- Email: [email protected]
Proof of the geometric diagram of Liapunov's theorem
PERGAMON
Applied Mathematics Letters 12 (1999) 103-107
Applied Mathematics letters
P r o o f of the Geometric Diagram of Liapunov's Theorem X.-Y. GE Department of Mathematics, The University of Queensland Brisbane, Qld, 4072 Australia xg~naths, uq. edu. au
(Received February 1998; accepted June 1998) A b s t r a c t - - l n this paper, three main resultsof the geometric diagram of Liapunov's theorem are presented by using some of the principal methods of differentialtopology. © 1999 Elsevier Science Ltd. All rights reserved.
Keywords--Liapunov's stability, Liapunov's function, Geometric diagram, Smooth mapping, Compact manifolds.
1. I N T R O D U C T I O N Let V : U -~ R be a differentiable function defined in a neighborhood U C W of ~. We use : U --~ R to denote the function defined by ? ( x ) = D V ( x ) ( f ( x ) ) . If ~ot(x) is the solution to a dynamic system ~ = f ( x ) passing through x when t -- 0, then ? ( x ) = dv(~ot(x))[t=o by the chain rule. Consequently, if V(x) is negative, then V decreases along the solution of ~ = f ( x ) through x. It is this criterion of stability that was founded by Liapunov in 1892. We state Liapunov's stability theorem as follows. LIAPUNOV THEOREM. Let ~2 E W be an equilibrium for ~ = f ( x ) . Let V : U --~ R be a continuous function defined on a neighborhood U C W of ~, differentiable on U \ {Z}, then we have the following. (i) I f V is the Liapunov's function in ~, then ~ is stable. (ii) I f V is the strict Liapunov's function in ~, then ~ is asymptotically stable. In [1], after giving the analytic proof of Liapunov's theorem, the geometric diagram of the intuitive obviousness of the theorem was also given and explained. Hirsch and Smale proposed t h a t in Figure la, the condition V < 0 means that when a trajectory crosses a "level surface" V - I ( C ) , it moves inside the set where V _< C and can never come out again. Unfortunately, Hirsch and Smale found it very difficult to justify the diagram, i.e., why should the sets V - I ( c ) shrink down to ~? Up to now, there have not been any results concerning this problem. If the level surfaces look like Figure 2b, it is hard to imagine such a V that fulfills all the requirements of a Liapunov's function. In this paper, we shall give some results of the geometric diagram of Liapunov's theorem using some of the principal methods of differential topology [2]. T h e author wishes to thank N.-L. D u for his helpful discussionsand suggestions.
0893-9659/99/$ - see front matter. (~) 1999 Elsevier Science Ltd. All rights reserved. PII: $0893-9659(99)00086-5
Typeset by AA4S-TEX
104
X.-Y. GE
v-'(c)
(a) (b) Figure 1. The level surface of Liapunov's function.
2. B A C K G R O U N D We first need the following simple fact. DEFINITION l. Let U C W be an open set in R n and 0 E U, ff V : U ~ [0, oo) satisfies
(i)
v E c ° in u , v E o n in u \
{0};
(ii) if Y(O) = 0 and V ( x ) > 0 for any x E V \ {0}; then V is the positively definite Liapunov's function.
DEFINITION 2. Let A > 0, i f a : [ O , A ] -* [0, oo) is a C o mapping of strict monotomous increase and a(O) = O, then a E K . LEMMA 1. Let V : BA (0) --* [0, oo) be a positively Liapunov's function, then there exist a, fl E K , such that
~(1~1) < v(x) < ~(Ixl),
x E BA(O),
where the equal sign is only possible when x = O.
PROOF. Let 0 < r < A. We define &(r) a__m i n { V ( x ) : r < Ixl ~ A},
J~(r) ~ max{V(x): Ixl ~ r},
then we have (i) ~(Ixl) <_ V ( x ) 0; (iii) &(r), j~(r), which are not decreasing in [0, A]. Suppose a(r) ~ _ A + r ~
~
~(r),
, ~ ( r ) -~ A~+ r ~ ,p~r~,
then a , ~ E K a n d
~(Ixl) ~ ~(Ixl) ~ V(x) ~/~(Ixl) ~ ~(Ixl). SARD'S LEMMA. (See [2].) Let f : U --* R K be a smooth mapping, where U C W is an open set in R n, and let C -- {x [ rank(o°-~x~) < min(n,k)} be the set of critical points o f f , then f ( C ) has measure zero in R K. JORDAN-BROUWER'S SEPARATION THEOREM. (See [2].) Let W be a connected supersufface in R n, then the complementary set R ~ \ W of W is not connected, i.e., it is formed by ~ro connected open sets (Do of "outside" and D1 of "inside"), where D1 is of compact manifolds, whose boundary OD1 = W .
Geometric Diagram
3. M A I N
105
RESULTS
In this section, we shall give several theorems for the geometric diagram of Liupunov's theorem. THEOREM 1. Let V be a positively definite Liapunov's function. Suppose A > 0 such that BA(O) C U, we define V. = V[B-~ : BA(O) --~ [0, oo), then we have the following. (i) There ex/sts a 6 (0, oo) such that V(BA(O)) = [0,a]. (ii) There exists a, ~ : [0, A] --* (0, oo) and a, ~ 6 K such that
cr(Ixl) < y,(x) < Z~(Ixl),
Vx 6 BA(O) \ {0}
and a(A) < a 2) whose radius is r, where r = min{V(x) : Ix[ = A}. PROOF. (i) Because V is continuous in BA(O) and BA(O) is a connected compact set, then V(BA(O)) is a connected compact set in [0,oo), and for any x 6 BA(O) \ {0}, where V(x) > O, then
there exists a E (0, oo) such that V(BA(O)) = [0,a]. (ii) According to Lemma 1, there exists c~, f~ : [0, A] --* (0, oo) and a,/~ 6 K such that
a(Ixl) < v(z) < Z(Izl),
' i x 6 BA(O) \ {0}.
Let Xo 6 BA(O) such that V(xo) = a, then a = V(xo) = fl(IXo]) = f~(A),
and there exists xl 6 BA(O) such that Ix1[ = A, we have a(A) = a(Ixll) < V ( x l ) < a,
thus, o < ~ ( A ) < a < ~(A).
THEOREM 2. Let V be positively definite Liapunov's function, suppose A > O, for any C 6 (0,a(A)], we let
then we have the following.
(i) Sc(a) U Sc($) C BA(O). (ii) If we have path 7 : [0, 1] -~ BA(O) such that 7(0) 6 8c(a), 7(1) 6 $c(~), then there exists r c 6 (0, 1) such that 7(rc) 6 V - I ( C ) . PROOF.
(i) For any C 6 [0, a(A)], we have
_-
c}
then
Sc(Ot) 13Sc(~) C BA(O).
<
o(A)} -.A(O),
106
GE
X.-Y.
(ii) Let 7 : [0, 1] --~ BA(O) be a path, where 7(0) e Sc(a), 7(1) 6 Sc(~). Considering the mapping VoT: [0,1] -* [0, a], then we have v o 7(0) = v ( 7 ( 0 ) ) >
where 7(0)
0, which 7(0) 6
Sc(a),
(17(0)1) = c ,
i.e., 17(0)1 = C # 0, and
V o 7(1) = V(7(1)) < fl([7(1)[) = C, i.e.,
Vo7(1) c~(Ixl) , then we have Ixl < ~ - 1 ( C ) , thus V - I ( C ) 6 B,-~(c)(0) C BA(0), if C = V(x) < j3(Ixl), then we have Ixl > f~-l(C), therefore, V - I ( C ) N S~-~(c)(0 ) = @and V - I ( C ) C B~-~(c)(0 ) \ B~-~(c)(0). Take account of •
\
=
=
let U1 = Ba-~(c)(O) \ B#-~(c)(O), considering VIa 1 : [/1 ~ [0, co) 6 C n, then we have (vIv,)-l(c)
=
where C is the regular value of (Vlu 1), i.e., there exists C such that for any x e (V[u,)-1 (C), we have D V ( x ) = D(VIuI)(x) # O. Then V -1 is the n - 1 dimensional C n connected compact manifolds, thus, V -1 (C) is local path connected. It is easy to verify all the path connected bifurcations of V -1 (C) are not only the open set but also the close set for V-1 (C), then all the connected bifurcations of V - 1 ( C ) are n - 1 dimensional compact manifolds. Moreover, from the compactness of V-1 (C), we know that the number of its path connected bifurcations can only be finite. In terms of Jordan-Brouwer's Separation Theorem, each of the finite n - 1 dimensional connected compact manifolds of V -1 (C) will separate R n into two domains, such that one connected open set is bounded, and another is unbounded. Note that all of these finite n - 1 dimensional connected compact manifolds of V - t (C) are included in the connected open set D = B~-~(c)(O ) \ B~-~(c)(O), therefore, by reduction to
Geometric Diagram
107
absurdity, we can verify that there at least exists a connected compact manifold, which separates a connected open set D into two connected open sets, such that Sc(a) and Sc(~) are, respectively, included in the boundary of one of the two connected open sets. If not, there should exist a path 7 t h a t could join the points in OBa-l(c)(O) = S c ( a ) and OB~-I(c)(O ) = Sc(~), such t h a t there would be no points of 7 included in V-I(C), it is contradictory to t h e proof of Theorem 2. Consequently, it is true that there exists a connected compact manifold, it can separate D into two connected open sets.
REFERENCES 1. M.W. Hirsch and S. Smale, Differential Equations, Dymamical Systems, and Linear Algebra, Academic Press, New York, (1974). 2. V. Cuillemin and A. Pollack, Differential Topology, Prentice Hall, Englewood Cliffs, NJ, (1974).
Applied Mathematics Letters 12 (1999) 103-107
Applied Mathematics letters
P r o o f of the Geometric Diagram of Liapunov's Theorem X.-Y. GE Department of Mathematics, The University of Queensland Brisbane, Qld, 4072 Australia xg~naths, uq. edu. au
(Received February 1998; accepted June 1998) A b s t r a c t - - l n this paper, three main resultsof the geometric diagram of Liapunov's theorem are presented by using some of the principal methods of differentialtopology. © 1999 Elsevier Science Ltd. All rights reserved.
Keywords--Liapunov's stability, Liapunov's function, Geometric diagram, Smooth mapping, Compact manifolds.
1. I N T R O D U C T I O N Let V : U -~ R be a differentiable function defined in a neighborhood U C W of ~. We use : U --~ R to denote the function defined by ? ( x ) = D V ( x ) ( f ( x ) ) . If ~ot(x) is the solution to a dynamic system ~ = f ( x ) passing through x when t -- 0, then ? ( x ) = dv(~ot(x))[t=o by the chain rule. Consequently, if V(x) is negative, then V decreases along the solution of ~ = f ( x ) through x. It is this criterion of stability that was founded by Liapunov in 1892. We state Liapunov's stability theorem as follows. LIAPUNOV THEOREM. Let ~2 E W be an equilibrium for ~ = f ( x ) . Let V : U --~ R be a continuous function defined on a neighborhood U C W of ~, differentiable on U \ {Z}, then we have the following. (i) I f V is the Liapunov's function in ~, then ~ is stable. (ii) I f V is the strict Liapunov's function in ~, then ~ is asymptotically stable. In [1], after giving the analytic proof of Liapunov's theorem, the geometric diagram of the intuitive obviousness of the theorem was also given and explained. Hirsch and Smale proposed t h a t in Figure la, the condition V < 0 means that when a trajectory crosses a "level surface" V - I ( C ) , it moves inside the set where V _< C and can never come out again. Unfortunately, Hirsch and Smale found it very difficult to justify the diagram, i.e., why should the sets V - I ( c ) shrink down to ~? Up to now, there have not been any results concerning this problem. If the level surfaces look like Figure 2b, it is hard to imagine such a V that fulfills all the requirements of a Liapunov's function. In this paper, we shall give some results of the geometric diagram of Liapunov's theorem using some of the principal methods of differential topology [2]. T h e author wishes to thank N.-L. D u for his helpful discussionsand suggestions.
0893-9659/99/$ - see front matter. (~) 1999 Elsevier Science Ltd. All rights reserved. PII: $0893-9659(99)00086-5
Typeset by AA4S-TEX
104
X.-Y. GE
v-'(c)
(a) (b) Figure 1. The level surface of Liapunov's function.
2. B A C K G R O U N D We first need the following simple fact. DEFINITION l. Let U C W be an open set in R n and 0 E U, ff V : U ~ [0, oo) satisfies
(i)
v E c ° in u , v E o n in u \
{0};
(ii) if Y(O) = 0 and V ( x ) > 0 for any x E V \ {0}; then V is the positively definite Liapunov's function.
DEFINITION 2. Let A > 0, i f a : [ O , A ] -* [0, oo) is a C o mapping of strict monotomous increase and a(O) = O, then a E K . LEMMA 1. Let V : BA (0) --* [0, oo) be a positively Liapunov's function, then there exist a, fl E K , such that
~(1~1) < v(x) < ~(Ixl),
x E BA(O),
where the equal sign is only possible when x = O.
PROOF. Let 0 < r < A. We define &(r) a__m i n { V ( x ) : r < Ixl ~ A},
J~(r) ~ max{V(x): Ixl ~ r},
then we have (i) ~(Ixl) <_ V ( x ) 0; (iii) &(r), j~(r), which are not decreasing in [0, A]. Suppose a(r) ~ _ A + r ~
~
~(r),
, ~ ( r ) -~ A~+ r ~ ,p~r~,
then a , ~ E K a n d
~(Ixl) ~ ~(Ixl) ~ V(x) ~/~(Ixl) ~ ~(Ixl). SARD'S LEMMA. (See [2].) Let f : U --* R K be a smooth mapping, where U C W is an open set in R n, and let C -- {x [ rank(o°-~x~) < min(n,k)} be the set of critical points o f f , then f ( C ) has measure zero in R K. JORDAN-BROUWER'S SEPARATION THEOREM. (See [2].) Let W be a connected supersufface in R n, then the complementary set R ~ \ W of W is not connected, i.e., it is formed by ~ro connected open sets (Do of "outside" and D1 of "inside"), where D1 is of compact manifolds, whose boundary OD1 = W .
Geometric Diagram
3. M A I N
105
RESULTS
In this section, we shall give several theorems for the geometric diagram of Liupunov's theorem. THEOREM 1. Let V be a positively definite Liapunov's function. Suppose A > 0 such that BA(O) C U, we define V. = V[B-~ : BA(O) --~ [0, oo), then we have the following. (i) There ex/sts a 6 (0, oo) such that V(BA(O)) = [0,a]. (ii) There exists a, ~ : [0, A] --* (0, oo) and a, ~ 6 K such that
cr(Ixl) < y,(x) < Z~(Ixl),
Vx 6 BA(O) \ {0}
and a(A) < a 2) whose radius is r, where r = min{V(x) : Ix[ = A}. PROOF. (i) Because V is continuous in BA(O) and BA(O) is a connected compact set, then V(BA(O)) is a connected compact set in [0,oo), and for any x 6 BA(O) \ {0}, where V(x) > O, then
there exists a E (0, oo) such that V(BA(O)) = [0,a]. (ii) According to Lemma 1, there exists c~, f~ : [0, A] --* (0, oo) and a,/~ 6 K such that
a(Ixl) < v(z) < Z(Izl),
' i x 6 BA(O) \ {0}.
Let Xo 6 BA(O) such that V(xo) = a, then a = V(xo) = fl(IXo]) = f~(A),
and there exists xl 6 BA(O) such that Ix1[ = A, we have a(A) = a(Ixll) < V ( x l ) < a,
thus, o < ~ ( A ) < a < ~(A).
THEOREM 2. Let V be positively definite Liapunov's function, suppose A > O, for any C 6 (0,a(A)], we let
then we have the following.
(i) Sc(a) U Sc($) C BA(O). (ii) If we have path 7 : [0, 1] -~ BA(O) such that 7(0) 6 8c(a), 7(1) 6 $c(~), then there exists r c 6 (0, 1) such that 7(rc) 6 V - I ( C ) . PROOF.
(i) For any C 6 [0, a(A)], we have
_-
c}
then
Sc(Ot) 13Sc(~) C BA(O).
<
o(A)} -.A(O),
106
GE
X.-Y.
(ii) Let 7 : [0, 1] --~ BA(O) be a path, where 7(0) e Sc(a), 7(1) 6 Sc(~). Considering the mapping VoT: [0,1] -* [0, a], then we have v o 7(0) = v ( 7 ( 0 ) ) >
where 7(0)
0, which 7(0) 6
Sc(a),
(17(0)1) = c ,
i.e., 17(0)1 = C # 0, and
V o 7(1) = V(7(1)) < fl([7(1)[) = C, i.e.,
Vo7(1)
\
=
=
let U1 = Ba-~(c)(O) \ B#-~(c)(O), considering VIa 1 : [/1 ~ [0, co) 6 C n, then we have (vIv,)-l(c)
=
where C is the regular value of (Vlu 1), i.e., there exists C such that for any x e (V[u,)-1 (C), we have D V ( x ) = D(VIuI)(x) # O. Then V -1 is the n - 1 dimensional C n connected compact manifolds, thus, V -1 (C) is local path connected. It is easy to verify all the path connected bifurcations of V -1 (C) are not only the open set but also the close set for V-1 (C), then all the connected bifurcations of V - 1 ( C ) are n - 1 dimensional compact manifolds. Moreover, from the compactness of V-1 (C), we know that the number of its path connected bifurcations can only be finite. In terms of Jordan-Brouwer's Separation Theorem, each of the finite n - 1 dimensional connected compact manifolds of V -1 (C) will separate R n into two domains, such that one connected open set is bounded, and another is unbounded. Note that all of these finite n - 1 dimensional connected compact manifolds of V - t (C) are included in the connected open set D = B~-~(c)(O ) \ B~-~(c)(O), therefore, by reduction to
Geometric Diagram
107
absurdity, we can verify that there at least exists a connected compact manifold, which separates a connected open set D into two connected open sets, such that Sc(a) and Sc(~) are, respectively, included in the boundary of one of the two connected open sets. If not, there should exist a path 7 t h a t could join the points in OBa-l(c)(O) = S c ( a ) and OB~-I(c)(O ) = Sc(~), such t h a t there would be no points of 7 included in V-I(C), it is contradictory to t h e proof of Theorem 2. Consequently, it is true that there exists a connected compact manifold, it can separate D into two connected open sets.
REFERENCES 1. M.W. Hirsch and S. Smale, Differential Equations, Dymamical Systems, and Linear Algebra, Academic Press, New York, (1974). 2. V. Cuillemin and A. Pollack, Differential Topology, Prentice Hall, Englewood Cliffs, NJ, (1974).