On Algebraic Proofs of Stabilityfor Homogeneous Vector Fields

On Algebraic Proofs of Stability for Homogeneous Vector Fields

Abstract

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sums of squares certificates and hence such a Lyapunov function can always be found by semidefinite programming. This generalizes the classical result that an asymptotically stable linear system admits a quadratic Lyapunov function which satisfies a certain linear matrix inequality, and can be of use to also show local asymptotic stability of non-homogeneous vector fields, by showing asymptotic stability of their lowest order homogeneous component.

We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a rational Lyapunov function, and that in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones.

Index Terms. Converse Lyapunov theorems, nonlinear dynamics, algebraic methods in control, semidefinite programming.

1 Introduction and outline of contributions

We are concerned in this paper with a continuous time dynamical system

(1)

where is continuously differentiable and has an equilibrium at the origin, i.e., . The problem of deciding asymptotic stability of equilibrium points of such systems is a fundamental problem in control theory. The goal of this paper is prove that if is a homogeneous vector field (see the definition below), then asymptotic stability is equivalent to existence of a Lyapunov function that is the ratio of two polynomials (i.e., a rational function). We also address the computational question of finding such a Lyapunov function in the case where the vector field is polynomial.

A scalar valued function is said to be homogeneous of degree if it satisfies for all and all . Similarly, we say that a vector field is homogeneous of degree if for all and all . Homogeneous systems have been extensively studied in the literature on nonlinear control; see e.g. [1], [2], [3, Sect. 57], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. These systems are not only of interest as is: they can also be used to study properties of related non-homogeneous systems. For example, if one can show that the vector field corresponding to the lowest-degree nonzero homogeneous component of the Taylor expansion of a nonlinear vector field is asymptotically stable, then the vector field itself will be locally asymptotically stable.

We recall that the origin of (1) is said to be stable in the sense of Lyapunov if for every , there exists a such that

We say that the origin is locally asymptotically stable if it is stable in the sense of Lyapunov and if there exists a scalar such that

The origin is globally asymptotically stable if it is stable in the sense of Lyapunov and for any initial condition in . A basic fact about homogeneous vector fields is that for these systems the notions of local and global asymptotic stability are equivalent. Indeed, the values that a homogeneous vector field takes on the unit sphere determines its value everywhere.

It is also well known that the origin of (1) is globally asymptotically stable if there exists a continuously differentiable Lyapunov function which is radially unbounded (i.e., satisfies as ), vanishes at the origin, and is such that

(2)
(3)

Throughout this paper, whenever we refer to a Lyapunov function, we mean a function satisfying the aforementioned properties. We say that is positive definite if it satisfies (2). When is a homogeneous function, the inequality (2) can be replaced by

where here denotes the unit sphere of . It is straightforward to check that a positive definite homogeneous function is automatically radially unbounded.

Our first contribution in this paper is to show that an asymptotically stable homogeneous vector field always admits a Lyapunov function which is rational function (Theorem 8). This is done by building up on a prior result in the literature on existence of homogeneous Lyapunov functions [10], [3], [13], [14] and proving a theorem on simultaneous approximation of homogeneous functions and their derivatives by homogeneous rational functions (Theorem 2.1).

1.1 Polynomial vectors fields

We will pay special attention in this paper to the case where the vector field is polynomial. Polynomial differential equations appear ubiquitously in applications—either as true models of physical systems or as approximations of other families of nonlinear dynamics—and have received a lot of attention in recent years because of the advent of promising analysis techniques using sum of squares optimization [15], [16], [17], [18], [19], [20]. In a nutshell, these techniques allow for an automated search over (a subset of) polynomial Lyapunov functions of bounded degree using semidefinite programming. However, there are comparatively few converse results in the literature (see [21], [22], [23], [24] for some examples) on guaranteed existence of such Lyapunov functions.

In [25], the authors prove that there are globally asymptotically stable polynomial vector fields (of degree as low as 2) which do not admit polynomial Lyapunov functions. We show in this paper that the same example in [25] does not even admit a rational Lyapunov function (Section 3.1). This counterexample justifies the homogeneity assumption of our main theorem.

Homogeneous polynomial vector fields of degree 1 are nothing but linear systems: . In this case, asymptotic stability of the origin can be tested in polynomial time by checking if the matrix is Hurwitz. As the following classical theorem demonstrates, this condition is equivalent to existence of a quadratic Lyapunov function. In the statement of this theorem, denotes the canonical inner product in , is the identity matrix of appropriate size, and the notation is used to denote that all eigenvalues of the symmetric matrix are nonnegative.

Theorem 1.1 (see e.g. Theorem 4.6 of [26]).

The dynamical system is asymptotically stable if and only if there exist symmetric matrices and such that

Moving up in the degrees, one can show that homogeneous vector fields of even degree can never be asymptotically stable [3, Sect. 17]. When the degree of is odd and , testing asymptotic stability of (1) is not a trivial problem. In fact, already when the degree of is equal to 3 (and even if we restrict to be a gradient vector field), the problem of testing asymptotic stability is known to be strongly NP-hard [27]. This result rules out the possibility of a polynomial time or even pseudo-polynomial time algorithm for this task unless P=NP. One difficulty that arises here is that tests of stability based on linearization fail. Indeed, the linearization of around the origin gives the identically zero vector field. This means (see e.g. [26, Thm. 4.15]) that homogeneous polynomial vector fields of degree are never exponentially stable. This fact is independently proven by Hahn in [3, Sect. 17].

The main contribution of this paper is to give a generalization of the classical result in Theorem 1.1 to homogeneous polynomial vector fields of higher degree:

Theorem (Main): A homogeneous polynomial dynamical system of degree is asymptotically stable if and only if there exist a nonnegative integer , a positive even integer , with , and symmetric matrices and , such that

(4)

where (resp. ) here denotes the vector of monomials in of degree (resp. ), and denotes the Jacobian of .

This theorem (which appears as Theorem 4.3 in Section 4) implies that a certificate of stability of homogeneous polynomial vector fields can always be found using semidefinite programming (SDP). This is because the constraint (4) is equivalent to the condition that the coefficients of the polynomials on the right and left hand sides of (4) match. This gives rise to a finite list of linear equations in the entries of and , which together with the required semidefinite constraints, leads to a semidefinite program when and are fixed.

The proof of this theorem can be found in Section 4 and also relies on results from Section 3. The statement of the above theorem is equivalent to existence of a rational Lyapunov function of the form

(5)

for system (1), where is a homogeneous positive definite polynomial of degree . Note that Theorem 1.1 corresponds to the case where the Lyapunov function in (5) has and .

Our next contribution in the paper is to show that the degree of in (5) cannot be bounded as a function of the dimension and degree of the vector field only (Section 5). This is done by presenting a family of asymptotically stable vector fields in dimension 2 and degree 3 that require rational Lyapunov functions of arbitrarily high degree (Theorem 5.2). We leave open the possibility that the degree of in (5) can be bounded as a computable function of the entries of . Such a statement (if true), together with the fact that semidefinite feasibility problems can be solved in finite time [28], would imply that the question of testing asymptotic stability for homogeneous polynomial vector fields is decidable. Decidability of asymptotic stability for polynomial vector fields is an outstanding open question of Arnlod; see [29], [30], [31].

Finally, in Section 6, we show a curious advantage of rational Lyapunov functions over polynomial ones. In Theorem 6.1, we give a family of homogeneous polynomial vector fields of degree 5 that all admit a low-degree rational Lyapunov function but require polynomial Lyapunov functions of arbitrarily high degree. We end the paper with some concluding remarks and future research directions in Section 7.

2 Approximation of homogeneous functions by rational functions

For a positive even integer , let denote the set of continuously differentiable homogeneous functions of degree . For a function , we define the norm as

This section proves the following theorem about approximation of functions in by rational functions.

Theorem 2.1.

For any and for any , there exist an even integer and a homogeneous polynomial of degree such that

Proof.

Fix and . For every integer , define the Bernstein polynomial of order as

The polynomial has degree , and has the property that for large enough, it satisfies

(6)

See [32, Theorem 4] for a proof. Let be fixed now and large enough for the above inequalities to hold. Since is an even function, the function

also satisfies (6). Because is even, the function

is of the form , where is a homogeneous polynomial and is an even integer. Also, by homogeneity, the degree of is .

It is clear that and are equal on the sphere, so

We argue now that the gradient of is close to the gradient of on the sphere. For that, fix . By Euler’s identity for homogeneous functions

Since , it is enough to control the part of the gradient that is orthogonal to . More precisely, let be the projection of a vector onto the hyperplane tangent to at the point . The following shows that and are equal when projected on :

Here, the second equation comes from the fact that and that the Jacobian of is equal to on , and the third equation relies on the fact that the projection of vector proportional to onto is zero.

Therefore,

We conclude by noting that

3 Rational Lyapunov functions

3.1 Nonexistence of rational Lyapunov functions

It is somewhat natural to wonder whether globally asymptotically stable polynomial vector fields always admit a rational Lyapunov function. In this subsection we show that this is not the case, hence also justifying the need for the homogeneity assumption in the statement of our main result (Theorem 1.1).

It has been shown in [25] that the polynomial vector field

(7)

is globally asymptotically stable (as shown by the Lyapunov function ) but does not admit a polynomial Lyapunov function. We show here that this vector field does not admit a rational Lyapunov functions either.

Suppose for the sake of contradiction that the system had a Lyapunov function of the form

where and are polynomials. Note first that the solution to system (7) from any initial condition can be written explicitly:

In particular, a solution that starts from for reaches the point after time

As , the function must satisfy

i.e.,

Fix and note that since as , then necessarily the degree of is larger than the degree of . We can see from this that the left-hand side of the above inequality grows exponentially in while the right-hand side grows polynomially, which cannot happen.

3.2 Rational Lyapunov functions for homogeneous dynamical systems

We now show that existence of rational Lyapunov functions is necessary for stability of homogeneous vector fields.

Theorem 3.1.

Let be a homogeneous, continuously differentiable function of degree . Then the system is asymptotically stable if and only if it admits a Lyapunov function of the type

(8)

where is a nonnegative integer and is a homogeneous (positive definite) polynomial of degree .

Proof.

The “if direction” of the theorem is a standard application of Lyapunov’s theorem; see e.g. [26, Thm. 4.2].

For the “only if” directionx, suppose is continuously differentiable homogeneous function of degree , and that the system is asymptotically stable. A result of Rosier [10, Thm. 2] (see also [3, Thm. 57.4] [13, Thm. 36] [14, Prop. p.1246]) implies that there exists a function such that

Since these inequalities are strict and involve continuous functions, we may assume that there exists a such that

Let

Theorem 2.1 proves the existence of a function of the form (8) that satisfies

Fix . An application of the Cauchy-Schwarz inequality gives

Therefore

4 An SDP hierarchy for searching for rational Lyapunov functions

For a rational function of the type in (5) to be a Lyapunov function, we need the polynomial and

to be positive definite. This condition is equivalent to the polynomials in the numerators of and being positive definite. In this section, we prove a stronger result that shows that one can always find a rational Lyapunov function whose two positivity requirements have sum of squares certificates. This makes the search for such a Lyapunov function a semidefinite program.

Recall that a homogeneous polynomial of degree is a sum of squares (sos) if it can be written as for some (homogeneous) polynomials . This is equivalent to existence of a symmetric positive semidefinite matrix that satisfies

(9)

where is the vector of all monomials of degree . We say that is strictly sos if it is in the interior of the cone of sos homogeneous polynomials of degree . This is equivalent to existence of a positive definite matrix that satisfies (9). Note that a strictly sos homogeneous polynomial is positive definite. We will need the following Positivstellensatz due to Scheiderer.

Theorem 4.1 (Scheiderer [33], [34]).

For any two positive definite homogeneous polynomials and , there exists an such that the polynomial is strictly sos for all integers .

Theorem 4.2.

If a homogeneous polynomial dynamical system admits a rational Lyapunov function of the form

where is a homogeneous polynomial, then it also admits a rational Lyapunov function

where the numerators of and are both strictly sos homogeneous polynomials.

Proof.

The condition that be positive definite is equivalent to being positive definite. The gradient of is equal to

If we let

then the condition that be positive definite is equivalent to being positive definite.

We claim that there exists a positive integer , such that

satisfies the conditions of the theorem. Indeed, by applying Theorem 4.1 with , there exists , such that is strictly sos for all integers .

Let us now examine the gradient of a function of the type . We have

Hence,

Since the homogeneous polynomials and are both positive definite, by Theorem 4.1, there exists an integer such that

is strictly sos for all . Taking finishes the proof as we can let

If we denote the dgree of by , then characterization (9) of strictly sos homogeneous polynomials applied to the numerator of and its derivative tells us that there exist positive definite matrices and such that

and

where (resp. ) here denotes the vector of monomials in of degree (resp. ). Notice that by multiplying by a positive scalar, we can assume without loss of generality that and .

Putting Theorem 8 and Theorem 4.2 together, we get the main result of this paper.

Theorem 4.3.

A homogeneous polynomial dynamical system of degree is asymptotically stable if and only if there exist a nonnegative integer , a positive even integer , with , and symmetric matrices and , such that

(10)

where (resp. ) here denotes the vector of monomials in of degree (resp. ), and denotes the Jacobian of .

For fixed integers and with , one can test for existence of matrices and that satisfy (10) by solving a semidefinite program. This gives rise to a hierarchy of semidefinite programs where one tries increasing values of , and for each , values of .

5 A negative result on degree bounds

The sizes of the matrices and that appear in the semidefinite programming hierarchy we just proposed depend on , but not . This motivates us to study whether one can bound as a function of the dimension and the degree of the vector field at hand. In this section, we show that the answer to this question is negative. In fact, we prove a stronger result which shows that one cannot bound the degree of the numerator of a rational Lyapunov function based on and only (even if one ignores the requirement that the Lyapunov function and its derivative have sos certificates of positivity).

To prove this statement, we build on ideas in reference [35] by Bacciotti and Rosier to construct a family of 2-dimensional degree-3 homogeneous polynomial vector fields that are asymptotically stable but necessitate rational Lyapunov function whose numerators have arbitrarily high degree.

Proposition 5.1.

Consider the vector field

(11)

parameterized by a scalar . For all values of , the origin is a center1 of (11), but for any irrational value of , there exist no two bivariate polynomials and such that the rational function

is nonzero, homogeneous, differentiable, and satisfies

for all .

Proof.

For the proof of the first claim see [35, Prop.5.2]. Our proof technique for the second claim is also similar to [35, Prop.5.2], except for some minor differences.

Suppose for the sake of contradiction that such a function exists. Let denote the degree of homogeneity of . We first observe that the function

satisfies

Therefore, on the level set

must be equal to a nonzero constant . A homogeneity argument shows that

Hence, by setting ,

(12)

Let be the largest nonnegative integer such that

where is a univariate polynomial. As a result, must satisfy , where . Then, from (12), we conclude that

(13)

The function can be prolonged to a holomorphic function on the open set

Furthermore, since , there exists an open neighborhood of where does not vanish. On the open set , the function

is holomorphic and does not vanish, and hence by (13), the function

is also holomorphic on . As a consequence, there exist an integer and a number such that

as . This implies that

and contradicts the assumption that is an irrational number. ∎

Theorem 5.2.

Let be a positive irrational real number and consider the following homogeneous cubic vector field parameterized by the scalar :

(14)

Then, for , the origin is asymptotically stable. However, for any positive integer , there exits a scalar such that the vector field in (14) does not admit a rational Lyapunov function with a homogeneous polynomial numerator of degree and a homogeneous polynomial denominator.

Proof.

Consider the positive definite Lyapunov function2

whose derivative along the trajectories of (14) is equal to

Since is negative definite for , it follows that for in this range, the origin of (14) is asymptotically stable.

To establish the latter claim of the theorem, suppose for the sake of contradiction that there exists an upper bound such that for all the system admits a rational Lyapunov function

where and are both homogeneous polynomials and is of degree at most independently of . Note that as a Lyapunov function, must vanish at the origin, and therefore the degree of is also bounded by . By rescaling, we can assume without loss of generality that the 2-norm of the coefficients of all polynomials and is 1.

Let us now consider the sequences and as . These sequences reside in the compact set of bivariate homogeneous polynomials of degree at most with the 2-norm of the coefficients equal to 1. Since every bounded sequence has a converging subsequence, it follows that there must exist a subsequence of (resp. ) that converges (in the coefficient sense) to some nonzero homogeneous polynomial (resp. ). Define . Since convergence of this subsequence also implies convergence of the associated gradient vectors, we get that

On the other hand, when , the vector field in (14) is the same as the one in (11) and hence the trajectories starting from any nonzero initial condition go on periodic orbits. This however implies that everywhere and in view of Proposition 5.1 we have a contradiction. ∎

Remark 5.1.

It is possible to establish the result of Theorem 5.2 without having to use irrational coefficients in the vector field. One approach is to take an irrational number, e.g. , and then think of a sequence of vector fields given by (14) that is parameterized by both and . We let the vector field in the sequence have and equal to a rational number representing up to decimal digits. Since in the limit as we have and , it should be clear from the proof of Theorem 5.2 that for any integer , there exists an asymptotically stable bivariate homogeneous cubic vector field with rational coefficients that does not have a Lyapunov where and are homogeneous and has degree less than .

6 Some advantages of rational Lyapunov functions over polynomial ones

In this section, we show that there are stable polynomial vector fields for which a polynomial Lyapunov function needs to have degree arbitrarily higher than the sum of the degrees of the numerator and the denominator of a rational Lyapunov function. This shows that for some vector fields, the SDP searching for a rational Lyapunov function can be much cheaper to solve than an SDP searching for a polynomial one. Note that the number of variables and constraints of the SDP arising form Theorem 4.3 does not depend at all on the degree of the denominator of the candidate rational Lyapunov function.

Theorem 6.1.

Consider the following homogeneous polynomial vector field parameterized by the scalar :

(15)

where

Then, for , the vector field admits the following rational Lyapunov function

and hence is asymptotically stable. However, for any positive integer , there exits a scalar such that does not admit a polynomial Lyapunov function of degree .

Our proof will use the following independent lemma about univariate polynomials.

Lemma 6.2.

There exist no two univariate polynomials and , with non-constant, that satisfy

Proof.

Assume for the sake of contradiction that such polynomials exist. For every nonnegative scalar , there exists a scalar such that . Therefore,

The expression above is an equality between two univariate rational functions valid on . Since both rational functions are well-defined on , the equality holds on that interval as well:

We get a contradiction by taking as the left hand side converges to , while the right hand side diverges to . ∎

Proof of Theorem 6.1: Let us first prove that is a rational Lyapunov function associated with the vector field whenever . It is clear that is positive definite and radially unbounded. A straightforward calculation shows that

hence

Note that the function is positive definite as

and is positive definite. This proves that when , the vector field is asymptotically stable with as a Lyapunov function.

To prove the latter claim of the theorem, suppose for the sake of contradiction that there exists an upper bound such that for all the system admits a polynomial Lyapunov function of degree at most .

By an argument similar to that in the proof of Theorem 5.2, there must exist some nonzero polynomial , with , that satisfies

We claim that must be constant on the level sets of . To prove that, consider an arbitrary positive scalar and the level set

Since is homogeneous and positive definite, is closed and bounded. In addition, is continuously differentiable and does not vanish on . Moreover, trajectories starting in remain in as

Hence, by the Poincaré-Bendixson Criterion [26, Lem 2.1], the set contains a periodic orbit of .

Let . We know that the trajectory starting from must visit . Since , we must have . Similarly, we must also have , and therefore .

Since we now know that is constant on the level sets of , there must exist a function such that

This proves that

Therefore, there exists a polynomial such that

(16)

Setting , we get that . Hence, for all . Taking , the second equality in (16) gives

Setting , we get that the polynomial satisfies

If we let

then in view of Lemma 6.2 and the fact that is not constant, we have a contradiction.

Example 6.1.

Consider the vector field in (15) with . One typical trajectory of this vector field is depicted in Figure 1. In this example, we use the modeling language YALMIP [36] and the SDP solver MOSEK [37] to search for rational and polynomial Lyapunov functions for this vector field.

Figure 1: A typical trajectory of the vector field in (15) with , together with the level sets of the Lyapunov functions and .

We know that for , the vector field is asymptotically stable. Therefore, by Theorem 4.3, the semidefinite programming hiearachy described in Section 4 is guaranteed to find a rational Lyapunov function. The first round to succeed corresponds to , and produces the feasible solution

If we look instead for a polynomial Lyapunov function, i.e. , the lowest degree for which the underlying SDP is feasible corresponds to . The Lyapunov function that our solver returns is the following polynomial:

Since all bivariate nonnegative homogeneous polynomials are sum of squares, the fact that our SDP is infeasible for means that admits no homogeneous polynomial Lyapunov function of degree lower than . Two level sets of and are shown in Figure 1 and they look quite similar.

7 Conclusions and future directions

We showed in this paper that existence of a rational Lyapunov function is necessary and sufficient for asymptotic stability of homogeneous vector fields. In the case where the vector field is polynomial, we constructed an SDP hiearachy that is guaranteed to find this Lyapunov function. The number of variables and constraints in this SDP hiearachy depend only on , the degree of the numerator of the candidate Lyapunov function, and not on , the degree of its denominator. To our knowledge, this theorem constitutes one of the few results in the theory of nonlinear dynamical systems which guarantees existence of algebraic certificates of stability that can be found by convex optimization (in fact, the only one we know of which applies to polynomial vector fields that are not exponentially stable). Regarding degree bounds, we proved that even for homogeneous polynomial vector fields of degree 3 on the plane, the degree of the numerator of such a rational Lyapunov function might need to be arbitrarily high. We also gave a family of homogeneous polynomial vector fields of degree 5 on the plane that all share a simple low-degree rational Lyapunov function, but require polynomial Lyapunov functions of arbitrarily high degree. Therefore, there are asymptotically stable polynomial vector fields for which a search for a rational Lyapunov function is much cheaper than a search for a polynomial one. We leave the following two questions for future research:

  • Can be upperbounded by a computable function of the coefficients of the vector field ? In particular, can