Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System

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


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.

Journal of Applied Mathematics
In this paper, we consider the following differential equations: . . .
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   (x) denote the Jacobian matrix of system { 1 , . . .,   } at point x.If all the eigenvalues of   (x) have negative real parts, then x is asymptotically stable.If the matrix   (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   (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  : U → R such that (1) : (0) = 0 and (x) > 0 whenever x ̸ = 0; (2) : (/)(0) = 0 and (/)(x) < 0 whenever x ̸ = 0.

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].
Theorem 3 (see [17]).Let (x) : R  → R  be a polynomial system, and x ∈ R  .Let IR be the set of real intervals, and IR  and IR × be the set of real interval vectors and real interval matrices, respectively.Given X ∈ IR  with 0 ∈ X and  ∈ IR × satisfies ∇  (x + X) ⊆   , for  = 1, 2, . . ., .Denote by   the identity matrix and assume where  x (x) is the Jacobian matrix of (x) at x. Then there is a unique x ∈  such that (x) = 0.Moreover, every matrix  ∈  is nonsingular, and the Jacobian matrix  x (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: Here n is a random vector in R  .The system has  +  equations and + variables; thus we can find real points through solving system (3).

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 (x), the following theorem gives a sufficient condition for the polynomial to be a Lyapunov function.
By the theory of linear algebra, one knows that the symmetric matrix ()| x=0 is positive definite if and only if all its eigenvalues are positive, and ((/))| x=0 is negative definite if and only if all its eigenvalues are negative. Let 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 ≤  ≤ , (−1)   − > 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 = { 1 > 0,  2 > 0, . . .,   > 0} . ( 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 = ( 1 ,  2 , . . .,   ), and construct the equation system as follows: 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 { 1 (a) > 0,  2 (a) > 0, . . .,   (a) > 0} , 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  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 .
Output: a Lyapunov function or UNKNOW.
In the following, we present a simple example to illustrate our algorithm.
Step 5. We obtain the first approximate real root of the system Substituting  = −2.407604610156789, = 4.633115716668555 into the left of the positive polynomial in (9), we obtain the following result: [11.26623143, 12.73590291, 17.71725365, 34.91943333] . ( This ensure the establishment of inequality in (9). Thus, is a Lyapunov function.
If the random hyperplanes {ℎ 1 , ℎ 2 } are as follows: we find that polynomial system {ℎ 1 = 0, ℎ 2 = 0,  = 0} has no real root; then we go to Step 7 in Algorithm 6 and obtain the following system: Solving the system { 1 = 0, 3 = 0}, we find the first approximate real root and substitute the value of  = 1.3053335232048229,  = 0.4314538107033688 into the left of the positive polynomial in (9) and we obtain the following result: This ensures the establishment of inequality in (9).Thus, is a Lyapunov function.

Experiments
In this section, some examples are given to illustrate the efficiency of our algorithm.
Example 8.This is an example from We assume that (, , ) =    Assume that (, , ) =  2 +  +  +  2 +  +  2 .With about 2.4 s, we got a real root for the parameters that form the coefficients of .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].Assume that  =  2 +  +  2 +  +  2 .For this program, our algorithm stops at Step 3, using about 1.24475 s.In [3], they use about 840 s.

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.