Local Lyapunov Functions for Nonlinear Stochastic

Differential Equations by Linearization

Hj

¨

ortur Bj

¨

ornsson

1

, Peter Giesl

2

, Skuli Gudmundsson

3

and Sigurdur Hafstein

1

Science Institute and Faculty of Physical Sciences, University of Iceland, Dunhagi 5, 107 Reykjav

´

ık, Iceland

2

Department of Mathematics, University of Sussex, Falmer, BN1 9QH, U.K.

3

Svensk Exportkredit, Klarabergsviadukten 61-63, 11164 Stockholm, Sweden

Keywords:

Stochastic Differential Equation, Lyapunov Function, Linearization, Asymptotic Stability in Probability.

Abstract:

We present a rigid estimate of the domain, on which a Lyapunov function for the linearization of a nonlinear

stochastic differential equation is a Lyapunov function for the original system. By using this estimate the

demanding task of computing a lower bound on the γ-basin of attraction for a nonlinear stochastic systems is

greatly simpliﬁed and the application of a resent numerical method for the same purpose facilitated.

1 INTRODUCTION

When analysing the stability of an equilibrium of a

nonlinear deterministic system

˙

x = f(x), f : R

d

→R

d

,

one often resorts to linearization around the equilib-

rium. Assuming, without restriction of generality, that

the equilibrium in question is at the origin, then one

analyzes the stability of the origin for the system

˙

x =

Ax, where A := Df(0) is the Jacobian of f at the origin.

Now, if the matrix A is Hurwitz, i.e. the real-parts of

the eigenvalues of A are all strictly negative, then one

can solve the Lyapunov equation A

>

P + PA = −Q,

where Q ∈ R

d×d

is an arbitrary symmetric and pos-

itive deﬁnite matrix. The solution P ∈ R

d×d

is then

symmetric and positive deﬁnite and V (x) = x

>

Px is

a Lyapunov function for the system, i.e. V has a min-

imum at the equilibrium at the origin and the deriva-

tive of V along solution trajectories of the linearized

system fulﬁlls

∇V (x)

•

Ax = −x

>

Qx

and is thus negative on R

d

\{0}. The function V will

also be a Lyapunov function for the original nonlinear

system

˙

x = f(x) on a neighbourhood N of the origin

where

V

0

(x) = ∇V (x)

•

f(x) < 0 for x ∈N \{0}.

Here V

0

denotes the orbital derivative of the system.

The size of the set N is of great importance because

compact sublevel sets of V that are within N are

lower bounds on the equilibrium’s basin of attraction,

i.e. the set of points which converge to the equilibrium

as time goes to inﬁnity. Explicit bounds for the size of

N are quite easily derived, cf. e.g. (Hafstein, 2004).

In this paper we will derive such an estimate, but for

the considerably more demanding case of stochastic

differential equations.

Notation: We denote by kxk the Euclidian norm

of a vector x ∈ R

d

and for A ∈ R

d×d

by kAk =

max

kxk=1

kAxk the matrix norm induced by the Eu-

clidian vector norm. Vectors are assumed to be col-

umn vectors.We denote by κ(A) := kAkkA

−1

k the

condition number with respect to the k · k norm of

the nonsingular matrix A ∈ R

d×d

. For a symmet-

ric and positive deﬁnite Q ∈ R

d×d

we deﬁne the en-

ergetic norm kxk

Q

:=

p

x

>

Qx and the correspond-

ing induced matrix norm kAk

Q

:= max

kxk

Q

=1

kAxk

Q

.

Recall that a symmetric and positive deﬁnite Q ∈

R

d×d

can be factorized as Q = ODO

>

where O ∈

R

d×d

is orthogonal, i.e. O

>

O = O

>

O = I and D =

diag(λ

1

, λ

2

, . . ., λ

d

) ∈ R

d×d

is a diagonal matrix with

0 < λ

1

≤ λ

2

≤ . . . ≤ λ

d

. For every a ∈ R we deﬁne

the matrix Q

a

= O diag(λ

a

1

, λ

a

2

, . . . , λ

a

d

)O

>

. It is not

difﬁcult to see that for a > 0 we have kQ

a

k = λ

a

d

and

kQ

−a

k = λ

−a

1

. Further,

kQ

−

1

2

k

−1

kxk ≤ kxk

Q

=

p

x

>

Qx

= kQ

1

2

xk ≤ kQ

1

2

kkxk.

We consider d-dimensional systems and in all sums

where the upper and lower bounds of the sum are

omitted they are assumed to be 1 and d respectively,

i.e.

∑

i

:=

∑

d

i=1

,

∑

i, j

:=

∑

d

i, j=1

etc.

A function α : R

+

→R

+

is said to be of class K

∞

Björnsson, H., Giesl, P., Gudmundsson, S. and Hafstein, S.

Local Lyapunov Functions for Nonlinear Stochastic Differential Equations by Linearization.

DOI: 10.5220/0006944505790586

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 579-586

ISBN: 978-989-758-321-6

579

if it is continuous, monotonically increasing, α(0) =

0, and lim

x→∞

α(x) = ∞.

We write and for probability and expecta-

tion respectively. The underlying probability spaces

should always be clear from the context. The abbrevi-

ation a.s. stands for almost surely, i.e. with probability

one, and

a.s.

= means equal a.s.

2 THE PROBLEM SETTING

We give a short discussion of the setup and the

problem at hand. For a more detailed discussion

of the setup see (Gudmundsson and Hafstein, 2018,

§2). The general d-dimensional stochastic differen-

tial equation (SDE) of It

ˆ

o type we consider is of the

form:

dX(t) = f(X(t))dt + g(X(t)) ·dW(t) (1)

or equivalently

dX

i

(t) = f

i

(X(t))dt +

U

∑

u=1

g

u

(X(t)) ·dW

u

(t)

for i = 1, 2, . . . , d. Thus f = ( f

1

, f

2

, . . . , f

d

)

>

, g =

(g

1

, g

2

, . . . , g

U

), and g

u

= (g

u

1

, g

u

2

, . . . , g

u

d

)

>

, where

f

i

, g

u

i

: R

d

→ R. We assume that the origin is an

equilibrium of the system, i.e. f(0) = 0 and g

u

(0) =

0 for u = 1, 2, . . . ,U and we consider strong solu-

tions to (1). For deterministic initial value solutions,

i.e. X(0) = x ∈ R

d

a.s., we write X

x

for the solution,

i.e.

X

x

(t) = x +

Z

t

0

f(X(s))ds +

Z

t

0

g(X(s))dW(s),

where the second integral is interpreted in the It

ˆ

o

sense. As shown in (Mao, 2008) it sufﬁces to consider

deterministic initial value solutions when studying the

stability of an equilibrium.

Numerous concepts are in use concerning the

stability of equilibria of SDEs. Here we will be

concerned with the so-called asymptotic stability in

probability of the zero solution (Khasminskii, 2012,

(5.15)), also referred to as stochastic asymptotic sta-

bility (Mao, 2008, Deﬁnition 4.2.1). For a more

detailed discussion of the stability of SDEs see the

books by Khasminskii (Khasminskii, 2012) or Mao

(Mao, 2008). We recall a few deﬁnitions:

Deﬁnition 2.1 (Stability in Probability (SiP)). The

null solution X(t)

a.s.

= 0 to the SDE (1) is said to be sta-

ble in probability (SiP) if for every r > 0 and 0 < ε < 1

there exists a δ > 0 such that :

kxk ≤ δ implies



sup

t≥0

kX

x

(t)k ≤ r



≥ 1 −ε.



Deﬁnition 2.2 (Asymptotic Stability in Probability

(ASiP)). The null solution X(t)

a.s.

= 0 to the SDE (1) is

said to be asymptotically stable in probability (ASiP)

if it is SiP and in addition for every 0 < ε < 1 there

exists a δ > 0 such that :

kxk ≤ δ implies

n

lim

t→∞

kX

x

(t)k = 0

o

≥ 1 −ε.



Our deﬁnitions of SiP and ASiP are equivalent to

the more common

lim

kxk→0

{

sup

t>0

kX

x

(t)k ≤ r

}

= 1 for all r > 0

for SiP and additionally

lim

kxk→0



limsup

t→∞

kX

x

(t)k = 0



= 1

for ASiP, which can be seen by ﬁxing r > 0 and writ-

ing down the deﬁnition of a limit: for every ε > 0

there exists a δ > 0.

The reason for our formulation is that we want to

look at a more practical concept related to such sta-

bility, namely a stochastic analog of the basin of at-

traction (BOA) in the stability theory for deterministic

systems, cf. (Gudmundsson and Hafstein, 2018). In-

stead of the limit kxk → 0 we consider: Given some

conﬁdence 0 < γ ≤1 how far from the origin can sam-

ple paths start and still approach the equilibrium as

t →∞ with probability greater than or equal to γ. This

is the motivation for the next deﬁnition.

Deﬁnition 2.3 (γ-Basin Of Attraction (γ-BOA)).

Consider the system (1) and let 0 < γ ≤ 1. We refer to

the set

n

x ∈ R

d

:

n

lim

t→∞

kX

x

(t)k = 0

o

≥ γ

o

(γ-BOA)

as the γ-basin of attraction, or short γ-BOA, of the

equilibrium at the origin.



Note that a 1-BOA corresponds to the usual BOA

for deterministic systems.

For the SDE (1) the associated generator is given

by

LV (x) := (2)

∇V (x)

•

f(x) +

1

2

∑

i, j



g(x)g(x)

>



i j

∂

2

V

∂x

i

∂x

j

(x)

for some appropriately differentiable V : U → R with

U ⊂ R

d

. Notice that this is just the drift term in the

expression for the stochastic differential of the pro-

cess t 7→V (X(t)). The generator for a stochastic sys-

tem corresponds to the orbital derivative of a deter-

ministic system.

CTDE 2018 - Special Session on Control Theory and Differential Equations

580

Deﬁnition 2.4 (Local Lyapunov function). Consider

the system (1). A function V ∈C(U) ∩C

2

(U \{0}),

where 0 ∈ U ⊂ R

d

is a domain, is called a (local)

Lyapunov function for the the system (1) if there are

functions µ

1

, µ

2

, µ

3

∈K

∞

, such that V fulﬁlls the prop-

erties :

(i) µ

1

(kxk) ≤V (x) ≤ µ

2

(kxk) for all x ∈ U

(ii) LV (x) ≤ −µ

3

(kxk) for all x ∈ U \{0}

Remark 2.5. It is of vital importance that V is not

necessarily differentiable at the equilibrium, because

otherwise a large number of systems with an ASiP

null solution do not possess a Lyapunov function,

cf. (Khasminskii, 2012, Remark 5.5).

The following theorem provides the ﬁrst center-

piece of Lyapunov stability theory for our application,

cf. (Khasminskii, 2012, Theorem 5.5 and Corollary

5.1):

Theorem 2.6. If there exists a local Lyapunov func-

tions as in Deﬁnition 2.4 for the system (1), then the

null solution is ASiP. Further, let V

max

> 0 and as-

sume that V

−1

([0,V

max

]) is a compact subset of U.

Then, for every 0 < β < 1 the set V

−1

([0, βV

max

]) is a

subset of the (1 −β)-BOA of the origin.

This concludes our discussion of the setup. In

the next section we discuss Lyapunov functions for

the linearization of (1) and prove the main contribu-

tion of this paper, a lower bound on the area where a

Lyapunov function for the linearization is also a Lya-

punov function for the nonlinear system.

3 MAIN RESULTS

We now consider the linearization of system (1). A

Lyapunov function for the linearized system can then

be constructed, e.g. with the method form (Hafstein

et al., 2018), much more easily than for the nonlinear

system (1). In addition to f and g satisfying the usual

sufﬁcient SDE solution-theory conditions locally Lip-

schitz and the linear-growth conditions, cf. e.g. (Mao,

2008, §2.3) or (Kallenberg, 2002, §21), we assume f

and g are C

2

on a convex neighbourhood U ⊂ R

d

of

the origin. The second order Taylor expansion for the

components f

i

of f at x ∈ U reads

f

i

(x) =

∑

j

x

j

F

i j

+

1

2

∑

j,k

x

j

x

k

R

i

jk

(x)

=



Fx)

i

+

1

2

x

>

R

i

(x) x,

and the components g

u

i

of g

u

,

g

u

i

(x) =

∑

j

x

j

G

u

i j

+

1

2

∑

j,k

x

j

x

k

R

ui

jk

(x)

=



G

u

x)

i

+

1

2

x

>

R

ui

(x) x

Here

F = (F

i j

)

i, j

∈ R

d×d

with F

i j

= ∂

j

f

i

(0)

and

G

u

=



G

u

i j



i, j

∈ R

d×d

with G

u

i j

= ∂

j

g

u

i

(0)

and the matrices R

i

(x) and R

ui

(x) are the Taylor re-

mainders

R

i

(x) =



R

i

jk

(x)



j,k

∈ R

d×d

and

R

ui

(x) =



R

ui

jk

(x)



j,k

∈ R

d×d

.

By abuse of notation we deﬁne the elements of up-

per bound matrices R

i

=



R

i

jk



j,k

∈ R

d×d

and R

ui

=



R

ui

jk



j,k

∈ R

d×d

as follows:



∂

2

jk

f

i

(x)



= |R

i

jk

(x)| ≤ R

i

jk

and (3)



∂

2

jk

g

u

i

(x)



= |R

ui

jk

(x)| ≤ R

ui

jk

, (4)

for all x ∈ N , where N is a neighbourhood of the

origin to be deﬁned later. Finally we ﬁx the constants

R

i

and R

ui

as

R

i

:= kR

i

k and R

ui

:= kR

ui

k. (5)

The action of the generator (2) of the system (1)

on some V ∈C(U)∩C

2

(U \{0}) can be written as

LV (x) =

1

2

∑

i, j

m

i j

(x)∂

2

i j

V (x) +

∑

i

f

i

(x)∂

i

V (x)

= L

0

V (x) +E(x)

where L

0

V (x) is the generator of the linearized sys-

tem deﬁned below and E(x) the rest (containing all

the Taylor remainders). We will now work out the

details, ﬁrst notice that:

m

i j

(x) =

U

∑

u=1

g

u

i

(x)g

u

j

(x)

=

∑

k,l

x

k

x

l

U

∑

u=1

G

u

ik

G

u

jl

+

1

2

∑

k,l,m

x

k

x

l

x

m

U

∑

u=1



G

u

ik

R

u j

lm

(x) + G

u

jk

R

ui

lm

(x)



+

1

4

∑

k,l,m,n

x

k

x

l

x

m

x

n

U

∑

u=1

R

ui

kl

(x)R

u j

mn

(x).

Local Lyapunov Functions for Nonlinear Stochastic Differential Equations by Linearization

581

We deﬁne L

0

as the generator associated to the lin-

earization of the system (1), i.e. the system

dX(t) = F X(t) dt +

U

∑

u=1

G

u

X(t) dW

u

(t) (6)

or equivalently

dX

i

(t) =

∑

j

F

i j

X

j

(t) dt +

U

∑

u=1

∑

j

G

u

i j

X

j

(t) dW

u

(t)

for i = 1, 2, . . . , d, which means that

L

0

V (x) = (7)

∑

i, j

F

i j

x

j

∂

i

V (x) +

1

2

∑

i, j

∑

k,l

x

k

x

l

U

∑

u=1

G

u

ik

G

u

jl

!

∂

2

i j

V (x).

We gather together the nonlinear parts of the full SDE

generator into the expression for E(x):

E(x) =

∑

s

E

s

(x)∂

s

V (x)

| {z }

E

F

(x)

+

1

2

∑

r,s

E

rs

(x)∂

2

rs

V (x)

| {z }

E

G

(x)

,

where

E

s

(x) =

1

2

∑

j,k

x

j

x

k

R

s

jk

(x) and

E

rs

(x) =

1

2

∑

k,l,m

x

k

x

l

x

m

U

∑

u=1

(G

u

rk

R

us

lm

(x) + G

u

sk

R

ur

lm

(x))

+

1

4

∑

k,l,m,n

x

k

x

l

x

m

x

n

U

∑

u=1

R

ur

kl

(x)R

us

mn

(x).

The plan for the rest of this section is as follows:

With LV (x) broken up into a linear part L

0

V (x) and a

nonlinear correction E(x), we take the explicit func-

tion

V (x) = kxk

p

Q

=



x

>

Qx



p

2

(8)

as the ansatz for the Lyapunov function candidate,

where Q ∈ R

d×d

is a symmetric and positive deﬁnite

matrix and p > 0. As argued in (Hafstein et al., 2018,

§4) this is the expected form of a Lyapunov function

for the linearized system (6) just as x 7→ x

>

Px for a

symmetric and positive deﬁnite P is the usual form for

a Lyapunov function for a linear deterministic system

˙

x = Ax. Note that typically p < 2 so V is not differen-

tiable at the origin. For this reason take x 6= 0 in the

calculations below. Assuming that we have ﬁxed Q

and p > 0 such that L

0

V (x) < 0 for all x ∈ R

d

\{0},

we derive a neighbourhood of the origin such that

|L

0

V (x)| > |E(x)|, which implies LV (x) < 0.

From (Hafstein et al., 2018, Lemma 4.1) we can

state the following: for V (x) = kxk

p

Q

we have

L

0

V (x) = −

1

2

pkxk

p−4

Q

H(x) for all x ∈R

d

\{0},

where

H(x) = −x

>

F

>

Q + QF +

U

∑

u=1

(G

u

)

>

QG

u

!

xkxk

2

Q

+ (2 −p)

U

∑

u=1



1

2

x

>

(QG

u

+ (G

u

)

>

Q)x



2

.

This V is a Lyapunov function for the linear system

(6) if there is a constant C > 0 such that

H(x) ≥Ckxk

2

Q

kxk

2

for all x ∈ R

d

,

because then

L

0

V (x) ≤ −

1

2

pCkxk

p−2

Q

kxk

2

(9)

for all x ∈ R

d

\{0}.

Before we state and prove our results we prove a

simple but useful lemma:

Lemma 3.1. Let A = (A

i j

),

e

A = (

e

A

i j

) ∈R

d×d

be such

that |A

i j

| ≤

e

A

i j

for i, j = 1, 2, . . . , d. Then

kAk ≤ k

e

Ak. (10)

In particular



∑

i, j

x

i

A

i j

y

j



≤ k

e

Akkxkkyk (11)

and



∑

i, j,k

x

i

Q

ik

A

k j

y

j



≤ k

e

AkkQ

1

2

kkxk

Q

kyk (12)

≤ k

e

Akκ(Q)

1

2

kxk

Q

kyk

Q

.

for every symmetric and positive deﬁnite Q ∈ R

d×d

.

If AQ

1

2

= Q

1

2

A we even have



∑

i, j,k

x

i

Q

ik

A

k j

y

j



≤ k

e

Akkxk

Q

kyk

Q

. (13)

Proof. For x = (x

1

, x

2

, . . . , x

d

)

>

set

˜

x =

(|x

1

|, |x

2

|, . . . , |x

d

|)

>

. Clearly kxk = k

e

xk. The

estimate (10) follows from

kAxk

2

= x

>

A

>

Ax =



∑

i, j,k

x

i

A

ki

A

k j

x

j



≤

∑

i, j,k

|x

i

|·|A

ki

|·|A

k j

|·|x

j

|

≤

∑

i, j,k

|x

i

|·

e

A

ki

e

A

k j

|x

j

| =

e

x

>

e

A

>

e

A

e

x = k

e

A

e

xk

2

≤ k

e

Ak

2

k

e

xk

2

= k

e

Ak

2

kxk

2

and thus

kAk := sup

x6=0

kAxk

kxk

≤ k

e

Ak.

CTDE 2018 - Special Session on Control Theory and Differential Equations

582

The inequality (12) follow from

|

∑

i, j,k

x

i

Q

ik

A

k j

y

j

| =



∑

i, j

x

i

∑

k

Q

ik

A

k j

!

y

j



= |x

>

QAy| = |(Q

1

2

x)

>

Q

1

2

Ay|

≤ kQ

1

2

xkkQ

1

2

Ayk = kxk

Q

kQ

1

2

Ayk

≤ kxk

Q

kQ

1

2

kkAkkyk

≤ k

e

AkkQ

1

2

kkQ

−

1

2

kkxk

Q

kyk

Q

and (11) follows form (12) with Q as the identity ma-

trix. To see (13) just note that if AQ

1

2

= Q

1

2

A we have

kQ

1

2

Ayk = kAQ

1

2

yk ≤ kAkkQ

1

2

yk ≤ k

e

Akkyk

Q

which can be used to improve the estimate above.

Remark 3.2. If A in (12) is symmetric we have

x

>

QAy =

∑

i, j,k

x

i

Q

ik

A

k j

y

j

=

∑

i, j,k

y

j

A

jk

Q

ki

x

i

= y

>

AQx.

Remark 3.3. For vectors x,

e

x ∈ R

d

, |x

i

| ≤

e

x

i

for

i = 1, 2, . . . , d, we obviously have kxk ≤ k

e

xk, but in

general kxk

Q

is not necessarily smaller than k

e

xk

Q

.

Take for example x = (1, −1)

>

,

e

x = (1, 1)

>

, and

Q =



2 −1

−1 2



. Then kxk

Q

=

p

x

>

Qx =

√

6 but

kyk

Q

=

√

2. For this reason one cannot expect |A

i j

|≤

e

A

i j

to imply kAk

Q

≤ k

e

Ak

Q

for matrices A,

e

A ∈ R

d×d

.

We now come to the main contribution of this pa-

per:

Theorem 3.4. Consider the system (1), assume that

V as in (8) is a Lyapunov function for its linearization

(6), and let C > 0 be a constant as in (9). Let ρ

∗

> 0

and assume the estimates (3), (4), and (5) hold true

on N = D

∗

:= {x ∈ R

d

: kxk

Q

≤ ρ

∗

}. Deﬁne

p

∗

:= 1 + |p −2|,

R

i

:= kR

i

k,

R

ui

:= kR

ui

k,

R

F

:= k(R

1

, R

2

, . . . , R

d

)k,

R

u

G

:= k(R

u1

, R

u2

, . . . , R

ud

)k,

R

G

:= k(R

1

G

, R

2

G

, . . . , R

U

G

)k

2

,

e

E

G

:= kQ

1

2

k

R

F

+ p

∗

U

∑

u=1

R

u

G

kQ

1

2

G

u

Q

−

1

2

k

!

,

e

E

∗

G

:=

1

4

p

∗

κ(Q)R

G

.

Then

LV (x) = L

0

V (x) +E(x)

where L

0

V is deﬁned in (7) and

|E(x)|≤

1

2

pkxk

p−2

Q

kxk

2

·kxk

Q



e

E

G

+

e

E

∗

G

kxk

Q



for x ∈ D

∗

:= {x ∈ R

d

: kxk

Q

≤ ρ

∗

}. In particular,

V is a Lyapunov function for the nonlinear system (1)

satisfying the condition of Deﬁnition 2.4 on

U = D := {x ∈ R

d

: kxk

Q

≤ ρ},

with

ρ < min

(

ρ

∗

,

1

2

e

E

∗

G



q

(

e

E

G

)

2

+ 4C

e

E

∗

G

−

e

E

G



)

.

Proof. Let us ﬁrst compute ∂

s

V (x) and ∂

2

rs

V (x),

∂

s

V (x) =

∑

j

Q

s j

x

j

+

∑

i

Q

is

x

i

!

p

2

∑

i, j

Q

i j

x

i

x

j

!

p

2

−1

= p

∑

i

x

i

Q

is

kxk

p−2

Q

and

∂

2

rs

V (x) = pQ

rs

kxk

p−2

Q

+ p

∑

j

x

j

Q

js

!



p

2

−1



×2

∑

i

x

i

Q

ir

!

∑

i, j

Q

i j

x

i

x

j

!

p

2

−2

= pkxk

p−2

Q

rs

+ p(p −2)

∑

i, j

x

i

x

j

Q

ir

Q

js

kxk

p−4

Q

= pkxk

p−4

Q

rs

kxk

2

Q

+ (p −2)

∑

i, j

x

i

x

j

Q

ir

Q

js

!

.

Now set z = (z

1

, z

2

, . . . , z

d

)

>

with z

s

:= x

>

R

s

(x)x and

then |z

s

| ≤ R

s

kxk

2

and kzk ≤ kxk

2

R

F

for x ∈ D

∗

.

Then

|E

F

(x)| ≤



∑

s

E

s

(x)∂

s

V (x)



≤

p

2

kxk

p−2

Q



∑

s,i, j,k

x

j

x

k

R

s

jk

(x)x

i

Q

is



=

p

2

kxk

p−2

Q



∑

s,i

x

i

Q

is

∑

j,k

x

j

R

s

jk

(x)x

k

!



=

p

2

kxk

p−2

Q



∑

s,i

x

i

Q

is



x

>

R

s

(x)x





=

p

2

kxk

p−2

Q



∑

s,i

x

i

Q

is

z

s



=

p

2

kxk

p−2

Q



x

>

Qz



=

p

2

kxk

p−2

Q

kQ

1

2

xkkQ

1

2

zk

≤

p

2

kxk

p−2

Q

kxk

Q

kQ

1

2

kkzk

≤

p

2

kxk

p−1

Q

kxk

2

kQ

1

2

kR

F

.

Local Lyapunov Functions for Nonlinear Stochastic Differential Equations by Linearization

583

Since E

G

(x) =

1

2

∑

r,s

E

rs

(x)∂

2

rs

V (x) and by using

our expressions for E

rs

and ∂

2

rs

V (x) we obtain:

|E

G

(x)| ≤

1

4

pkxk

p−4

Q



∑

r,s,k,l,m

x

k

x

l

x

m

×

U

∑

u=1

(G

u

rk

R

us

lm

(x) + G

u

sk

R

ur

lm

(x))

×

Q

rs

kxk

2

Q

+ (p −2)

∑

i, j

x

i

x

j

Q

ir

Q

js

!



+

1

8

pkxk

p−4

Q



∑

r,s,k,l,m,n

x

k

x

l

x

m

x

n

×

Q

rs

kxk

2

Q

+ (p −2)

∑

i, j

x

i

x

j

Q

ir

Q

js

!

×

U

∑

u=1

R

ur

kl

(x)R

us

mn

(x)



.

We now estimate the expression on the right-hand

side term by term: Set z

u

= (z

u

1

, z

u

2

, . . . , z

u

d

)

>

, where

z

u

i

:= x

>

R

ui

(x)x, and then |z

u

i

|≤ R

ui

kxk

2

and kz

u

k≤

kxk

2

R

u

G

for x ∈ D

∗

. Then

∑

r,s,k,l,m

x

k

x

l

x

m

U

∑

u=1

G

u

rk

R

us

lm

(x)Q

rs

kxk

2

Q

= kxk

2

Q

U

∑

u=1

∑

r,s,k

x

k

∑

l,m

x

l

R

us

lm

x

m

!

Q

sr

G

u

rk

= kxk

2

Q

U

∑

u=1

∑

r,s,k

x

k



x

>

R

us

x



Q

sr

G

u

rk

= kxk

2

Q

U

∑

u=1

∑

r,s,k

z

u

s

Q

sr

G

u

rk

x

k

= kxk

2

Q

U

∑

u=1

(z

u

)

>

QG

u

x

= kxk

2

Q

U

∑

u=1

(z

u

)

>

QG

u

Q

−

1

2

Q

1

2

x

≤ kxk

3

Q

kxk

2

U

∑

u=1

kQG

u

Q

−

1

2

kR

u

G

≤ kxk

3

Q

kxk

2

kQ

1

2

k

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

kR

u

G

and similarly

∑

r,s,k,l,m

x

k

x

l

x

m

U

∑

u=1

G

u

sk

R

ur

lm

(x)Q

rs

kxk

2

Q

= kxk

2

Q

U

∑

u=1

∑

r,s,k



x

>

R

ur

x



Q

rs

G

u

sk

x

k

= kxk

2

Q

U

∑

u=1

∑

r,s,k

z

u

r

Q

rs

G

u

sk

x

k

= kxk

2

Q

U

∑

u=1

(z

u

)

>

QG

u

x

≤ kxk

3

Q

kxk

2

U

∑

u=1

kQG

u

Q

−

1

2

kR

u

G

≤ kxk

3

Q

kxk

2

kQ

1

2

k

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

kR

u

G

.

Further

∑

r,s,k,l,m

x

k

x

l

x

m

U

∑

u=1

G

u

rk

R

us

lm

(x)(p −2)

∑

i, j

x

i

x

j

Q

ir

Q

js

= (p −2)

U

∑

u=1

∑

j,s

∑

i,k,r

x

i

Q

ir

G

u

rk

x

k

!

×

∑

l,m

x

l

R

us

lm

(x)x

m

!

Q

s j

x

j

= (p −2)

U

∑

u=1

∑

j,s



x

>

QG

u

x



x

>

R

us

x



Q

s j

x

j

= (p −2)

U

∑

u=1

∑

j,s



x

>

QG

u

Q

−

1

2

Q

1

2

x



x

>

R

us

x



Q

s j

x

j

≤ |p −2|

U

∑

u=1

kxk

2

Q

kQ

1

2

G

u

Q

−

1

2

k



∑

j,s

z

u

s

Q

s j

x

j



≤ |p −2|kxk

2

Q

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

k



(z

u

)

>

Qx



≤ |p −2|kxk

2

Q

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

kkz

u

kkQ

1

2

kkxk

Q

≤ |p −2|kxk

3

Q

kxk

2

kQ

1

2

k

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

kR

u

G

CTDE 2018 - Special Session on Control Theory and Differential Equations

584

and similarly

∑

r,s,k,l,m

x

k

x

l

x

m

U

∑

u=1

G

u

sk

R

ur

lm

(x)(p −2)

∑

i, j

x

i

x

j

Q

ir

Q

js

= (p −2)

U

∑

u=1

∑

i,r

∑

j,k,s

x

i

Q

js

G

u

sk

x

k

!

×

∑

l,m

x

l

R

ur

lm

(x)x

m

!

Q

ri

x

i

= (p −2)

U

∑

u=1

∑

i,r



x

>

QG

u

x



x

>

R

ur

x



Q

ri

x

i

≤ |p −2|

U

∑

u=1

kxk

2

Q

kQ

1

2

G

u

Q

−

1

2

k



∑

i,r

z

u

r

Q

ri

x

i



≤ |p −2|kxk

3

Q

kxk

2

kQ

1

2

k

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

kR

u

G

.

Further

∑

r,s,k,l,m,n

x

k

x

l

x

m

x

n

Q

rs

kxk

2

Q

U

∑

u=1

R

ur

kl

(x)R

us

mn

(x)

= kxk

2

Q

U

∑

u=1

∑

r,s

∑

k,l

x

k

R

ur

kl

(x)x

l

!

Q

rs

×

∑

m,n

x

m

R

us

mn

(x)x

n

!

= kxk

2

Q

U

∑

u=1

∑

r,s

z

u

r

Q

rs

z

u

s

= kxk

2

Q

U

∑

u=1



(z

u

)

>

Qz

u



≤ kxk

2

Q

U

∑

u=1

kQkkz

u

k

2

≤ kxk

2

Q

kxk

4

kQk

U

∑

u=1

(R

u

G

)

2

= kxk

2

Q

kxk

4

kQkR

G

≤ kxk

4

Q

kxk

2

kQ

−1

kkQkR

G

= kxk

4

Q

kxk

2

κ(Q)R

G

.

Finally

∑

r,s,k,l,m,n

x

k

x

l

x

m

x

n

(p −2)

∑

i, j

x

i

x

j

Q

ir

Q

js

U

∑

u=1

R

ur

kl

(x)R

us

mn

(x)

= (p −2)

U

∑

u=1

∑

i, j,r,s

x

i

Q

ir

∑

k,l

x

k

R

ur

kl

(x)x

l

!

x

j

Q

js

×

∑

m,n

x

m

R

us

mn

(x)x

n

!

= (p −2)

U

∑

u=1

∑

i, j,r,s

x

i

Q

ir

z

u

r

x

j

Q

js

z

u

s

= (p −2)

U

∑

u=1

∑

i,r

x

i

Q

ir

z

u

r

!

∑

i,r

x

j

Q

js

z

u

s

!

= (p −2)

U

∑

u=1



x

>

Qz

u



2

≤ |p −2|

U

∑

u=1

kxk

2

Q

kQ

1

2

k

2

kz

u

k

2

≤ |p −2|kxk

2

Q

kxk

4

kQkR

G

≤ |p −2|kxk

4

Q

kxk

2

kQ

−1

kkQkR

G

= |p −2|kxk

4

Q

kxk

2

κ(Q)R

G

.

By combining the results from these estimates we get

|E

G

(x)| ≤

1

4

pkxk

p−4

Q



2kxk

3

Q

kxk

2

kQ

1

2

k

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

kR

u

G

+ 2|p −2|kxk

3

Q

kxk

2

kQ

1

2

k

U

∑

u=1

kQ

1

2

G

u

Q

−

1

2

kR

u

G



+

1

8

pkxk

p−4

Q



kxk

4

Q

kxk

2

κ(Q)R

G

+ |p −2|kxk

4

Q

kxk

2

κ(Q)R

G



=

1

2

pkxk

p−1

Q

kxk

2

(1 + |p −2|)

×

kQ

1

2

k

U

∑

u=1

R

u

G

kQ

1

2

G

u

Q

−

1

2

k+

1

4

κ(Q)R

G

kxk

Q

!

and we can estimate

|E(x)|≤ |E

F

(x)|+ |E

G

(x)|

≤

1

2

pkxk

p−2

Q

kxk

2

·kxk

Q



e

E

G

+

e

E

∗

G

kxk

Q



,

which proves the ﬁrst stated inequality.

Local Lyapunov Functions for Nonlinear Stochastic Differential Equations by Linearization

585

Since

LV (x) = L

0

V (x) +E(x)

≤ −

1

2

pCkxk

p−2

Q

kxk

2

+ E(x)

≤ −

1

2

pkxk

p−2

Q

kxk

2

h

C −kxk

Q



e

E

G

+

e

E

∗

G

kxk

Q

i

we have LV (x) < 0 if

kxk

Q



e

E

G

+

e

E

∗

G

kxk

Q



< C,

i.e.

kxk

Q

<

−

e

E

G

+

q

(

e

E

G

)

2

+ 4C

e

E

∗

G

2

e

E

∗

G

.

Thus for

x ∈ D = {x ∈R

d

: kxk

Q

≤ ρ}

with

ρ < min

(

ρ

∗

,

1

2

e

E

∗

G



q

(

e

E

G

)

2

+ 4C

e

E

∗

G

−

e

E

G



)

,

we have LV (x) < 0, which concludes the proof.

4 CONCLUSIONS

We derived rigid bounds on a domain, on which a

Lyapunov function for a linearized stochastic differ-

ential equation is also a Lyapunov function for the

original nonlinear system. This allows for the deriva-

tion of a lower bound on the equilibrium’s γ-basin of

attraction, i.e. the area in which all started solutions

converge to the equilibrium with probability no less

than γ. Another application is the facilitation of a nu-

merical method to compute Lyapunov functions for

nonlinear stochastic differential equations on a larger

domain as discussed in (Gudmundsson and Hafstein,

2018), because one ﬁrst needs a local Lyapunov func-

tion at the equilibrium.

ACKNOWLEDGEMENT

The research done for this paper was supported by the

Icelandic Research Fund (Rann

´

ıs) in the project ‘Lya-

punov Methods and Stochastic Stability’ (152429-

051), which is gratefully acknowledged.

REFERENCES

Gudmundsson, S. and Hafstein, S. (2018). Probabilistic

basin of attraction and its estimation using two Lya-

punov functions. Complexity, Article ID:2895658.

Hafstein, S. (2004). A constructive converse Lyapunov the-

orem on exponential stability. Discrete Contin. Dyn.

Syst. Ser. A, 10(3):657–678.

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