Lecture 5 - how to find a Lyapunov function?€¦ · R. M. Murray Lyapunov Stability Analysis 17...
Transcript of Lecture 5 - how to find a Lyapunov function?€¦ · R. M. Murray Lyapunov Stability Analysis 17...
CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical Systems
CDS 110a
R. M. Murray Lyapunov Stability Analysis 17 October 2007
This lecture provides an overview of Lyapunov stability for time-invariant systems. We presenta survey of the main results that we shall need in the sequel; proofs are omitted, but can befound in standard texts such as Vidyasagar [2] or Khalil [1].
Reading:
• Astrom and Murray, Section 4.4
1 Basic definitions
Consider a closed loop dynamical system of the form
x = F (x) x ! Rn (1)
with equilibrium point xe ! Rn. We recall the following definition from Monday’s lecture:
Definition 1. An equlibrium point xe = 0 is locally asymptotically stable if
1. xe = 0 is stable in the sense of Lyapunov: for any ! > 0there exists " > 0 such that
"x(0)#xe" < " =$ "x(t)#xe" < ! for all t > 0.
2. xe = 0 is locally attractive: there exists " > 0 such that
"x(t0)" < " =$ limt!"
x(t) = 0
B!
B"
2 Lyapunov Stability
Definition 2. A continuous function V : Rn % R is (locally) positive definite if for some r > 0
V (0) = 0 and V (x) > 0 for all x &= 0 and "x" < r.
V is (locally) positive semi-definite if for some r > 0
V (0) = 0 and V (x) ' 0 for all "x" < r.
V is globally positive (semi-) definite if these statements are true for all x ! R.
Review
4
5
x1
x2
x1 = 2x1x2
x2 = x22 � x2
1
Why capture the notion of stability?
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical Systems
CDS 110a
R. M. Murray Lyapunov Stability Analysis 17 October 2007
This lecture provides an overview of Lyapunov stability for time-invariant systems. We presenta survey of the main results that we shall need in the sequel; proofs are omitted, but can befound in standard texts such as Vidyasagar [2] or Khalil [1].
Reading:
• Astrom and Murray, Section 4.4
1 Basic definitions
Consider a closed loop dynamical system of the form
x = F (x) x ! Rn (1)
with equilibrium point xe ! Rn. We recall the following definition from Monday’s lecture:
Definition 1. An equlibrium point xe = 0 is locally asymptotically stable if
1. xe = 0 is stable in the sense of Lyapunov: for any ! > 0there exists " > 0 such that
"x(0)#xe" < " =$ "x(t)#xe" < ! for all t > 0.
2. xe = 0 is locally attractive: there exists " > 0 such that
"x(t0)" < " =$ limt!"
x(t) = 0
B!
B"
2 Lyapunov Stability
Definition 2. A continuous function V : Rn % R is (locally) positive definite if for some r > 0
V (0) = 0 and V (x) > 0 for all x &= 0 and "x" < r.
V is (locally) positive semi-definite if for some r > 0
V (0) = 0 and V (x) ' 0 for all "x" < r.
V is globally positive (semi-) definite if these statements are true for all x ! R.
Review
6
Type of stability: Lyapunov theory
7
Remarks:
1. To see the di!erence between positive definite and positive semi-definite, suppose that x ! R2
and letV1(x) = x2
1 V2(x) = x21 + x2
2.
Both V1 and V2 are always non-negative. However, it is possible for V1 to be zero even ifx "= 0. Specifically, if we set x = (0, c) where c ! R is any non-zero number, then V1(x) = 0.On the other hand, V2(x) = 0 if and only if x = (0, 0). Thus V1(x) is positive semi-definiteand V2(x) is positive definite.
Theorem 1. Let V be a non-negative function on Rn and let V represent the time derivative of V
along trajectories of the system dynamics (1):
V =!V
!x
dx
dt=
!V
!xF (x).
Let Br = Br(0) be a ball of radius r around the origin. If there exists r > 0 such that V ispositive definite and V is negative semi-definite for all x ! Br, then x = 0 is locally stable in thesense of Lyapunov. If V is positive definite and V is negative definite in Br, then x = 0 is locallyasymptotically stable.
Remarks
1. A function V satisfying the conditions of the theorem is called a Lyapunov function.
2. V (x) is an “energy like” function that bounds the size of x.
V (x) = c1
dx
dt
!V
!x
V (x) = c1
Example 1.
x1 = #x1 # x2 V (x) = x21 + x2
2 > 0 $x "= 0
x2 = #x2 V (x) = 2x1x1 + 2x2x2
= #2x21 # 2x1x2 # 2x2
2
= #(x1 + x2)2 # x2
1 # x22 < 0 $x "= 0
=% globally asymptotically stable.
2
stable in the
How to find a Lyapunov function?• In general (linear + non-linear systems):
- Before guess (before 2000) - Now use SOS-tools (Matlab toolbox) for polynomial
systems with known coefficients
How about for linear systems?
8
• For linear systems: • Let • V positive definite ⇔ P positive definite matrix (P > 0)
• Compute its derivative:
• Let
• Solve for P in
To fix this problem, we skew the level sets slightly, so that the flow of the system crosses the levelsurfaces transversely. Define
V (x, t) = 12
!
q
q
"T !
k !m
!m m
" !
q
q
"
= 12 qmq + 1
2qkq + !qmq,
where ! is a small positive constant such that V is still positive definite. The derivative of theLyapunov function becomes
V = qmq + qkq + !mq2 + !qmq
= (!c + !m)q2 + !(!kq2 ! cqq) = !
!
q
q
"T !
!k 12!c
12!c c ! !m
" !
q
q
"
.
The function V can be made negative definite for ! chosen su!ciently small (exercise) and hencewe can conclude exponential stability.
Remarks
1. As the previous example shows, a Lyapunov function need not be unique and di"erent Lya-punov functions can give stronger stability results.
2. Lyapunov functions can also be used to prove that a system is unstable: search for V positivedefinite with V positive definite.
3 Lyapunov Functions for Linear Systems
Consider a linear system of the formx = Ax.
Search for a quadratic Lyapunov function
V (x) = xT Px
Compute the derivativedV
dt=
"V
"x
dx
dt= xT (AT P + PA)x.
The requirement that V is positive definite is equivalent to P > 0 and the requirement that V isnegative definite becomes a condition that
Q = AT P + PA < 0 (2)
(as a matrix).
Trick: equation (2) is linear in P . So we can choose Q < 0 and the solve for P .
4
To fix this problem, we skew the level sets slightly, so that the flow of the system crosses the levelsurfaces transversely. Define
V (x, t) = 12
!
q
q
"T !
k !m
!m m
" !
q
q
"
= 12 qmq + 1
2qkq + !qmq,
where ! is a small positive constant such that V is still positive definite. The derivative of theLyapunov function becomes
V = qmq + qkq + !mq2 + !qmq
= (!c + !m)q2 + !(!kq2 ! cqq) = !
!
q
q
"T !
!k 12!c
12!c c ! !m
" !
q
q
"
.
The function V can be made negative definite for ! chosen su!ciently small (exercise) and hencewe can conclude exponential stability.
Remarks
1. As the previous example shows, a Lyapunov function need not be unique and di"erent Lya-punov functions can give stronger stability results.
2. Lyapunov functions can also be used to prove that a system is unstable: search for V positivedefinite with V positive definite.
3 Lyapunov Functions for Linear Systems
Consider a linear system of the formx = Ax.
Search for a quadratic Lyapunov function
V (x) = xT Px
Compute the derivativedV
dt=
"V
"x
dx
dt= xT (AT P + PA)x.
The requirement that V is positive definite is equivalent to P > 0 and the requirement that V isnegative definite becomes a condition that
Q = AT P + PA < 0 (2)
(as a matrix).
Trick: equation (2) is linear in P . So we can choose Q < 0 and the solve for P .
4
To fix this problem, we skew the level sets slightly, so that the flow of the system crosses the levelsurfaces transversely. Define
V (x, t) = 12
!
q
q
"T !
k !m
!m m
" !
q
q
"
= 12 qmq + 1
2qkq + !qmq,
where ! is a small positive constant such that V is still positive definite. The derivative of theLyapunov function becomes
V = qmq + qkq + !mq2 + !qmq
= (!c + !m)q2 + !(!kq2 ! cqq) = !
!
q
q
"T !
!k 12!c
12!c c ! !m
" !
q
q
"
.
The function V can be made negative definite for ! chosen su!ciently small (exercise) and hencewe can conclude exponential stability.
Remarks
1. As the previous example shows, a Lyapunov function need not be unique and di"erent Lya-punov functions can give stronger stability results.
2. Lyapunov functions can also be used to prove that a system is unstable: search for V positivedefinite with V positive definite.
3 Lyapunov Functions for Linear Systems
Consider a linear system of the formx = Ax.
Search for a quadratic Lyapunov function
V (x) = xT Px
Compute the derivativedV
dt=
"V
"x
dx
dt= xT (AT P + PA)x.
The requirement that V is positive definite is equivalent to P > 0 and the requirement that V isnegative definite becomes a condition that
Q = AT P + PA < 0 (2)
(as a matrix).
Trick: equation (2) is linear in P . So we can choose Q < 0 and the solve for P .
4
9
V (x) = x>Px
How to find a Lyapunov function?
x>Px > 0 for all x 6= 0
Q < 0
ExampleExample 3. Consider the linear system
dx1
dt= !ax1
dx2
dt= !bx1 ! cx2.
with a, b, c > 0, for which we have
A =
!
!a 0!b !c
"
P =
!
p11 p12
p21 p22
"
.
We choose Q = !I " R2!2 and the corresponding Lyapunov equation is
!
!a !b
0 !c
" !
p11 p12
p21 p22
"
+
!
p11 p12
p21 p22
" !
!a 0!b !c
"
=
!
1 00 1
"
and solving for the elements of P yields
P =
#
b2+ac+c2
2a2c+2ac2"b
2c(a+c)"b
2c(a+c)12
$
or
V (x) =b2 + ac + c2
2a2c + 2ac2x2
1 !b
c(a + c)x1x2 +
1
2x2
2.
It is easy to verify that P > 0 (check its eigenvalues) and by construction P = !I < 0. Hence thesystem is asymptotically stable.
4 Krasolvskii-Lasalle Invariance Principle (optional)
The Krasolvskii-Lasalle invariance principle provides a way to prove asymptotic stability when V
is negative semi-definite (which is usually much easier to find).
Denote the solution trajectories of the time-invariant system
dx
dt= F (x) (3)
as x(t; x0, t0), which is the solution of equation (3) at time t starting from x0 at t0. We writex(·; x0, t0) for the set of all points lying along the trajectory.Definition 3 (! limit set). The ! limit set of a trajectory x(·; x0, t0) is the set of all points z " Rn
such that there exists a strictly increasing sequence of times tn such that
s(tn; x0, t0) # z
as n # $.
5
Example 3. Consider the linear system
dx1
dt= !ax1
dx2
dt= !bx1 ! cx2.
with a, b, c > 0, for which we have
A =
!
!a 0!b !c
"
P =
!
p11 p12
p21 p22
"
.
We choose Q = !I " R2!2 and the corresponding Lyapunov equation is
!
!a !b
0 !c
" !
p11 p12
p21 p22
"
+
!
p11 p12
p21 p22
" !
!a 0!b !c
"
=
!
1 00 1
"
and solving for the elements of P yields
P =
#
b2+ac+c2
2a2c+2ac2"b
2c(a+c)"b
2c(a+c)12
$
or
V (x) =b2 + ac + c2
2a2c + 2ac2x2
1 !b
c(a + c)x1x2 +
1
2x2
2.
It is easy to verify that P > 0 (check its eigenvalues) and by construction P = !I < 0. Hence thesystem is asymptotically stable.
4 Krasolvskii-Lasalle Invariance Principle (optional)
The Krasolvskii-Lasalle invariance principle provides a way to prove asymptotic stability when V
is negative semi-definite (which is usually much easier to find).
Denote the solution trajectories of the time-invariant system
dx
dt= F (x) (3)
as x(t; x0, t0), which is the solution of equation (3) at time t starting from x0 at t0. We writex(·; x0, t0) for the set of all points lying along the trajectory.Definition 3 (! limit set). The ! limit set of a trajectory x(·; x0, t0) is the set of all points z " Rn
such that there exists a strictly increasing sequence of times tn such that
s(tn; x0, t0) # z
as n # $.
5
Example 3. Consider the linear system
dx1
dt= !ax1
dx2
dt= !bx1 ! cx2.
with a, b, c > 0, for which we have
A =
!
!a 0!b !c
"
P =
!
p11 p12
p21 p22
"
.
We choose Q = !I " R2!2 and the corresponding Lyapunov equation is
!
!a !b
0 !c
" !
p11 p12
p21 p22
"
+
!
p11 p12
p21 p22
" !
!a 0!b !c
"
=
!
1 00 1
"
and solving for the elements of P yields
P =
#
b2+ac+c2
2a2c+2ac2"b
2c(a+c)"b
2c(a+c)12
$
or
V (x) =b2 + ac + c2
2a2c + 2ac2x2
1 !b
c(a + c)x1x2 +
1
2x2
2.
It is easy to verify that P > 0 (check its eigenvalues) and by construction P = !I < 0. Hence thesystem is asymptotically stable.
4 Krasolvskii-Lasalle Invariance Principle (optional)
The Krasolvskii-Lasalle invariance principle provides a way to prove asymptotic stability when V
is negative semi-definite (which is usually much easier to find).
Denote the solution trajectories of the time-invariant system
dx
dt= F (x) (3)
as x(t; x0, t0), which is the solution of equation (3) at time t starting from x0 at t0. We writex(·; x0, t0) for the set of all points lying along the trajectory.Definition 3 (! limit set). The ! limit set of a trajectory x(·; x0, t0) is the set of all points z " Rn
such that there exists a strictly increasing sequence of times tn such that
s(tn; x0, t0) # z
as n # $.
5
10
�a �b0 �c
� p11 p12p21 p22
�+
p11 p12p21 p22
� �a 0�b �c
�=
�1 00 �1
�
Example 3. Consider the linear system
dx1
dt= !ax1
dx2
dt= !bx1 ! cx2.
with a, b, c > 0, for which we have
A =
!
!a 0!b !c
"
P =
!
p11 p12
p21 p22
"
.
We choose Q = !I " R2!2 and the corresponding Lyapunov equation is
!
!a !b
0 !c
" !
p11 p12
p21 p22
"
+
!
p11 p12
p21 p22
" !
!a 0!b !c
"
=
!
1 00 1
"
and solving for the elements of P yields
P =
#
b2+ac+c2
2a2c+2ac2"b
2c(a+c)"b
2c(a+c)12
$
or
V (x) =b2 + ac + c2
2a2c + 2ac2x2
1 !b
c(a + c)x1x2 +
1
2x2
2.
It is easy to verify that P > 0 (check its eigenvalues) and by construction P = !I < 0. Hence thesystem is asymptotically stable.
4 Krasolvskii-Lasalle Invariance Principle (optional)
The Krasolvskii-Lasalle invariance principle provides a way to prove asymptotic stability when V
is negative semi-definite (which is usually much easier to find).
Denote the solution trajectories of the time-invariant system
dx
dt= F (x) (3)
as x(t; x0, t0), which is the solution of equation (3) at time t starting from x0 at t0. We writex(·; x0, t0) for the set of all points lying along the trajectory.Definition 3 (! limit set). The ! limit set of a trajectory x(·; x0, t0) is the set of all points z " Rn
such that there exists a strictly increasing sequence of times tn such that
s(tn; x0, t0) # z
as n # $.
5
c
c
Example 3. Consider the linear system
dx1
dt= !ax1
dx2
dt= !bx1 ! cx2.
with a, b, c > 0, for which we have
A =
!
!a 0!b !c
"
P =
!
p11 p12
p21 p22
"
.
We choose Q = !I " R2!2 and the corresponding Lyapunov equation is
!
!a !b
0 !c
" !
p11 p12
p21 p22
"
+
!
p11 p12
p21 p22
" !
!a 0!b !c
"
=
!
1 00 1
"
and solving for the elements of P yields
P =
#
b2+ac+c2
2a2c+2ac2"b
2c(a+c)"b
2c(a+c)12
$
or
V (x) =b2 + ac + c2
2a2c + 2ac2x2
1 !b
c(a + c)x1x2 +
1
2x2
2.
It is easy to verify that P > 0 (check its eigenvalues) and by construction P = !I < 0. Hence thesystem is asymptotically stable.
4 Krasolvskii-Lasalle Invariance Principle (optional)
The Krasolvskii-Lasalle invariance principle provides a way to prove asymptotic stability when V
is negative semi-definite (which is usually much easier to find).
Denote the solution trajectories of the time-invariant system
dx
dt= F (x) (3)
as x(t; x0, t0), which is the solution of equation (3) at time t starting from x0 at t0. We writex(·; x0, t0) for the set of all points lying along the trajectory.Definition 3 (! limit set). The ! limit set of a trajectory x(·; x0, t0) is the set of all points z " Rn
such that there exists a strictly increasing sequence of times tn such that
s(tn; x0, t0) # z
as n # $.
5
Verify that P > 0
• A is Hermitian (or symmetric in the case of real matrices) matrix is positive definite if:
- All the diagonal entries are positive - Each diagonal entry is greater than the sum of the
absolute values of all other entries in the same row.
11
How to determine positive-definiteness
Sufficient but not necessary!
• A is Hermitian (or symmetric in the case of real matrices) matrix is positive definite if and only if:
- M is positive-definite if and only if all the following matrices have a positive determinant:
- the upper left 1-by-1 corner of M, - the upper left 2-by-2 corner of M, - ... - M itself.
- All the eigenvalues of M are positive.
12
Necessary and sufficient criteriaSylvester’s criterion
Definition (! limit set). The ! limit set of a trajectory x(· ; x0, t0) is the set ofall points z 2 R such that there exists a strictly increasing sequence of time tnthat satisfies
limn!1
tn =1
and as n !1x(tn; x0, t0) = z
<latexit sha1_base64="pccmApurO01imMqzBE2Q6x6+NCY=">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</latexit><latexit sha1_base64="pccmApurO01imMqzBE2Q6x6+NCY=">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</latexit><latexit sha1_base64="pccmApurO01imMqzBE2Q6x6+NCY=">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</latexit><latexit sha1_base64="pccmApurO01imMqzBE2Q6x6+NCY=">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</latexit>
x1 = 4x2 � x1�x21 + x
22 � 4
�
x2 = �4x1 � x2�x21 + x
22 � 4
�<latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit><latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit><latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit><latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit>
x1 = x2
x2 = �x1<latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit><latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit><latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit><latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit>
x1 = x2
x2 = � sin x1 � 0.5x2<latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit><latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit><latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit><latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit>
x1 = 4x2 � x1�x21 + x
22 � 4
�
x2 = �4x1 � x2�x21 + x
22 � 4
�<latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit><latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit><latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit><latexit sha1_base64="X8dI3zyxSAEaW5DVrt7CJGD57IY=">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</latexit>
x1 = x2
x2 = �x1<latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit><latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit><latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit><latexit sha1_base64="IgCscotfUUmtUuF1uz3l/H9N6cE=">AAACNXicbZDLSsNAFIYn9VbjLerSzWBR3FiSUtCNUHDjsoK9QBvCZDpph04uzJxIS+jWp3GrPosLd+LWNxCnbRa29cCBn+/cOL+fCK7Att+Nwtr6xuZWcdvc2d3bP7AOj5oqTiVlDRqLWLZ9opjgEWsAB8HaiWQk9AVr+cPbab31yKTicfQA44S5IelHPOCUgEaehbu9GLKR50zOb0Zepds1c1DR4FJzzyrZZXsWeFU4uSihPOqe9aM30DRkEVBBlOo4dgJuRiRwKtjE7KaKJYQOSZ91tIxIyJSbzT6Z4DNNejiIpc4I8Iz+nchIqNQ49HVnSGCglmtT+G9NsAB4b7R0HoJrN+NRkgKL6Px6kAoMMZ56hXtcMgpirAWhkusHMB0QSShoRxfW09CXvD+AiWlqw5xle1ZFs1J2tL6vlmrV3LoiOkGn6AI56ArV0B2qowai6Ak9oxf0arwZH8an8TVvLRj5zDFaCOP7F0MWqyo=</latexit>
x1 = x2
x2 = � sin x1 � 0.5x2<latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit><latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit><latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit><latexit sha1_base64="CuXvyGt5nQE0FFMe+dam2epzzMc=">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</latexit>
Definition (invariant set). The set M ⇢ Rn is said to be an invariant set if forall y 2 M and t0 � 0, we have
x(t; y , t0) 2 M for all t � t0.<latexit sha1_base64="Xcxwqq1erIhNgU38ru7ZdF57b+U=">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</latexit><latexit sha1_base64="Xcxwqq1erIhNgU38ru7ZdF57b+U=">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</latexit><latexit sha1_base64="Xcxwqq1erIhNgU38ru7ZdF57b+U=">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</latexit><latexit sha1_base64="Xcxwqq1erIhNgU38ru7ZdF57b+U=">AAAC63icbVJNj9MwEHXC1xI+tsCRy4gGqSutqnQvgOCwEhy4rLSg7e5KTalsZ9JYdZxgO6VVlF/BDXHlLyHxZxBOtkK0y5ye3nszYz+blVIYG0W/PP/GzVu37+zdDe7df/Bwv/fo8bkpKs1xzAtZ6EtGDUqhcGyFlXhZaqQ5k3jBFm9b/WKJ2ohCndl1idOczpVIBafWUbPez1gVQiWoLMQWV5al9TtMhRKtDAOhllQL6lSD9qAZwlmGLYTwJDYVa1GcU5sxVn9sPqkQhAFDRQK2AIZAFcSYl1m9NacBkUJaaKBSQriOhYKT0HkTCO0sgniOnyEKD+ELQkaXGMSTYDWwr9eHTj3o3G4pK1Y1/J0CDdiuz1mGQTyd9frRMOoKroPRBvTJpk5nvd9xUvAqd0FwSY2ZjKLSTmuqreASmyCuDJaUL+gcJw4qmqOZ1l3+DTx3TNKdJS3cFTv2346a5sasc+acbVhmV2vJ/2oSUyuS1c56m76cukTLyqLiV9vTSraRty8MidDIrVw7QLl2D8mBZ1RTbt0/2BrPc6bFPLNNELjARrvxXAfnR8ORwx+O+sevNtHtkafkGRmQEXlBjsl7ckrGhHtvPOYtPOnn/lf/m//9yup7m54nZKv8H38AcQHnGg==</latexit>
Remarks:
• The word “largest” refers to the union of all invariant sets within S
• If r = , then the origin is globally asymptotically stable
15
Theorem (Krasovskii-Lasalle). Let V : Rn 7! R be a locally positive definitefunction such that, on the compact set ⌦r = {x 2 R : V (x) r}, we haveV (x) 0. Define
S = {x 2 ⌦r : V (x) = 0}.
As t ! 1, any trajectory starting inside ⌦r converges to the largest invariantset inside S. That is, the ! limit set of any such trajectory is contained insidethe largest infant set in S. In particular, if S contains no invariant set except theequilibrium xe = 0, then the origin is asymptotical stable.
<latexit sha1_base64="uePnm8h+Rm6Gtxi85YAyrJIBGxY=">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</latexit><latexit sha1_base64="uePnm8h+Rm6Gtxi85YAyrJIBGxY=">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</latexit><latexit sha1_base64="uePnm8h+Rm6Gtxi85YAyrJIBGxY=">AAAEhXicbVNbaxNBFN6mqa3x1uqjLwe7Qgtp2BTxUihW9EGxYLVtWujGMDt7Nhk7O7OdmU0TQv6m4H8R8ewmITZ1YOHsuX3fuUWZFNYFwa+lynJ15c7q2t3avfsPHj5a33jcsjo3HE+5ltqcR8yiFApPnXASzzODLI0knkWX7wv7WR+NFVqduGGG7ZR1lUgEZ45UnY2KDpUWKkblIHQ4cFEyOumhNpjC1mfDrO7bSyF2DpllUuL2uAGH6MBv7YUpc70oGn0bf1ckZ9ZpmOt8iBAYSM0pbAiZtsKJPkKMiVDCISS54gUFsDnvgesxVwf6cz0ErtOMcQe2AAq/pNhlHbMfjgahUHOEvdbWYDuUeAUmHPt1uEboMULww1i7UWtMVijNgd+ADwUs1sKL2vE0EcwSwx7MI/aDcNyohW14Z8F3oRHdnmPG6GugkMQNCYepITjDfiB32gzBkt0J1QWhrIhxTtinOhS1vosWqDVFYZIVf45c+8wIpiYlzgKPiecJ9QGErZfufqiLVD5IkYqJr05K/EnP5iSELcAcoxrjWb6bgMkcbYL0SUFWMOc5OdVBJIV6lsWC0gssccAxc2VWvMqFFJEReQr+oIP7gV8SnkxPU9MIhCgxO0wzpwmDyaJPtJMN6KxvBo2gfHBbaE6FTW/6jjrrf2g8PE9pQ7lk1l40g8y1RyV3ieNamFukdblkXbwgUbEUbXtUXsYYnpMmhkQb+qiQUvtvxIilllhG5Fksll20Fcr/2iQmTsSDBXiXvG6PhMpyh4pP0JNcluOn24NYGBoYnUMsGDd0EBx4jxnadbrQG+l5GpW7N67VqGHNxfbcFlq7jSbJX3c3D95MW7fmPfWeeVte03vlHXgfvSPv1OOVn5XfyyvLd6qr1Z3qi+rLiWtlaRrzxLvxqm//AvmfeTk=</latexit><latexit sha1_base64="uePnm8h+Rm6Gtxi85YAyrJIBGxY=">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</latexit>
1<latexit sha1_base64="KfLbUTn34Eqn29GllEC0Gydp8qw=">AAACG3icbZC7TsMwFIadcivhVmBksWiRmKqkC7BVYmEsEr1IbVQ5jtNadZzIPkFEUV+CFXgaNsTKwMsg3DYDbTmSpU//f46P/fuJ4Boc59sqbWxube+Ud+29/YPDo8rxSUfHqaKsTWMRq55PNBNcsjZwEKyXKEYiX7CuP7md+d1HpjSP5QNkCfMiMpI85JSAkXq1AZchZLVhperUnXnhdXALqKKiWsPKzyCIaRoxCVQQrfuuk4CXEwWcCja1B6lmCaETMmJ9g5JETHv5/L1TfGGUAIexMkcCnqt/J3ISaZ1FvumMCIz1qjcT//UEC4EHTyvrIbz2ci6TFJiki+1hKjDEeJYIDrhiFERmgFDFzQcwHRNFKJjclq6nka/4aAxT2zaBuavxrEOnUXcN3zeqzZsiujI6Q+foErnoCjXRHWqhNqJIoGf0gl6tN+vd+rA+F60lq5g5RUtlff0Css6hzQ==</latexit><latexit sha1_base64="KfLbUTn34Eqn29GllEC0Gydp8qw=">AAACG3icbZC7TsMwFIadcivhVmBksWiRmKqkC7BVYmEsEr1IbVQ5jtNadZzIPkFEUV+CFXgaNsTKwMsg3DYDbTmSpU//f46P/fuJ4Boc59sqbWxube+Ud+29/YPDo8rxSUfHqaKsTWMRq55PNBNcsjZwEKyXKEYiX7CuP7md+d1HpjSP5QNkCfMiMpI85JSAkXq1AZchZLVhperUnXnhdXALqKKiWsPKzyCIaRoxCVQQrfuuk4CXEwWcCja1B6lmCaETMmJ9g5JETHv5/L1TfGGUAIexMkcCnqt/J3ISaZ1FvumMCIz1qjcT//UEC4EHTyvrIbz2ci6TFJiki+1hKjDEeJYIDrhiFERmgFDFzQcwHRNFKJjclq6nka/4aAxT2zaBuavxrEOnUXcN3zeqzZsiujI6Q+foErnoCjXRHWqhNqJIoGf0gl6tN+vd+rA+F60lq5g5RUtlff0Css6hzQ==</latexit><latexit sha1_base64="KfLbUTn34Eqn29GllEC0Gydp8qw=">AAACG3icbZC7TsMwFIadcivhVmBksWiRmKqkC7BVYmEsEr1IbVQ5jtNadZzIPkFEUV+CFXgaNsTKwMsg3DYDbTmSpU//f46P/fuJ4Boc59sqbWxube+Ud+29/YPDo8rxSUfHqaKsTWMRq55PNBNcsjZwEKyXKEYiX7CuP7md+d1HpjSP5QNkCfMiMpI85JSAkXq1AZchZLVhperUnXnhdXALqKKiWsPKzyCIaRoxCVQQrfuuk4CXEwWcCja1B6lmCaETMmJ9g5JETHv5/L1TfGGUAIexMkcCnqt/J3ISaZ1FvumMCIz1qjcT//UEC4EHTyvrIbz2ci6TFJiki+1hKjDEeJYIDrhiFERmgFDFzQcwHRNFKJjclq6nka/4aAxT2zaBuavxrEOnUXcN3zeqzZsiujI6Q+foErnoCjXRHWqhNqJIoGf0gl6tN+vd+rA+F60lq5g5RUtlff0Css6hzQ==</latexit><latexit sha1_base64="KfLbUTn34Eqn29GllEC0Gydp8qw=">AAACG3icbZC7TsMwFIadcivhVmBksWiRmKqkC7BVYmEsEr1IbVQ5jtNadZzIPkFEUV+CFXgaNsTKwMsg3DYDbTmSpU//f46P/fuJ4Boc59sqbWxube+Ud+29/YPDo8rxSUfHqaKsTWMRq55PNBNcsjZwEKyXKEYiX7CuP7md+d1HpjSP5QNkCfMiMpI85JSAkXq1AZchZLVhperUnXnhdXALqKKiWsPKzyCIaRoxCVQQrfuuk4CXEwWcCja1B6lmCaETMmJ9g5JETHv5/L1TfGGUAIexMkcCnqt/J3ISaZ1FvumMCIz1qjcT//UEC4EHTyvrIbz2ci6TFJiki+1hKjDEeJYIDrhiFERmgFDFzQcwHRNFKJjclq6nka/4aAxT2zaBuavxrEOnUXcN3zeqzZsiujI6Q+foErnoCjXRHWqhNqJIoGf0gl6tN+vd+rA+F60lq5g5RUtlff0Css6hzQ==</latexit>