Homogeneous State Feedback Stabilization of Homogenous Systems

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HOMOGENEOUS STATE FEEDBACK STABILIZATION OF HOMOGENEOUS SYSTEMS

LARS GRUNE^y

Abstract. We show that for any asymptotically controllable homogeneous system in euclidian space (not necessarily Lipschitz at the origin) there exists a homogeneous control Lyapunov function and a homogeneous, possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system. If the system satises the usual local Lipschitz condition on the whole space we obtain semi-global stability of the sampled closed loop system for each suciently small xed sampling rate, if the system satises a global Lipschitz condition we obtain global exponential stability for each suciently small xed sampling rate. The control Lyapunov function and the feedback are based on the Lyapunov exponents of a suitable auxiliary system and admit a numerical approximation.

Key words. Homogeneous system, state feedback stabilization, control Lyapunov functions, Lyapunov exponents

AMSsubject classications.93D15 (93D22, 93D30, 93D20)

1. Introduction.

In this paper we consider the problem of state feedback stabilization of homogeneous control systems in^Rⁿ. This problem has been considered by a number of authors during the last years, see e.g. [15, 16, 17, 20, 21, 22, 25], to mention just a few examples. Stability in this context will always mean asymptotic stability.

Homogeneous systems appear naturally as local approximations to nonlinear systems, which inherit some local properties of their homogeneous approximations, e.g.

asymptotic controllability [14]. In order to make use of this property in the design of locally stabilizing feedbacks for nonlinear systems the main idea lies in the construction of homogeneous feedbacks, i.e. feedback laws that preserve homogenity for the resulting closed loop system. Utilizing a corresponding homogeneous Lyapunov function, those laws can then be shown to be locally stabilizing also for the approx- imated nonlinear system, cf. [14, 17, 19]. Regarding the existence of homogeneous stabilizing feedback laws, it was shown in [15] that if the system admits a continuous, but not necessarily homogeneous, stabilizing state feedback law, then there exists a homogeneous dynamic feedback stabilizing the system. Unfortunately, if we are look- ing for state feedback laws, it is in general not true that any continuously stabilizable homogeneous system is stabilizable by a continuous and homogeneous state feedback law, as the examples in [22] show. Even worse, there exist homogeneous systems, e.g.

Brockett's classical example [2], which|although asymptotically controllable|do not admit a stabilizing continuous state feedback law at all.

Especially Brockett's results inspired the search for alternative feedback concepts.

In the present paper we are going to use discontinuous state feedback laws for which the corresponding closed loop systems are dened as sampled systems. Although this is not a new concept, see e.g. [12, 13, 23], it has recently received new attention, see e.g. the survey [24]. In particular, it was shown in [4] that (global) asymptotic controllability is equivalent to the existence of a (globally) stabilizing discontinuous

yFachbereich Mathematik, J.W. Goethe-Universitat, Postfach 111932, 60054 Frankfurt a.M., Germany, E-Mail: gruene@math.uni-frankfurt.de. This paper has been written while the author was visiting the Dipartimento di Matematica, Universita di Roma \La Sapienza", Italy, supported by DFG-Grant GR1569/2-1.

1

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state feedback law for the sampled closed loop system. Stability in this context means asymptotic stability for the sampled trajectories, where|in general|the intersampling times have to tend to zero close to the equilibrium and far away from it. A related but slightly dierent concept of a discontinuous feedback is the notion of discrete feedback introduced in [7]; here also sampled trajectories are considered, but with xed intersampling times. With this approach it was possible to show in [10]

that for semilinear systems asymptotic controllability is equivalent to (exponential) discrete feedback stabilizability.

The goal of the present paper is to provide a link between these two concepts in the framework of homogeneous systems. As in [10] we use a spectral characterization of asymptotic controllability by means of Lyapunov exponents, and obtain stability results for xed sampling rates; as in [4] we construct the feedback based on a suitable (and here also homogeneous) control Lyapunov function, and obtain stability not only for xed intersampling times but for all suciently small ones. Furthermore, and this is a key feature of our construction, the resulting stabilizing state feedback law is homogeneous, thus rendering the corresponding closed loop system homogeneous. All this will be done just under the assumption that the corresponding homogeneous system is asymptotically controllable.

The organization of this paper is as follows. After dening the setup and the concepts we pursue, in Section 3 we introduce a class of auxiliary systems we call homogeneous-in-the-state. In some sense these systems have a built in homogenity for each control value. These systems will be simplifyed by suitable coordinate and time transformations, and for the resulting system we will characterize asymptotic controllability by means of its Lyapunov exponents. In Section 4 we will use this characterization in order to construct a suitable control Lyapunov function which will then be used for the construction of the stabilizing feedback law. After giving some hints about a numerical approximation of these feedback laws in Section 5, we will return to the homogeneous systems in Section 6 and prove the stabilization result by showing that these systems can easily be transformed into systems homogeneous-in- the-state without loosing the asymptotic controllability property. Finally, in Section 7 we discuss two examples.

2. Setup.

We consider a class of systems x_(t) =g(x(t);w(t)) (2.1)

on ^Rⁿ where w()² ^W, and ^W denotes the space of measurable and locally essen- tially bounded functions from ^R to W ^R^m. We assume that the vector eld g is continuous,g(;w) is locally Lipschitz on^Rⁿⁿ^f0^gfor eachw²W, and satises the following property.

Definition 2.1. We call g homogeneous if there exist ri > 0, i = 1;:::;n, sj>0, j= 1;:::;m and ²( miniri;¹) such that

g(x;w) =g(x;w) for all w²W; 0 (2.2)

where

=

0

B

@

^r¹ 0 0 0 ... ... ...

... ... ... 0 0 ::: 0 ^rⁿ

1

C

A

and =

0

B

@

^s¹ 0 0 0 ... ... ...

... ... ... 0 0 ::: 0 ^s^m

1

C

A

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are called dilation matrices. With k= miniri we denote the minimal power (of the state dilation) and the value ²( k;¹) is called the degree of the system.

This denition generalizes the one given in [22] to the case of a multidimensional control input, see e.g. cite [14] for an alternative denition (equivalent on ^Rⁿ) for vector elds on arbitrary manifolds. The use of dilation matrices instead of the usual dilation functions allows a more compact notation in what follows. Observe that g is Lipschitz in the origin i 0 and globally Lipschitz i = 0, furthermore the denition impliesg(0;0) = 0.

Corresponding to the dilation matrix we dene a function N :^Rⁿ ^![0;¹) which can be interpreted as a \dilated norm" w.r.t. . Denotingd= 2^Qⁿ_i⁼¹ri we deneN(x) by

N(x) := ^Xⁿ

i⁼¹x_i^dri

! 1d

(2.3)

implyingN(0) = 0,N(x)>0 ifx⁶= 0, andN(x) =N(x).

Note that the trajectories of (2.1) may tend to innity in nite time if >0 and that uniqueness of the trajectory may not hold if <0, however it holds away from the origin. As long as uniqueness holds (i.e. if 0 or the trajectory does not cross the origin) we denote the (open loop) trajectories of (2.1) by x(t;x⁰;w()) for each x⁰ ²^Rⁿ and eachw()²^W, wherex(0;x⁰;w()) =x⁰, Then from Denition 2.1 we obtain

x(t;x⁰;w()) = x(t;x⁰;w()) (2.4)

for x⁰ ² ^Rⁿ. If uniqueness fails to hold x(;x⁰;w()) shall denote one possible trajectory; in this case we implicitely assume the following denitions to be valid for all possible trajectories.

The following denition gives the meaning of asymptotic controllability.

Definition 2.2. We call the systemasymptotically controllable (to the origin), if for eachx⁰²^Rⁿ there existswx⁰()²^W such that^kx(t;x⁰;wx⁰())^k^!0 as t^!¹.

We now discuss the concept of homogeneous state feedbacks. A state feedback law is a mapF :^Rⁿ ^!W. A homogeneous state feedback law satisesF(x) = F(x) for allx²^Rⁿ and all0, thus implyingg(x;F(x)) =^tg(x;F(x)), i.e. the closed loop system usingF becomes homogeneous. Observe thatW needs to satisfy some structural condition in order to allow nontrivial homogeneous feedbacks; in what follows we will assume

W W for all 0; where W :=^fw^jw²W^g

which gives a necessary and sucient condition for the fact that given some c > 0 any homogeneous mapF :^Rⁿ ^!^R^m satisfying F(x) ²W on ^fx ²^R^d^jN(x) = c^g satisesF(x)²W for allx²^Rⁿ.

Note that we do not require any continuity property of F. This is due to the fact, that in many examples stabilizing continuous feedbacks cannot exist, cf. e.g. [24, Section 2.2] where also Brockett's classical example [2] is discussed which|in suitable coordinates|is in fact a homogeneous system. Furthermore, even if stabilizing continuous feedback laws exist, it is possible that no such law is homogeneous, as the examples in [22] show (Brockett's example and the rst example from [22] will be discussed in Section 7). However, using discontinuous feedbacks for the solutions

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of the classical closed loop system _x=g(x;F(x)) the usual existence and uniqueness results might not hold. In order to obtain a meaningful solution for the closed loop system we use the following concept of a sampled closed loop system.

Definition 2.3. (Sampled closed loop system) Consider a feedback lawF :^Rⁿ ^! W. An innite sequence = (ti)i^2N0 of times satisfying

0 =t⁰< t¹< t²< ::: and ti^!¹asi^!¹ is called asampling schedule. The values

ti:=ti⁺¹ ti and d() := sup_i

2N

0

ti

is called theintersampling times and the sampling rate, respectively. For any sampling schedule the corresponding sampled or -trajectory x(t;x⁰;F) with initial value x⁰²^Rⁿ at initial timet⁰= 0 is dened inductively by

x(t;x⁰;F) =x(t ti;xi;F(xi)); for all t²[ti;ti⁺¹];i²^N⁰

wherexi =x(ti;x⁰;F) and x(t;xi;F(xi)) denotes the (open loop) trajectory of (2.1) with constant control valueF(xi) and initial value xi.

Observe that this denition guarantees the existence and uniqueness of trajectories in positive time on their maximal intervals of existence (except possibly at the origin if <0, in which case we use the same convention as for open loop trajectories). Moreover, the sampled-trajectories have a meaningful physical interpretation, as they correspond to an implementation of the feedback law F using a digital controller.

The next denition introduces control Lyapunov functions which will be vital for the construction of the feedback.

Definition 2.4. A continuous function V : ^Rⁿ ^! [0;¹) is called a control Lyapunov function (clf), if it is positive denite (i.e.V(0) = 0 iV = 0), proper (i.e.

V(x)^!¹as^kx^k^!¹), and there exists a continuous and positive denite function P : ^Rⁿ ^! [0;¹) such that for each bounded subset G ^Rⁿ there exists a compact subsetWG W with

v^2coming⁽x;WG⁾DV(x;w) P(x) for all x²G:

Here DV(x;v) denotes the lower directional derivative DV(x;v) := liminf_t

&0;v⁰^!v1

t⁽V(x+tv⁰) V(x));

g(x;WG) :=^fg(x;w)^jw²WG^g, andcog(x;WG) denotes the convex hull ofg(x;WG).

The following denition now describes the stability concepts we will use in this paper. For this denition recall that a function : [0;¹) ^! [0;¹) is of class ^K, if it satises (0) = 0 and is continuous and strictly increasing, and a function : [0;¹)² ^![0;¹) is of class^{K L}, if it is decreasing to zero in the second and of class

K in the rst argument.

Definition 2.5. We call the sampled closed loop system from Denition 2.3 (i) semi-globally practically stable with xed sampling rate, if there exists a class

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K L function such that for each open set B ^Rⁿ and each compact set K ^Rⁿ satisfying0²BK there existsh >0 such that

x(t;x⁰;F)⁶²B ⁾ ^kx(t;x⁰;F)^k(^kx⁰^k;t) for allx⁰²K and all withd()h,

(ii) semi-globally stable with xed sampling rate, if (i) holds and the sampling rate h >0 can be chosen independently ofB,

(iii)globally practically stable with xed sampling rate if (i) holds and the sampling rateh >0 can be chosen independently ofK,

(iv)globally stable with xed sampling rate if (i) holds and the sampling rate h >0 can be chosen independently ofK andB.

We call the stability in (i){(iv) exponential if can be chosen asCe ^t^kx⁰^kfor constants C; > 0 which may depend onK, and uniformly exponential if C; >0 can be chosen independently ofK.

Note that each of the concepts (ii){(iv) implies (i) which is equivalent to the s- stability property as dened in [4], cf. also [24, Sections 3.1 and 5.1]. Hence any of these concepts implies global stability for the (possibly nonunique) limiting trajectories ash^!0. The dierence \only" lies in the performance with xed sampling rate. From the applications point of view, however, this is an important issue, since e.g. for an implementation of a feedback using some digital controller arbitrary small sampling rates in general will not be realizable. Furthermore if the sampling rate tends to zero the resulting stability may be sensitive to measurement errors, if the feedback is based on a non-smooth clf, see [18, 24]. In contrast to this it is quite straightforward to see that for a xed sampling rate the stability is in fact robust to small errors in the state measurement (small, of course, relative to the norm of the current state of the system) if the corresponding clf is Lipschitz, cf. [24, Theorem E].

The main result we will prove in this paper is the following theorem on the existence of a homogeneous clfV and a homogeneous stabilizing feedbackF.

Theorem 2.6. Consider system(2.1) satisfying Denition 2.1 with dilation matrices and , minimal power k > 0, and degree ² ( k;¹), and assume asymptotic controllability. Then there exists > 0 and a clf V being Lipschitz on

Rnnf0^g, satisfying

V((x)) =²^kV(x) and

v^2coming⁽x;Wx⁾DV(x;v) 2N(x)V(x)

for the function N from (2.3) and Wx = _N⁽_x⁾U for some suitable compact subset U W.

Furthermore there exists a feedback law F : ^Rⁿ ^!W satisfying F(x)²Wx and F(x) = F(x) for allx²^Rⁿ and all0 such that the corresponding sampled closed loop system is either

(i) semi-globally stable (if >0), or

(ii) globally uniformly exponentially stable (if = 0), or (iii) globally practically exponentially stable (if <0) with xed sampling rate.

The proof is given in Section 6.

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3. Systems homogeneous-in-the-state.

In this section we dene a class of auxiliary systems which are homogeneous in the state and will turn out to be useful for our analysis. By suitable coordinate and time transformations we will then simplify this class of systems and and characterize controllability of the simplied system by means of its Lyapunov exponents.

We consider the class of systems

x_(t) =f(x(t);u(t)) (3.1)

on ^Rⁿ where u()² ^U, and^U denotes the space of measurable functions from ^R to some compact set U ^R^m. We assume that the vector eldf is continuous,f(;u) is locally Lipschitz on^Rⁿⁿ^f0^gfor eachu²U, and satises the following property.

Definition 3.1. We call f homogeneous-in-the-state if there existri >0, i = 1;:::;nand ²( miniri;¹) such that

f(x;u) =f(x;u) for all u²U (3.2)

whereis thedilation matrix as in Denition 2.1,k= minir_i is called theminimal power and the value ²( k;¹) is called the degree of the system.

Note that this denition implies f(0;u) = 0 for all u ² U. We denote the trajectories of (3.1) with initial value x⁰ at the time t = 0 and control function u()²^U again byx(t;x⁰;u()). Observe that also the trajectories of (3.1) may escape in nite time if >0 and that uniqueness of the trajectory may not hold in the origin if <0 (here again we use the convention as for the trajectories of (2.1)). As long as the trajectories exist and uniqueness holds we obtain from Denition 3.1 that

x(t;x⁰;u()) = x(t;x⁰;u()) (3.3)

for allx⁰²^Rⁿ.

Besides being useful auxiliary systems for our stabilization problem for homogeneous systems, homogeneous-in-the-state systems themselves form an interesting class of systems. They generalize homogeneous bilinear and semilinear systems (see e.g. [5, 6, 7, 10]). Generally speaking they model systems in which the control aects parameters of the system rather that representing some force acting on the system, cf. the examples in [8, 9]. Also for this class of systems there exist examples which are stabilizable but not with a continuous feedback law, see [24, Example after The- orem A]. Note that this class can be generalized analogously to the generalization of semilinear systems made in [10]; all results in this paper can easily be adapted to that case.

Applying suitable coordinate and time transformations we can considerably simplify the class of systems to be considered: Using the dilated normN from (2.3) the function

P(x) := _N¹⁽_x⁾x

denes a projection from^Rⁿ^nf0^gontoN ¹(1) satisfyingP(x) =P(x) for all >0.

We denote then 1 dimensional embedded unit sphere^fx²^Rⁿ^j^kx^k= 1^gby^Sⁿ ¹. Then, since N(tx) is strictly increasing in t 0 the function S : N ¹(1) ^! ^Sⁿ ¹, S(x) =x=^kx^kis a dieomorphism between these two manifolds, thus we can dene a coordinate transformationy= (x) by

(x) =N(x)^kS(P(x)); ¹(y) = ^pkky^kS ¹

y

ky^k

;

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and (0) = 0, ¹(0) = 0, which is continuous on ^Rⁿ and C¹ on ^Rⁿ ⁿ^f0^g. This denition implies

(x) =^k (x); ¹(^ky) = ¹(y) and by dierentiation of (x) and^k (x) one sees

D (x) =^k¹D (x): Thus dening

f~(y;u) =D ( ¹(y))f( ¹(y);u) we obtain (withx= ¹(y))

f~(^ky;u) =D (x)f(x;u) =^k¹D (x)f(x;u) =^kf^~(y;u) implying

f~(y;u) =⁺¹f^~(y;u);

with==k, i.e. ~f is homogeneous-in-the-state with respect to the standard dilation =Id, with mimimal powerk= 1, and with degree =.

Furthermore setting f(y;u) = ~f(y;u)^ky^k (which denes a time transformation for ~f) we obtain a system with degree = 0. In what follows we will therefore assume

f(x;u) =f(x;u) for all x²^Rⁿ; 0 (3.4)

and will retranslate the results to the general case in Theorem 4.3. Observe that the newf is now globally Lipschitz with a uniform constant which we will denote byL.

In order to obtain a way to characterize asymptotic controllability of (3.4) we introduce the nite time exponential growth rate (cf. [10, 11])

^t(x⁰;u()) = 1t^ln^kx(t;x⁰;u())^k

kx⁰^k :

It follows immediately from (3.4) that x(t;x⁰;u()) = x(t;x⁰;u()) and thus the growth rates satisfy ^t(x⁰;u()) =^t(x⁰;u()) for allx⁰ ²^R^dⁿ^f0^gand all >0.

The meaning of^t is described by the following proposition.

Proposition 3.2. System(3.4) is asymptotically controllable if and only if there exists a time T > 0 and some > 0 such that for each x ² ^Rⁿ ⁿ^f0^g there exists ux()²^U with

^t(x;ux())< for all tT (3.5)

Proof. Obviously (3.5) implies exponential controllability, thus in particular asymptotic controllability.

For the converse implication since^t(x;u()) =^t(x;u()) it is sucient to show (3.5) for ^kx^k = 1, i.e. x ² ^Sⁿ ¹. Asymptotic controllability implies that for each x²^Sⁿ ¹there exist ~ux()²^U, ~tx>0, andCx>0 such that^k'(~tx;x;u~x())^k<1=2, and ^k'(t;x;u~x())^k< Cx for all t ²[0;^~tx]. By compactness of^Sⁿ ¹and continuous dependence on the initial value we can choose the controls such thatT¹= sup_x^2Sn ¹~tx

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and C = sup_x^2Sn ¹Cx are nite. Now for each x ² ^Sⁿ ¹ we dene ux() and a sequenceti inductively byt⁰= 0 and

ti⁺¹=ti+ ~txi; ux(t) = ~uxi(t ti); t²[ti;ti⁺¹]

where xi = '(ti;x;ux())=^k'(ti;x;ux())^k. Choosing ti maximal with ti t (i.e.

t ti< T¹andti> t T¹) this implies ^t(x;ux()) =ti

t ^tⁱ⁽x;ux()) +t ti

t ^{t t}ⁱ⁽xi;ux(ti+)) t T¹

t ln 12 +T¹ t ^lnC where the last expression is independent of x and negative for all t T for T > 0 suciently large, which yields the assertion.

In fact, we can show something more than just the negativity of the nite time exponential growth rates. We dene the Lyapunov exponent of each trajectory by

(x;u()) := limsup_t

!1

^t(x;u())

and the supremum w.r.t. the state and inmum w.r.t. the control over these exponents by

:= sup_x

2Rn^nf0g_uinf

()2U

(x;u()):

Lyapunov exponents for control systems have been utilized in the analysis of bilinear systems (see e.g. [5] for some basic concepts and [6] for a detailled exposition) and for the global stabilization of semilinear and the local stabilization of dierentiable nonlinear systems at singular points [10]. In the homogeneous setup we obtain the following characterization.

Proposition 3.3. Consider the system(3.4) and its sup-inf Lyapunov exponent . Then for each ²(0;) there exists T >0 such that for each x ²^Rⁿ ⁿ^f0^gthere existsux()²^U with

^t(x;ux())< for all tT Proof. Exactly as [10, Proof of Proposition 3.4].

Since by Proposition 3.2 for our class (3.4) of homogeneous systems asymptotic controllability immediately implies <0, Proposition 3.3 establishes a spectral condition for the asymptotic controllability of (3.4).

4. Stabilization of systems homogeneous-in-the-state.

In this section we will construct a Lyapunov function and a stabilizing feedback for system (3.4). Af- terwards we retranslate this stabilization result to general systems homogeneous-in- the-state from Denition 3.1.

We begin with the construction of a homogeneous Lyapunov function for system (3.4). First observe that the projection

s(t;s⁰;u()) := x(t;x⁰;u())

kx(t;x⁰;u())^k; s⁰= x⁰

kx⁰^k

of (3.4) onto ^Sⁿ ¹ is well dened due to the homogenity of the system. A simple application of the chain rule shows thatsis the solution of

s_(t) =f^S(s(t);u(t)); f^S(s;u) =f(s;u) ^hs;f(s;u)ⁱs

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and that fors⁰=x⁰=^kx⁰^kthe exponential growth rate^t satises ^t(x⁰;u()) =^t(s⁰;u()) = 1t

Z t

0

q(s(;s⁰;u());u())d withq(s;u) =^hs;f(s;u)ⁱ. Thus dening the discounted integral

J(s⁰;u()) :=^Z ¹

0

e q(s(;s⁰;u());u())d and the corresponding optimal value function

v(s⁰) := inf_u

()2U

J(s⁰;u())

from Propositions 3.2 and 3.3, and [10, Lemma 3.5(ii)] we obtain that if system (3.4) is asymptotically controllable then for each ² (0;) there exists >0 such that for all²(0;] and alls⁰²^Sⁿ ¹the inequality

v(s⁰)<

holds. Note thatv is Holder continuous and bounded for each >0, cp. e.g. [1]. We now x some²(0;) and some²(0;] and dene

V⁰(x) :=e²^v⁽^x=^k^x^k)^kx^k²:

Lemma 4.1. The function V⁰ is a clf which is homogeneous with degree 1 (with respect to the standard dilation) and satises

v^2cominf⁽x;U⁾DV⁰(x;v) 2V⁰(x):

Proof. Homogeneity, positive deniteness and properness follow immediately from the denition. Now for eacht >0 the functionv satises the dynamic programming principle

v(s⁰) = inf_u

()2U Z t

0

e q(s(;s⁰;u());u())d+e ^tv(s(t;s⁰;u()))

; see e.g. [1]. Abbreviatingq(t;s⁰;u()) =q(s(t;s⁰;u());u(t)) and usinge ^t 1 t we obtain for the integral part of this equality

Z t

0

e q(;s⁰;u())d ^Z ^t

0

q(;s⁰;u()) + (e ^t 1)Mqd =t^t(s⁰;u()) Mqt₂² whereMq denotes a bound of^jq^j. Thus withs⁰=x⁰=^kx⁰^kwe obtain

V⁰(x⁰)_uinf

()2U

exp[2t^t(x;u()) Mqt²+ 2e ^tv(s(t;s⁰;u()))]^kx⁰^k²

= inf_u

()2U

e²^t^t⁽^x;u⁽⁾⁾e ^M^q^t²e²⁽^e ^t ¹⁾^v⁽^s⁽^t;s⁰^;u⁽⁾⁾⁾e²^v⁽^s⁽^t;s⁰^;u⁽⁾⁾⁾^kx⁰^k²

= inf_u

()2U

kx(t;x⁰;u())^k²

kx⁰^k² e ^M^q^t²e²⁽^e ^t ¹⁾^v⁽^s⁽^t;s⁰^;u⁽⁾⁾⁾e²^v⁽^s⁽^t;s⁰^;u⁽⁾⁾⁾^kx⁰^k²

= inf_u

()2U

e ^M^q^t²⁺²⁽^e ^t ¹⁾^v⁽^s⁽^t;s⁰^;u⁽⁾⁾⁾V⁰(x(t;x⁰;u()))

_uinf

()2U

e ^M^q^t²⁺²⁽¹ ^e ^t⁾⁼V⁰(x(t;x⁰;u())):

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Now for eacht >0 we chooseut()²^U such that the inmum of the last expression is attained up tot². Usingb b²1 e ^bb forb >0 we can conclude

V⁰(x(t;x⁰;ut())) V⁰(x⁰)(1 e ^M^q^t²⁺²⁽¹ ^e ^t⁾⁼)V⁰(x(t;x⁰;ut())) +t²

(1 e ^M^q^t²⁺²^t ²^t²)V⁰(x(t;x⁰;ut())) +t²

( 2t+ (Mq+ 2)t²)V⁰(x(t;x⁰;u())) +t² for allt >0 suciently small. Denotingvt= (x(t;x⁰;ut()) x⁰)=twe obtain

1t ⁽V⁰(x⁰+tvt) V⁰(x⁰)) 2V⁰(x(t;x⁰;u())) + (Mq+ 2)tV⁰(x(t;x⁰;u())) +t and since by compactness ofU there exists a v ² cof(x;U) and a sequenceti ^! 0 such thatvti^!vasi^!¹the assertion follows by the denition ofDV⁰.

Based onV⁰and using the techniques from [4] we can now construct the stabilizing feedback law for system (3.4). To this end for >0 we consider the approximation ofV⁰via the inf-convolution

V(x) = inf_y

2Rn

V⁰(y) +^kx y^k² 2²

: (4.1)

Observe thatV is locally Lipschitz andV ^!V⁰ as^!0.

Proposition 4.2. For each ²(0;) there exists >0 such that the function V is a Lipschitz continuous clf which is homogeneous with degree 1 (with respect to the standard dilation) and satises

v^2cominf⁽x;U⁾DV(x;v) 2V(x):

Furthermore there exists a feedback law F : ^Rⁿ ^! U satisfying F(x) = F(x) for all x ² ^Rⁿ, > 0 and constants h > 0 and C > 0 such that any -trajectory corresponding to some partition with d()hsatises

kx(t;x⁰;F)^kCe ^kx⁰^k: (4.2)

Proof. By its denitionV is obviously positive denite. Now for eachx²^Rⁿ we denote byy(x) a point realizing the minimum on the right hand side of (4.1). Since V⁰is homogeneous with degree 1 we have that

V⁰(y) +^kx y^k² 2²

=²V⁰(y) +^kx y^k² 2²

and thus in particularV is also homogeneous with degree 1, hence proper, and we can choosey(x) in such a way that y(x) =y(x). SinceV⁰is strictly increasing along the raysxin >0 it follows that^ky(x)^k^kx^k.

Now we dene

(x) := x y(x) 2² which implies(x) =(x).

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By [4, Lemma III.1 and III.2] (or by straightforward calculations) for this vector we can deduce the inequalities

V(x+v)V(x) +^h(x);vⁱ+²^kv^k² 2² (4.3)

and

V⁰(y(x) +v)V⁰(y(x)) +^h(x);vⁱ ²^kv^k² 2² ; (4.4)

i.e. (x) is a proximal supergradient ofV inx and a proximal subgradient ofV⁰ in y(x) (see e.g. [3] for an exposition of these concepts). We choose the feedbackF(x) in such a way that

h(x);f(x;F(x))ⁱ= inf_u

2U^h(x);f(x;u)ⁱ

andF(x) =F(x) for allx²^Rⁿ ⁿ^f0^gand all >0. The valueF(0) can be chosen arbitrary.

Now consider points x ² ^Rⁿ with ^kx^k= 1, i.e. x ² ^Sⁿ ¹. For these points the Holder continuity ofV⁰(which is inherited from the Holder continuity ofv) and the denitions ofV and imply

21²^ky(x) x^k²V⁰(x) V⁰(y(x))H^ky(x) x^k and thus

k(x)^kky(x) x^kH²² (4.5)

where H > 0 and ² (0;1] denote the Holder constant and exponent of V⁰ on

fx²^Rⁿ^j^kx^k1^g. From (4.5) and the denition ofV we immediately obtain

jV⁰(y(x)) V(x)^jH²² (4.6)

Now the Lipschitz continuity off implies that

h(x);f(x;F(x))ⁱmin_u

2U^h(x);f(y(x);u)ⁱ+L^k(x)^kky(x) x^k

and by (4.4) and the denition of DV⁰ it follows that ^h(x);vⁱDV⁰(y(x);v) for allv ²^Rⁿ. Thus by the linearity of the scalar product and Proposition 4.1 we can conclude

minu²U^h(x);f(y(x);u)ⁱ=_v min

2cof⁽y⁽x⁾;U⁾^h(x);vⁱ 2V⁰(y(x)): Combining these inequalities with (4.5) and (4.6) yields

h(x);f(x;F(x))ⁱ 2V(x) + 2H²² +LH²²: (4.7)

Dening

f_x := 1

Z

0

f(x(t;x;F(x));F(x))dt

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and using M := sup^k_x^k2_;u²_Uf(x;u) and the Lipschitz continuity of f for > 0 suciently small we obtain

kf_x f(x;F(x))^kML; ^kf_x^kM:

Thus by (4.3), (4.7), and the fact that ^k(x)^k C for all x ² ^Sⁿ ¹ and some suitableC >0 we can conclude

V(x(;x;F(x))) V(x) =V(x+f_x) V(x)

^h(x);f_x)ⁱ+²^kf_x^k² 2²

^h(x);f(x;F(x))ⁱ+ML²^k(x)^k+²M² 2²

( 2V(x) + (2+L)H²²) +²

MLC+M² 2²

Denoting

:= sup_x

2Sn ¹(2+L)H²²

V(x) ; C^~:= sup_x

2Sn ¹MLC

V(x) + M² 2²V(x); and exploiting homogenity ofx(;x;F(x)) andV we obtain for arbitraryx⁶= 0

V(x(;x;F(x))) V(x)( 2+)V(x) +²C^~V(x) which immediately implies both assertions since^!0 as^!0.

This proposition shows the stabilization for systems of type (3.4). The following theorem shows how this result can be translated to the general homogeneous-in-the- state system from Denition 3.1.

Theorem 4.3. Consider system(3.1) satisfying Denition 3.1 with dilation matrix , minimal power k > 0, and degree ² ( k;¹), and assume asymptotic controllability. Then there exists > 0 and a clf V being Lipschitz on ^Rⁿ ⁿ^f0^g, satisfying

V((x)) =²^kV(x) and

v^2cominf⁽x;U⁾DV(x;v) 2N(x)V(x) for the functionN from(2.3).

Furthermore there exists a feedback law F :^Rⁿ ^! U satisfying F(x) = F(x) for allx²^Rⁿ, >0 such that the corresponding sampled closed loop system is either

(i) semi-globally stable (if >0), or

(ii) globally uniformly exponentially stable (if = 0), or (iii) globally practically exponentially stable (if <0) with xed sampling rate.

Proof. Obviously if the system dened byf is asymptotically controllable, then the transformed system dened by f is asymptotically controllable. Thus from Propo- sition 4.2 we obtain V = V and F = F satisfying the assertion for f which is homogeneous-in-the-state with =Id,k= 1 and = 0.

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We start by showing the result for the system dened by ~f(x;u) = f(x;u)^kx^k being homogeneous-in-the-state with with =Id, k= 1 and =. Let ~V(x) = V(x). Then we immediately obtain

v^2cominf^~⁽x;U⁾DV^~(x;v) =^kx^k_v min

2co

f⁽x;U⁾DV^~(x;v) ^kx^k2V^~(x):

Now observe that for each control functionu()²^U the trajectories ~x and xof these systems satisfy

x~(t;x⁰;u()) = x(t(t);x⁰;u(~t())) (4.8)

where ~t(t) denotes the inverse of t(t) which is dened by t(t) =^Z ^t

0

kx~(;x⁰;u()))^kdt

and thus is well dened as long as the solution ~x(t;x⁰;u()) exists. If both ~x and x uniquely exist for allt0 it is immediate that t(t)^!¹ast^!¹.

Setting ~F(x) = F(x) a ~-trajectory ~x^~(t;x⁰;F^~) of _~

x= ~f(~x;F^~(~x)) (4.9)

on some interval [0;T] on which ~x exists becomes a -trajectory x(t(t);x⁰;F) of _

x= f(x;F(x)) (4.10)

where = (ti)i^2N⁰ is given by ti = t(~ti) with ~ = (~ti)i^2N⁰. Now we distinguish the three cases:

(i) >0: By the choice of F there existC;;h >0 such that inequality (4.2) holds for each -trajectory x of (4.10) with d()h and eachx ²^Rⁿ. Now consider a compact set K^Rⁿ with 0²intK. Let CK := sup_x²_K^kx^k, consider a ~-trajectory x~^~(t;x⁰;F^~) of (4.9) with d(~) h(CCK) and x ² K, and assume that there exists a (minimal) time t > 0 such that^kx~^~(t;x⁰;F^~)^k =C^kx^k. W.l.o.g. we may assume t = ~tl ² ~ for some l > 0, otherwise we may reduce the sampling interval containing t. Then since ^kx~^~(t;x⁰;F^~)^k CCK for all t ² [0;^~tl] the rescaled satises ti ti ¹hfor alli= 1;:::;l, thus we obtain

kx~^~(~tl;x⁰;F^~)^k=^kx(tl;x⁰;F)^kCe ^t^l^kx⁰^k< C^kx⁰^k

contradicting the choice of t = ~tl. Thus^kx~^~(t;x⁰;F^~)^kC^kx^kholds for all t 0, and henced()h, implying

k~x^~(t;x⁰;F^~)^kCe ^t⁽^t⁾^kx⁰^k

which implies the desired stability estimate with(^kx^k;t) =Ce ^~^t⁽^t⁾^kx^kwhich is of class ^{K L} because the corresponding trajectories stay inside some compact set, thus exist for allt0, and are unique since >0, hence t(t)^!¹ast^!¹.

(ii)= 0: In this case the assumption follows immediately from Proposition 4.2.

(iii) <0: As in case (i) there existC;;h >0 such that inequality (4.2) holds for each -trajectory x of (4.10) with d()hand each x²^Rⁿ. Consider a compact setK^Rⁿ and an open set B^Rⁿ with 0²BK. LetCK = sup_x²_K^kx^k,CB = infx⁶²B^kx^k=2>0. By continuous dependence on the initial value and compactness we