The Method of Liapunov Functions for Nonlinear

The Method of Liapunov Functions for Nonlinear Input Systems Andrea Bacciotti Dipartimento di Matematica del Politecnico 10129 Torino, Italy and Lionel Rosier Laboratoire d'Analyse Numerique Universite Paris-Sud 91405 Orsay, France

Abstract. The purpose of this work is to survey some possible de nitions of external

stability for nonlinear systems and to provide the basic tools for investigating their relationship and dependence on internal stability properties. In particular, we review some early and more recent results about converse Liapunov theorems. To this respect, some improvements are pointed out without proof.

1. Introduction During the last few years, new directions of studies are emerged in the literature about nonlinear systems. In particular, many papers have been devoted to the investigation and characterization of certain input/output stability properties in the framework of the classical Liapunov functions method. The present work is an attempt of reviewing some important results obtained on this subject and exposing the basic material in an organized way. We are interested in physical input systems modeled by continuous time, 1

time-dependent, nite dimensional ordinary dierential equations (1)

x_ = f (t; x; u)

where x = (x1 ; : : : ; xn) 2 Rn represents the state variables, u = (u1; : : : ; um ) 2 Rm represents the input variables and t  0. The regularity assumptions about the function f = (f1 ; : : : ; fn ) : [0; +1)  Rn  Rm ! Rn are crucial for the following developments and will be speci ed later. Together with (1), we will often consider the unforced associated system (2)

x_ = f (t; x; 0) :

Basically, (2) accounts for the \internal" behavior of the system. More precisely, (2) describes the natural dynamics of (1) when no energy is supplied through the input channels. The analysis of the \external" behavior is rather concerned with the eect of the inputs (disturbances or exogenous signals) on the evolution of the state response of (1). Looking at the linear case, we see that there is an intimate connection between the internal and the external behavior. On the contrary, dealing with nonlinear systems, the connection becomes weaker and it needs a more delicate description. A rst aim of this paper is therefore to survey possible de nitions of internal and external stability in a nonlinear setting and to discuss their relationships by means of the Liapunov functions method. We will also consider the problem of achieving a more desirable stability behavior (both from the internal and the external point of view) by means of properly designed feedback laws. To this end, it is convenient to think of the input as a sum u = ue + uc. The term ue represents the external forces, while uc is actually available for control action. As well known, many applications of stabilization theory are based on the so-called Jurdjevic-Quinn method for the design of feedback uc = k(t; x), which in turn depends on the knowledge of a (weak) Liapunov function for the unforced associated system. Thus, for the need of both stability analysis and feedback design, we are led to enlighten the importance of having at our disposal a variety of theorems which state, under minimal assumptions, the existence of Liapunov functions with appropriate properties. These theorems are usually called \converse Liapunov theorems". The second aim of this paper is to illustrate the state of the art on this subject, and to present some recent developments. 2

The plan of exposition is as follows. In Sect. 2 we recall some well known facts concerning the linear case: since the linear case has been studied for a long time, it is indeed natural to take it as a starting point. Sect. 3 is devoted to nonlinear systems without inputs: in particular, we review here the converse theorems for asymptotic stability, local stability and Lagrange stability, and we state without proof some more recent and new results. In Sect. 4 we shall deal with dierential inclusions of the form (3)

x_ 2 F (t; x)

where for each pair (t; x), F (t; x) denotes a subset of Rn. These equations are very close to control systems: indeed, both closed-loop systems with discontinuous feedback and systems with free inputs can be interpreted as dierential inclusions. In Sect.s 5 and 6 we consider some possible extensions of the classical stability concepts to systems with inputs and their characterizations by means of suitable Liapunov functions. Finally, Sect. 7 deals with the stabilization problem. For reader's convenience, and in order to render the paper self-contained, the basic de nition of stability and Liapunov functions are reported in the Appendix. The subject of external stability, and especially Sontag's notion of input-tostate stability, has recently attracted the interest of many authors. The list of references reported in this work is far from being exhaustive; however, it can provide some orientative indications. As well known, the analysis of input/output behavior is related to very important developments of modern control theory, such as robust and adaptive control, behavior of interconnected systems, H1-control and so on. However, these developments will be not considered in this paper. We conclude this introduction by some agreements about basic assumptions and notation. Let t0  0 and x0 2 Rn. Throughout this paper, we will always assume that for each measurable input function u : [0; +1) ! Rm there exists at least one local solution '(t) of (1) such that '(t0 ) = x0 . When we need to emphasize the dependence of any such solution on the initial pair (t0 ; x0 ) and the input u(t), we shall use the notation '(t) = '(t; t0 ; x0 ; u(
)). Note that we do not make standing assumptions about uniqueness of solutions. The euclidean norm of a nite dimensional vector v is denoted by jvj. For each measurable, essentially bounded function u : [0; +1) ! Rm , we denote 3

kuk1 = ess sup ju(t)j < +1 : t0

For locally Lipschitz continuous functions V (t; x) : [0; +1)  Rn ! R, we sometimes use the notion of upper Dini's directional derivative. Given t, x and v 2 Rn, this is denoted by

V (t + h; x + hv) ? V (t; x) : D+ V (t; x; v) = lim sup h + h!0

2. Linear systems The relationship between external and internal stability is well understood in the case of the time-invariant linear system (4)

x_ = Ax + Bu

(here, A and B are real matrices of appropriate dimensions). To describe the internal stability properties of the unforced associated system (5)

x_ = Ax

the classical concepts of (global) asymptotic stability and (local) stability suf ce: indeed, in the linear case Lagrange stability is equivalent to local stability. Moreover, there is no distinction between local and global aspects. When (5) is asymptotically stable at the origin, it is usual to say, in short, that A is Hurwitz. Similarly, when (5) is locally stable, we simply say that A is stable. If A is Hurwitz and if u : [0; +1) ! Rm is a measurable input function such that kuk1 < +1, then there exist positive numbers ; 1; 2 such that (6)

j'(t; 0; x0; u(
))j  1jx0 je?t + 2kuk1

for each t  0. This inequality admits the following interpretation: for large t, the eect of the initial conditions is negligible, and the solutions are ultimately 4

bounded by a term which is related to the input energy (measured by its L1 norm) by means of the constant \gain" 2. This re ects the distinction between transient and steady state in the classical engineering literature. We want now to introduce the intuitive idea of bounded input bounded state (in short, BIBS) stability(1). To this end, it is convenient to assume (without loss of generality) that (4) has been put in Kalman's canonical form. Proposition 1 Let A11 and A22 be respectively the diagonal blocks of A corresponding to the controllable part and the uncontrollable part of (4). Then, there exist 1; 2 > 0 such that (7)

j'(t; 0; x0; u(
))j  1jx0 j + 2kuk1

holds for each t  0, each x0 and each u(
), if and only if A11 is Hurwitz and A22 is stable. Inequality (7) can be interpreted by saying that when the input is bounded, then the solutions are bounded by a term which depends linearly on the energy initially stored in the system (measured by the norm of the initial state) and the energy due to the input supply. This precisely corresponds to the intuitive meaning of BIBS stability (the formal de nition will be given later, in a more general context). Of course, (7) is weaker than (6). Hence, if A is Hurwitz then (4) is BIBS stable, but the converse is true only under additional assumptions; for instance, if (4) is completely controllable (that is when A = A11 ). The analysis of the internal stability properties of a linear system can be carried out in terms of quadratic Liapunov functions, that is functions of the form V (x) = xtPx where t denotes transposition and P is a positive de nite, symmetric real matrix. For instance, it is well known that A is Hurwitz if and only if 2xtPAx = ?jxj2 (1) Since the stability analysis is usually performed with respect to outputs, instead of state evolution as we are doing in the present paper, the acronym BIBO is more popular than BIBS, but the underlying idea is similar. 5

for some P , while A is stable if and only if xt PAx is negative semi-de nite for some P . Unfortunately, the characterization of external stability by means of quadratic Liapunov functions is not as much plain. For instance, it is not dicult to prove that (6) holds if and only if there exists a positive de nite, symmetric real matrix P which enjoys the following property: for each R > 0 there exists  > 0 such that

xt P (Ax + Bu) < 0

(8)

for each x 2 Rn, u 2 Rm , subject to the conditions juj  R and jxj  . At this point, it could be tempting to conjecture that a necessary and sucient condition for BIBS stability is obtained in a similar way, by simply substituting the strict inequality in (8) by a weak one. But this is false, as shown by the extremely simple two-dimensional example



x_ 1 = ?x1 + u x_ 2 = 0 : It is clear that this system has the BIBS property. Setting for instance u = 1  and given a matrix P = pp11 pp12 , the expression in (8) is easily computed: 12

22

(p11 x1 + p12 x2 )(1 ? x1 ) : It is clear that this expression takes both positive and negative values for j(x1 ; x2 )j arbitrarily large, and for any choice of p11 ; p12. 3. Systems without inputs In this section we shall review some converse Liapunov theorems for nonlinear systems without inputs. In view of the main purpose of this paper, a system of this type should be always thought of as originated by some system of the form (1), when the input is set to be zero. However, for simplicity, throughout this section we shall adopt the notation (9)

x_ = f (t; x)

t  0 ; x 2 Rn 6

and assume that f (t; 0) = 0, so that the origin is an equilibrium position. First of all, we resume in the following statements well known versions of rst and second Liapunov's theorem. Theorem 1 If there exists a weak Liapunov function in the small for (9), then the origin is uniformly stable. Theorem 2 If there exists a strict Liapunov function in the small for (9), then the origin is uniformly asymptotically stable. Studies on the invertibility of rst and second Liapunov's theorem have a long history. Apart from the linear case, that has been shortly reviewed in the previous section, the more familiar situation arises when (9) is time-invariant (i.e., f (t; x) = f (x)) and there is just one solution for each initial condition. In [28], Massera proved that if the origin is locally asymptotically stable for the system (10)

x_ = f (x)

and f is of class C 1, then there exists a strict Liapunov function, independent of t and of class C 1. More generally, one can assume that (10) de nes a dynamical system (in the topological sense, see [9]). Under this weaker assumption, it is possible to prove ([9], p. 66) that the origin is locally asymptotically stable if and only if there exists a positive de nite function V (x) which is continuous in a neighborhood of the origin and satis es the following property: V ('(t)) is strictly decreasing for each non trivial solution '(t). For more recent contributions, including the case where the attracting set is not reduced to a singleton, see [30], [46]. Under the assumption that (10) de nes a dynamical system, a necessary and sucient condition for local stability is represented by the existence of a weak Liapunov function in the small which is independent of t but which needs not be continuous, in general ([9], p. 84). The existence of continuous Liapunov functions is related to a more restrictive notion of stability which accounts not only for the behavior of solutions, but also for the behavior of their prolongations ([4]; see [19] for an extension to polysystems). 7

Now, we come back to the time-dependent case. The rst studies about converse Liapunov theorems are due to Persidskii (1936). Later, around 1955, Krasovski ([20]), Kurzweil ([22]) and Yoshizawa ([49]) obtained independently the converse theorem for uniform stability. It should be emphasized that Kurzweil's construction applies when f is of class C 1, while for Yoshizawa's construction, f continuous and Lipschitz continuous with respect to x suces. In [24], Kurzweil and Vrkoc addressed the problem of the existence of a weak Liapunov function for stable and uniformly stable systems of the form (9), assuming that f is merely continuous. They showed by an example that the lack of uniqueness may constitute an obstruction to the existence of such a continuous Liapunov function. Indeed, even in this case one can de ne \approximate" solutions which exhibit an unstable behavior. However, they proved that the existence of a continuous weak Liapunov function becomes necessary and sucient when the de nition of stability is conveniently strengthened. We report here the de nition of [24], which will be referred to as \robust stability". De nition 1 We say that (9) is robustly locally stable at the origin if there exist a sequence fGi gi=0;1;2;::: of open sets in R+  Rn and two sequences of real numbers fai gi=0;1;2;::: and fbi gi=0;1;2;::: such that (i) 0 < bi+1 < ai  bi for each i = 0; 1; 2; : : :, and bi ! 0 for i ! +1 (ii) R+  fx : jxj < ai g  Gi  R+  fx : jxj < bi g for each i = 0; 1; 2; : : : (iii) for each i = 0; 1; 2; : : :, each initial pair (t0 ; x0 ) 2 Gi and each solution '(t; t0 ; x0) one has (t; '(t; t0 ; x0)) 2 Gi for each t  t0 . It should be noted that the new restrictions introduced in the de nition of robust stability, when applied to the case of time-invariant systems, turn out to be weaker than the conditions about prolongations imposed in [4]. For instance, for a center-focus con guration (see again [9], p. 87) it is possible to construct a time-dependent continuous weak Liapunov function, but not a continuous timeinvariant one. More or less in the same years, several converses of the second Liapunov's theorem for time-dependent systems were published ([23], [29] and the references therein). To this respect, the crucial remark is that the second Liapunov's theorem in the time-dependent case cannot be inverted in general, if one assume that the 8

origin is merely asymptotically stable. However, it is known that the existence of a strict Liapunov function in the small implies in fact that the asymptotic stability of the origin is of the uniform type. Actually, the second Liapunov's theorem is invertible under the stronger assumption of uniform asymptotic stability, and provided that the right hand side of (9) is suciently regular. The more general converse of second Liapunov's theorem obtained in that period, is due to Kurzweil ([23]). He proved that if the origin is locally uniformly asymptotically stable and f (t; x) is merely continuous, then there exists a strict Liapunov function in the small V (t; x) of class C 1. Kurzweil's result improves Massera's theorem of 1949, as well. Indeed, he proved that if f does not depend on t, then it is possible to nd a strict Liapunov function in the small which does not depend on t and preserves the same properties as before. So long, we have discussed only local results. As far as the global aspect is concerned, we limit ourselves to point out that corresponding (direct and converse) results for asymptotic stability can be recovered by replacing the notion of strict Liapunov function in the small by that of global strict Liapunov function (see again [9], [23], [29]). The notion of global stability is studied in [1]. In this work we are specially interested in Lagrange stability (often referred to also as boundedness of solutions). This property can be somewhat reviewed as a \large state" counterpart of local stability. For instance, it is not dicult to prove an analogous of rst Liapunov's theorem. Theorem 3 If there exists a weak Liapunov function in the large for (9), then the system is uniformly Lagrange stable. Lagrange stability was deeply studied by Yoshizawa ([50]). He proved that if f is continuous, and Lipschitz continuous with respect to x, then Theorem 3 can be inverted. Moreover, Yoshizawa's construction gives rise to a Liapunov function V (t; x) which is Lipschitz continuous with respect to both t and x. Finally, he showed that if f is Lipschitz continuous with respect to both variables, a Lipschitz continuous weak Liapunov function can be \smoothed out" to any preassigned order of regularity. If the system is Lagrange stable and f is merely continuous, Yoshizawa's construction can be adapted and gives rise again to a weak Liapunov function in the large: however, in general, there exists no continuous such a Liapunov function. 9

This can be seen by an easy modi cation of the aforementioned counterexample by Kurzweil and Vrkoc ([8]). In analogy with the case of uniform stability, the existence of a continuous weak Liapunov function turns out to be equivalent to a stronger type of Lagrange stability, that we agree to call \robust Lagrange stability". The de nition requires the same modi cations as in the case of robust stability and it is left to the reader. This was, more or less, the state of the art at the end of the 50's. The further step has been addressed very recently. It involves systems of equations with discontinuous right hand side. The motivation for such a development is provided by some achievements of modern control theory. Indeed, discontinuous equations arise in many applications as a result of the implementation of discontinuous feedback (synthesis of the minimal time problem, variable structure control, internal and external stabilizability). We recall in particular that the existence of a discontinuous feedback for the asymptotic stabilization problem does not imply in general the existence of a continuous one ([10]). Note that when f is not continuous with respect to x, the classical notion of solution does not apply. According to the well known Filippov's solutions approach, systems of equations with discontinuous right hand side are equivalent to a dierential inclusion of the form (3), with = F (t; x) = Kxf (t; x) def

\ \

>0 (N )=0

co ff (t; B(x; )nN )g

where B(x; ) is the ball of center x and radius , co denotes the convex closure and  is the usual Lebesgue measure of Rn. We recall that if f (t; x) is measurable and locally bounded, then the multivalued map F (t; x) = Kxf (t; x) enjoys the following properties ([2], [3], [12], [15]): H1 ) F (t; x) is a nonempty, compact, convex subset of Rn , for a.e. t  0 and each x 2 Rn H2 ) F (t; x), as a multivalued map of x, is upper semi-continuous for xed t a.e. in [0; +1) H3 ) F (t; x), as a multivalued map of t, is measurable for each x H4 ) for each R > 0 and each T > 0 there exists M > 0 such that

F (t; x)  fv : jjvjj  M g for a.e. t 2 [0; T ] and 0  jjxjj  R. 10

We have so established an important link between control theory and the theory of dierential inclusions. 4. Dierential inclusions A second, important link to the theory of dierential inclusions is given by the fact that a system with free inputs can be actually reviewed as a dierential inclusion of a particular type. Consider a system of the form (1) and assume that f (t; x; u) satis es Caratheodory conditions (local boundedness, measurability with respect to t and continuity with respect to x and u). Let U be a given subset of Rm , and assume that an input function u(
) is admissible only if it ful lls the constraint u(t) 2 U a.e. t  0. Then, it is evident that every solution of (1) corresponding to an admissible input is a solution of a dierential inclusion (3) where the right hand side is now de ned by (11)

F (t; x) = f (t; x; U ) :

A celebrated theorem by Filippov ([14]) states that also the converse is true, provided that f (t; x; u) is continuous and U is a compact set. We recall that under the same assumptions on f (t; x; u) and U , F (t; x) turns out to be Hausdor continuous. On the other hand, if f (t; x; u) is continuous with respect to t and locally Lipschitz continuous with respect to x (uniformly with respect to u) then F (t; x) is (Hausdor continuous) and locally Lipschitz Hausdor continuous with respect to x. The purpose of this section is to illustrate certain stability concepts for differential inclusions. However, we should not forget that dierential inclusions are here a way to represent a system with free inputs or a system with discontinuous feedback. Hence, the interest of possible de nitions depends, for us, on their interpretation from the point of view of control theory. Let us recall that in the literature about dierential inclusions, there are two possible way to extend the classical notions of stability. The notions labeled \weak" (local, asymptotic, Lagrange stability) are deduced from De nitions A.5, A.6, A.7 by asking that the respective conditions are satis ed for at least one solution corresponding to prescribed initial data. These notions are not irrelevant from a control theory point of view: indeed, they are related to controllability problems, feedback stabilization, viability theory and so on. 11

On the contrary, the notions labeled \strong" (local, asymptotic, Lagrange stability) imply that all the solutions corresponding to the prescribed initial data satisfy the respective conditions. From our point of view, this type of stability is the ideal one we can look for, when the inputs are interpreted as disturbances. Indeed, it is obviously desirable that the eect of a disturbance is quickly absorbed and that it does not aect too much the evolution of the system. In the spirit of the present work, from now on we focus therefore on the strong notions, which can be reviewed as some forms of external stability. A further remark about the literature is appropriate. While generalizations of direct Liapunov's theorems for strong stability and strong asymptotic stability for dierential inclusions are reported by several authors ([2], [12], [15], [34]), the subject of converse theorems is rarely and partially treated (see for instance [33]). Moreover, apparently there are no results about Lagrange stability in this context. However, recently there has been a growth of interest. In the remaining part of this section, we deal with Liapunov functions for dierential inclusions. The adaptation of De nitions A.1, A.2, A.3 and A.4 to the present situation is straightforward and it is left to the reader. In [26] the authors prove a converse theorem which is applicable to dierential inclusions (12) x_ 2 F (x) whose right-hand-side is independent of t. Theorem 4 Assume that the multivalued map in (12) is compact valued and locally Lipschitz Hausdor continuous. Assume further that 0 2 F (0). If the origin is globally asymptotically stable, then there exists a strict Liapunov function which is independent of t and of class C 1. It should be pointed out that Theorem 4 is proved in [26] not only for equilibrium positions, but more generally for closed sets. In [32], the author proves a converse theorem for global or local asymptotic stability, by extending Kurzweil's method to Filippov's solutions of equations with time-dependent, discontinuous right-hand-side. This result is not a consequence of [26] since, as already mentioned, the dierential inclusion associated to a discontinuous system is upper semi-continuous, but not Lipschitz continuous, in general. 12

In fact, by using the same method of [32] and Lemma 4.1 of [8], it is possible to achieve the following stronger result. Theorem 5 Assume that the right hand side of (3) satis es the assumptions H1), H2 ), H3 ), H4 ). Assume also that 0 2 F (t; 0), for a.e. t  0, so that the origin is an equilibrium position. If it is (resp., globally or locally) uniformly asymptotically stable, then there exists a strict Liapunov function (resp., global or in the small). Such a Liapunov function turns out to be locally Lipschitz continuous with respect to both variables. As far as local or Lagrange stability are concerned, Yoshizawa's approach is generalized in [5]. Theorem 6 Assume that the right hand side of (3) is compact valued, Hausdor continuous and locally Lipschitz Hausdor continuous with respect to x. Assume also that (3) is locally uniformly stable at the origin (resp., Lagrange stable). Then there exists a weak Liapunov function in the small (resp., in the large). Such a Liapunov function turns out to be locally Lipschitz continuous with respect to both variables. Finally, dierential inclusions with upper semi-continuous right hand side are considered again in [8], where the method of Kurzweil and Vrkoc is extended. Theorem 7 Assume that the right hand side of (3) satis es the assumptions H1), H2 ), H3 ), H4 ). If it is locally robustly stable at the origin (resp., robustly Lagrange stable), then there exists a weak Liapunov function in the small (resp., in the large). Such a Liapunov function turns out to be locally Lipschitz continuous with respect to both variables. Proving the existence of more regular Liapunov functions in the circumstances of Theorems 5, 6 and 7 is, for the moment, an open problem. 13

5. External stability for nonlinear systems: De nitions The idea underlying the notions of strong stability discussed in the previous section is, as already mentioned, that some conditions must be ful lled by all the solutions corresponding to certain initial data. This is not exactly the same idea underlying the external stability concepts typical of the linear case illustrated in Sect. 2. In the present section, we shall see that both (6) and (7) can be extended to nonlinear systems. De nition 2 We say that system (1) possesses the input-to-state stability (in short, ISS) property if there exist maps 2 LK, 2 K such that, for each initial pair (t0 ; x0 ), each measurable essentially bounded input u : [0; +1) ! Rm and each t  t0 (13)

j'(t; t0; x0 ; u(
))j  (t ? t0; jx0 j) + (kuk1) :

The previous de nition was introduced by Sontag in [37] for the time independent case and studied in a number of subsequent papers ([39], [41], [42]). The present form of the de nition, including time-dependence, is taken from [21]. ISS stability implies global asymptotic stability of the associated unforced system (but the converse is not true: see [37] for an example) and it is weaker than strong asymptotic stability (referred to the dierential inclusion de ned by (1) via (11)). A ISS stable system shares certain qualitative properties of internally stable linear systems: for instance, there is a unique equilibrium position and it is possible to distinguish transient and steady state in its evolution. De nition 3 We say that system (1) possesses the uniform bounded-input bounded state (in short, UBIBS) stability property if there exist maps 1; 2 2 K1 such that, for each initial pair (t0 ; x0 ), each measurable essentially bounded input u : [0; +1) ! Rm and each t  t0 (14)

j'(t; t0; x0 ; u(
))j  1(jx0 j) + 2 (kuk1) 14

This de nition is classical ([43], [45], [46], [47]). It extends (7) and has been recently re-considered in [1] for the time-invariant case. It is clear that every ISS system is UBIBS. However, a system with the UBIBS property may exhibit a more large variety of nonlinear phenomena: multiple equilibrium points, limit cycles etc. The one-dimensional example

x_ = ?(cos u)2x (see [39]) shows that a system may be internally stable (in the sense that the associated unforced system is globally asymptotically stable) and UBIBS stable, but not ISS stable. Furthermore, it is immediate to see that if (1) is UBIBS stable, then the associated unforced system is Lagrange stable (the converse is false). As mentioned in Sect. 3, Lagrange stability looks like a \dual" property of local stability for systems without inputs. Even for UBIBS stability, it is possible to introduce a \dual" property which extends local stability to systems with inputs ([1]). The de nition is as follows. De nition 4 We say that system (1) is uniformly locally input-to-state stable (in short ULIS stable) if for each " > 0 there exists  > 0 such that for each t0  0, each x0 with jx0 j < , and each measurable input such that kuk1 < , the following holds: (15)

j'(t; t0 ; x0 ; u(
))j < " ; t  t0 :

The de nition of ULIS stability is very close to the classical de nition of total stability (see for instance [16]). Let us recall that a system of the form (9) with an equilibrium position at the origin is said to be (uniformly) totally stable if for each " > 0 there exists  > 0 such that for each t0  0, each x0 and each function g(t; x) : [0; +1)  Rn ! Rn the following fact holds: if jx0j <  and jf (t; x) ? g(t; x)j <  for t  t0 and jxj < ", then j (t)j  " for each t  t0 and each solution of the problem



y_ = g(t; y) y(t0 ) = x0 : 15

It was proved by Malkin that if (9) is uniformly asymptotically stable at the origin then it is totally stable, but the converse is false even in the time-invariant case. On the other hand, it is not dicult to verify that if f (t; x; u) is continuous with respect to u at u = 0 (uniformly for t  0 and for x in some neighborhood of the origin) and if the associated unforced system is totally stable, then the system is ULIS. Hence, we conclude that if (1) is internally stable (in the sense that it is locally uniformly asymptotically stable) then it is ULIS. Systems which are at the same time UBIBS stable and ULIS stable, or even systems which are UBIBS stable and internally stable at the origin, form classes of systems which include properly the class of ISS systems, but which have been scarcely investigated. 6. External stability for nonlinear systems: Characterizations Several characterizations of the ISS property are given in [42] for the particular case of time-invariant systems. We limit ourselves to report the following statement, whose proof is based on the reduction to the converse theorem for \strong" asymptotic stability proved in [26] and already illustrated in Sect. 4. Theorem 8 For a time-invariant system of the form (16)

x_ = f (x; u)

the following statements are equivalent: (i) the system possesses the ISS property (ii) there exist a positive de nite, radially unbounded C 1 function V : Rn ! R and a function  2 K such that (17)

rV (x)
f (x; u) < 0

for all x 2 Rn (x 6= 0) and u 2 Rm , provided that jxj  (juj) (iii) there exist a positive de nite, radially unbounded C 1 function V : Rn ! R and two functions !;  2 K01 such that 16

(18)

rV (x)
f (x; u)  !(juj) ? (jxj)

for all x 2 Rn and u 2 Rm . In fact, in a recent paper ([44]) Tsinias shows that a system has the ISS property if and only if all the trajectories which satisfy certain state constraints are attracted by the origin. As far as the UBIBS property is concerned, we recall the following result, due to [45]. Theorem 9 Assume that f satis es Caratheodory conditions and that it is locally Lipschitz continuous with respect to x (uniformly in u). System (1) is UBIBS stable if and only if for each a  0 there exist a number r = r(a) > 0, a function Va(t; x) : [0; +1)  Br ! R and functions 1 ; 2 2 K1 (depending on a) such that: (i) 1 (jxj)  Va(t; x)  2(jxj) for each t  0 and each x with jxj > r (ii) Va is Lipschitz continuous and D+ V(t; x; f (t; x; u))  0 for each t  0, each x with jxj > r and each u 2 Rm with juj  a This theorem involves whole a family of Liapunov functions. In fact, it is not dicult to give a sucient condition in terms of only one Liapunov function. How the following statement shows (see [8]), it is also possible to relax the assumptions about f . Theorem 10 Assume that f (t; x; u) in (1) is locally bounded, measurable with respect to the variables t, x and continuous with respect to u. Assume that there exist functions V (t; x), a(r), b(r) and a number L > 0 such that: (I) V is locally Lipschitz continuous with respect to both variables t and x (II) a and b are of class K1 (III) a(jxj)  V (t; x)  b(jxj) for each t  0 and x with jxj > L. Assume further that: 17

(IV) for each R > 0 there exists  > L such that the following holds for a.e. t  0

juj < R; jxj > ; v 2 Kxf (t; x; u) =) D+ V (t; x; v)  0

(note that in the construction of Kxf (t; x; u), u is reviewed as a parameter). Then, (1) is uniformly bounded input bounded state stable.

7. Stabilizability Stabilizability theory has been basically developed for time-invariant systems. Let us recall that (16) is said to be (locally or globally) asymptotically stabilizable at the origin if there exists a function u = k(x) such that the closed-loop system

x_ = f (x; u(x)) is (locally or globally) asymptotically stable at x = 0. Such a function u = k(x) is called a stabilizing feedback. Asymptotic stabilization at the origin will be simply referred to also as internal stabilizability. We can also consider external stabilizability notions. For instance, we shall say that (16) is ISS-stabilizable (or UBIBS-stabilizable) if there exists a feedback law uc = k(x) such that the system x_ = f (x; k(x) + ue) is ISS-stable (or UBIBS stable), with respect to the input ue. The same de nitions apply also to the time-varying system (1), but in this case the stabilizing feedback laws should be allowed to depend on t. An obvious necessary condition for asymptotic stabilizability is that the system should be (locally or globally) asymptotically zero-controllable. In other words, for a given initial pair (t0; x0 ) there must exist an open-loop control law u(t0;x0)(t) which drives the corresponding solutions asymptotically toward the origin. The dierence between asymptotic zero-controllability and stabilizability can be emphasized very well in terms of control Liapunov functions. Let us limit ourselves, for the moment, to the time invariant case (16) and assume that f 2 C 1. According to [38], a control Liapunov function for (16) is a positive de nite, radially unbounded C 1 function V (x) such that 18

8x 6= 0 9u : rV (x)f (x; u) < 0 : Moreover, a control Liapunov function satis es the small control property if

8" 9 : 0 < jxj <  =) rV (x)f (x; u) < 0 for some u with juj < " : We also can consider control Liapunov functions which are not of class C 1. In such a case, the directional derivative should be replaced by the Dini's derivative lim sup V ('(t; x; u(t
))) ? V (x) : t!0+

In [36], [40] Sontag shows that (16) is globally asymptotically zero-controllable if and only if it admits a continuous (but in general not everywhere dierentiable) control Liapunov function with the small control property. On the contrary (see [38], [40]), the system is asymptotically stabilizable by a feedback law which is continuous everywhere and C 1 in Rnnf0g if and only if it admits a C 1 control Liapunov function with the small control property. Asymptotic stabilization by smooth feedback is related to the ISS property, as well. We have already noticed that an internally (globally and asymptotically) stable system is not necessarily an ISS system. However, in [37] Sontag proved the following result, which applies to ane systems of the form (19)

x_ = A(x) +

m X i=1

uiBi(x)

A(x); B1 (x); : : : ; Bm(x) being smooth vector elds of Rn. Theorem 11 Every internally stable (or stabilizable) time-invariant ane system (19) is ISS stabilizable. By analogy with the previous theorem, it is natural to ask whether it is possible to stabilize in the UBIBS sense a system whose unforced associated system is Lagrange stable. For general (non ane) systems the answer is negative (see [6] for an example). However, it is possible to state a positive result for ane (time-varying) systems of the form 19

(20)

x_ = A(t; x) +

m X i=1

uiBi(t; x) :

Theorem 12 Consider the system (20), where the functions A and Bi are locally bounded and measurable for t  0, x 2 Rn . Assume also that for each t  0 and i = 1; : : : ; m, Bi (t; x) is continuous with respect to x. Let L, V (t; x), a(r) and b(r) be such that conditions (I), (II) and (III) of Theorem 10 are ful lled. In addition, assume that V (t; x) is dierentiable for each t  0 and x 2 Rn, and that for a.e. t  0, the maps

x ! rxV (t; x)
Bi(t; x) (i = 1; : : : ; m) are continuous, where rx denotes the partial gradient with respect to x, and the dot denotes the usual scalar product. Finally, we assume that for each t; x and y 2 KxA(t; x) we have

@V (t; x) + r V (t; x)
y  0 : x @t Then, system (20) is UBIBS stabilizable. The proof of Theorem 12 can be obtained by repeating the reasoning in [7], and taking into account some ideas of [8]. Appendix: Prerequisites We start by introducing certain classes of functions which will be used as comparison or \gain" functions (see [16]). A function  : [0; r1 ) ! [0; +1) is said to belong to the class K0 if it is continuous, strictly increasing and (0) = 0. Here, r1 may be a positive number or +1 and may depend on . When  2 K0 and r1 = +1, we shall say that  is of class K. Moreover, a function  : [0; +1) ! [0; +1) is said to belong to the class L if it is continuous, decreasing and satis es limr!+1 (r) = 0. 20

Further, a function : [0; +1)  [0; +1) ! [0; +1) is said to belong to the class LK if it is of class L with respect to the rst variable and of class K with respect to the second one. Finally, we shall say that a function  : [r2 ; +1) ! [0; +1) (with r2  0) is of class K1 if it is continuous, strictly increasing and limr!+1 (r) = +1. We shall also set K01 = K0 \ K1 = K \ K1. Let h > 0, and let Bh = B(0; h) = fx 2 Rn : jxj < hg, Bh = fx 2 Rn : jxj > hg = RnnB(0; h). De nition A.1 A weak Liapunov function in the small is a real map V (t; x) which is de ned on [0; +1)  Bh for some h, and full ls the following properties: (i) there exist a; b 2 K0 such that

a(jxj)  V (t; x)  b(jxj)

for t  0; x 2 Bh

(ii) for each solution '(
) of (9) and each interval I one has

t1; t2 2 I; t1 < t2 =) V (t1 ; '(t1 ))  V (t2 ; '(t2)) provided that ' is de ned on I and '(t) 2 Bh for t 2 I . De nition A.2 A function V (t; x) which full ls the same properties as in De nition A.1, but with a; b 2 K1 instead of K0, and with Bh replaced by Bh, will be called a weak Liapunov function in the large. De nition A.3 A strict Liapunov function in the small is a real map V (t; x) which is de ned on [0; +1)  Bh for some h, and full ls the following properties: (i) there exist a; b 2 K0 such that

a(jxj)  V (t; x)  b(jxj)

for t  0; x 2 Bh

(ii) there exists c 2 K0 such that for each solution '(
) of (9) and each interval I one has 21

t1; t2 2 I; t1 < t2 =) V (t2 ; '(t2)) ? V (t1 ; '(t1))  ? provided that ' is de ned on I and '(t) 2 Bh for t 2 I .

Z t2 t1

c('(t)) dt

De nition A.4 A function V (t; x) which ful lls the same properties as in De nition A.3, but which in addition is de ned for all x 2 Rn, with a; b 2 K01 instead of K0, c 2 K instead of K0, and with Bh replaced by Rn, will be called a global strict Liapunov function. We warn the reader that the de nitions above are intended for the use of the present paper: they correspond in the spirit, but not necessarily in the terminology, to those more widely used in the literature. When V (t; x)  a(jxj) for some a 2 K [resp. a 2 K1], one usually says that V (t; x) is positive de nite [resp. radially unbounded]. Note that if V is continuous and it does not depend on t (i.e., V (t; x) = V (x)), positive de niteness is equivalent to V (x) > 0 for x 6= 0 in a neighborhood of the origin, while radial unboundedness is equivalent to limjxj!+1 V (x) = +1. The condition V (t; x)  b(jxj) for b 2 K0, which implies V (t; 0) = 0, is sometimes referred to by saying that V admits an in nitesimal upper bound, or that it is decrescent. Note also that if V is dierentiable everywhere, then condition (ii) of De nitions A.1 and A.3 can be checked by looking at the sign of the function n @V @V (t; x) + X (t; x)fi (t; x) @t @x i i=1 (which reduces to rV (x)
f (x) in the time-invariant case). As well known, for time invariant systems there is a large variety of de nitions of stability and asymptotic stability. Here, we limit ourselves to recall the de nitions of the so-called \uniform" properties. De nition A.5 Let the origin be an equilibrium position for (9), that is f (t; 0) = 0 for t  0. We say that (9) is (locally) uniformly stable at the origin (or that the origin is

22

locally uniformly stable for (9)) if for each " > 0 there exists  > 0 such that for each t0  0, each jx0j <  and each solution '(
; t0; x0 ) one has that ' is continuable on [t0; +1) and that

j'(t; t0; x0 )j < " for each t  t0 : De nition A.6 Let the origin be an equilibrium position. We say that system (9) is locally uniformly asymptotically stable at the origin (or that the origin is locally uniformly asymptotically stable for (9)) if it is uniformly stable at the origin and, in addition, the following condition holds: there exists 0 > 0 such that for each  > 0 there exists T = T (; 0 ) > 0 such that ()

j'(t; t0 ; x0 )j < 

for each jx0 j < 0 , each t0  0, each t  t0 + T , and each solution '(
; t0; x0 ). The origin is globally uniformly asymptotically stable if it is uniformly stable and for each  > 0 there exists T = T () > 0 such that () holds for each x0 2 Rn, each t0  0, each t  t0 + T , and each solution '(
; t0; x0 ). De nition A.7 System (9) is said to be uniformly Lagrange stable if for each R > 0 there exists S > 0 such that for each pair (t0 ; x0 ) with t0  0 and each solution '(
; t0 ; x0 ) one has that ' is continuable on [t0; +1) and that

jx0 j < R =) j'(t; t0; x0 )j < S for each t  t0 : These de nitions can be re-formulated in terms of comparison functions. The following propositions are not dicult, and can be proved by standard arguments ([1], [16]). Proposition A.1 The following statements are equivalent: (i) the origin of (9) is uniformly stable 23

(ii) there exists
2 K such that for each " > 0 and each t0, if jx0 j <
(") then j'(t; t0; x0 )j < " for t  t0 (iii) there exists 2 K0 de ned on [0; r0) such that for each R 2 (0; r0 ) and each t0, if jx0j < R then j'(t; t0; x0 )j < (R) for t  t0 (iv) there exists  2 K0, de ned on [0; r1 ), such that for each t0 and each x0 with jx0 j < r1 , one has j'(t; t0 ; x0 )j < (jx0 j), for t  t0 (similar characterizations apply to Lagrange stability, with the obvious modi cations). Proposition A.2 The origin of (9) is globally uniformly asymptotically stable if and only if there exists a function of class LK such that

j'(t; t0; x0 )j  (t ? t0; jx0 j) for each x0 2 Rn, each t0  0, each t  t0 , and each solution '(
; t0 ; x0 ).

24

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