎟⎟⎠⎞⎜⎜⎝⎛ ++≤⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎟⎠⎞⎜⎜⎝⎛ +< , for every ⎟⎟⎠⎞⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛ +++++∈ jiji NsljtN sljt ,max, and for all }1,...,2,1,0{ , −−∈ lNs ji with max, tN slj ji <++ and the latter implies the desired (3.58). This completes the proof of property P4. We are now in a position to establish (3.36). Let Dn Mdx ×ℜ∈ })0{\(),( and ⎟⎟⎠⎞⎢⎢⎣⎡ +++∈ ++ jiji NljN ljt ,1,10 for some }1,...,1,0{ , −∈ ji Nl with ],[),( ii rrxtV − ∈ . Then, exploiting inequality (3.48b) we obtain: ii axtVdxttxtVaxtV +≤≤− , ⎥⎥⎦⎤⎢⎢⎣⎡ ++∈∀ + ji Nljtt ,10 (3.66) Since ++ −<+ iiii arar , by virtue of (3.66) and (3.59) of P4 we get ( )( ) ii ardxtttV +≤< φ , for all ( ) [ ) max0 +∈ jttt and this establishes (3.36). The proof of Lemma 3.2 is complete. < Lemma 3.3:
Under the same hypotheses imposed in Lemma 3.1 for system (1.1), there exists a continuous mapping Uk n →ℜ×ℜ + })0{\(:~ , being continuously differentiable with respect to }0{\ n x ℜ∈ , which satisfies ),(~),(~ xtbxtk ≤ , })0{\(),( n xt ℜ×ℜ∈∀ + (3.67) where ),(~ ⋅⋅ b is defined by (3.4) and in such a way that that the following property holds for all ( ) Dn Mdxt ×ℜ×ℜ∈ + }0{\),,( : ),(9));,,(,( xtVdxttxtV ≤ , [ ] ( ) [ ) max00 ,2min, tttt +∈∀ (3.68) where );,,( dxtx ⋅ denotes the unique solution of )),(~,,,( xtkxdtfx = & , })0{\(),( n xt ℜ×ℜ∈ + (3.69) with initial condition }0{\)( n xtx ℜ∈= , corresponding to input D Md ∈ , and ),,(: tdxttt >= denotes its maximal existence time. Moreover, there exists );(~ ++ ℜℜ∈ C ρ being positive definite with ss ≤ )(~ ρ for all ≥ s , such that ( )( ) ( ) ),2(~),2(;,2,22,22 xjVxjVdxjjxjV ρ −≤++ , for all ( ) + ××ℜ∈ ZMjdx Dn }0{\),,( with max tj <+ (3.70) where jt max > in (3.70) is the maximal existence time of the solution );,2,( dxjx ⋅ of (3.69). Proof of Lemma 3.3:
Let { }
Zirr i ∈= : be a set with > i r and such that ii rr ≤ + and +∞= +∞→ ii r lim , = −∞→ ii r (3.71) Consider the set ⎭⎬⎫⎩⎨⎧ ∈+=′=′ + Zirrrr iii :2 (3.72) which by virtue of (3.72) satisfies >′ i r and further ii rr ′≤′ + and +∞=′ +∞→ ii r lim , =′ −∞→ ii r (3.73) Define { } ],[:)(min41: iii rrss − ∈= ρμ , { } ],[:)(min41: iii rrss ′′∈=′ − ρμ (3.74) and let { } Ziaa i ∈= : , { } Ziaa i ∈′=′ : be a pair of sets satisfying: > i a ; >′ i a (3.75a) − ≤ ii ra ; − ′≤′ ii ra (3.75b) ++ −<+ iiii arar ; ++ ′−′<′+′ iiii arar (3.75c) 8 iiii rraa −≤′+ + ; 8 −− −≤′+ iiii rraa (3.75d) ii a μ′≤ ; + ≤′ ii a μ (3.75e) By Lemma 3.2, there exist continuous mappings Uk nar →ℜ×ℜ + })0{\(: , , Uk nar →ℜ×ℜ +′′ })0{\(: , being continuously differentiable with respect to }0{\ n x ℜ∈ with 0),(),( ,, == ′′ xjkxjk arar , ( ) + ×ℜ∈∀ Zjx n }0{\),( (3.76a) ,, =∂∂=∂∂ ′′ xjxkxjxk arar , ( ) + ×ℜ∈∀ Zjx n }0{\),( (3.76b) satisfying properties (3.35), (3.36) and (3.38) . Finally, consider the map Uk n →ℜ×ℜ + })0{\(:~ defined as: ⎪⎩⎪⎨⎧ ℜ×∈++∈ ℜ×∈+∈= +′′ + })0{\(),(,)22,12[,),( })0{\(),(,)12,2[,),(),(~ , , nar nar Zxjjjtforxtk Zxjjjtforxtkxtk (3.77) By taking into account (3.76a,b), (3.77) and regularity properties of ),( , ⋅⋅ ar k and ),( , ⋅⋅ ′′ ar k , it follows that ),(~ ⋅⋅ k is continuous, continuously differentiable with respect to }0{\ n x ℜ∈ and satisfies = xjk , })0{\(),( n Zxj
ℜ×∈∀ + (3.78) Moreover, (3.67) is an immediate consequence of definition (3.77) and inequality (3.35). Also, by (3.36), (3.71) and (3.75b), it follows that for all ( ) ZMidxt Dn ××ℜ×ℜ∈ + }0{\),,,( it holds: ),(3));,,(,(],[),( xtVdxttxtVrrxtV arii ≤⇒∈ − , for all [ ] ( ) [ ) ar tttt ,max00 ,1min, +∈ (3.79a) ),(3));,,(,(],[),( xtVdxttxtVrrxtV arii ≤⇒∈ ′′− , for all [ ] ( ) [ ) ar tttt ′′ +∈ ,max00 ,1min, (3.79b) where );,,( dxtx ar ⋅ denotes the (unique) solution of )),(,,,( , xtkxdtfx ar = & , })0{\(),( n xt ℜ×ℜ∈ + (3.80a) and );,,( dxtx ar ⋅ ′′ is the (unique) solution of )),(,,,( , xtkxdtfx ar ′′ = & })0{\(),( n xt ℜ×ℜ∈ + (3.80b) with same initial condition }0{\)( n xtx ℜ∈= and D Md ∈ , and tt ar > and tt ar > ′′ , respectively denote their maximal existence times. The desired inequality (3.68) is a direct consequence of (3.79a,b), definition (3.77) and the following obvious fact: Fact:
The solution of (3.69) with initial condition }0{\)( n xtx ℜ∈= , corresponding to input D Md ∈ , is identical for [ ] ( ) [ ) ar tttt ,max00 ,1min, +∈ to the solution );,,( dxttx ar of (3.80a) if [ ] t is even, and is identical for [ ] ( ) [ ) ar tttt ′′ +∈ ,max00 ,1min, to the solution );,,( dxttx ar ′′ of (3.80b), if [ ] t is odd. In order to show (3.70), let ( ) + ××ℜ∈ ZMjdx Dn }0{\),,( such that the unique solution );,2,( dxjx ⋅ of (3.69) with initial condition }0{\)2( n xjx ℜ∈= , corresponding to input D Md ∈ is well-defined on ]22,2[ + jj (notice that if there is no such ( ) + ××ℜ∈ ZMjdx Dn }0{\),,( then property (3.70) trivially holds for every positive definite function );(~ ++ ℜℜ∈ C ρ ). Let Zi ∈ be the smallest integer with iiii arxjVar −≤<− −− (3.81) whose existence is guaranteed from (3.71), (3.75c). By virtue of (3.38), (3.81) and previous fact, it follows that ( )( ) ( ) iii xjVardxjjxjV μ−+≤++ −− ),2(,2max;,2,12,12 (3.82) Notice that by virtue of (3.75d), we have ( )( ) ii ardxjjxjV ′−′≤++ . Consequently, there exists an integer ik ≤ with ( )( ) kkkk ardxjjxjVar ′−′≤++<′−′ −− (3.83) We distinguish the following cases: Case 1: ik < In this case it follows from (3.83) that ( )( ) −− ′−′≤++ ii ardxjjxjV . By virtue of (3.38) and the fact above we then obtain ( )( ) ( ) −−−−−− ′−+−′−′+′≤++ iiiiiii arxjVardxjjxjV μμμ (3.84) We now take into account (3.75d) which implies −−−−−− −−−≤′+′ iiiiii rrarar (3.85) From (3.84), (3.85) and the left hand side inequality in (3.81) we get ( )( ) ⎟⎟⎠⎞⎜⎜⎝⎛ ′−−−+≤++ −−−− iiii arrxjVdxjjxjV μ (3.86) which by virtue of (3.75e) implies: ( )( ) ( ) −−− ′−−≤++ iii rrxjVdxjjxjV μ (3.87) Case 2: ik = . Notice that, since −−−− +>′−′ iiii arar (which is a consequence of (3.75d)), we conclude from (3.82) and using the left hand side inequality (3.83) with ik = : ),2(2 xjVar iii <+′−′ −− μ (3.88) Also, by (3.38) and the fact above we get ( )( ) ( ) iiii xjVardxjjxjV μμ −′−′+′≤++ −− ),2(,2max;,2,22,22 , which in conjunction with (3.88) gives ( )( ) ii axjVdxjjxjV μ−′+≤++ − and the latter by virtue of (3.75e) implies: ( )( ) i xjVdxjjjV μφ −≤++ (3.89) We conclude from (3.87) and (3.89) that in both cases we have: ⇒−≤<− −− iiii arxjVar ( )( ) i xjVdxjjxjV γ−≤++ ),2(;,2,22,22 (3.90a) ( ) iiiii rr μμγ −−− ′−= (3.90b) Now let ( ) ( )( )( )( ) ( ) ⎪⎪⎩⎪⎪⎨⎧ = −−∈++−− +−−= −−−−− −−−+ sfor ararsforarar arss iiiiiiiiii iiiiii γγγγγγρ (3.91) Notice that (3.74), (3.90b) and (3.91) imply that ( ) iiii s γργγγ ≤≤< +− )(,,min0 for ]2,2( iiii arars −−∈ −− and further ==′= −∞→−∞→−∞→ iiiiii γμμ . Thus, we may easily verify that ++ ℜ→ℜ : ρ is positive definite and continuous. Finally, define { } sss ),(min:)(~ ρρ = (3.92) Property (3.90a,b) in conjunction with (3.91) imply the desired (3.70) is satisfied and the proof is complete. < We are now in a position to prove Theorem 2.8.
Proof of Theorem 2.8:
By virtue of Lemma 3.3 there exists a continuous mapping Uk n →ℜ×ℜ + })0{\(:~ , being continuously differentiable with respect to }0{\ n x ℜ∈ , satisfying (3.67), (3.68) and (3.70). We define: ),(~),( xtkxtK = for ≥ t , ≠ x (3.93a) = tK for ≥ t (3.93b) It follows from (2.3), (3.4), (3.67) and definition (3.93) that UK n →ℜ×ℜ + : is a continuous and continuously differentiable mapping with respect to n x ℜ∈ on the set })0{\( n ℜ×ℜ + . Fact 1:
For every Dn Mdxt ×ℜ×ℜ∈ + ),,( , the solution );,,( dxtx ⋅ of (2.6) with initial condition )( xtx = , corresponding to input D Md ∈ is unique and is defined for all tt ≥ . Proof of Fact 1:
Consider the resulting system (2.6) with ),( ⋅⋅ K as above and notice that its solution with initial condition }0{\)( n xtx ℜ∈= , corresponding to some D Md ∈ coincides with the unique solution of (3.69) evolving on })0{\( n ℜ×ℜ + with same initial condition }0{\)( n xtx ℜ∈= , and same D Md ∈ on the interval ),[ max0 tt , where tt > is the maximal existence time of the solution of (3.69). For the case +∞= max t , the statement of Fact 1 is a direct consequence of previous argument. Suppose next that +∞< max t . To establish the desired claim, we need the following implication, which is a consequence of (2.1) and (3.68): ⇒+∞< max t max = − → tx tt (3.94) In order to show (3.94), let Dn Mdxt ×ℜ×ℜ∈ + })0{\(),,( and suppose that the maximal existence time tt > of the (unique) solution of (3.69) with initial condition }0{\)( n xtx ℜ∈= corresponding to D Md ∈ is finite, i.e., +∞< max t . Repeated use of (3.68) implies that ),(9))(,( xtVtxtV i ≤ , ),[ max0 ttt ∈∀ where + ∈ Zi is the smallest integer with the property max ti ≥ . The above inequality in conjunction with (2.1) with ≡ t β gives ( ) +∞<=≤ −∈ )(9)(min1:)( max xaaMtx it τμ τ , ),[ max0 ttt ∈∀ (3.95) Definition of max t and (3.95) implies (3.94). By applying standard arguments we may also establish show that for every D Mdt ×ℜ∈ + ),( , the solution of (2.6) with initial condition = tx , corresponding to input D Md ∈ is unique and satisfies = tx for all tt ≥ . Indeed, suppose on the contrary that there exists a nonzero solution of (2.6) with initial condition = tx , defined on ),[ htt + for some 0 > h and let ),[ httt +∈ with ≠ tx and { } =∈= txttta . Then = ax and ≠ tx for all ),( tat ∈ . Without loss of generality we may assume that +≤ at (if +> at then we may use + a instead of t , which in this case satisfies ≠+ ax ). Define ( ) >= txta με and let ),( tat ∈ such that 9))(,( ε ≤ txtV for all ],[ tat ∈ . By taking into account (3.68) and the fact +≤ tt it then follows that ε ≤ ))(,( txtV for all ],[ ttt ∈ and the latter in conjunction with (2.1) yields ( ) εμ ≤ )()( txta . But this contradicts the definition of ε , hence, we conclude that = tx for all tt ≥ . The previous discussion in conjunction with (3.94) asserts that the solution )( ⋅ x of (2.6) with initial condition }0{\)( n xtx ℜ∈= , corresponding to D Md ∈ coincides with the solution of (3.69) with same initial condition, and same D Md ∈ on the interval ),[ max0 tt , tt > being the maximal existence time of the solution (3.69); moreover, if +∞< max t , the corresponding solution of (2.6) satisfies = tx for all max tt ≥ and the proof of Fact 1 is complete. Fact 1 asserts that, if for some Dn Mdxt ×ℜ×ℜ∈ + ),,( we consider the maximum existence time ),,(: dxttt = of the corresponding solution of (2.6), then +∞= max t and the latter in conjunction with (3.68) and (3.70) assert that the following properties are fulfilled for every ++ ××ℜ×ℜ∈ ZMjdxt Dn ),,,( : ),(9));,,(,( xtVdxttxtV ≤ , for all [ ][ ] +∈ ttt (3.96) ( )( ) ( ) ),2(~),2(;,2,22,22 xjVxjVdxjjxjV ρ −≤++ (3.97) where );(~ ++ ℜℜ∈ C ρ is the positive definite function involved in (3.70), );,,( dxttx denotes the solution of (2.6) with initial condition n xtx ℜ∈= )( , corresponding to input D Md ∈ and [ ] t is the integer part of t . The following inequality is a straightforward consequence of inequalities (3.96), (3.97): ),(81));,,(,( xtVdxttxtV ≤ , for all tt ≥ and Dn Mdxt ×ℜ×ℜ∈ + ),,( (3.98) Inequality (3.98) in conjunction with inequality (2.1) with ≡ t β , implies Robust Forward Completeness, Uniform Robust Lagrange Output Stability and Uniform Robust Lyapunov Output Stability. Therefore, in order to establish URGAOS for (2.6), it remains to show Uniform Output Attractivity on compact sets of initial states. Let ≥ R , > ε and Dn Mdxt ×ℜ×ℜ∈ + ),,( with Rx ≤ and consider the smallest non-negative integer j , which satisfies jt ≤ . Then we have: Fact 2:
For every > ε , it holds: )(91));,,2(,2( ε adxtixiV ≤ (3.99) for every + ∈ Zi with r Raji )(9 +≥ ; ⎭⎬⎫⎩⎨⎧ ⎥⎦⎤⎢⎣⎡ +∈= )(9)(91),(91:)(~min: Raaassr εερ (3.100)
Proof of Fact 2:
Suppose on the contrary that there exists > ε , r Raji )(9 +≥ with )(91));,,2(,2( ε adxtixiV > . Using (3.97), it follows that )(91));,,2(,2( ε adxtkxkV > for ijk ,..., = . Also (3.97) implies that ));,,2(,2());,,2(,2( dxtjxjVdxtkxkV ≤ , for ijk ,..., = . Consequently, from (2.1) (with ≡ t β ), (3.96) we get ⎥⎦⎤⎢⎣⎡ +∈ )(9)(91),(91));,,2(,2( RaaadxtkxkV εε , for ijk ,..., = (3.101) On the other hand, by recalling (3.97) and using (3.100), (3.101) it follows: rdxtkxkVdxtkxkV −≤++ ));,,2(,2());,),1(2(),1(2( , for ijk ,..., = which in turns gives: )());,,2(,2());,,2(,2( jkrdxtjxjVdxtkxkV −−≤ , for ijk ,..., = (3.102) Using (3.96) and (2.1) with ≡ t β we get: )(9));,,2(,2( RadxtjxjV ≤ (3.103) Inequalities (3.102), (3.103) in conjunction with the fact that r Raji )(9 +≥ , give ≤ dxtixiV , which contradicts the hypothesis )(91));,,2(,2( ε adxtixiV > and the proof of Fact 2 is complete. Applying again (2.1) with ≡ t β and (3.96), (3.99) of Fact 2, it follows that for every ≥ R , > ε , Dn Mdxt ×ℜ×ℜ∈ + ),,( with Rx ≤ , it holds that: ε≤ );,,(,( dxttxtH for all r Ratt )(182 ++≥ where ⎭⎬⎫⎩⎨⎧ ⎥⎦⎤⎢⎣⎡ +∈= )(9)(91),(91:)(~min: Raaassr εερ and this establishes Uniform Output Attractivity on compact sets of initial states. The proof is complete. < For the proof of Theorem 2.9 we need an additional lemma, which provides sufficient conditions for (non-uniform in time) RGAOS. It is important to mention here the paper [17], where, under different hypotheses than those imposed below, asymptotic stability for time-varying system is explored by estimating the difference between values of an appropriate Lyapunov function along the trajectories of system at a given sequence of times.
Lemma 3.4:
Consider system (2.4) , where nn Df ℜ→ℜ××ℜ + : , kn H ℜ→ℜ×ℜ + : are continuous and for every bounded interval + ℜ⊂ I and every compact set n S ℜ⊂ there exists ≥ L such that yxLydtfxdtf −≤− ),,(),,( for all DIdt ×∈ ),( , Syx ∈ , . Moreover, assume that the set l D ℜ⊂ is compact and = dtf , = tH for all Ddt ×ℜ∈ + ),( . Suppose that there exists a function + ∈ K γ satisfying +∞< ∑ +∞= )2( j j γ (3.104a) = +∞→ t t γ (3.104b) and further there exist functions );( ++ ℜℜ×ℜ∈ n CV , ∞ ∈ Kaaa ,, , + ∈ K γβμ ,, , );( ++ ℜℜ∈ C ρ being positive definite such that (2.1) holds and the following properties are fulfilled for all ++ ××ℜ×ℜ∈ ZMjdxt Dn ),,,( : ( ) )(),());,,(,(sup txtVadxttxtV ttt γ +≤ +∈ (3.105a) ( ) )2(),2(),2());,2,22(,22( jxjVxjVdxjjxjV γρ +−≤++ (3.105b) where );,,( dxtx ⋅ denotes the unique solution of (2.4) with initial condition n xtx ℜ∈= )( , corresponding to input D Md ∈ . Then system (2.4) is RGAOS. Proof of Lemma 3.4:
Let Dn Mdxt ×ℜ×ℜ∈ + ),,( and let + ∈ Zj be the smallest integer, which satisfies jt ≤ . Inequality (3.105b) implies that ∑ = +≤ ik kdxtjxjVdxtixiV )2()),,,2(,2()),,,2(,2( γ , for all integers ji ≥ (3.106) Let ∑ +∞= = )2(: k kM γ and )(sup: tB t γ ≥ = . Then by (2.1), (3.105a) and (3.106) we get: ( )( ) BMBxtaaadxttxtV +++≤ ))(()),,,(,( β , for all tt ≥ (3.107) Inequality (3.107) in conjunction with (2.1) implies RFC and Robust Lagrange Output Stability. Therefore, according to Lemma 3.5 in [12], in order to establish RGAOS, it suffices to show that system (2.4) satisfies the property of Uniform Output Attractivity on compact sets of initial data. To establish this property, consider arbitrary constants >ε , ≥ R , ≥ T and let Dn Mdxt ×ℜ×ℜ∈ + ),,( with ],0[ Tt ∈ and Rx ≤ . Define BMBtRaaaK Tt +⎟⎟⎠⎞⎜⎜⎝⎛ ++⎟⎠⎞⎜⎝⎛= ∈ ))(max(: ],0[2 β . Then by (3.107) it holds that KdxttxtV ≤ )),,,(,( , tt ≥∀ (3.108) Define )(min:)(~ ys Kys ρρ ≤≤ = (3.109) which obviously is a non-decreasing and continuous function and let ≥ J be an integer with )2()(~21 iK γρ ≥ for all integers Ji ≥ , whose existence is guaranteed from (3.104b). Define the sequence ⎭⎬⎫⎩⎨⎧ ≥∈= )2()(~21:],0[inf: isKsq i γρ for Ji ≥ (3.110) Notice, by virtue of (3.104b) and (3.110) that → i q and consequently, there exists an integer JKNN ≥= ),(: ε such that )()2( εγ Siq i ≤+ and )(21)2( εγ ai ≤ , for all Ni ≥ (3.111) where ∞ ∈ Ka is the function involved in (2.1) and >ε S is defined by ⎟⎠⎞⎜⎝⎛= − )(21:)( εε aaS (3.112) Notice next that (3.105b) asserts that for all integers ( ) jNi ,max ≥ the following holds: ( ) ⎟⎠⎞⎜⎝⎛ −≤++ ));,,2(,2(21));,,2(,2(,)(max));,),1(2(),1(2( dxtixiVdxtixiVSdxtixiV ρε (3.113) Indeed, to establish (3.113) we may distinguish two cases. First assume that i qdxtixiV ≥ )),,,2(,2( . Then it follows from (3.108), (3.109) and (3.110) that ( ) )2()),,,2(,2(21 idxtixiV γρ ≥ and the latter in conjunction with (3.105b) implies (3.113). The other case is i qdxtixiV ≤ )),,,2(,2( . Then the latter in conjunction with (3.105b) and (3.111) implies again (3.113). The following is a consequence of (3.113): )()),,,2(,2( ε SdxtixiV ≤ for all integers ( ) ( )( )
1~ 2,max ++≥ ερ SKjNi (3.114) To show (3.114), suppose on the contrary that there exists integer ( ) ( )( )
1~ 2,max ++≥ ερ SKjNi with )()),,,2(,2( ε SdxtixiV > . Then (3.113) implies that )()),,,2(,2( ε SdxtkxkV > for all ( ) ( ) ijNjNk ,...,1,max,,max += (3.115) From (3.108), (3.109), (3.113), (3.115) it follows that ( ) )(~21)),,,2(,2()),,),1(2(),1(2( ερ SdxtkxkVdxtkxkV −≤++ , for all ( ) ( ) −+= ijNjNk which directly implies ( )( ) ( ) )(~21),,(max)),,,2(,2( ερε SjKNkKdxtkxkV −−≤ , for all ( ) ( ) ijNjNk ,...,1,max,,max += . The previous inequality for ik = gives < dxtixiV , which is a contradiction, hence (3.114) is established. Using (3.105a) and (3.114) we obtain ( ) )2()());,,(,(sup iSadxtttV iit γεφ +≤ +∈ , for all integers ( ) ( )( )
1~ 2,max ++≥ ερ SKjNi . This in conjunction with (3.111) and (3.112) gives: )());,,(,(sup εφ adxtttV it ≤ ≥ , for all integers ( ) ( )( )
1~ 2,max ++≥ ερ SKjNi
Using the inequality above and (2.1), we may conclude that the property of Uniform Output Attractivity on compact sets of initial data holds for system (2.4). This completes the proof of Lemma 3.4. < We are now in a position to prove Theorem 2.9.
Proof of Theorem 2.9:
According to the statement of Lemma 3.3 there exists a continuous mapping Uk n →ℜ×ℜ + })0{\(:~ , being continuously differentiable with respect to }0{\ n x ℜ∈ , which satisfies (3.67), (3.68), (3.70). We define: ),(~)exp( )exp(),(:),( xtkt txtVhxtK ⎟⎟⎠⎞⎜⎜⎝⎛ − −−= , for )exp(),( txtV −> (3.116a) = xtK , for )exp(),( txtV −≤ (3.116b) where ]1,0[: →ℜ h is a smooth non-decreasing function with = sh for ≤ s and = sh for ≥ s . It can be easily verified that, according to definition (3.116) and the properties of Uk n →ℜ×ℜ + })0{\(:~ , the map K takes values in U and satisfies = tK for all ≥ t . Moreover, UK n →ℜ×ℜ + : , is a continuous and continuously differentiable mapping with respect to n x ℜ∈ on n ℜ×ℜ + . In order to prove Theorem 2.9 we will make use of Lemma 3.4 and three facts below concerning certain properties of the solution of (2.6). Let { } { }{ } ⎪⎩⎪⎨⎧ ∅=<≥∞+ ∅≠<≥<≥== txtVtttif txtVtttiftxtVtttdxtTT (3.117) where );,,()( dxtxx ⋅=⋅ denotes the unique solution of (2.6) with initial condition n xtx ℜ∈= )( corresponding to some D Md ∈ . The following fact is an immediate consequence of (3.116), (3.117) and continuity of the mapping ))(,( txtVt → . Fact 1:
The unique solution );,,()( dxtxx ⋅=⋅ of (2.6) with initial condition }0{\)( n xtx ℜ∈= , satisfying )exp(2),( txtV −≥ , corresponding to some D Md ∈ coincides with the unique solution of (3.69) with same initial condition and same D Md ∈ on the interval ],[ Tt , where ),,( dxtTT = is defined by (3.117) and )exp(2))(,( TTxTV −= if { } ∅≠<≥ txtVttt (3.118) Next, we prove the following: Fact 2:
For the system (2.6), the following property holds for all Dn MZdxj ×ℜ×∈ + ),,( : ( )( ) ( ) )2exp(18),2(~),2(;,2,22,22 jxjVxjVdxjjxjV −+−≤++ ρ (3.119) Proof of Fact 2:
Obviously, the desired (3.119) holds for = x . Next, assume that ≠ x . Let jt max > the maximal existence time of );,2,( dxjx ⋅ . We distinguish two cases. The first case is { } ∅=<+∈ dxjtxtVtjtjt (3.120) In this case, Fact 1 in conjunction with inequalities (2.1), (3.68) and (3.70) guarantee that max +> jt and that (3.119) holds. The second case is { } ∅≠<+∈ dxjtxtVtjtjt (3.121) Let { } <+∈= dxjtxtVtjtjtt (3.122) Clearly, we have from (3.122) ≤ − → dxjtxtVt tt (3.123) and this by virtue of (2.1) implies tt > . If += jt the desired (3.119) follows from (3.123) holds. If +< jt , definition (3.122) guarantees that ≥ dxjtxtVt for all ))22,min(,[ max1 +∈ jttt and the latter in conjunction with (3.123) gives = dxjtxtVt (3.124) Using Fact 1 together with (3.68) and (3.124) we get )exp(18));,2,(,( tdxjtxtV −≤ for all ))22,min(,[ max1 +∈ jttt . By exploiting (2.1) we conclude that max +> jt and therefore the estimate )exp(18));,2,(,( tdxjtxtV −≤ is fulfilled for every ]22,[ +∈ jtt . The latter implies (3.119) and this completes proof of Fact 2. Finally we show the following fact. Fact 3:
The following property holds for system (2.6): ( )( ) )exp(18),(9;,,, txtVdxttxtV −+≤ , for all [ ][ ] +∈ ttt , Dn Mdxt ×ℜ×ℜ∈ + ),,( (3.125) Proof of Fact 3:
Obviously, (3.125) holds for = x . Suppose next that ≠ x and let us on the contrary assume that there exists [ ][ ] +∈ ttt with ( )( ) )exp(18),(9;,,ˆ,ˆ txtVdxttxtV −+> (3.126) We distinguish two cases. First assume that { } ∅=<∈ dxtsxsVstts In this case, (3.68) guarantees that ( )( ) ),(9;,,ˆ,ˆ xtVdxttxtV ≤ , which contradicts (3.126). Consider the remaining case { } ∅≠<∈ dxtsxsVstts and let { } <∈= dxtsxsVsttst . If tt ˆ = , we would have ( )( ) )exp(2;,,ˆ,ˆ tdxttxtV −≤ , which contradicts (3.126). If tt ˆ < , then we would have ≥ dxtsxsVs for all ]ˆ,[ tts ∈ and = dxjtxtVt . Therefore (3.68) gives )exp(18));,,(,( tdxtssV −≤ φ for all ]ˆ,[ tts ∈ , which again contradicts (3.126) and we conclude that (3.125) holds. This completes proof of Fact 3. Inequalities (3.119), (3.125) in conjunction with Lemma 3.4 show that (2.6) is RGAOS and the proof of Theorem 2.9 is complete. <
4. Conclusions
For general time-varying systems, it is established that existence of an “Output Robust Control Lyapunov Function” implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result towards feedback stabilization due to J. M. Coron and L. Rosier in [7] concerning stabilization of autonomous systems by means of time-varying periodic feedback. Further extensions towards same subject, including stabilization of time-varying systems (1.1) by means of discontinuous time-varying feedback in the Fillipov sense (see [3,8,24]) and existence of appropriate control Lyapunov functions will be a subject of forthcoming research.
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