1

Differential dissipativity theory
for dominance analysis

arXiv:1710.01721v2 [] 5 Oct 2017

Fulvio Forni, Rodolphe Sepulchre

Abstract—High-dimensional systems that have a lowdimensional dominant behavior allow for model reduction and
simplified analysis. We use differential analysis to formalize
this important concept in a nonlinear setting. We show that
dominance can be studied through linear dissipation inequalities
and an interconnection theory that closely mimics the classical
analysis of stability by means of dissipativity theory. In this
approach, stability is seen as the limiting situation where the dominant behavior is 0-dimensional. The generalization opens novel
tractable avenues to study multistability through 1-dominance
and limit cycle oscillations through 2-dominance.

I. I NTRODUCTION
The analysis of a system is considerably simplified when
it is low-dimensional. Linear system analysis frequently exploits the property that a few dominant poles capture the
main properties of a possibly high-dimensional system. Lowdimensional models are even more critical in nonlinear system
analysis. Multistability or limit cycle analysis is difficult
beyond the phase plane analysis of two-dimensional systems.
In this paper we seek to formalize the property that a
nonlinear system has low-dimensional dominant behavior. Our
approach is differential: we characterize the property for linear
systems and then study the nonlinear system differentially, that
is, along the linearized flow in the tangent bundle. The seminal
example of differential analysis in control theory is contraction
analysis [26], [34], [15], [37], [43], which we interpret as
a differential analysis of exponential stability. The property
is the contraction of a ball, characterized via a Lyapunov
dissipation inequality. A nonlinear system is contractive when
this dissipation inequality holds infinitesimally along any of its
trajectories. In the present paper, the corresponding property
is 0-dominance: it ensures that the dominant behavior of the
nonlinear system is 0-dimensional. A more recent example of
differential analysis is differential positivity [12], the differential analysis of positivity. The linear property is the contraction
of a cone, also characterized by a dissipation inequality. A
nonlinear system is differentially positive when this dissipation
inequality holds infinitesimally along any of its trajectories. In
the context of the present paper, the corresponding property
is 1-dominance: it guarantees that the dominant behavior of
the nonlinear system is 1-dimensional. More generally, we
study the property of p-dominance differentially. The property
F. Forni and R. Sepulchre are with the University of Cambridge,
Department of Engineering, Trumpington Street, Cambridge CB2 1PZ
f.forni@eng.cam.ac.uk|r.sepulchre@eng.cam.ac.uk. The
research leading to these results has received funding from the European
Research Council under the Advanced ERC Grant Agreement Switchlet
n.670645.

is characterized via a dissipation inequality, which is then
required to hold infinitesimally along the trajectories of the
nonlinear system. We prove a general theorem that formalizes
that the dominant behavior of a p-dominant system is pdimensional.
Differential analysis is general in that it allows to extend
a linear system property to an arbitrary flow defined on a
differentiable manifold. An important restriction in this paper
is that we only consider dissipation properties characterized by
linear matrix inequalities. Furthermore, we impose the linear
dissipation inequality to be uniform on the tangent bundle.
In geometric terms, this endows the differentiable manifold
with a flat non-degenerate metric structure, characterized by a
constant quadratic form.
The linear-quadratic restriction allows for a fruitful bridge
with the linear-quadratic theory of dissipativity. Many of the
available computational tools of dissipativity theory become
available for the analysis of p-dominance. The analysis of
p-dominance is reformulated as the search of a quadratic
differential storage that decreases along the solutions of the
linearized dynamics. The classical interconnections theorems
of (linear-quadratic) dissipativity theory are reformulated as
tools that facilitate that construction. The only substantial
difference with the classical theory is that the quadratic form
that characterizes the storage is no longer required to be a
Lyapunov function, i.e. positive definite. Instead, it is required
to have a fixed inertia, that is, p negative eigenvalues and n−p
positive eigenvalues. Stability corresponds to p = 0, whereas
p-dominance allows for any integer 0 ≤ p ≤ n.
The paper is organized as follows. Section II characterizes
p-dominance and p-dissipativity for linear time-invariant models. Section III defines p-dominance for nonlinear systems with
two main results that characterize their asymptotic behavior.
The connection with related results in the literature is discussed in Section IV and dominance analysis via the solution
of LMIs is illustrated in Section IV. Differential p-dissipativity
is addressed in Section VI, where we primarily illustrate
the role of differential passivity and differential small-gain
theorems in the design and analysis of multistable systems
(p = 1) and limit cycle oscillations (p = 2). The proofs of the
main theorems are provided in appendix.
II. D OMINANT LTI SYSTEMS
Definition 1: A linear system ẋ = Ax is p-dominant with
rate λ ≥ 0 if and only if there exist a symmetric matrix P
with inertia (p, 0, n − p) such that
AT P + P A ≤ −2λP − εI .

(1)

2

for some ε ≥ 0. The property is strict if ε > 0.
y
In terms of the quadratic form V (x) = xT P x, the (strict)
dissipation inequality (1) reads
V̇ (x)

=

xT (AT P + P A)x

≤

−2λV (x) − ε | x |2

For ε > 0 this implies that the two cones
K− = {x ∈ Rn | V (x) ≤ 0}, K+ = {x ∈ Rn | V (x) ≥ 0}
are strictly contracting either in forward or in backward time:
∀t > 0 : e−At K+ ⊂ K+ , eAt K− ⊂ K−
Equivalent characterizations of p-dominance are provided in
the following proposition, whose proof is in the preliminary
version of this paper [13].
Proposition 1: For ε > 0, the Linear Matrix Inequality (1)
is equivalent to any of the following conditions:
1) The matrix A + λI has p eigenvalues with strictly
positive real part and n − p eigenvalues with strictly
negative real part.
2) there exists an invariant subspace splitting Rn = H ⊕ V
such that AH ⊂ H and AV ⊂ V. The dimension of V is
n − p and the dimension of H is p. Furthermore, there
exist constants 0 < C ≤ 1 ≤ C and λ < λ < λ such
that
At

∀x ∈ H : | e x |≥ C e

−λt

| x |, t ≥ 0

∀x ∈ V : | eAt x |≤ C e−λt | x |, t ≥ 0.
y
The property of p-dominance ensures a splitting between
n − p transient modes and p dominant modes. Only the
p dominant modes dictate the asymptotic behavior. Because
λ ≥ 0 and V ⊂ K+ , the quadratic form V (x) is a Lyapunov
function for the transient behavior, that is, for the restriction
of the flow in V.
For p = 0, p-dominance is the classical property of exponential stability: all modes are transient and the asymptotic
behavior is 0-dimensional.
The matrix inequality (1) is equivalent to the conic constraint
 T 
 
ẋ
0
P
ẋ
≤ 0.
(2)
x
P 2λP + εI
x
Dissipativity theory extends p-dominance to open systems
by augmenting the internal dissipation inequality with an
external supply. The external property of p-dissipativity is
captured by a conic constraint between the state of the system
x, its derivative ẋ, and the external variables y and u of the
form
 T
   T
 
Q L y
ẋ
0
P
ẋ
y
(3)
≤
x
P 2λP + εI
x
u
LT R u
where P is a matrix with inertia (p, 0, n − p), λ ≥ 0, L, Q, R
are matrices of suitable dimension, and ε ≥ 0. The property
is strict if ε > 0. We call supply rate s(y, u) := y T Qy +
y T Lu + uT LT y + uT Ru the right-hand side of (3). An open
dynamical system is p-dissipative with rate λ if its dynamics
ẋ = Ax + Bu, y = Cx + Du satisfy (3) for all x and u.

p-dissipativity has a simple characterization in terms of
linear matrix inequalities.
Proposition 2: A linear system ẋ = Ax+Bu, y = Cx+Du
is p-dissipative with rate λ ≥ 0 if and only if there exists a
symmetric matrix P with inertia (p, 0, n − p) such that
h T
i
A P +P A+2λP −C TQC+εI
P B−C TL−C TQD
≤ 0 . (4)
T
T
T
T
T
T
B P −L C−D QC
−D QD−L D−D L−R
y
Proof: [⇒] Just replace ẋ = Ax+Bu and y = Cx+Du
in (3) and rearrange. [⇐] Multiply (4) by [xT uT ] on the left,
and by [xT uT ]T on the right. Then we get ẋT P x + xT P ẋ +
2λxT P x < s(y, u) as desired.
An interconnection theorem can be easily derived. A proof
is provided in the preliminary version of this paper [13].
Proposition 3: Let Σ1 and Σ2 p1 -dissipative and p2 dissipative systems respectively, with uniform rate λ and with
supply rate
 T 
 
Qi Li
y
y
si (yi , ui ) =
(5)
u
u
LTi Ri
for i ∈ {1, 2}. The closed-loop system given by negative
feedback interconnection
u1 = −y2 + v1

u 2 = y 1 + v2

(6)

is (p1 + p2 )-dissipative with rate λ from v = (v1 , v2 ) to y =
(y1 , y2 ) with supply rate


L1
R2  
 T Q1 + R2 −L1 + LT2
T
y  −L1 + L2 Q2 + R1 −R1 L2  y
.
s(y, v) =
LT
−R1
R1
0
v
v
1
R2

LT
2

0

R2

(7)

The closed loop is (p1 + p2 )-dominant with rate λ if


Q1 + R2 −L1 + LT2
≤0.
−LT1 + L2 Q2 + R1

(8)
y
Mimicking classical dissipativity theory, there are two important particular cases of supply rates : the passivity supply
 T 
 
y
0 I
y
s(y, u) =
.
(9)
u
I 0
u
and the gain supply:
s(y, u) =

 T 
y
−I
u
0

 
y
.
γ2I
u
0

(10)

Hence, Proposition 3 provides an analog of the small-gain
theorems and passivity theorems for p-dominance of a linear
time-invariant system.
III. D IFFERENTIAL ANALYSIS OF
DOMINANT NONLINEAR SYSTEMS
For a nonlinear system
ẋ = f (x)

x∈X

(11)

we define dominance differentially, that is, through the linear
dissipation inequality
 T 
 
˙
˙
0
P
δx
δx
≤0
(12)
P 2λP + εI
δx
δx

3

for every δx ∈ Tx X , where P is a matrix with inertia (p, 0, n−
p) and ε ≥ 0.
In what follows we assume that X is a smooth Riemannian
manifold of dimension n. Given any δx ∈ Tx√X , |δx| denotes
the Riemanian metric on X represented by δxTδx in local
coordinates. ψ t (x) denotes the flow of (11) at time t passing
through x ∈ X at time 0. ∂ψ t (x) denotes the differential of
ψ t (x) with respect to x. Note that (ψ t (x), ∂ψ t (x)δx) is a flow
in the tangent bundle. It is the solution of the prolonged system
[6]

ẋ = f (x)
(x, δx) ∈ T X
(13)
˙ = ∂f (x)δx
δx
at time t passing through (x, δx) ∈ T X at time 0. In local
coordinates ∂f (x) denotes the differential of f at x and P is
the local representation of a metric tensor P with fixed inertia.
Following the approach of Section II, we define the property
of dominance for nonlinear systems as follows.
Definition 2: A nonlinear system ẋ = f (x) is p-dominant
with rate λ ≥ 0 if and only if there exist a symmetric matrix
P with inertia (p, 0, n − p) such that (12) is satisfied by the
solutions of the prolonged system (13) for some ε ≥ 0. The
property is strict if ε > 0.
y
In terms of the quadratic function V (δx) := δxT P δx, the
differential dissipation inequality (12) reads

V̇ (δx) = δxT ∂f (x)TP + P ∂f (x) δx
≤

−2λV (δx) − ε | δx |2

(14)

As a consequence, for ε > 0, the two cone fields
K− (x) := {δx ∈ Tx X | V (δx) ≤ 0}

are strictly contracting either in forward time or in backward
time, respectively:
t

−

−

t

∂ψ (x)K (x) ⊂ K (ψ (x))

∀t > 0

∀t ∈ R ,
∀t ∈ R .

(15)

Hx and Vx are distributions of dimension p and n − p,
respectively. Furthermore, there exist constants C ≤ 1 ≤ C
and λ < λ < λ such that
|∂ψ t (x)δx| ≥ Ce−λt |δx|

|∂ψ t (x)δx| ≤ Ce−λt |δx|

∀x ∈ A, ∀δx ∈ Hx

∀x ∈ A, ∀δx ∈ Vx .

The proofs of Theorem 1 and Theorem 2 are given in
Appendix. Below we detail the connections of those results
with earlier results in the literature of dynamical systems and
control theory.
A. Dominated splittings and dominance

∀t > 0

The following result provides the differential analog of
Proposition 1.
Theorem 1: Let A ⊆ X be a compact invariant set and let
(11) be a strictly p-dominant system with rate λ ≥ 0 and tensor
P . Then, for each x ∈ A, there exists an invariant splitting
Tx X = Hx ⊕ Vx
∂ψ t (x)Hx ⊆ Hψt (x)
∂ψ t (x)Vx ⊆ Vψt (x)

T
V̇ (x−y) = (x−y)TP(f (x)−f (y))+(f (x)−f (y))
 P (x−y)
R1
= 2(x − y) 0 P ∂f (sx + (1−s)y)ds (x − y)
R

1
≤ −ε(x − y)T 0 −2λP + εI ds (x − y)
≤ −2λV (x − y) − ε | x − y |2 .
(17)
The following result is based on the incremental dissipation
inequality (17). We denote by Ω(x) the ω-limit set of x, that
is, the set of all ω-limit points of x.
Theorem 2: For X = Rn , let (11) be a strictly p-dominant
system with rate λ ≥ 0. Then, the flow on any compact ωlimit set is topologically equivalent to a flow on a compact
invariant set of a Lipschitz system in Rp .
y
For small values of p, Theorem 2 severly constrains the
possible attractors of the system.
Corollary 1: Under the assumptions of Theorem 2, every
bounded solution asymptotically converges to
• a unique fixed point if p = 0;
• a fixed point if p = 1;
• a simple attractor if p = 2, that is, a fixed point, a set of
fixed points and connecting arcs, or a limit cycle.
y

IV. C ONNECTIONS WITH THE LITERATURE

K+ (x) := {δx ∈ Tx X | V (δx) ≥ 0},

∂ψ −t (x)K+ (x) ⊂ K+ (ψ −t (x))

is a differential Lyapunov function in the invariant distribution
V ⊂ TX.
For X = Rn the characterization of the asymptotic behavior
of dominant systems can be further refined. This is because,
by integration, the differential dissipation inequality (14) leads
to the incremental inequality

(16a)
(16b)

y
The interpretation of the theorem is that the linearized flow
∂ψ t (·) admits an invariant splitting between n − p transient
modes and p dominant modes. The p dominant modes dictate
the long-term behavior of the flow. The quadratic form V (δx)

The property of dominance studied in this paper is closely
related to the cousin concepts of dominated splittings and partial hyperbolicity. Both concepts have appeared in dynamical
systems theory as part of the extensive research to generalize
the key concept of hyperbolicity pioneered by Smale and
Anosov in the 60’s. The common theme of that research line
is that robust features of smooth dynamical systems should
be captured by robust features of their linear approximations.
Robust is to be understood here in the sense of structural
stability, that is, robustness to small perturbations of the vector
field.
The subject is vast but we refer the interested reader to
the recent survey [38] for an orientation map. Both dominated
splittings and partial hyperbolicity continue to be an important
subject in dynamical systems theory [7], [38], [36], [19], [24],
[35]. We refer the reader to [48], [35] for the implications
of Theorem 1 on normal hyperbolicity and structural stability
of compact attractors. Dominated splittings have also received
attention in control theory [5].
Theorem 1 and its proof are grounded in the results and
proofs of [33, Theorem 1.2] and [12, Theorem 3]. Theorem
2 and its proof are grounded in the results and proofs of [22,

4

Theorem 3.17] and [39, Proposition 3]. The proof merges these
approaches with the techniques in [41].
The decomposition T X = H⊕V in Theorem 1 is a splitting
that is invariant under the linearized flow ∂ψ t . The splitting
is called dominated because the flow satisfies
C
|δxv |
|∂ψ t (x)δxv |
≤ e−(λ−λ)t
t
|∂ψ (x)δxh |
C
|δxh |

for any δxh 6= 0 ∈ Hx and δxv ∈ Vx . The dominated splitting
is a consequence of the contraction of the cone fields K+ (x)
and K− (x).
The contraction of a cone is projective, that is, it expresses
a contraction of the non-dominant directions relative to the
dominant directions of the flow. Because of the assumption
of a nonnegative dissipation rate λ ≥ 0, dominance further
imposes vertical contraction, that is, contraction of the flow
∂ψ t in the vertical distribution V, see (16b). The requirement
of vertical contraction is an extra requirement of dominance
with respect to the property of dominated splitting. Theorem
2 does not hold without this extra requirement.

vertical contraction. Differential positivity does not imply
contraction in the 1-dimensional vertical subspace Vx . As a
consequence, the statement of Corollary 1 for p = 1 only holds
for generic initial conditions of strictly differentially positive
systems, see [12, Corollary 5].
For X = Rn , 1-dominance is also tightly related to monotonicity [40], [21], [1], [22]. A monotone system preserves a
partial order : any pair of trajectories x(·), y(·) from ordered
initial conditions x(0)  y(0) satisfy x(t)  y(t) for all t ≥ 0.
Monotonicity is implied by 1-dominance. The partial order
is the usual partial order associated to a pointed convex
cone: x  y iff y − x ∈ K∗ . Monotonicity requires that
trajectories y(0) − x(0) ∈ K∗ satisfy y(t) − x(t) ∈ K∗
for all t ≥ 0, which is guaranteed by 1-dominance by the
invariance V (x(t) − y(t)) ≤ 0 for all t ≥ 0 and by the fact
that if x(0) − y(0) = 0 then x(t) − y(t) = 0, for all t ≥ 0.
Here also, monotonicity is independent of λ. The assumption
λ ≥ 0 makes 1-dominance stronger than monotonicity and
the statement of Corollary 1 for p = 1 only holds for generic
initial conditions of monotone systems, see [18], [22].

B. Contraction, differential stability, and 0-dominance
A strict 0-dominant system is a contractive system [26],
[34], [37], [11]. For a linear system, the property is simply
exponential stability, meaning hyperbolicity and contraction of
the n transient modes to the 0-dimensional attractor. Because
P is positive definite, the dissipation inequality implies the
contraction of an ellipsoid. The quadratic form V (δx) is a
differential Lyapunov function in the terminology of [11].
On vector spaces X = Rn , its integration along geodesic
curves leads to the incremental Lyapunov function V (x − y).
From (17), it implies exponential contraction of the difference
between any two trajectories. The attractor of a 0-dominant
system is necessarily a unique fixed point.
C. Monotonicity, differential positivity, and 1-dominance
Strictly 1-dominant systems are strictly differentially positive systems [12], [9]. The contractive cone K− is an ellipsoidal cone, that is, it is the union of two solid pointed convex
cones K− = −K∗ ∪ K∗ as illustrated in Figure 1.
δx(0)
δx(t)

K∗
δxT P δx ≤ 0

Figure 1. For ε > 0 (12) guarantees that any trajectory moves from the
boundary of K∗ towards the interior.

A linear system that contracts a solid pointed convex cone is
a positive system [27], [4]. Differential positivity thus induces
a dominated splitting between one dominant direction and
n − 1 dominated directions. The reader is referred to [12],
[9] for a comprehensive analysis of the asymptotic behavior
of differentially positive systems.
The distinction between 1-dominance and differential positivity is again the property of projective contraction versus

D. Contraction of rank 2 cones and 2-dominance
The property of 2-dominance provides the following generalization of Poincaré-Bendixson theorem:
Corollary 2: For p = 2, under the assumptions of Theorem
2, let U ⊆ X be a compact forward invariant set that does not
contain fixed points. Then, the ω-limit set of any point in U
is a closed orbit.
y
Proof: Take any ω-limit set contained in U. By Theorem
2 the flow restricted to this set is topologically equivalent to
the flow of a planar system. By Poincaré-Bendixson theorem
[20, Chapter 11, Section 4], a nonempty compact limit set of a
planar system which contains no fixed points is a closed orbit.
A similar generalization was developped in the papers [41],
[42], [39] by generalizing the concept of monotonicity to rank2 cones. It is this generalization that motivated the results
in the present paper. Note that the satement of Corollary 2
only holds for generic initial conditions of rank 2 monotone
systems. The stronger conclusion of Corollary 2 is again due
to the assumption of a nonnegative dissipation rate λ ≥ 0.

E. Invariant cone fields
The assumption of a constant matrix P makes the cone
fields K+ (x) and K− (x) constant, that is, the same cone is
attached to every x ∈ X . A more intrinsic characterization
for X = Rn is that the cone field is invariant by translation,
the natural group action on a vector space. This geometric
interpretation allows for extensions on Lie groups and, more
generally, homogeneous spaces [29], [30], [3]. The definition
of dominance thus assumes an invariant cone field in the
present paper. The invariance of the cone field is an important
source of tractability for the search of the storage.

5

V. A LGORITHMIC TEST FOR p- DOMINANCE

C. Example

AND A SIMPLE EXAMPLE

A. Dominant spectral splitting
A necessary condition for dominance is that the spectrum of
the family of matrices ∂f (x) + λI, x ∈ X , admits a uniform
splitting, see Figures 2 (left) and 3 (left) for an illustration.
Theorem 3: Let (11) be a strictly p-dominant system. Then,
there exists a maximal interval (λmin , λmax ) such that ∂f (x)+
λI has p unstable eigenvalues and n − p stable eigenvalues
(negative real part) for each λ ∈ (λmin , λmax ), for every x ∈
X.
y
Spectral analysis of the Jacobian matrix ∂f (x) is thus useful
to select p and λ in dominance analysis. The uniform splitting
of the spectrum is necessary but of course not sufficient
for dominance. This is well-known even for p = 0. For
instance, classical counterexamples to Kalman’s conjecture
[23] illustrate that a system can fail to be contractive even
when the spectrum of its Jacobian is uniformly in the left half
complex plane.

B. Convex relaxations
Sufficient conditions for dominance are provided by the
inequality
∂f (x)T P + P ∂f (x) + 2λP + εI ≤ 0

∀x ∈ X

(18)

whose solutions P , for some ε ≥ 0, must also satisfy a fixed
inertia constraint.
If the spectrum of ∂f (x) admits a stable splitting for a given
λ, then all the solutions P of (18) must share the same inertia,
meaning that the inertia condition can be dropped. One is then
left with solving an infinite family of LMIs.
It is common practice to reduce an infinite family of
LMIS to a finite family through convex relaxation. Let
A := {A1 , . . . , AN } be a family of matrices such that
∂f (x) ∈ ConvexHull (A) for all x. Then, by construction,
any (uniform) solution P to
ATi P + P Ai + 2λP + εI ≤ 0

1≤i≤N

(19)

is a solution
PN to (18). For instance, at each x,
∂f (x) =
i=1 ρi (x)Ai for a given set of ρi (x) such
PN
that
ρ
(x)
P i=1 i = 1. Thus,
Pthe left-handside of (18) reads
N
N
T
P +P
i=1 ρi (x)Ai
i=1 ρi (x)Ai + 2λP + εI =

PN
T
i=1 ρi (x) Ai P + P Ai + 2λP + εI ≤ 0, where the last
inequality follows from (19).
The algorithmic steps of dominance analysis of a given
nonlinear system ẋ = f (x) can thus be summarized as
follows:
1) Estimate p and λ from the spectrum analysis of ∂f (x)
2) Reduce the infinite family of LMIs to a finite family by
convex relaxations.
3) Test the feasability of the relaxed LMI with a LMI
solver.

We illustrate the theory on a classical textbook example:
a one degree of freedom mechanical system with nonlinear
spring (Duffing model), actuated by a DC motor with a
PI feedback control. While this example is elementary, it
illustrates the tractability of dominance analysis on a fourdimensional model, for which a global analysis of the attractors is a nontrivial problem.
The mechanical model is given by
ẋp = xv

ẋv = −α(xp ) − cxv + u

(20)

where xp and xv are position and velocity of the mass
respectively, u is the force input to the system, c is the damping
coefficient, and α(xp ) := ∂U (xp ) is the force deriving from
the mechanical potential U : R → R. Contraction, or 0dominance is expected with sufficient damping if the potential
is strictly convex. A differential quadratic storage is easily
found for the numerical value c = 5 and the assumption
1 ≤ ∂α(xp ) ≤ 5 .
 0 1 
P is computed
convex relaxation (19) for A1 := −1
−5

 0 via
1
. For λ = 0 and ε = 0.01, the LMI solver
and A2 := −5 −5
(Yalmip [25]) returns


0.8696 0.1482
P :=
.
0.1482 0.1304
which is positive definite.
The same approach is repeated for the non-convex potential
−2 ≤ ∂α(xp ) ≤ 5 ,
which allows for several minima (including the classical
double-well potential of the nonlinear Duffing model [17]).
Figure 2 (left) suggests a stable splitting between the two
eigenvalues. The differential storage
by

 1 is computed again
0 1
and A2 := −5
convex relaxation (19) for A1 := 02 −5
−5 .
For λ = 2 and ε = 0.01, the LMI solver returns


−5.1987 3.6260
P =
.
3.6260 6.1987
which has inertia (1, 0, 1). Figure 2 (right) shows the non
positive level sets of δxT P δx. The system is 1-dominant,
meaning that every bounded trajectory asymptotically converges to some fixed point.
Suppose now that the mechanical system is driven by a DC
motor modelled by the electrical equation
u = kf xi

Lẋi = −Rxi − ke xv + V .

(21)

xi is the current of the circuit, kf is a static approximation
of the current to force characteristic, L and R are inductance
and resistance respectively, ke is the back electromotive force
coefficient, and the voltage V is an additional input.
It is easy to verify that 1-dominance is preserved for R = 1,
kf = 1, ke = 1, and 0 < L < 0.05, since the time-scale separation between electrical and mechanical dynamics introduces
a mild perturbation on the dominant/slow dynamics of the
system. For L = 0.1 the reduced time scale separation allows
for interaction between electrical and mechanical dynamics.

2.5

1

2

0.8

1.5

0.6

1

0.4

0.5

0.2

/ x2

Im

6

0

0

-0.5

-0.2

-1

-0.4

-1.5

-0.6

-2

-0.8

-2.5
-6

-4

-2

ε = 0.01, in which case the LMI solver returns the storage


−4.3713 1.7901 −0.5507 0.0216
 1.7901
5.6483
0.3768 −0.9320 
 .
P =
 −0.5507 0.3768
1.0521 −0.4363 
0.0216 −0.9320 −0.4363 −1.3291
with inertia 2, 0, 2). For r = 0 the unique fixed point at 0
is unstable. We conclude that every bounded trajectory must
converge to a periodic orbit, as illustrated in Figure 4 (right).

-1
-1

0

-0.5

0

0.5

1

3

/ x1

Re

2
2

Figure 2. Left: roots of the Jacobian of the mechanical systems for different
xp . The value of λ is emphasized by the vertical dashed line. Right: negative
level sets of δxT P δx where P has inertia (1, 0, 1).

The distribution of the eigenvalues of the Jacobian in Figure 3
(left) suggests that strict 1-dominance still holds. Indeed, for
L = 0.1, λ = 2, and ε = 0.01, the LMI solver returns


−3.0942 0.8985 −0.5355
P =  0.8985 3.3771 0.1935  .
−0.5355 0.1935 0.7171
which has inertia (1, 0, 2). For constant inputs V , every
bounded trajectory necessarily converges to some fixed point,
as illustrated in Figure 3 (right), where we considered the
double well potential U (xp ) := x2p /4 + cos(xp ).
2
2
1.5

0

xp

Im

1

1

-1
0.5
-2
-10

-5

0

Re

0

-1

-1

-2

-2
-10

0

-5

0

-3
0

50

Re

100

150

200

t

Figure 4. Left: roots of the Jacobian of the closed loop given by (20), (21)
and (22) by sampling −2 ≤ ∂α(xp ) ≤ 5 at different xp . Right: Trajectory
of the mass position in time for the closed loop (20), (21), (22) with potential
U (xp ) := x2p /4 + cos(xp ) and PI control gains kP = 1 and kI = 5, from
the initial condition xp = 0.05, xv = 0, xi = 0 and xc = 0, at constant
r = 0.

Remark 1: A variant of the example is to replace the
mechanical model with the nonlinear pendulum ẋp = xv ,
ẋv = − sin(xp ) − cxv + u with (xp , xv ) ∈ S × R. Onedominance is proven as in Duffing example but the statespace is now nonlinear. From Theorem 1, every attractor of
the pendulum exhibits at least one direction of contraction,
locally. The shape of S × R makes 1-dominance compatible
with fixed point attractors, for small constant torque u, and
with attractors defined by periodic orbits, for large constant
torque u, [44, Section 8.5]. See also the analysis in [12] using
differential positivity.
y

0
0

50

100

VI. I NTERCONNECTIONS

t

Figure 3. Left: roots of the Jacobian of (20) and (21), by sampling −2 ≤
∂α(xp ) ≤ 5 at different xp . Right: Trajectory of the mass position in time
for the interconnected system (20), (21) with potential U (xp ) := x2p /4 +
cos(xp ), from the initial condition xp = 0.05, xv = 0, xi = 0, at constant
V = 0.

Finally, we close the loop with a PI controller
V = kP (r − xp ) + kI xc

1

xp

Im

1

ẋc = r − xp

(22)

where xc is the integrator variable, kP and kI are proportional
and integral gains, respectively, and r is the reference.
The degree of dominance of the closed loop can be modulated via PI control. A detailed analysis of PI control for pdominance is beyond the scope of this paper. We just observe
that with gains kP = 1 and kI = 5 the eigenvalues of the
Jacobian in Figure 4 (left) exhibits a stable splitting into two
groups of two eigenvalues. 2-dominance is verified with λ = 2,

A. Differential dissipativity theory
Dissipativity theory is a fundamental complement to stability theory in systems and control [49], [50], [51]. Stability,
and more generally, dominance, is the property of a closed
system. Dissipativity is the property of an open system, i.e.
a system with inputs and outputs. By decomposing a closed
system as an interconnection of open subsystems, the search
of quadratic storage for the analysis of the closed system is
converted into the solution of linear matrix inequalities for the
open subsystems.
The recent papers [10], [14], [46] have proposed to use dissipativity theory differentially in order to study contraction (i.e.
0-dominance) via interconnections. We pursue this approach
to study p-dominance. We restrict to open systems of the form

ẋ = f (x) + Bu
x ∈ X , (y, u) ∈ W
(23)
y = Cx + Du

7

where X and W are smooth manifolds of dimension n and
m, respectively. The associated prolonged system reads

ẋ = f (x) + Bu


 ˙
δx = ∂f (x)δx + Bδu
(24)
y = Cx + Du



δy = Cδx + Dδu
where (x, δx) ∈ T X and (w, δw) := ((y, u), (δy, δu)) ∈ T W.
We also assume that W is covered by a single chart and
that the matrix L and the symmetric matrices Q and R below
have suitable dimensions. All restrictions above relate to the
restriction of constant tensors P in this paper for the analysis
of dominance.
Definition 3: A nonlinear system (23) is differentially pdissipative with rate λ and differential supply rate
 T
 
Q L δy
δy
s(w, δw) :=
(25)
δu
LT R δu
if for some symmetric matrix P with inertia (p, 0, n − p) and
some constant ε ≥ 0, the prolonged system (24) satisfies the
conic constraint
 T
   T
 
˙
˙
Q L δy
0
P
δy
δx
δx
≤
(26)
P 2λP +εI δx
δu
LT R δu
δx
for all (x, δx) ∈ T X and all (w, δw) ∈ W. (23) is strictly
differentially p-dissipative if ε > 0.
y
Differential dissipativity is guaranteed by the feasibility of
the inequality
h
i
∂f (x)TP+P ∂f (x)−C TQC +2λP +εI P B −C TL − C TQD
≤0
T
T
T
T
T
T
B P −L C − D QC
−R−D L−L D−D QD
for some symmetric matrix P with inertia (p, 0, n − p) and
some constant ε ≥ 0. A necessary condition for the feasibility
of this inequality is that ∂f (x)TP +P ∂f (x)−C TQC +2λP +
εI ≤ 0 which corresponds to (18) for Q = 0 and clarifies the
connection between differential dissipativity and dominance.
This infinite family of LMIs can reduced to a finite family
through relaxations, following the approach in Section V-B.
B. A dissipativity theorem for p-dominance
Suppose that a nonlinear system can be decomposed as the
interconnection of two subsystems:

ẋ1 = f1 (x1 ) + B1 u1
(27a)
y1 = Cx1 + D1 u1

ẋ2 = f2 (x2 ) + B2 u2
(27b)
y2 = Cx2 + D2 u2
u = Hy + v

(27c)

where the matrix H specifies the interconnection pattern
between inputs u = [ uT1 uT2 ]T and outputs y = [ y1T y2T ]T .
v = [ v1T v2T ]T is an additional input. Then the dissipativity
theorem below provides conditions for the differential pdissipativity of the interconnected system from the differential
p-dissipativity of its components.
The formulation of the theorem follows the approach of
[31]. The theorem can be easily adapted to the interconnection of several systems [32]. In what follows we will use

h
i
h
i
h
i
Q
L
R
Q̄ := 1 Q , L̄ := 1 L , and R̄ := 1 R for readability,
2
2
2
where for i ∈ {1, 2} the matrices Qi , Li and Ri characterize
the differential supply rate fo each subsystem.
Theorem 4: Let (27a) and (27b) be (strictly) differentially
p1 -dissipative and p2 -dissipative respectively, with uniform
rate λ and differential supply rate
 T 
 
Qi Li
δy
δy
i ∈ {1, 2} .
(28)
δu
δu
LTi Ri
Then, the interconnected system (27) is (strictly) differentially
p-dissipative with degree p = p1 + p2 , rate λ, and differential
supply rate
 T
 
Q L
δy
δy
(29)
δv
δv
LT R
where
Q := Q̄ + L̄H + H T L̄T + H T R̄H
L := L̄ + H T R̄
R := R̄ .

(30)

For v = 0, the interconnected system (27) is (strictly) pdominant if Q ≤ 0.
y
Theorem 4 is a general result to study dominance via
interconnections. In particular, it provides an interconnection
theorem for the analysis of contraction (0-dominance) and
monotonicity (1-dominance).
C. Differential passivity analysis
The passivity supply
 T
 
δy
0 I δy
δu
I 0 δu

(31)

plays an important role in dissipativity theory because it
connects the theory to the physical property that a passive
system can only store the energy supplied by its environment.
The theory of port-Hamiltonian systems encompasses a broad
modeling framework of physical models with passivity properties [47].
The passivity theorem states that the feedback interconnection of passive systems is passive. A differential version of
this important theorem is provided by Theorem 4: any system
that is differentially p-passive preserves that property under
the feedback interconnection with a differentially passive (i.e.
0-passive) system.
Passivity theory has proven useful in identifying robust controller structures that preserve stability. An important particular
case is the class of proportional-integral controllers, that we
now revisit in the light of dominance analysis.
We first consider the proportional feedback controller
(P ) ȳ = kP (ū) .

(32)

The controller is differentially 0-passive (for arbitrary nonnegative rate) from ū to ȳ provided that the mapping kp (·)
is monotone, or differentially positive: ∂kP (ū) ≥ 0 for all
ū ∈ R. For instance,
 T
 
δ ȳ
0 I δ ȳ
= 2δ ūT ∂k(ū)δ ū ≥ 0.
δ ū
I 0 δ ū

8

Likewise, the proportional-integral feedback controller
(P I) ẋc = ū, ȳ = kP (ū) + kI xc

(33)

from ū to ȳ is
• differentially 0-passive if kP (·) is monotone and if kI ≥
0, with rate λ = 0.
• differentially 1-passive if kP (·) is monotone and if kI <
0, with rate λ ≥ 0 (strictly for λ > 0).
This is because the storage S(δxc ) := k2I δx2c satisfies Ṡ ≤
δ ūδ ȳ. Furthermore, for λ > 0 and kI < 0 we have Ṡ +
λkI δx2c + εδx2c ≤ δ ūδ ȳ for ε = λ|kI |.
As an illustration, we revisit the nonlinear (Duffing) massspring-damper system in Section V-C. For any nonlinear spring
satisfying −3 ≤ ∂α(xp ) ≤ 3, the system is strictly differential
1-passive with rate λ = 2 from u to y = −xp : defining the
˙ = A(x)δx +
state x := [ xp xv ], the variational dynamics δx
Bδu, δy = Cδx


 


0
1
0
A(x) :=
B :=
C := −1 0 .
−∂α(xp ) −5
1

v

+

u

Plant

+

xp

u

−xc

Plant

xp

− 1s

Figure 5. Left: Closed loop of the plant given by (20) with α(xp ) = xp
and u = tanh(2xp ) + v (saturated proportional feedback). Right: Closed
loop given by the interconnection of the linear mass-spring-damper system
and nonlinear proportional-integral controller.

0.6

0.6

0.4

0.4

0.2

0.2

ū = −y .

The closed loops with the proportional controller and with the
proportional-integral controller are illustrated in Figure 5, for
the the particular case kP (ū) := tanh(2ū) and kI := −1.
By Theorem 4, the proportional feedback u = −kP (y) +
v = kP (xp ) + v preserves 1-differential passivity. Figure
6 (left) illustrates a situation where the original system is
monostable (linear spring, i.e. a quadratic potential) and then
converted to a bistable system with the static output feedback
u = − tanh(2y) + v = tanh(2xp ) + v. The example is
elementary but illustrative of a general principle: the feedback controller shapes the potential energy of a contractive
mechanical system to convert the system from monostable to
bistable.
Likewise,
the proportional-integral
control u = −k
R
R
R P (y) −
kI y + v = kP (xp ) − kI y + v = tanh(2xp ) − xp + v
makes the closed-loop system differentially 2-passive because
it corresponds to the negative feedback interconnection of two
differentially 1-passive systems. Figure 7 illustrates that, in
this new configuration, the role of the proportional controller
is to convert a linear stable system (kP = 0) into a 2-dominant
system with a stable limit cycle oscillation (kP = 1).
Again the example is elementary but illustrates the general
principle that PI control can turn a contractive mechanical system into a system with a limit cycle attractor. Such conclusions
are usually drawn from a local analysis of the linearization
around the stable equilibrium of the contractive system. The
first scenario corresponds to a saddle node bifurcation whereas
the second scenario corresponds to a Hopf bifurcation. The
differential approach in this paper makes this analysis non
local.

xv

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6
-1

0

1

-1

0

xp

1

xp

Figure 6.
Left: trajectories from different initial conditions of the open
loop system (20) with α(xp ) = xp and u = 0. Right: trajectories from
different initial conditions of the closed loop (20),u = tanh(2xp ) + v, with
α(xp ) = xp and v = 0.

0.3

0.8

0.2

0.6
0.4

0.1

0.2
0

xv

u = ȳ + v

0

xv

A(x)T P + P A(x) + 2λP + εI ≤ 0
P B = CT
 −2 −1 
for P = −1 0 and ε = 0.01, for all x ∈ R2 .
Assume for simplicity that kP (·) is odd and monotone and
consider the interconnection

xv

satisfies

0

-0.1
-0.2
-0.2

-0.4

-0.3

-0.6

-0.4

-0.8
-1

-0.5

0

xp

0.5

1

-2

-1

0

1

2

xp

Figure 7.
Left: Trajectory of the linear closed-loop system (kP = 0).
Right: Trajectory of the nonlinear closed-loop system (kP = 1) . Parameters:
α(x) = xp . Initial condition xp = 1, xv = 0, xc = 0.

9

D. Differential small gain analysis
The supply
T
 
δy
−I 0
δy
.
δu
0 γ 2 I δu



(34)

also has a special status in dissipativity theory because of its
connection to the small gain theorem, a cornerstone of robust
control theory [53], [52], [8], [54], [45].
When specialized to the supply (34), Theorem 4 provides a
differential version of the small gain theorem: p-dominance is
preserved under feedback with a 0-dominant system provided
that their finite differential gains γ1 and γ2 (of degree p and
0 respectively) satisfy the small gain condition γ1 γ2 < 1.
The differential small gain theorem opens the way to a
differential approach of nonlinear robust control. The link with
an operator-theoretic definition of the differential gain of a
dominant system is beyond the scope of this paper (see e.g.
[16] for a concept of differential gain for stable systems), but
we briefly illustrate how the theorem can be used in nonlinear
robustness analysis.
∆
r

Nominal
+ v closed
loop

y

Figure 8. A nonlinear plant with parametric uncertainties represented as the
feedback interconnection of nominal plant and uncertain dynamics ∆.

Consider them nominal closed loop in Figure 5 (right)
where the mass-spring-damper system (20) with linear spring
α(xp ) = xp is interconnected to aR saturated proportionalintegral feedback u = tanh(2xp ) − xp + v. We study the
robustness of the oscillations to perturbations affecting the
spring constant and damping coefficient. We use the usual
representation of parametric uncertainties through feedback
interconnections as shown in Figure 8.
For c = 5 and λ = 2 the nominal closed loop system is
2-dominant with differential gain γ := 0.5636 from the input
v to the position output xp , obtained for


−0.5522 0.0498 −0.0171
P :=  0.0498 1.4946 0.3068  .
−0.0171 0.3068 0.0576
Nonlinear perturbations on the mechanical spring of the form
α(xp ) = xp + ∆(xp )
are captured by the feedback interconnection in Figure 8,
taking v = ∆(xp ). The differential small-gain theorem implies
that 2-dominance is preserved for any |∂∆(xp )| ≤ 1/γ.
Furthermore, for any perturbation ∆(xp ) that preserves the
origin as a unique and unstable fixed point of the closed loop
system, every bounded trajectory of the perturbed closed loop
will converge to a periodic orbit, like in the nominal case.
The analysis of perturbations affecting the damping coefficient c = 5 is similar. For c = 5 and λ = 2 the nominal closed

loop system is 2-dominant with differential gain γ := 0.5468
from the input v to the velocity output xv , obtained for


−0.2859 −0.0028 −0.0131
0.2977  .
P :=  −0.0028 1.2328
−0.0131 0.2977
0.0532
Therefore oscillations will persist when the linear damping
coefficient cxv is replaced by a nonlinear coefficient c(xv )
given by
c(xv ) = 5xv + ∆(xp )
provided that |∆(xv )| < 1/γ and that the origin remains an
unstable fixed point.
VII. C ONCLUSION
This paper illustrated that linear-quadratic dissipativity theory, a cornerstone of stability theory, generalizes with surprising ease to the analysis of dominance. Dominance analysis
in turn is relevant to capture the frequent property that the
asymptotic behavior of a nonlinear dynamical model is lowdimensional. In particular, the theory seems relevant to generalize the theory of stability to a theory of multistability and
limit cycle analysis.
The approach in this paper is differential, meaning that the
usual linear matrix inequalities of dissipativity theory are considered in the tangent bundle. They characterize a dominated
splitting of the linearized flow between p dominant directions
and n − p transient directions. This property is captured with
the usual linear matrix inequalities of dissipativity theory, with
the only difference that the solution matrix P is required to
have a fixed inertia (p negative eigenvalues and n − p positive
eigenvalues), the standard stability framework corresponding
to the case p = 0.
An important restriction throughout this paper is to analyze
p-dominance with a constant quadratic storage (i.e. a constant
P ). This restriction is the price to be paid for tractability.
Standard LMI solvers can then be used to construct the storage.
A number of generalizations deserve further attention.
Those include the construction of differential storages that
are non quadratic, or/and state-dependent (i.e. non constant
P (x)), as well as the study of dominance with state-dependent
rate λ(x), or systems with a different degree of dominance in
different parts of the state-space, or the analysis of dominance
in non smooth systems. Such generalizations have received
considerable attention in the analysis of contraction, i.e. 0dominance, suggesting clear avenues to study p-dominance.
Finally, differential dissipativity theory offers an opportunity to
revisit classical results from robust control theory and absolute
stability theory in the context of multistable and oscillatory
systems, see e.g. [28] for a first step in that direction.
A PPENDIX
Proof of Theorem 1.
Invariant splitting. For any p > 0 and for any δx ∈ K− , the
dissipation inequality (14) implies
V̇ (δx) ≤ −2λV (δx) − ε | δx |2 ≤ −ε | δx |2

(35)

10

for all x ∈ X and all δx on the boundary of K− , which
guarantees that
∀t ≥ 0 : ∂ψ t K− ⊆ K−
t

(36a)

−

−

∀t > 0 : ∂ψ (K \ {0}) ⊂ K .

(36b)

The dissipation inequality (14) also implies
V̇ (δx) ≤ −2λV (δx) − ε | δx |2 ≤ −(2λ − ε1 )V (δx)

(37)

ε

for ε1 := |λmin (P )| > 0. Time-integration of this inequality
yields the estimate
∀t ≥ 0 :

e2λt V (∂ψ t δx)
≥ e ε1 t
V (δx)

(38)

which holds uniformly for all x ∈ X and all δx in the interior
of K− . (36) and (38) guarantee that there exist T > 0 and
λt
∂ψ t δx|
µ > 1 such that |e |δx|
≥ µ for all t ≥ T , all x ∈ X and
all δx ∈ K− .
Likewise, for any n − p > 0 and for any δx ∈ K+ , the
dissipation inequality (14) implies
V̇ (δx) ≤ −2λV (δx) − ε | δx |2 ≤ −(2λ + ε2 )V (δx)

(39)

ε

for ε2 = λmax (P ) > 0. Integration of the first inequality
backward time guarantees that
∀t ≥ 0 : ∂ψ −t K+ ⊆ K+

∀t > 0 : ∂ψ

−t

+

(40a)
+

(K \ {0}) ⊂ K .

(40b)

Integration of the second inequality backward time also yields
the estimate
∀t ≥ 0 :

e−2λt V (∂ψ −t δx)
≥ eε2 t
V (δx)

(41)

which holds uniformly for all x ∈ X and all δx in the interior
of K+ . As above, (40) and (41) guarantee that there exist
−λt
∂ψ −t δx|
T > 0 and µ > 1 such that |e |δx|
≥ µ for all t ≥ T ,
+
all x ∈ X and all δx ∈ K .
From here, we proceed as in the proof of [33, Theorem 1.2]
(see also [2, Chapter 3]) to show that
T
λt
t
−
−t
−
Hx :=
t≥0 e ∂ψ (x)K (ψ (x)) ⊂ K (x)
T
−λt
Vx :=
∂ψ −t (ψ t (x))K+ (ψ t (x)) ⊂ K+ (x)
t≥0 e
are invariant distributions of dimension p and n − p respectively, that is,
eλt ∂ψ t (x)Hx ⊆ Hψt (x)
eλt ∂ψ t (x)Vx ⊆ Vψt (x)

∀t ∈ R ,
∀t ∈ R .

Since eλt is just a scalar factor, (15) follows.
Exponential estimates. Observe that δx ∈ H implies that
δx belongs to the interior of K− . The estimate (16a) with
λ = λ − ε21 follows from the fact that −V (δx) is positive
definite in H and that V (δx) satisfies (37). For instance, there
exist 0 < ρ1 ≤ ρ2 such that ρ1 δxTδx ≤ −V (δx) ≤ ρ2 δxTδx
for all δx ∈ H and (37) gives ρ2 |∂ψ t δx|2 ≥ −V (∂ψ t δx) ≥
−e−(2λ−ε1 )t V (δx) ≥ e−(2λ−ε1 )t ρ1 |δx|2 for all t ≥ 0, from
which (16a) follows.
Likewise, δx ∈ V implies that δx belongs to the interior of
K+ . The estimate (16b) with λ = λ + ε22 follows from the

fact that V (δx) is positive definite in V and satisfies (39). 
Proof of Theorem 2. From the dissipation inequality (17) we
d 2λt
e V (x − y) = e2λt V̇ + 2λe2λt V (x −
derive the inequality dt
2λt
y) ≤ −e ε | x − y |2 which, by time integration, implies
the following estimate for any pair of solutions initialized at
x0 , y0 ∈ X :
∀t ≥ 0 : V (ψ t (x0 ) − ψ t (y0 )) ≤ e−2λt V (x0 − y0 ) −
Z t
−ε e2λ(τ −t) |ψ τ(x0 ) − ψ τ(y0 )|2 dτ

(42)

0

For large t ≥ 0, the first term on the right hand side vanishes.
This implies that the difference between any two solutions
either asymptotically vanishes or eventually remains in the
cone K− . We conclude that if x and y are distinct ω-limit
points, then necessarily V (x − y) < 0.
Consider any compact set Ω of ω-limit points. Let HP and
VP the invariant subspaces of the matrix P associated to the
p negative and n − p positive eigenvalues, respectively. Define
the linear projection Π : X → HP parallel to VP . We claim
that Π restricted to Ω is one-to-one. This is because x 6= y
and Π(x − y) = 0 imply V (x − y) > 0, which was proved to
contradict (42).
The remaining argument follows the proof of [22, Theorem
3.17]. If y ∈ ΠΩ(x) then y = Πz for a unique z ∈ Ω(x) and
the flow Πψ t (z) on VP is generated by the vector field
F (y) := Πf (Π−1 (y))

y ∈ Ω(x) ,

which is Lipschitz by construction.

Proof of Theorem 3
By Proposition 1, the feasibility of ∂f (x)T P + P ∂f (x) +
2λP ≤ −εI, for ε > 0 and for P of inertia (p, 0, n − p)
guarantees that each matrix ∂f (x) + λI has p unstable eigenvalues and n − p stable eigenvalues (negative real part) at
each x ∈ X . The splitting of the eigenvalues of ∂f (x) at
each x is preserved for every value of λ within some given
spectral gap (λmin (x), λmax (x)). Thus, by the uniformity of
the strict inequality above, λmin := supx∈X λmin (x) < λ and
λmax := inf x∈X λmax (x) > λ.

Proof of Theorem 4
Let P1 and P2 be solutions to (3) respectively for (27a)
and (27b). Take P := P1 + P2 , x := [ xT1 xT2 ]T , and δx :=
[ δxT1 δxT2 ]T . Note that P1 has inertia (p1 , 0, n1 − p1 ) and P2
has inertia (p2 , 0, n2 −p2 ), where n1 and n2 are the dimensions
of the two state manifolds, respectively. It follows that P has
inertia (p1 + p2 , 0, n1 + n2 − p1 − p2 ). Furthermore, by (strict)
differential dissipativity of (27a) and (27b), (3) can be written
˙ T P δx + δxT P δx
˙ + δxT (2λP +
in the aggregated form δx
T
T T
T
εI)δx ≤ δy Qδy + δy Lδu + δu L δy + δuT Rδu, for
some ε ≥ (ε > 0). Since δu = Hδy + δv, the expression
˙ T P δx + δxT P δx
˙ + δxT (2λP + εI)δx ≤
above reads δx
T
T
T T
δy Qδy+δy Lδv+δv L δy+δuT Rδu which shows (strict)
differential p-dissipativity of the interconnected system.
˙ T P δx + δxT P δx
˙ + δxT (2λP +
Finally, for v = 0 we get δx
T
εI)δx ≤ δy Qδy ≤ 0 where the last inequality follows from

11

the condition Q ≤ 0. This shows that P is a solution of (12).

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