SOLUTIONS TO LYAPUNOV STABILITY PROBLEMS : NONLINEAR SYSTEMS WITH CONTINUOUS MOTIONS

The necessary and sufficient conditions for accurate construction of a Lyapunov function and the necessary and sufficient conditions for a set to be the asymptotic stability domain are algorithmically solved for a nonlinear dynamical system with continuous motions. The conditions are established by utilizing properties of o-uniquely bounded sets, which are explained in the paper. They allow arbitrary selection of an o-uniquely bounded set to generate a Lyapunov function. Simple examples illustrate the theory and its applications.


INTRODUCTION
In his fundamental dissertation 1] Lyapunov referred to papers by Poincar6 [2], [3] as those inspiring him to establish a method that has become fundamental for qualitative and stability analysis of motions of a very general class of nonlinear systems.
The promising methodological effectiveness of the Lyapunov method has not been fully achieved due to the need to construct a system Lyapunov function.Significant results on a Lyapunov function generation were initiated by Zubov 14].The literature on the Lyapunov method is too vast [9]- [11], [13], [14] to be referred to herein.
The problem of the necessary and sufficient conditions for constructing a Lyapunov function and the problem of the necessary and sufficient conditions for a set to be the asymptotic stability domain have not yet been solved.Solutions to these problems will be established by using properties of o-uniquely bounded sets.Their features will be explained briefly by referring to [7], [8], where they were discovered and studied.

C(S)
an open connected neighborhood ofx -0, an open hyperball, the closure ofBa, the boundary of both Ba and Ba, the set of all functions of x continuous on S, Do,D,,D,R D Dt., the domain of attraction, of stability, of asymptotic stability, respectively, of x 0, the Dini derivative of v along the system motion (Yoshizawa 13]), a family of functions determined by Definition 5, a given nonlinear vector function, the largest subinterval of R/ over which a motion x(t; Xo) exists, the dimension of the system, an open connected neighborhood ofx -0, the interior of N (in fact N), the set of real numbers, [0,+oo[-{a: a eR,0 +o}, an open neighborhood of x 0, an o-uniquely bounded set, the generating function of the o-uniquely bounded set U, a set generated by the function u and a positive number t;, a tentative Lyapunov function of the system, the system motion (solution), x(t;Xo),-x(t), x(O;xo) Xo, Euclidean norm on R', the empty set.
Systems to be analyzed are described by the following equation They are assumed to possess either of the following two features:

Weak Smoothness Property: (i)
There is an open neighborhood S ofx 0, R" _.D S, such that for every Xo S (a) the system (1) has the unique solution x(t;Xo) through x0 at 0, and (b) the motion x(t;Xo) is defined and continuous in (t,xo) Io S. (ii) For every Xo tE (R" -S) every motion x(t;xo) of the system (1) is continuous in Io.

Strong Smoothness Property:
(i) The system (1) has Weak Smoothness Property.(ii) Ifthe boundary 0S ofS is non-empty then every motion of the system (1) passing through Xo G 0S at 0 obeys inf[I x(t;Xo)l[ I0] > 0 for every Xo /oS.

DEFINITIONS
ON THE DEFINITIONS OF STABILITY DOMAINS For the definitions of the attraction domain D,, see [4]- [6], [9], [11], [14].The stability domain D, and the asymptotic stability domain D ofx 0 are defined in [5], [6].We shall refer to those definitions in the sequel.
For the system (1) with Weak Smoothness Property, the stability domains are mutually related as follows: LEMMA 1.If the state x 0 of the system (1) possessing Weak Smoothness Property has both the domain of attraction Do, S 2)Do, and the domain of stability D,, then they and the asymptotic stability domain D are interrelated by D, 2)D, D-D,.

ON THE DEFINITION OF A POSITIVE DEFINITE FUNCTION
The notion of a positive definite function is used in a broader Lyapunov sense [1].DEFINITION 1.A function v: R" R is a positive definite if and only if there is an open connected neighborhood A of x 0, R" 2) A, such that 1) v(x) is uniquely determined by x CA and v is continuous onA" v(x) C(A), 2) v(0)-0, and 3) v(x) > 0 for every (x 0) 6A.

DEFINITIONS AND PROPERTIES OF O-UNIQUELY BOUNDED SETS
O-uniquely bounded sets were introduced, defined and studied in [7], [8].DEFINITION 2. A set U, R" 2) U, is o-uniquely bounded if and only if it is bounded and for every (x # 0) R" there is exactly one positive number L, k-(x;U), such that (kx) OU.DEFINITION 3. A function u: R" R is radially increasing on an open neighborhood N of x 0 if and only if for every (x ,, 0) N and any Ix, 1, 2, obeying both 0 Ix1 < Ih and Ix.,x N it satisfies u Ctx < u (o,:. ).
PROPERTY U. Let N be an open neighborhood of x 0 and U, N 2) U, be a given bounded set.There is a function u" R" --, R that obeys the following: (a) u is continuous onN:u(x)C(N), Ca) ifN-R" then u(x) +oo as Ilxll +, (c) u(0)-0, (d) u(x) > O for all (x , O) N, (e) there is positive number , (U), such that both 1. and 2. hold: 1. u (x) for x N if and only if x U, 2. u(x) for x N if and only ifx OU, (f) u(kix) , 1,2, holds for any (x ,, 0) N if and only ifk L2 x; U) ]0, +oo], (g) u is radially increasing on N. (ii) The function u is the generating function of the uniquely bounded set U if and only if they obey (i) for N R".Lemma 2 and Definition 4 imply the following corollary [8].
COROLLARY 1.If a function u is the generating function on N of an o-uniquely bounded set U then for any > 0 for which N _ _ .N; the subset U; of N is a connected open neighborhood of x 0 that is also an o-uniquely bounded set with the generating function u on N.

SOLUTIONS VIA O-UNIQUELY BOUNDED SEI
We shall make use of the family E(S;f) defined as follows.
DEFINITION 5. A function u" R" R belongs to the family E(S;.f)if and only if 1) u is continuous on S; u (x) E C(S), and 2) the following equations along the motions of the system (3.1),D/v(x) -u(x), (5.la) have a solution v that is well defined in R and continuous for every x -, for some t el0, +, t t(u,/3. TItEOREM 1.In order for the state x -0 of the system (1) with Strong Smoothness Property to have the domain D of asymptotic stability and for a set N, R" __.N, to be the domain of its asymptotic stability, N -D, it is both necessary and sufficient that 1) the set N is an open connected neighborhood of x 0 and S _D N, 2) f(x) 0 for x E N if and only if x 0, and 3) for arbitrarily selected o-uniquely bounded set U, S D U, with the generating function u on S obeying u E(S;.f), the equations (5.1) have a unique solution function v on Nwith the following properties: (i) v is positive definite on N, and (ii) if the boundary ON of N is non-empty then v(x) +oo as x ON, x tEN.PROOF.Necessity.Let x-0 of the system (3.1) with Strong Smoothness Property have the asymptotic stability domain D. Definitions of Do and D [5], [6] show that it has also the attraction domain Do, Do D_D.It is a neighborhood of x-0 due to Definition of D, and S is a neighborhood of x -0 in view of the smoothness property.Hence, D,, fqS ,, O. Let us prove S D D. If OS O, then S R" and S _D D due to R" :3 Do.If OS 0, then we shall consider both x 0 tE OS and x ft.(R" ).Ifx .O S, then x0 D,, due to (ii) of Strong Smoothness Property.Therefore, OS f'lD -0.If x e (R"-), then for x(t:x) 0 as +oo it is necessary that there is t" tER/ such that x(t*;x) tE 0S, because D and S are neighborhoods ofx 0,x and the motion x(t; x0) is continuous in R/ due to (ii) of Weak Smoothness Property ensured by (i) of Strong Smoothness Property.However, x(t';x) tE OS implies that x(t;Xo) does not converge to x-0 because of (ii) of Strong Smoothness Property.This yields x D and (R" -S) fqD O.By connecting the above results, that is Do tqS , 0, D f30S 0 and Do t"I(R" -S) O, we conclude that S _D Do.Therefore, D D (Lemma 1) and S D. Let N D so that S _ N. Hence, N is open connected neighborhood of x 0 due to (i-b) of Weak Smoothness Property, N D Do, and invariance olD,, with respect to system motions (Theorem 1.5.14 by Bhatia and Szegti [4], Theorem 33.3 by Hahn [9]).This proves necessity of the condition 1).From N -D -D,,, D, _D D,,, and Definitions of Do and D it results that x 0 is the unique equilibrium state of the system (1) in N, which implies f(x) 0 for x E N D if and only if x 0 (Proposition 7 in [6]) and proves necessity of the condition 2).
From N D it follows that the interval I0 of existence of x(t;Xo) equals R/, I0 R/, for every x C N, due to Definitions ofD,,, Ds and D [5], [6].Let U be arbitrarily selected open o-uniquely bounded set such that N U and its generating function u on S obeys u 6/E(S;f).Such a set U exists because S is open neighborhood ofx 0 CLemma 2).Definition 3, Property U, and Lemma 2 show that the function u is also positive definite on S. Since S _2) N D then the function u is the positive definite generating function on N, too.The property of u E(S;f) ensures existence of Ix > 0 such that there exists a solution function v to the equations (5.1), which is well defined in R and continuous for every x 6/B,, that is that vCx)l +oo for every x', and v(x)C(B'-. (5.2) Let ]0, +oo[ be such that B,IqU D U. (5.3) The existence of such is assured by Corollary 1.Let x [0, +oo[, x-X(Xo;f;u; ), be such that for any Xo N the following condition holds, x(t;xo) U s for every Ix, +oo[.(5.5) After integrating (5.1a) from 6R/ to +oo we derive v[x(+oO;Xo)]-v[x(t;xo) f, u[x(o;xo)]do for every (t,xo) U.R/xN. (5.6)Since u 6 E(S;f) then the following holds, v(0)-0. (5.7) Now, (5.5)-(5.7)yield v[x(t;Xo) f, u[x(o;x0)]do for every (t,Xo) R/xN (5.8) This can be written in the following form, v[x(t;Xo)] f. utx(o;x for every (t,X0) R/xN. (5.9) Positive invariance of D with respect to system motions, N -D, continuity of the motions x due to the smoothness property, continuity of u on N, the definition of' (5.4) and (5.2), and compactness of [;, t] for any R/ prove u[x(O;Xo)]do < +oo for every (t,xo) _R.xN.(5.9)-(5.11)together prove boundedness of v[x(t;xo)] expressed as v[x(t;Xo)]l < +oo for every (t,Xo)R/xN.
Hence, by setting -0 and x -x in (5.12) we derive v(x)l < +oo for every x N. (5.11) (5.12) (5.13) Continuity of the motion x in Xo e N, continuity of u in x e N, and of v in x e ff,, -,_D U--;, positive invariance ofN -D wiih respect to system motions, (5.4), (5.9) and (5.12) prove continuity of v in x N v(x) eC(N). (5.14) Positive invariance of N with respect to system motions, positive definiteness of u on N and (5.8) imply v(x)>O for all (x ,0)N. (5.15) Now, (5.7), (5.14) and (5.15) prove necessity of the positive definiteness of v on N.
To prove uniqueness of the solution v to (5.1.ab)we shall suppose that there are two solutions v and v to (5.1).Hence, Vl(X)-v2(x)" fo {U[Xl(;x)]-u[x2(;x)]}d for every x0eN. (5.16)Since u(x) is uniquely determined by x @N, due to (a) of Property U and Definition 4, and the motion of the system is unique through ] so that vl(x0) v:,(xo) 0 for every x0 N.This proves uniqueness of the solution v to (5.1) and completes the proof of 3(i).
Let ON be non-empty, x,x xk, be a sequence converging to x', xk x' as k +=, where x, N, for all k-1,2 and x' ON.Let ]0, +oo[ be arbitrarily chosen so that U {x" u(x)< }, U _D U s.Such exists because the set U is o-uniquely bounded and the function u is its generating function on N (Definitions 2 and 3, Property U, Lemma 2 and Definition 4).The set U is a connected open neighborhood ofx -0 (Corollary 1).Let T, T-T(x,)[0, +oo[, be the first instant obeying the fol- lowing x(t;xk)U forall [T,+oo[.(5.17) The existence of such T is guaranteed by x, N andN -D (Definitions of Do and D [5], [6]).Continuity of the motions x in (t,x0)_RN due to Strong Smoothness Property and -N-D (Theorem 33.1 by Hahn [9]) and S _.D D imply T +oo as k +oo (Theorem 33.2 by Hahn [9]).Let m be a natural number such that x, G (N Us) for all k m,m + Such rn exists because N is open, N D U and x, ON as k +=.Let a' be defined by (18), ct' min[u(x):x (N Us) (
Sufficiency.Let all the conditions of Theorem hold.Then, S _D N. Two possible cases will be considered separately: a) N is a bounded set, b) N is an unbounded set.a) Let N be a bounded set.Then, under the conditions of the theorem to be proved all the conditions of Theorem by Vanelli and Vidyasagar [12] are satisfied, which proves N D.,.Since Do D (in view of Weak Smoothness Property implied by Strong Smoothness Property and Lemma 1), N -D.b) Let N be an unbounded set.Under the conditions of the theorem to be proved the zero state x 0 of the system (1) is asymptotically stable (cf.Yoshizawa  [13]).Hence, it has the domain of asymptotic stability D. Since  (5.23)Since S _ _ .N, S is also unbounded.If OS is empty, then S R", which implies S :) D. If 0S is non-empty, then OS ND 0 due to (ii) of Strong Smoothness Property and Definitions ofDo, D, and D [5], [6].This result impliesS _D D because both D and S are neighborhoods ofx 0 and D is also connected.Altogether, in both cases S _D D. We shall treat separately the cases of non-empty OD and of empty OD.The definition of the function v, S D D, and the proof of the necessity part prove continuity of v on D and v(x) +oo as x OD, which together with continuity of v also on N, S _D N and v(x) + as x ON [the condition 3(ii)] imply both OD fN and D ON These equations and (5.23) prove both OD ON and D N due to the fact that both D and N are open connected neighborhoods ofx 0. Let now OD be empty.Then D R'.Hence, v is positive definite on R" (see the proof of the necessity part).Thus, it is continuous on R", which implies v(x) < +oo for every x R".Therefore, ON R" -0 due to the conditions 3(ii), which yields N-R" so that N-D.Finally, N -D in all the cases, which completes the proof.| The conditions slightly change if the system (3.1)possesses Weak Smoothness Property rather than Strong Smoothness Property.
THEOREM 2. For the state x 0 of the system (1) possessing Weak Smoothness Property to have the domain D of asymptotic stability and that a subset N of S, S _D N, equals D" N D, it is both necessary and sufficient that 1) the set N is an open connected neighborhood of x 0, 2) .f(x) 0for x N if and only if x 0, and 3) for arbitrarily selected o-uniquely bounded set U, S D U, with the generating function u on obeying u E(S;.f), the equations (5.1) have a unique solution function v on Nwith the following properties: (i) v is positive definite on N, and (ii) if the boundary ON ofN is non-empty then v(x) +oo as x ON, x N. PROOF.Necessity.Let the system (3.1)possess Weak Smoothness Property.Let x -0 have the asymptotic stability domain D, S _.D D, and let N, S _D N, be equal to D. Let an o-uniquely bounded set U, S D U, with the generating function u obeying u E(S;f), be arbitrarily selected.From this point on we have to repeat the proof of the necessity part of Theorem 1 to show that the conditions 1)-3) of Theorem 2 hold.In that way we complete the proof of the necessity part.
Sufficiency.Let the system (3.1)possess Weak Smoothness Property and the conditions 1)-3) be valid.Then x-0 of the system (3.1) is asymptotically stable [1].Therefore, x-0 has the domain of asymptotic stability (Definitions of D,,, D, and D [5], [6]).Let x0 C (R" -N).Since x(t;x0) is continuous in C I0, then it can enter N iff it passes through ON.But v(x) +oo as x ON, x N [the condition 3(ii)].This and D /v(x) < 0 for x C (R -N) in view of positive definiteness of u on R" and (5.1a), show that x(t;x0) cannot reach ON.Hence, x(t;x0)(R"-N) for all Cl0.Therefore, N DD.Furthermore, (5.1a) and positive definiteness of u on R" imply (see the proofofthe necessity part ofTheorem 1) v(x as x OD, x D, which together with the condition 3(i) proves dD ("IN O.This result, N D D, and the fact that D and N are non-empty open connected neighborhoods of x 0 imply D N and complete the proof. The properties of the generating function u of an o-uniquely bounded set U are essential for the accurate one-shot determination of the asymptotic stability domain.However, such properties are not needed for asymptotic stability of x 0 only.This is clarified by the next result.THEOREM 3.For the state x 0 of the system (3.1)possessing Weak Smoothness Property to be asymptotically stable it is both necessary and sufficient that for any positive definite function p E(S;f) there exists a unique solution function v to (5.24) with (5.24a) determined along system motions, O /v(x -p (x (5.24a) v(0) 0, (5.24b) which is also positive definite.
PROOF.Necessity.Let the system (3.1)possess the Weak Smoothness Property.Let x-0 be asymptotically stable.Then it has Do, D, and D, and D,, S , , , O, D, f"lS ,, O and D f'lS , O, because D,,, D,, D and S are neighborhoods ofx O. Let p E(S;f) be an arbitrarily selected positive definite function (Definition 1).Such properties of p and its membership to E(S;f) guarantee existence of a solution v to the equations (5.24), which is well defined in R and continuous (see the proof of the necessity part of Theorem 1) on the setA determined in Definition 1.The setL -A OD,D D_ L, is also an open connected neighborhood of x ,-0 (see the proof of Theorem 1 for such a property of D).Let e satisfying L D B, be arbitrarily selected.ThenD D_ B,.Let p ]0, e[ obeyingD,(e) _ B, be also arbitrarily selected, where D,(e) is defined [5], [6] as the neighborhood ofx 0 such that x(t;xo)ll < e for all R/ holds iffxo D,(e).By following the proofs of (5.13) and (5.14), we prove that v, defined by (5.24), has the following properties since A _ L _ B, _ _ _ D,(e) _ B,, v(x)l < + for every x B,, Notice that D,(e)D_Bp and the definitions of D,(e)and D guarantee [5], [6] x(t;x0)B, for every (t,xo) R,xB This result, A I'qD D_B,, positive definiteness of the functionp onA, x(+;xo) 0 for every xo B, (because D 2) B,) and (5.24a), integrated from 0 to +, together with (5.24b) prove (5.26), V(Xo) > 0 for every (x o # (0)) B,. (5.26) Now, (5.24b) through (5.26) prove positive definitness of the solution v to (5.24) on B,.Its uniqueness is proved in the same way as in the proof of Theorem 1, which completes the proof of the necessity part.
The solution v to this equation is v(x) -In(1 Ix I), x S. (6.
Integrating this equation along motions of the system (6.3)we derive v(x)-m(-Ixl), xN, which is negative definite on N and, thus, does not satisfy the necessary and sufficient conditions for asymptotic stability of x -0 of the system (6.3).Hence, x -0 of the system (6.3) is not asymptotically stable and does not have the asymptotic stability domain.(6.4) The function j'is globally Lipschitz continuous.The system has Strong Smoothness Property on R2.The set S, of its equilibrium states is determined by S,-{x'xR2,(x-O) or (Ix l-1,11-1)}.
The function v and the setN S obey all the conditions of Theorem 2. Therefore, x 0 of the system (6.4) is asymptotically stable with the domain D of its asymptotic'stability obtained as D -N -S, that is that D {x:x S:', Ix1 I< 1, Ix I< 1}.

CONCLUSION
The necessary and sufficient conditions for asymptotic stability of the zero equilibrium state and for a set to be the domain of its asymptotic stability are proved in an algorithmic form that enables accurate construction of a system Lyapunov function.If a function v obtained from D/v --u for an arbitrarily chosen u, which is a generating function of an o-uniquely bounded set, is not positive definite then the zero state is not asymptotically stable.There is no sense to try with another function u.However, if so derived function v is positive definite then the zero state is asymptotically stable.In this way the problem of an algorithm to construct accurately and directly a system Lyapunov function has been solved.However, it imposes other very complex mathematical problems: the problem of finding conditions on u guaranteeing existence of well defined and continuous v satisfying (5.1)on anyhow small neighborhood ', ofx 0, and the problem of solving (5.1).These problems have not been solved.
Theorems of the paper open and initiate new directions in the Lyapunov stability analysis.

(5. 4 )
Such : exists in view of Definitions of D,, and D, D,, D, N D and xo N. Notice that xo N implies also xC+o;xo) o.
both N and D are open connected neighborhoods of x 0, N ND , O. | { I ) x a for ]xl[0,1], (6.1)
[8]inition 2 implies the next result due to Definition 2, Corollary 1 and Proposition 4 in[8].LEMMA 2. For a bounded subset U of an open neighborhood N of x 0 to be o-uniquely bounded it is both necessary and sufficient that it possesses Property U.DEFINITION4.(i) A function u is the generating function on N of an o-uniquely bounded set U if and only if they have Property U.
[11ice that f(x)[ -]x[11-Ix ,x , is not a generating function on N of any o-uniquely bounded set because it is not radially increasing on N. Example 2. Let the function h be defined as in Example and It is clear that the system possesses Strong Smoothness Property on R and has the equilibrium states