Global Stability Analysis of Fluid Flows using Sum-of-Squares

Abstract

This paper introduces a new method for proving global stability of fluid
flows through the construction of Lyapunov functionals. For finite dimensional
approximations of fluid systems, we show how one can exploit recently developed
optimization methods based on sum-of-squares decomposition to construct a
polynomial Lyapunov function. We then show how these methods can be extended to
infinite dimensional Navier-Stokes systems using robust optimization
techniques. Crucially, this extension requires only the solution of
infinite-dimensional linear eigenvalue problems and finite-dimensional
sum-of-squares optimization problems.
We further show that subject to minor technical constraints, a general
polynomial Lyapunov function is always guaranteed to provide better results
than the classical energy methods in determining a lower-bound on the maximum
Reynolds number for which a flow is globally stable, if the flow does remain
globally stable for Reynolds numbers at least slightly beyond the energy
stability limit. Such polynomial functions can be searched for efficiently
using the SOS technique we propose.

Full text

Page 1

arXiv:1101.1043v1 [math.OC] 5 Jan 2011
Global Stability Analysis of Fluid Flows using Sum-of-Squares
P. Goulart∗, S. Chernyshenko
Department of Aeronautics, Imperial College London,
Prince Consort Road, London, SW7 2AZ, United Kingdom
Abstract
This paper introduces a new method for proving global stability of fluid flows through
the construction of Lyapunov functionals. For finite dimensional approximations of fluid
systems, we show how one can exploit recently developed optimization methods based on
sum-of-squares decomposition to construct a Lyapunov function. We then show how these
methods can be extended to full infinite dimensional Navier-Stokes systems using robust
optimization techniques. Crucially, this extension requires only the solution of infinite-
dimensional linear eigenvalue problems and finite-dimensional sum-of-squares optimization
problems. The resulting method is guaranteed to provide results at least as good as classical
energy methods in determining a lower-bound on the maximum Reynolds number for which
a flow is globally stable.
Keywords:
Navier-Stokes; Flow stability; Sum-of-squares; Lyapunov methods
1. Background and problem statement
In this paper we propose a new analytical method for determining whether a fluid flow is
globally stable. This new approach has its origins in two hitherto distinct research areas.
The first of these is the classical energy approach of [13, 5], which provides conservative
lower bounds on the stability limits of flows by analyzing the time evolution of the energy
of flow perturbations. The other is the emerging field of sum-of-squares (SOS) optimization
over polynomials, which can be used to prove global stability of finite-dimensional systems
of ordinary differential equations with polynomial right-hand sides [10, 11]. It is our hope
that the present text is written in such a way that it will be understandable to researchers
∗Corresponding author. Tel: +442075945045.
Email addresses: p.goulart@imperial.ac.uk (P. Goulart), s.chernyshenko@imperial.ac.uk
(S. Chernyshenko)
Preprint submitted to ElsevierJanuary 6, 2011

Page 2

from either of these two areas, which until now have remained almost completely isolated.
We are aware of only two other publications where SOS methods have been used to analyze
the behavior of systems governed by partial differential equations, namely [8, 16].
The proposed approach is applied to the problem of assessing global stability of an incom-
pressible flow. The velocity w and pressure q of a flow of viscous incompressible fluid,
evolving inside a bounded domain Ω with boundary ∂Ω under the action of body force f,
is governed by the Navier-Stokes and continuity equations
∂w
∂t+ w · ∇w = −∇q +1
∇ · w = 0,
R∇2w + f
(1a)
(1b)
with a boundary condition w = 0 on ∂Ω. Here R is the Reynolds number, which is a
dimensionless parameter indicating the relative influence of viscous and inertial forces in
the flow. In what follows we will make extensive use of an inner product of vector fields
defined as
?u,v? :=
?
Ω
u · vdΩ
with the usual L2norm ?.? defined as ?u?2:= ?u,u?. Similarly, we define
?∇u,∇v? :=
?
Ω
∂ui
∂xj
∂vi
∂xjdΩ
and ?∇u?2:= ?∇u,∇u?.
We say that a steady solution w = ¯ u, q = ¯ p of the system (1) is globally stable if, for each
ǫ > 0, there exists some δ > 0 such that ?w − ¯ u? ≤ δ at time t0implies that ?w − ¯ u? ≤ ǫ
for all time t ≥ t0. We say that it is globally asymptotically stable if in addition w → ¯ u as
time t → ∞ for any initial conditions. Our principal aim is to identify the largest value R
for which these conditions can be guaranteed to hold for the system (1).
The results described in this paper can be extended to other types of boundary conditions,
most notably to the frequently encountered case of periodic boundary conditions. A useful
property of systems with such boundary conditions is that, for any solenoidal vector field
or fields satisfying these boundary conditions and for any scalar function φ, the following
hold true: ?v,∇φ? = 0,?v1,∇2v2
?v1,v2· ∇v3? = −?v3,v2· ∇v1?,
?=?v2,∇2v1
?= −?∇v1,∇v2? and
(2)
hence
?v,v · ∇v? = 0. (3)
2

Page 3

They can be proved by applying well known identities from vector calculus or simply
by integrating by parts, using incompressibility (∇ · v = 0) and applying the boundary
conditions. These properties are, of course, well known.
In particular, the identity (3) plays an important role in the energy approach to proving
global stability [13]. Introducing velocity perturbations u = w − ¯ u and pressure perturba-
tions p = q − ¯ p, the system (1) can be written as
∂u
∂t+ u · ∇u + S(u, ¯ u) = −∇p +1
∇ · u = 0,
where S(u,v) := u·∇v+v·∇u is introduced for compactness of notation. The time rate
of change of the velocity perturbation energy can be obtained by taking the inner product
of both sides of (4a) with u, to obtain the energy equation
R∇2u
(4a)
(4b)
∂ ?u?2/2
∂t
=1
R
?u,∇2u?− ?u,S(¯ u,u)?. (5)
Note that the nonlinear term u·∇u in (4a) does not feature in the energy equation because
of the identity (3). This is particularly useful because it allows one to obtain immediately
an (albeit conservative) method for checking stability of the system (4), which we now
describe briefly.
There exists a real constant κ such that for all solenoidal u,
1
R
?u,∇2u?− ?u,S(¯ u,u)? ≤ κ?u?2. (6)
The smallest such κ can be found as the solution of the following optimization prob-
lem [3, p. 33-34]
?
?u?2
subject to the incompressibility condition ∇ · u = 0 and the boundary conditions. Since
the ratio that appears in this optimization problem is homogeneous in u, one is free to
optimize over the numerator only, with additional constraint ?u? = 1. This leads to the
eigenvalue problem
λu = (e −1
∇ · u = 0, u|∂Ω= 0,
where p is the Lagrange multiplier for the incompressibility condition and λ is the Lagrange
multiplier for the unit norm condition ?u? = 1. Here e is the base-flow rate of strain tensor
with components
eij(x) :=1
2
sup
−1
R?∇u?2−?
Ωu · (∇¯ u) · udV
?
,
R∇2)u + ∇p
(7)
?∂¯ ui
∂xj+∂¯ uj
∂xi
?
.
3

Page 4

Note the identities used in arriving at (7); ?u,S(¯ u,u)? =????? ?
u ·(∇¯ u)· u = u · e ·u. Since the operator in (7) is self-adjoint [3], all the eigenvalues λkof
(7) are real. If these eigenvalues are ordered by decreasing value with λ1being the largest,
then the inequality (6) is tight with κ = λ1. If the largest eigenvalue λ1< 0, then (5) is
always negative for ?u? ?= 0 and hence the energy ?u?2/2 is a Lyapunov functional for (4),
thus proving the global stability.
?u, ¯ u · ∇u?
=0+ ?u,u · ∇¯ u? and
A particularly nice feature of the energy approach is that proving global stability requires
solutions of a linear eigenvalue problem only, even though (4) is a system of nonlinear partial
differential equations. This is a direct consequence of the unique advantages of using ?u?
as a Lyapunov functional, in particular the opportunity to exploit (3). Note that any other
choice of Lyapunov functional would generally result in a stability condition featuring the
nonlinear inviscid term u · ∇u, in contrast to the more benign energy stability condition
(5).
On the other hand, the energy approach can give very conservative results, in the sense
that the largest R for which global stability can be proven by this method is generally well
below the maximum R for which the flow is generally observed to be globally stable, either
numerically or experimentally.
The approach proposed in the present study aims to improve this bound, by using a partial
Galerkin decomposition of the infinite dimensional system (4). Finite dimensional meth-
ods, based on recently developed techniques in polynomial optimization, are used to define
a Lyapunov functional that is nonlinear in a finite number of terms, while otherwise main-
taining some of the attractive numerical advantages of energy methods for the remaining
(infinite dimensional) dynamics. We stress that our results suggest a way of computing a
Lyapunov functional verifying stability of the full infinite dimensional system (4), and not
some truncated finite dimensional approximation thereof.
1.1. Finite Dimensional Systems and the Sum-of-Squares Decomposition
We first comment briefly on the state of the art in direct methods for computing Lyapunov
functions for finite dimensional nonlinear systems. Suppose that the evolution of a finite-
dimensional system with state vector a ∈ Rnis governed by a set of ordinary differential
equations (ODEs)
˙ a = f(a) (8)
with equilibrium point a = 0. We will use ∇athroughout to indicate the gradient of a
functions defined on this n-dimensional state space, and otherwise use ∇ to indicate the
gradient or divergence of functions in physical space, as in (1).
The origin of the system (8) is globally asymptotically stable if there exists a continuously
differentiable Lyapunov function V : Rn→ R such that V (0) = 0, V (a) > 0 for all
4

Page 5

a ∈ R \ {0} and˙V (a) = ∇aV ·f(a) < 0 for all a ∈ R\{0} [6]. Given a Lyapunov candidate
function V (a) and associated˙V (a), these conditions amount to checking global positivity
or negativity of functions. There is of course no general method for performing such a
check, nor any systematic way of constructing Lyapunov functions for general systems of
ODEs.
A truncated Galerkin approximation reduces the Navier-Stokes equations (4) to a system
of ODEs in exactly the form (8), but with polynomial (in fact quadratic) right hand side
f(a). In this particular case, checking that a polynomial function V (a) serves as a Lyapunov
function reduces to verifying the positive-definiteness of the two related polynomials V (a)
and −˙V (a). However, verifying positive-definiteness of a general multivariate polynomial
is still NP-hard in general, and is a classical problem in algebraic geometry.
Nevertheless, there has been significant recent progress in stability analysis of polyno-
mial systems using sum-of-squares optimization methods, which were first employed in
the context of dynamical systems in [10]. These methods are based on a recognition that
a sufficient condition for a polynomial function to be positive-definite is that it can be
rewritten as a sum-of-squares (SOS) of lower order polynomial functions1. Verifying this
stronger condition, and solving other problems related to such representations, is signifi-
cantly simpler than verifying global positivity in general. Therefore the general approach
of sum-of-squares optimization in control applications is to search for a Lyapunov function
V (a) and associated function˙V (a) that satisfy sum-of-squares conditions.
Every polynomial V (a) of order 2k can be represented as a quadratic form of monomials
of order less than k + 1, i.e. in the form V = Cijmi(a)mj(a). The monomials mi(a) in
this factorization are expressions of the form ak1
satisfying?n
monomial set. Since all the diagonal elements in the resulting expression will be positive,
this gives a representation of the polynomial as a sum of squares of polynomials of lower
order. Such a representation is known as a sum-of-squares decomposition.
1ak2
2...akn
n, with integer exponents ki≥ 0
iki≤ k. If the matrix Cij is positive-definite and symmetric (the latter is
always possible), then it can be diagonalized by a suitable linear transformation of the
Hence, the problem of finding the Lyapunov function is reduced to finding the coefficients
Cij= Cjisuch that this matrix is positive-definite and the corresponding matrix factoriza-
tion representing −˙V (a) is also positive-definite. The relationship between the coefficients
of V (a) and the matrix Cijamounts to a set of linear equality constraints, with similar lin-
ear equality constraints relating the coefficients of˙V (a) and its factorization. A further set
1This condition is, however, not a necessary one, except in certain exceptional cases of involving relatively
few variables or low-order polynomials. An example of a polynomial function that is not a sum-of-squares
is the Motzkin polynomial y4z2+ y2z4− 3y2z2+ 1. A nice account of the history of this problem, the 17th
of Hilbert’s 23 famous problems posed at the turn of the 20th century, can be found in [12].
5

Page 6

of equality constraints couple the coefficients of the polynomial functions V (a) and˙V (a).
Problems such as that described above can be solved efficiently since the set of positive-
definite matrices is convex. The general field of optimization theory and numerical methods
related to such problems is known as semidefinite programming2, and such problems are
solvable in an amount of time that is polynomial in the size of their problem data[15, 14, 2].
Standard software tools are freely available for posing and solving sum-of-squares problems
[7, 9] as semidefinite programs.
1.2. Application of SOS methods to fluid systems
With respect to the SOS approach, ODE systems obtained via finite-dimensional approx-
imation of the Navier-Stokes equations require special treatment for two reasons. First,
since the ODEs describing the dynamics are quadratic, if V (a) is of even degree (as it
must to be positive-definite), then −˙V (a) is formally of odd degree, and hence will not
be positive-definite in general. Second, for three-dimensional flows the Lyapunov function
(or, rather, functional) should tend to a function of ?w? as ?u? → ∞, as can be inferred
from the fact that ?w? is the only invariant for three-dimensional inviscid flows and that
it decays monotonically due to the action of viscosity if ?w? is large enough.
We describe in Sections 2 and 3 how one can apply SOS methods to truncated ODE
approximations to (4) to obtain numerical estimates of the maximum value R for which
global stability can be proven. A preliminary version of this section has appeared in [4],
to which we also refer the reader for a related numerical example, where a stability limit
approximately seven times larger than the value demonstrable via energy methods was
obtained, and which was close to the global stability limit estimated by direct numerical
simulation.
There remains the question of convergence of global stability results obtained in this way
as the number of modes in the truncated Galerkin approximation tends to infinity; global
stability of a truncated approximation does not imply global stability of the Navier-Stokes
solution. This problem is particularly acute since one cannot realistically expect to apply
SOS methods to high resolution approximations of the Navier-Stokes equations, since the
size of the related optimization problems quickly becomes unmanageable.
We therefore demonstrate in Sections 4 and 5 how these difficulties can be overcome by
seeking a Lyapunov functional of the Navier-Stokes system in the form V = V (a,q2),
where a is a finite-dimensional vector of the amplitudes of several Galerkin modes, and q2
2Strictly speaking, the problem described here has positive-definite matrix constraints, rather than
semidefinite constraints as in standard semidefinite programming. The conditions described above can be
recast as semidefinite constraints via inclusion of appropriate terms, e.g. V (a) > 0 if V (a) − ǫ?a?2=
Cijmi(a)mj(a) ≥ 0 for some small ǫ > 0. In this case the semidefiniteness condition Cij ? 0 is sufficient.
6

Page 7

is the collective energy of all of the remaining modes. Such an approach requires estimates
via q for the terms stemming from the nonlinearity of the Navier-Stokes system, which is
immediately reminiscent of the difficulties preventing proof of existence of solutions to the
Navier-Stokes equations.
In the present context it turns out, however, that the required estimates are available
since they are needed only for the effect of higher-order modes on the finite set of Galerkin
modes. As a result, the required estimates can be obtained by solving only linear eigenvalue
problems in infinite dimensions and certain maximization problems in finite dimensions,
and the resulting system can be treated using the SOS approach. It will also be demon-
strated that with a suitable basis for the Galerkin approximation the proposed approach
is guaranteed to give results at least as good as the standard energy approach.
2. Finite Dimensional Flow Models
We will assume throughout that the perturbation velocity u can be written as
u(x,t) = ai(t)ui(x) + us(x,t),i = 1,...,n (9)
where the basis functions uiare mutually orthogonal, solenoidal, and satisfy the boundary
conditions. Likewise, usis assumed to be solenoidal, to satisfy the boundary conditions,
and to be orthogonal to the bases ui. Assume also that each of the basis functions has unit
norm, i.e. ?ui? = 1. For brevity, we will denote by S the set of all possible vector fields
that are solenoidal, satisfy the boundary conditions and are orthogonal to all ui, so that
us∈ S.
In order to address the global stability of the nonlinear Navier-Stokes system (4), we will
partition its dynamics into the interaction of an ODE, representing the evolution of the
basis weights a, and a PDE, representing the remaining unmodeled modes of the system
us. We work initially with the ODE part only, and hence assume initially that us= 0.
We first substitute (9) into (4a) and take an inner product of both sides with each of the
basis functions uiin turn, yielding an ODE in the form
˙ ai+ ?ui,uj· ∇uk?ajak+ ?ui,S(uj, ¯ u)?aj=1
R
?ui,∇2uj
?aj. (10)
Defining matrices Λ, W and?Qj?
Λij:=?ui,∇2uj
with L :=1
j∈1,...,nsuch that
?, Wij:= −?ui,S(uj, ¯ u)?, Qj
ik:= −?ui,uj· ∇uk?,
RΛ + W, and defining a linear matrix-valued operator N : Rn→ Rn×nas
N(a) := ajQj,
7

Page 8

one arrives at a compact representation of the ODE (10)
˙ a = f(a) := La + N(a)a.(11)
Two useful general observations about this system are that the matrix Λ is negative-definite
(in particular, it is diagonal if the basis functions uiare chosen as eigenfunctions of (7)),
and that aTN(a)a = 0 for all a. The latter assertion is a restatement of the energy
conservation relation (3) in finite dimensions.
3. Stability of Finite Dimensional Models using SOS
For simplicity of exposition, we will assume in this section that the base flow ¯ u is spanned
by the basis functions ui, i.e. that there exist some real constants cisuch that ¯ u = uici.
In this case, one can rewrite the dynamics of the finite dimensional system (11) as
˙ a =1
RΛa + N(a + c)(a + c) − N(c)c. (12)
We wish to find the largest value of R for which the above system is globally stable. To this
end, we first recall that an ODE system ˙ a = f(a) is stable if one can find a continuously
differentiable Lyapunov function V satisfying each of the following conditions [6]:
V (0) = 0(L1)
V (a) > 0
∀a ?= 0
∀a ?= 0.
(L2)
∇aV (a) · f(a) < 0 (L3)
There is unfortunately no known method to construct such a function for an arbitrary
system of nonlinear ODEs. However, in the case of a system described exclusively by
polynomial functions such as (11), the situation is more hopeful.
To this end, we first define the energy-like functions Eθ: Rn→ R as
Eθ(a) :=1
2?a + θc?2.
Of special interest will be the perturbation energy function E0and the total energy func-
tion E1. In particular, a useful observation is that
∇aE0(a) · N(a)a = aTN(a)a = 0,
i.e. the nonlinear part of the dynamics of the system (11) is invariant with respect to the
perturbation energy. Selecting as a candidate Lyapunov function V = E0, stability of the
system (11) is therefore assured for all R such that
∇aV (a) · f = aT(1
RΛ + W)a < 0
∀x ?= 0. (13)
8

Page 9

Calculation of the maximum value of R for which (13) holds is then straightforward, since
one needs only to find the largest R such that the matrix Λ + R/2 · (W + WT) remains
negative-definite. Of course, this mirrors exactly the situation in the infinite dimensional
case.
We next consider whether it is possible to establish stability of the system (11) using some
alternative polynomial Lyapunov function. In order to restrict the overall size of our search
space, we first consider the essential features of such a function.
3.1. System behavior for extreme values of ?a?
Consider first the linear part of the system (11) in isolation, i.e.
˙ a = La.(14)
If L has any nonnegative eigenvalues then the system (14) is unstable, implying immediately
that the nonlinear system (11) is also unstable. If the system (14) is stable, then there
exists some P ≻ 0 such that V (a) = aTPa is positive-definite and
∇aV (a) · f(a) = aT(LTP + PL)a < 0
∀a ?= 0, (15)
see [6, Thm. 4.6]. Such a function also ensures stability of the nonlinear system (11) for
some region around the origin, since the linear component of (11) dominates when ?a? ≪ 1.
Considering the nonlinear term of (11) in isolation, one typically expects that for any
V (a) = aTPa with P ≻ 0,
{a | ∇aV (a) · N(a)a > 0} ?= ∅,
unless P ∝ I. Since the nonlinear component of (11) is the dominant term when ?a? ≫ 1,
we should therefore not expect to find a second-order positive-definite polynomial Lyapunov
function V other than the perturbation energy function V = E0, or some monotone function
thereof. On the other hand, using the system representation (12) it follows that
∇aE1(a) · f(a) = (a + c) ·
?1
RΛa + [N(a + c)](a + c) − N(c)c
?
=1
R· (aTΛa + aTΛc) − aTN(c)c. (16)
Consequently, ∇aE1· f < 0 for all ?a? sufficiently large with respect to a fixed Reynolds
9

Page 10

number3, though the choice V = E1 would not satisfy condition (L1).
approach therefore is to search for a candidate Lyapunov function in the form V = A+B,
where the components A : Rn→ R and B : Rn→ R have the following properties:
[A + B](0) = 0
∇a[A(a) + B(a)] ≈ aTP
∇aB(a) ≈ σ(a) · ∇aE1(a)
?∇aA(a)? ≪ ?∇aB(a)?
where P ≻ 0 and σ : Rn→ R is a nondecreasing positive function. The condition (17b)
ensures that ∇aV · f satisfies approximately the linear Lyapunov condition (15) in a lo-
calized region about the origin. The conditions (17c)–(17d) ensure that ∇aV · f < 0 for
states sufficiently far from the origin, in accordance with (16).
A reasonable
(17a)
∀?a? ≪ 1
∀?a? ≫ 1
∀?a? ≫ 1,
(17b)
(17c)
(17d)
We restrict our attention to cases where both A and B are polynomial functions and
degA < degB. A useful observation is that any choice of B in the form
B(a) =
k?
i=1
Eθi(a) with
1
k
k
?
i=1
θi= 1 and θ1= 0 (18)
satisfies the condition (17c). In searching for a Lyapunov function in the form V = A+B,
we will view the function A as a term to be optimized, and therefore refer to it as the
variable term. We will restrict the function B to be some combination of energy-like
functions in the form (18), and hence refer to it as the energy term.
3.2. Lyapunov Function Generation Using Sum-of-Squares
If we restrict our attention to polynomial functions V with no constant term (so that
V (0) = 0), then the Lyapunov conditions (L1)–(L3) can be rewritten as
{a | V (a) ≤ 0, ℓ1(a) ?= 0} = ∅
(19a)
{a | −∇aV (a) · f(a) ≤ 0, ℓ2(a) ?= 0} = ∅, (19b)
3This effect is not exclusive to the total energy function E1. If one defines the energy-like function
Ed := (x + d)T(x + d), then ∇aEd· f is negative-definite whenever the Reynolds number R and vector d
are contained in the set?
2
(R,d)
????Λ +R
??
W + WT?
+
?˜ W(d) +˜ WT(d)
??
≺ 0
?
,
where˜ W(d) is linear in d and defined such that˜ W(d)a = N(d)a+N(a)d. The above set is convex in R for
fixed d and vice-versa. In the case that ¯ u = ciui, one can make the particularly convenient choice d = c, so
that the above set is unbounded in R.
10

Page 11

where positive-definite polynomial functions ℓiare used in place of the vector-valued con-
dition a ?= 0. For simplicity, we can define the functions ℓias
ℓi(a) :=
n
?
j=1
ǫija2
j,
and impose a strict positivity constraint on the values ǫij. Straightforward application of
the Positivstellensatz (see [10] and the references therein) shows that satisfaction of the
conditions (19) is assured if one can identify polynomial functions (s1,s2) ∈ Σnsuch that
V (a) − ℓ1(a) = s1
−∇aV (a) · f(a) − ℓ2(a) = s2,
where Σndenotes the set of all sum-of-squares polynomials in Rn.
(SOS)
The problem of determining whether (SOS) can be satisfied for some (s1,s2) ∈ Σn can
be reformulated as a convex optimization problem in the form of a semidefinite program
(SDP) using standard software tools [7, 9]. If degV = 2d (note that the degree of V must
be even for (L1) to be satisfied), then the general form of our problem is:
(SDP)



























min
V,H1,H2,{ǫij}
0(20a)
subject to:
V (x) − ℓ1(a) = m(a)TH1m(a)
∂a· f(a) − ℓ2(a) = m(a)TH2m(a)
(H1,H2) ? 0
(ǫ1,j,ǫ2,j) ≥ ¯ ǫ
(20b)
−∂V
(20c)
(20d)
∀j ∈ {1,...,n}, (20e)
where m(a) is a vector of all monomials in a with degree less than or equal to d. The
objective function in our optimization problem is zero since we are interested only in
feasibility. Note that any solution to the problem (SDP) will satisfy the original sum-of-
squares condition (SOS), since the semidefiniteness constraint (20d) ensures that (20b)–
(20c) can be expressed as sums-of-squares following a suitable similarity transformation.
The lower bounding constant ¯ ǫ for the terms ǫijin (20e) is also arbitrary (though it must
be strictly positive).
3.3. Determining Stable Values for R
One can estimate an upper bound on the value of R for which a solution to (SOS) can
be found via a straightforward binary search strategy. However, in all but the trivial case
11

Page 12

V = E0, there is no reason to suppose a priori that if a solution to (SOS) can be found for
some¯R, then a solution can be found for all R ∈ (0¯R]. Provision of such an assurance is
possible by augmenting (SOS) with additional constraints. First note that the Lyapunov
condition (L3) can be written as
∇aV (a) ·
?
Wa + N(a)a
?
+1
R·
?
∇aV (a) · Λa
?
< 0
∀a ?= 0. (21)
If (21) is satisfied for some¯R, then it is satisfied for all R ∈ [0,¯R] provided that the
second term ∇aV (a) · Λa ≤ 0 for all a. Satisfaction of this condition can be imposed as a
sum-of-squares constraint,
− ∇aV (a) · Λa = s3∈ Σn,
and included as an additional condition to (SOS) (or checked a posteriori).
(22)
Note also that, given a Lyapunov function V for some value¯R, it is possible to compute
directly the smallest and largest value R for which V is a Lyapunov function, since (21) is
affine in 1/R; e.g. one can compute an upper bound by solving the sum-of-squares problem
min
R≥0
1/R
subject to:
− ∇aV (a)?Wa + N(a)a?−1
R
?∇aV (a) · Λa?− ℓ2(a) ∈ ΣN
and taking the inverse of its minimum value.
3.4. Computational Complexity
We next consider the computational effort required to solve the problem (SDP) for various
degrees of Lyapunov candidate function V . If we assume that V is a polynomial function
with arbitrary coefficients and degV = 2d, then the monomial vector m(a) is composed of
D distinct monomial terms, where
D =(n + d)!
n!d!
.
Standard results from semidefinite programming ensure that one can solve the problem
(SDP) in O(√D) iterations using a primal-dual interior point method4, with each itera-
tion requiring O(D3) operations. In practice, it is generally the case that the number of
4More precisely, one can guarantee that a primal-dual interior point algorithm will reduce the duality
gap of its solution iterate to a multiple ǫ of its original value within O(ln(1/ǫ)√D) iterations. The reader
is referred to [15, 14, 2] and references therein for an overview of algorithms and complexity results for
semidefinite programming.
12

Page 13

iterations required to solve a semidefinite programming problem is roughly constant with
respect to increasing problem size, so the computation time is determined almost entirely
by the per-iteration computation cost.
The rapid increase in computational burden with increasing system dimension means that
SOS methods are likely to be applicable for relatively low dimensional models only, even if
one assumes that the considerable degree of problem-specific structure inherent in (20) can
be somehow exploited (e.g. using a structured approach such as (17)). In particular, it is
not advisable to attempt to estimate the maximum stable Reynolds number in the infinite
dimensional Navier-Stokes system (4) via solution of a succession of problems in the form
(20) with increasing dimension. We therefore require a more indirect approach, whereby
the finite-dimensional techniques of this section can be extended to the infinite-dimensional
system (4) without excessive additional computation. We propose such an approach in the
remainder of the paper.
4. Infinite Dimensional Flow Models
We now return to the general case where us?= 0, which we will view as an uncertain forcing
term in our ODE. In this case substituting (9) into (4a) and taking an inner product of
both sides results in a model similar to the ODE (11), but with additional perturbation
terms in us, i.e.
˙ a = f(a) + Θa(us) + Θb(us,a) + Θc(us), (23)
where the additional perturbation terms are defined as
Θai(us) :=
1
R
?ui,∇2us
?− ?ui,S(us, ¯ u)?
(24a)
Θbi(us,a) := −?ui,S(uj,us)?aj
Θci(us) := −?ui,us· ∇us?,
(24b)
(24c)
and f(a) is as defined in (11). In the above, a subscript i indicates that the expression
is the ith element of a vector quantity. The perturbation term Θa represents a linear
disturbance in us, Θb represents a bilinear disturbance in (us,a), and Θc represents a
quadratic disturbance in us.
We would like to bound the influence of each of these perturbation terms on our ODE
in terms of ?us? and ?a?. In order to do so, we apply (2) repeatedly to eliminate the
13

Page 14

appearance of terms ∇us, so that (24) can be rewritten as5
Θai(us) = ?us,hi?,
Θbi(us,a) = ?us,hij?aj,
Θci(us) = ?us,us· ∇ui?.
hi:=1
R∇2ui− ∇¯ u · ui+ ¯ u · ∇ui,
hij:= uj· ∇ui− ∇uj· ui,
(25a)
(25b)
(25c)
We are of course left with an ODE in the form (23) which still features the perturbations us.
We next bound the influence of this term by modeling only the evolution of its energy q,
which we model as q2= ?us?2/2. In the process we add a single ODE to supplement (23),
representing the time evolution of the squared energy term q2.
Substituting (9) into (4a) and taking an inner product of both sides with the total velocity
field u = uiai+ usprovides the additional ODE in term of the perturbation energy q2,
˙
(q2) = aTf(a) − aT˙ a + Γ(us) + χ(us,a), (26)
where
Γ(us) :=1
R
?us,∇2us
?− ?us,S(¯ u,us)?,
gj:=
R∇2− e
(27)
χ(us,a) := ?us,gj?aj,
?1
?
uj. (28)
Verification of the above relies on the aforementioned assumptions about the subspace S
and on application of the various identities described in Section 1. In particular, these
allow one to establish the relations
?
u,∂u
∂t
?
= aT˙ a +
˙
(q2) and aTf(a) =
?1
R
?ui,∇2uj
?− ?ui,S(¯ u,uj)?
?
aiaj.
Note that in (26), the terms aTf(a) and Γ(us) represent the self-contained dissipation or
generation of energy depending on aiui and us, while the term χ(us,a) represents the
generation or dissipation of energy containing cross terms between these velocity fields.
4.1. Description as an Uncertain System
The complete system of interest can now be written as
˙ a = f(a) + Θa(us) + Θb(us,a) + Θc(us)
˙
(q2) = aTf(a) − aT˙ a + Γ(us) + χ(us,a).
(29a)
(29b)
5The notation used can be clarified by the equivalent expression for the Cartesian components of the
vector hi: hm
i =
1
R∇2um
i −
∂¯ uk
∂xmuk
i+ ¯ u · ∇um
i.
14

Page 15

We are now free to treat usas an uncertain term driving the ODE system (29), whose
time evolution is known to satisfy the subspace constraint us ∈ S and the energy con-
straint q2= ?us?2/2. The worst-case effect of this uncertainty can then be bounded via
appropriate norm bounds.
The first of these bounds relates to the uncertain terms in (29a). There exist constants
ci≥ 0 and a polynomial function p1(a,q) ≥ 0 such that
?Θa(us) + Θb(us,a) + Θc(us)?2≤ p1(a,q) = c1q2+ c2q2?a?2+ c3q4
for any a and us. A rigorous proof of the existence of these constants is given in Ap-
pendix Appendix A. Critically, estimation of the coefficients ciinvolves the solution only
of linear problems for partial differential equations and optimization over finite-dimensional
polynomials.
(30)
A second bound relates to the uncertain term Γ(us) in (29b). Comparing (27) with (6)
shows that Γ(us) ≤ κ?us?2with κ = λ1. However, ussatisfies the additional constraints
?us,ui? = 0 and therefore may admit a stronger bound. Note that the number of positive
eigenvalues of (7) is always finite [1]. Hence, if uiare chosen as the first n eigenfunctions
of (7) and n is large enough, then
Γ(us) ≤ κs?us?2= 2κsq2
(31)
for all us∈ S, where κs= λn+1< 0. If uiare not eigenfunctions of (7), then κsis the
largest eigenvalue of the following problem
λu + µkuk= (e −1
?uk,u? = 0, ∇ · u = 0, u|∂Ω= 0.
R∆)u + ∇p
In what follows we will assume that κs< 0 in (31).
A final bound relates to the uncertain term χ(us) in (29b). If ui are eigenfunctions of
(7) then χ = 0, because in this case gi = −λiui+ ∇φi with some scalar functions φi
and because usis orthogonal to both ui(by definition) and to gradients of any scalars
(since ∇ ·us= 0). In the general case there exists a constant d and a polynomial function
p2(a,q) ≥ 0 such that
?χ(us,a)?2≤ p2(a,q) = dq2?a?2.
The proof is very similar to the proof of (30).
(32)
5. Stability of Infinite Dimensional Models using SOS
Given the (uncertain) ODE system (29), we can now search for a Lyapunov function
verifying stability of the composite state vector (a,q2). We therefore would like to construct
15