Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 859578, 5 pages
http://dx.doi.org/10.1155/2013/859578
Research Article
Constructing the Lyapunov Function through Solving Positive
Dimensional Polynomial System
Zhenyi Ji,1,2
Wenyuan Wu,2
Yong Feng,2
and Guofeng Zhang3
1
Laboratory of Computer Reasoning and Trustworthy Computation, School of Computer Science and Engineering,
University of Electronic Science and Technology of China, Chengdu 611731, China
2
Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology,
Chinese Academy of Science, Chongqing 401120, China
3
L.A.S Department of ChengDu College, University of Electronic Science and Technology of China, Chengdu 611731, China
Correspondence should be addressed to Zhenyi Ji; zyji001@163.com
Received 24 July 2013; Accepted 21 November 2013
Academic Editor: Bo-Qing Dong
Copyright © 2013 Zhenyi Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose an approach for constructing Lyapunov function in quadratic form of a differential system. First, positive polynomial
system is obtained via the local property of the Lyapunov function as well as its derivative. Then, the positive polynomial system is
converted into an equation system by adding some variables. Finally, numerical technique is applied to solve the equation system.
Some experiments show the efficiency of our new algorithm.
1. Introduction
Analysis of the stability of dynamical systems plays a very
important role in control system analysis and design. For
linear systems, it is easy to verify the stability of equilibria. For
nonlinear dynamical systems, proving stability of equilibria
of nonlinear systems is more complicated than linear systems.
One can use the Lyapunov function at the equilibria to
determine the stability.
For an autonomous polynomial system of differential
equations, how to compute the Lyapunov function at equilibria
is a basic problem. In [1, 2], the author transformed
the problem of computing the Lyapunov function into a quantifier
elimination problem. The disadvantage of the method
is that the computation complexity of quantifier elimination
is doubly exponential in the number of total variables. In
order to avoid this problem, She et al. [3] propose a symbolic
method; they first construct a special semialgebraic system
using the local properties of a Lyapunov function as well as
its derivative and solving these inequations using cylindrical
algebraic decomposition (CAD) introduced by Collins in
[4]. The algorithm in [5] uses semidefinite programming
to search for Lyapunov function. There are also other algorithms,
see [6, 7] for more details.
In this paper, we suppose Lyapunov function has quadratic
form and some coefficients of Lyapunov function are
unknown numbers. Some positive polynomials are obtained
using the technique mentioned in [3] first, then a positive
dimensional polynomial system is constructed by adding
some new variables. The parameter in Lyapunov function is
computed through solving the real root of the positive dimensional
system using the numerical method.
The rest of this paper is organized as follows: Definitions
and preliminaries about the Lyapunov function and the asymptotic
stability analysis of differential system are given in
Section 2. Section 3 reviews some methods for solving the
real root of positive dimensional polynomial system. The new
algorithm to compute the Lyapunov function and some experiments
are shown in Section 4. In Section 5, some examples
are given to illustrates the efficiency of our algorithm. Finally,
Section 6 draws a conclusion of this paper.
2. Stability Analysis of Differential Equations
In this section, some preliminaries on the stability analysis of
differential equations are presented.
2 Journal of Applied Mathematics
In this paper, we consider the following differential
equations:
̇𝑥1 = 𝑓1 (x)
̇𝑥2 = 𝑓2 (x)
...
̇𝑥 𝑛 = 𝑓𝑛 (x) ,
(1)
where x = (𝑥1, 𝑥2, . . . , 𝑥 𝑛), 𝑓𝑖 ∈ R[x], and 𝑥𝑖 = 𝑥𝑖(𝑡),
̇𝑥𝑖 = 𝑑𝑥𝑖/𝑑𝑡. A point x = (𝑥1, 𝑥2, . . . , 𝑥 𝑛) in the 𝑛-dimensional
real Euclidean space R 𝑛
is called an equilibrium of differential
system (1) if 𝑓𝑖(x) = 0 for all 𝑖 ∈ {1, 2, . . . , 𝑛}. Without loss of
generality, we suppose the origin is an equilibrium of the given
system in this paper.
In general, there exists two techniques to analyze the stability
of an equilibrium: the Lyapunov’s first method with the
technique of linearization which considers the eigenvalues of
the Jacobian matrix at equilibrium.
Theorem 1. Let 𝐽 𝐹(x) denote the Jacobian matrix of system
{𝑓1, . . . , 𝑓𝑛} at point x. If all the eigenvalues of 𝐽 𝐹(x) have
negative real parts, then x is asymptotically stable. If the matrix
𝐽 𝐹(x) has at least one eigenvalue with positive real part, then x
is unstable.
For a small system, it is easy to obtain the eigenvalues
of the matrix 𝐽 𝐹(x); then one can analyze the stability of the
equilibrium using Theorem 1. For a high-dimensional system,
solving the characteristic polynomial to get the exact zeros is
a difficult problem. Indeed, to answer the question on stability
of an equilibrium, we only need to know whether all the
eigenvalues have negative real parts or not. Therefore, the
theorem of Routh-Hurwitz [8] serves to determine whether
all the roots of a polynomial have negative real parts.
Another method to determine asymptotic stability is to
check if there exists a Lyapunov function at the point x, which
is defined in the following.
Definition 2. Given a differential system and a neighborhood
U of the equilibrium, a Lyapunov function with respect to the
differential system is a continuously differential function 𝐹 :
U → R such that
(1) : 𝐹(0) = 0 and 𝐹(x) > 0 whenever x ̸= 0;
(2) : (𝑑/𝑑𝑡)𝐹(0) = 0 and (𝑑/𝑑𝑡)𝐹(x) < 0 whenever x ̸= 0.
3. Solving the Real Roots of
Positive Dimensional Polynomial System
Solving polynomial system has been one of the central topics
in computer algebra. It is required and used in many scientific
and engineering applications. Indeed, we only care about the
real roots of a polynomial system arising from many practical
problems. For zero dimensional system, homotopy continuation
method [9, 10] is a global convergence algorithm. For
positive dimensional system, computing real roots of this
system is a difficult and extremely important problem.
Due to the importance of this problem, many approaches
have been proposed. The most popular algorithm which
solves this problem is CAD; another is the so-called critical
point methods, such as Seidenberg’s approach of computing
critical points of the distance function [11]. The algorithm in
[12] uses the idea of Seidenberg to compute the real root of
a positive dimensional defined by a signal polynomial; and
extends it to a random polynomial system in [13]. Actually,
these algorithms depend on symbolic computations, so they
are restricted to small size systems because of the high
complexity of the symbolic computation. In order to avoid
this problem, homotopy method has been used to compute
real root of polynomial system in [14, 15].
Recently, Wu and Reid [16] propose a new approach,
which is different from the critical point technique. In order
to facilitate the description of this algorithm, we suppose
polynomial system 𝑔 = {𝑔1, 𝑔2, . . . , 𝑔 𝑘}; the system has 𝑘
polynomials, 𝑛 variables, and 𝑘 < 𝑛. First, 𝑛 − 𝑘 hyperplanes
ℎ = {ℎ1, . . . , ℎ 𝑛−𝑘} in R[x] are chosen randomly. Note that
{𝑔1, . . . , 𝑔 𝑘, ℎ1, . . . , ℎ 𝑛−𝑘} is a square system; then witness points
are computed by homotopy method and verified by the
following theorem.
Theorem 3 (see [17]). Let 𝑓(x) : R 𝑛
→ R 𝑛
be a polynomial
system, and x ∈ R 𝑛
. Let IR be the set of real intervals, and IR 𝑛
and IR 𝑛×𝑛
be the set of real interval vectors and real interval
matrices, respectively. Given X ∈ IR 𝑛
with 0 ∈ X and 𝑀 ∈
IR 𝑛×𝑛
satisfies ∇𝑓𝑖(x + X) ⊆ 𝑀𝑖, for 𝑖 = 1, 2, . . . , 𝑛. Denote by
𝐼𝑛 the identity matrix and assume
−𝐹−1
x (x) 𝐹 (x) + (𝐼𝑛 − 𝐹x (x) 𝑀) X ⊆ int (X) , (2)
where 𝐹x(x) is the Jacobian matrix of 𝐹(x) at x. Then there is a
unique ̂x ∈ 𝑋 such that 𝑓(̂x) = 0. Moreover, every matrix 𝑀 ∈
𝑀 is nonsingular, and the Jacobian matrix 𝐹x(x) is nonsingular.
There may exist some components which have no intersection
with these random hyperplanes. Some points on
these components must be the solutions of the Lagrange
optimization problem:
𝑓 = 0,
𝑘
∑
𝑖=1
𝜆𝑖∇𝑓𝑖 = n. (3)
Here n is a random vector in R 𝑛
. The system has 𝑛 + 𝑘 equations
and 𝑛+𝑘 variables; thus we can find real points through
solving system (3).
4. Algorithm for Computing the
Lyapunov Function
In this section, we will present an algorithm for constructing
the Lyapunov function. Our idea is to compute positive
polynomial system which satisfies the definition of Lyapunov
function first. Then we solve the polynomial system deduced
from the positive polynomial system using homotopy algorithm;
at this step, we use the famous package hom4ps2 [18].
Given a quadratic polynomial 𝐹(x), the following theorem
gives a sufficient condition for the polynomial to be a
Lyapunov function.
Journal of Applied Mathematics 3
Theorem 4 (see [3]). Let 𝐹(x) be a quadratic polynomial,
for a given differential system; if 𝐹(x) satisfies the fact that
𝐻𝑒𝑠𝑠(𝐹)|x=0 is positive definite and 𝐻𝑒𝑠𝑠((𝑑/𝑑𝑡)𝐹)|x=0 is negative
definite, then 𝐹(x) is a Lyapunov function.
By the theory of linear algebra, one knows that the symmetric
matrix 𝐻𝑒𝑠𝑠(𝐹)|x=0 is positive definite if and only if all
its eigenvalues are positive, and 𝐻𝑒𝑠𝑠((𝑑/𝑑𝑡)𝐹)|x=0 is negative
definite if and only if all its eigenvalues are negative.
Let
ℎ = 𝑠 𝑛
+ 𝑡 𝑛−1 𝑠 𝑛−1
+ ⋅ ⋅ ⋅ + 𝑡0 (4)
be a characteristic polynomial of a matrix; the following theorem
deduced from the Descartes’ rule of signs [19] can be used
to determine whether ℎ has only positive roots or not.
Theorem 5 (see [3]). Suppose all the roots of a real polynomial
ℎ are real; then its roots are all positive if and only if for all
1 ≤ 𝑖 ≤ 𝑛, (−1)𝑖
𝑡 𝑛−𝑖 > 0.
Combine Theorems 4 and 5, finding that the Lyapunov
function in quadratic form can be converted into solving the
real root of some positive polynomial system, denoting it by
Inequ = {𝑔1 > 0, 𝑔2 > 0, . . . , 𝑔 𝑛 > 0} . (5)
Suppose we have obtained the positive polynomial system
as in (5), and denote the variable in the system by a. In order
to obtain one value of a using numerical technique, we first
convert the positive equation into equation. A simple ideal is
to add new variable set x = (𝑥1, 𝑥2, . . . , 𝑥 𝑛), and construct the
equation system as follows:
𝑝𝑠 = {𝑔1 − 𝑥2
1, 𝑔2 − 𝑥2
2, . . . , 𝑔 𝑛 − 𝑥2
𝑛} . (6)
If we find one real point (a, x) of system (6) such that there
has nonzero element in x, then it is easy to see that the point
a satisfies
{𝑔1 (a) > 0, 𝑔2 (a) > 0, . . . , 𝑔 𝑛 (a) > 0} , (7)
which means the differential system exists a Lyapunov function
at the equilibrium.
Note that the number of variable is more than the number
of equation in system (6); then the system 𝑝𝑠 must be a
positive dimensional polynomial system.
Recall the algorithm mentioned in Section 3; all of the
algorithms obtain at least one real point in each connect
component, and they use Theorem 3 to verify the existence of
real root which deduces the low efficiency. However, in this
paper, we only need one real point of system (6) to ensure
the establishment of these inequalities in (7), so we verify
the establishment of these inequalities using the residue of
inequalities at the real part of every approximate real root of
the system (6).
In the following we propose an algorithm to determine if
there exists a Lyapunov function at the equilibrium.
Algorithm 6. Input: a differential system as defined in (1) and
a tolerance 𝜖.
Output: a Lyapunov function or UNKNOW.
(1) Construct the positive polynomial.
(2) Convert the positive polynomial system into positive
dimensional system defined in system (6).
(3) We choose 𝑛 random point (̂x1, ̂x2, . . . , ̂x 𝑛) and 𝑛 random
vector k1, k2, . . . , k 𝑛; then construct 𝑛 hyperplane
in R 𝑛
through ̂x𝑖 with normal k𝑖 for 𝑖 = 1, 2, . . . , 𝑛.
Denote the set of this hyperplane by 𝑝𝑠2.
(4) Let 𝑝𝑠 = {𝑝𝑠1, 𝑝𝑠2}, and solve the square system using
homotopy continuation algorithm, denoting solution
of 𝑝𝑠 by 𝑟𝑜𝑜𝑡𝑠.
(5) for 𝑠 = 1 : 𝑙𝑒𝑛𝑔𝑡ℎ(𝑟𝑜𝑜𝑡𝑠)
(a) if the norm of imaginary part of 𝑟𝑜𝑜𝑡𝑠{𝑠} is
smaller than 𝜖, then substitute the real part of
𝑟𝑜𝑜𝑡𝑠{𝑠} into {𝑔1, . . . , 𝑔 𝑛}, and denote the value
by {V1, V2, . . . , V 𝑛}. If V𝑖 > 0 for all 𝑖 ∈ {1, 2, . . . , 𝑛},
then return the real part of 𝑟𝑜𝑜𝑡𝑠{𝑠} and break
the program.
(6) End for.
(7) Construct polynomial system 𝑝𝑠3 = ∑ 𝑛
𝑖=1 𝜆𝑖∇𝑓𝑖 =
k, where 𝜆𝑖 is new variable and k are chosen from
{k1, . . . , k 𝑛} randomly.
(8) Solve {𝑝𝑠1, 𝑝𝑠3} using homotopy continuation algorithm,
denote its solution by 𝑟𝑜𝑜𝑡𝑠, and go to Step 4.
(9) return UNKNOW.
In the following, we present a simple example to illustrate
our algorithm.
Example 7. This is an example from [20]
̇𝑥 = −𝑥 + 2𝑦3
− 2𝑦4
̇𝑦 = −𝑥 − 𝑦 + 𝑥𝑦.
(8)
Let Lyapunov function 𝐹(𝑥, 𝑦) = 𝑥2
+ 𝑎𝑥𝑦 + 𝑏𝑦2
.
Step 1. We obtain the positive polynomial using Theorems 4
and 5 as follows:
[2𝑏 + 2 > 0, −𝑎2
+ 4𝑏 > 0,
2𝑎 + 4𝑏 + 4 > 0, 4𝑎2
+ 4𝑏2
− 16𝑏 > 0] .
(9)
Step 2. Convert system (9) into the following system:
𝑝𝑠1 =
{{{
{{{
{
2𝑏 + 2 − 𝑥2
1 = 0
−𝑎2
+ 4𝑏 − 𝑥2
2 = 0
2𝑎 + 4𝑏 + 4 − 𝑥2
3 = 0
4𝑎2
+ 4𝑏2
− 16𝑏 − 𝑥2
4 = 0.
(10)
4 Journal of Applied Mathematics
Step 3. Construct two hyperplanes {ℎ1, ℎ2} in R6
randomly,
where
ℎ1 = 0.09713178123584754𝑎 + 0.04617139063115394𝑏
+ 0.27692298496089𝑥1 + 0.8234578283272926𝑥2
+ 0.694828622975817𝑥3 + 0.3170994800608605𝑥4
+ 0.9502220488383549,
ℎ2 = 0.3815584570930084𝑎 + 0.4387443596563982𝑏
+ 0.03444608050290876𝑥1 + 0.7655167881490024𝑥2
+ 0.7951999011370632𝑥3 + 0.1868726045543786𝑥4
+ 0.4897643957882311.
(11)
Step 4. Compute the roots of the augmented system {𝑝𝑠1 =
0, ℎ1 = 0, ℎ2 = 0} using homotopy method, and we find the
system has only 16 roots.
Step 5. We obtain the first approximate real root of the system
x = [−2.407604610156789, 4.633115716668555,
3.356520733339377, 3.568739680591174,
−4.209186815331512, −5.909266734956268] .
(12)
Substituting 𝑎 = −2.407604610156789, 𝑏 =
4.633115716668555 into the left of the positive polynomial
in (9), we obtain the following result:
[11.26623143, 12.73590291, 17.71725365, 34.91943333] .
(13)
This ensure the establishment of inequality in (9).
Thus,
𝐹 (𝑥, 𝑦) = 𝑥2
+ 4.633115716668555𝑦2
− 2.407604610156789𝑥𝑦
(14)
is a Lyapunov function.
If the random hyperplanes {ℎ1, ℎ2} are as follows:
ℎ1 = −3𝑎 − 𝑏 + 𝑥1 + 2𝑥2 − 2𝑥3 − 2𝑥4 − 3,
ℎ2 = 3𝑎 − 3𝑏 − 𝑥1 − 2𝑥2 + 𝑥3 + 2𝑥4 − 2,
(15)
we find that polynomial system {ℎ1 = 0, ℎ2 = 0, 𝑝𝑠 = 0} has
no real root; then we go to Step 7 in Algorithm 6 and obtain
the following system:
𝑝𝑠3 =
{{{{{{{
{{{{{{{
{
−2𝜆2 𝑎 + 2𝜆3 + 8𝜆4 𝑎 − 1 = 0
2𝜆1 + 4𝜆2 + 4𝜆3 + 𝜆4 (8𝑏 − 16) − 3 = 0
−2𝜆1 𝑥1 + 1 = 0
−2𝜆2 𝑥2 + 2 = 0
−2𝜆3 𝑥3 − 2 = 0
−2𝜆4 𝑥4 − 3 = 0.
(16)
Solving the system {𝑝𝑠1 = 0, 𝑝𝑠3 = 0}, we find the
first approximate real root and substitute the value of 𝑎 =
1.3053335232048229, 𝑏 = 0.4314538107033688 into the left
of the positive polynomial in (9) and we obtain the following
result:
[2.862907621406738, 0.021919636011159,
8.336482289223121, 0.656931019037197] .
(17)
This ensures the establishment of inequality in (9).
Thus,
𝐹 (𝑥, 𝑦) = 𝑥2
+ 0.4314538107033688𝑦2
+ 1.3053335232048229𝑥𝑦
(18)
is a Lyapunov function.
5. Experiments
In this section, some examples are given to illustrate the
efficiency of our algorithm.
Example 8. This is an example from [7]
̇𝑥 = 𝑦,
̇𝑦 = 𝑧,
̇𝑧 = −4𝑥 − 3𝑦 − 2𝑧 + 𝑥2
𝑦 + 𝑥2
𝑧.
(19)
We assume that 𝐹(𝑥, 𝑦, 𝑧) = 𝑥2
+𝑦2
+𝑧2
+𝑎𝑥𝑦+𝑏𝑥𝑧+𝑐𝑦𝑧.
Algorithm 6 returns a Lyapunov function
𝐹 (𝑥, 𝑦, 𝑧) = 𝑥2
+ 𝑦2
+ 𝑧2
+ 1.370502803658027𝑥𝑦
+ 0.655753434727512𝑥𝑧
+ 0.632220465746607𝑦𝑧,
(20)
at Step 4 using only 1.085175 s. If the algorithm does not
terminate at Step 4, it returns
𝐹 (𝑥, 𝑦, 𝑧) = 𝑥2
+ 𝑦2
+ 𝑧2
+ 0.566986159377122𝑥𝑦
+ 1.934844270891010𝑥𝑧
+ 0.065341301862036𝑦𝑧,
(21)
using about 21.285095 s.
Example 9. This is an example from a classic ODE’s textbook:
̇𝑥 = −𝑥 − 3𝑦 + 2𝑦 + 𝑦𝑧,
̇𝑦 = 3𝑥 − 𝑦 − 𝑧 + 𝑥𝑧,
̇𝑧 = −2𝑥 + 𝑦 − 𝑧 + 𝑥𝑦.
(22)
Assume that 𝐹(𝑥, 𝑦, 𝑧) = 𝑥2
+ 𝑎𝑥𝑦 + 𝑥𝑧 + 𝑐𝑦2
+ 𝑑𝑦𝑧 +
𝑒𝑧2
. With about 2.4 s, we got a real root for the parameters
that form the coefficients of 𝐹. Indeed, this point was obtained
from Step 4. If there is no real point at Step 4, this program
returns one real root using about 267 s, which is also more
efficient than 1800 s in [3].
Journal of Applied Mathematics 5
Example 10. This is another example from an ODE’s text-
book:
̇𝑥 = −𝑥 + 𝑦 + 𝑥𝑧2
− 𝑥3
,
̇𝑦 = 𝑥 − 𝑦 + 𝑧2
− 𝑦3
,
̇𝑧 = −𝑦𝑧 − 𝑧2
.
(23)
Assume that 𝐹 = 𝑥2
+ 𝑏𝑥𝑧 + 𝑐𝑦2
+ 𝑑𝑦𝑧 + 𝑒𝑧2
. For this
program, our algorithm stops at Step 3, using about 1.24475 s.
In [3], they use about 840 s.
6. Conclusion
For a differential system, based on the technique of computing
real root of positive dimensional polynomial system,
we present a numerical method to compute the Lyapunov
function at equilibria. According to the relationship between
the positive dimensional system and the Lyapunov function,
we know we just need only one real root of this system, so we
convert the algorithm into two steps. At each step, rather than
using interval Newton’s method to verify the existence of real
root, we use the residue of the positive polynomial system at
approximate real root to verify the correctness of the positive
polynomial system.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgments
This research was partially supported by the National Natural
Science Foundation of China (11171053) and the National
Natural Science Foundation of China Youth Fund Project
(11001040) and cstc2012ggB40004.
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