Dual Seminorms, Ergodic Coefficients and Semicontraction Theory

Dynamical systems that are contracting on a subspace are said to be semicontracting. Semicontraction theory is a useful tool in the study of consensus algorithms and dynamical flow systems such as Markov chains. To develop a comprehensive theory of semicontracting systems, we investigate seminorms on vector spaces and define two canonical notions: projection and distance semi-norms. We show that the well-known lp ergodic coefficients are induced matrix seminorms and play a central role in stability problems. In particular, we formulate a duality theorem that explains why the Markov-Dobrushin coefficient is the rate of contraction for both averaging and conservation flows in discrete time. Moreover, we obtain parallel results for induced matrix log seminorms. Finally, we propose comprehensive theorems for strong semicontractivity of linear and non-linear time-varying dynamical systems with invariance and conservation properties both in discrete and continuous time.


Problem description and motivation
Before Stefan Banach proved his famous contraction principle in 1922 [2], Andrey Markov started in 1906 [25] the study of stochastic processes.As documented by Eugene Seneta [31], Markov established a key contraction inequality and a corresponding contraction factor now known with the name of ergodic coefficient of a Markov chain.This paper aims to provide a modern semicontraction theory approach to explain and generalize ergodic coefficients.
To be concrete, let the matrix A be row-stochastic and consider the discrete-time dynamical systems Similarly, let L be a Laplacian matrix and consider the continuous-time counterparts: Submitted on 2022/12/22.This work was supported in part by AFOSR grant FA9550-22-1-0059 and by Fondazione Ing.Aldo Gini.
These systems are perhaps the simplest examples of general averaging-based dynamics (e.g., robotic coordination and distributed optimization) and dynamical flow systems (e.g., compartmental and traffic systems).Important generalizations include systems of the form ẋ = f (t, x), where f satisfies invariance properties (generalizing A1 n = 1 n ) or conservation properties (generalizing 1 T n A T = 1 T n ); in all these (linear and nonlinear) cases, the system is at most marginally stable.
Markov and later scientists essentially showed that, under a certain connectivity assumption, maps of the form π → A T π are contraction maps with respect to the total variation distance on the simplex.To be specific, define the simplex and the total variation distance on ∆ n by d TV (π, σ) = 1 2 i |π i − σ i |.Then any two solutions π(k), σ(k) to (1b) satisfy where τ 1 (A) is the so-called Markov-Dobrushin ergodic coefficient defined by In short, when τ 1 (A) < 1, existence, uniqueness and global exponential stability of an equilibrium π * ∈ ∆ n for system (1b) is ensured.Now comes a remarkable similarity.If one defines the seminorm |||x||| dist,∞ = 1 2 (max i {x i } − min j {x j }), the following fact is also known [13,Theorem 1.1] about averaging systems of the form (1a): Despite the extensive research in this field, numerous known related facts remain somehow mysterious and numerous related mathematical questions remain open.For example, why is the same ergodic coefficient τ 1 relevant for the contraction properties of both dynamical flow systems and averaging systems?And is it the tightest such bound?How does one generalize the bounds (3) and (5) to ergodic coefficients τ p defined with respect to arbitrary p norms (instead of the 1 norm in (4))?How does one provide a unified robust stability analysis for both systems?What are the canonical Lyapunov functions for both systems (1a)-(1b), whose discrete-time variation along the flow is described by τ p (A)? How does one define ergodic coefficients for continuous-time systems?Is there a contraction theoretic framework that applies to timevarying and nonlinear systems with generalized invariance or conservation properties?

Contributions
This paper provides a comprehensive answer to all the open research questions outlined above.
In order to define Lyapunov functions for averaging, flow systems and their generalizations to nonlinear dynamical systems with invariant subspaces, we study seminorms, induced matrix seminorms for discrete time systems and logarithmic seminorms for continuous-time systems.A key contribution of this paper is to explain precisely in what sense ergodic coefficients are induced matrix seminorms and, when less than unity, contraction factors for discrete-time systems.This equality is the fundamental reason why ergodic coefficients play a critical role in robust stability theory for discrete-time dynamical systems with invariance properties.It is surprising that induced norms are widely studied in the matrix theory literature, but induced seminorms much less (e.g., see [15]).
After characterizing various seminorms' properties, we define two canonical sets of seminorms, namely, distance and projection seminorms, and establish remarkable duality properties between the two.Our first result generalizes and strengthens the so called Markov contraction inequality as a duality result between the aforementioned seminorms.Our duality result precisely explains why the induced matrix seminorms for both A and A T are identical, when computed with respect to dual seminorms.Particular emphasis is given to the case of consensus seminorms, that is, seminorms whose kernel is the consensus space (i.e., seminorms that are positive definite about the consensus space).Consensus seminorms appear naturally in averaging algorithms and surprisingly in systems with conservation property (such as Markov chains and dynamical flow systems).
It is an elementary algebraic observation that the total variation distance on the simplex arises from the restriction of the 1 projection consensus seminorm.
We then leverage all these notions to provide a general nonlinear semicontraction theory, grounded in two key theorems both for continuous and discrete time varying dynamical systems.The semicontraction theory we develop is tailored to systems with invariance or conservation properties.More in detail, when either the system's Jacobian leaves invariant the seminorm kernel (invariance property) or its orthogonal complement (conservation property), there is a well defined notion of perpendicular dynamics which is strictly contracting.For both systems, in the linear time varying case, we show how canonical Lyapunov functions (some of which partly known in the literature) naturally arise from seminorms.For the non-linear case, our first key theorem establishes conditions and features of strong semicontracting continuous time, time varying systems that enjoy the invariance property.The theorem extends Theorem 13 in [17] through the formulation of a cascade decomposition and by establishing a strong contractivity property on the orthogonal complement to the seminorm kernel.The second key theorem is entirely novel and pertains semincontraction conditions for continuous time, time varying, dynamical systems that enjoy the conservation property.A discrete time version of these two theorems is also provided.

Literature review
Interest in contractivity of dynamical systems via matrix measures can be traced back to Demidovič [7] and Krasovskiȋ [20].Logarithmic norms have been exploited in control theory later on by Desoer and Vidyasagar in [8] and applied in the study of contraction theory for dynamical systems for the first time by Lohmiller and Slotine [24].In the context of control theory, this literature inspired many generalizations of contraction theory such as partial contraction [36], weak-and semi-contraction [17], horizontal contraction on Riemannian and Finsler manifolds [12], [32], etc.
In particular, partial contraction refers to convergence of systems trajectories to a specific behavior, or a manifold [33], see also [9] for a survey on this theory.While partial contraction establishes convergence to a manifold, semicontraction ensures contractivity on the subspace perpendicular to the kernel of the seminorm.For a characterization of partial contraction in the 2 -norm for the study of synchronization in networked systems, see [36].The notion of partial contraction is closely related to the one of semicontraction and weak contraction proposed and investigated in [17].Semicontraction theory relies on a relaxed concept of matrix measure, known as matrix semimeasure.For this reason, contractivity of a dynamical system is only ensured on a certain subspace and the distance between trajectories is allowed to increase along certain directions.
A relevant behavior, to which semicontraction theory applies, is the one of consensus for dynamical systems.Strictly related to consensus, when it comes to stochastic systems, is the concept of (weak) ergodicity [34].The concept of weak ergodicity was first formalized in 1931 by Kolmogorov [18], who stated that a sequence of stochastic matrices is weakly ergodic if the rows of the matrix product tend to become identical as the number of factors increases.The study of ergodicity coefficients is traced back to the pioneering work of Markov [25], in 1906, in which a first expression of ergodicity coefficients was provided in the context of the Weak Law of Large Numbers.Subsequent works from Doeblin [11] and Dobrushin [10] provided conditions for weak ergodicity.The key results in this research area were extended and then reviewed by Seneta in the 80's, see, e.g., [29].A survey of ergodicity coefficients is given by Ipsen and Selee [16].a historical discussion is given by Hartfiel [13, Chapter 1], and a recent treatment on their connection with spectral graph theory is given by Marsli and Hall [26].A characterization of "convergability" [23], namely the convergence of a product of an infinite number of stochastic matrices, is studied by Liu et.al in [23], where a different approach, based on optimally deflated matrices, is proposed.Despite the evident relation between ergodicity coefficients, contraction factors and induced matrix seminorms, especially in the context of stochastic and averaging systems [1], to the best of our knowledge none in the past has shed full light on their connections (see [6] for some preliminary work in this direction).This manuscript aims to bridge the existing gap in the scientific literature between semicontraction and ergodicity of dynamical systems.

Paper organization
Section II presents notation and preliminary results.Section III introduces the projection and distance seminorms and establishes their duality relationship.Section IV pertains with induced matrix seminorms and induced matrix log-seminorms.In Section V semicontraction theory is applied to dynamical systems.Finally, Section VII concludes the manuscript.
All theorems in this manuscript are new.Lemmas and Corollaries are either new or simple derivations from known results.This manuscript extends the submitted version in the IEEE Transaction on Automatic Control and includes the proofs of Lemma 13 and Theorem 22, explicit expressions for projection seminorms of columns stochastic matrices ( 17)- (19), Corollary 24, the property (ii) from Lemma 17, and Remark 3.

A. Notation
The set R ≥0 is the set of nonnegative real numbers.Let I n ∈ R n×n denote the identity matrix of size n.Let 1 n and 0 n denote the n dimensional column vectors all whose entries equal 1 and 0, respectively.Let e i denote the i-th vector of the canonical basis in R n .For a matrix A ∈ R n×n , let A T denote its transpose, [A] i,j its (i, j)th entry.The matrix A is nonnegative if all its entries are nonnegative, it is row stochastic if it is nonnegative and denotes the standard inner product on R n .We let Π ⊥ denote the orthogonal projection matrix onto K ⊥ , where the symbol K ⊥ denotes the orthogonal complement of K.Note that Π ⊥ = Π T ⊥ , and if K = span{1 n }, then Π ⊥ = I n − 1 n 1 T n /n =: Π n .Given x ∈ R n , the perpendicular and parallel components of x to K are denoted by x ⊥ = Π ⊥ x and x = (I n −Π ⊥ )x, respectively.Define the n-simplex as and the sign function, sign : R → {−1, 0, 1}, as sign(x) = x |x| if x = 0, and sign(0) = 0. Given two matrices A, B ∈ R n×n we use the notation A B to indicate that A−B is a negative semidefinite matrix.
A directed, weighted graph is a triple [5], G = (V, E, A), where V = {1, . . ., n} is the set of vertices, E ⊆ V × V is the set of arcs and A is the adjacency matrix.An arc (i, j) belongs to G if and only if [A] ij = 0. Two nodes i, j ∈ V are weakly adjacent if either (i, j) ∈ E or (j, i) ∈ E.
Given a real vector space V , the dual space V is the vector space of linear maps from V into R.If V = R n , then V is the vector space of row vectors in R n .In this case, it is typical to make a slight abuse of notation and assume V = R n .

B. Basic concepts
We start with some basic useful concepts.For x ∈ R n and p ∈ N, the p -norm of x is For A ∈ R n×m and p ∈ N, the p -induced norm of A is Proof.The result is a direct consequence of the reverse triangle inequality and the sub additivity property of seminorms applied to the orthogonal decomposition x = x ⊥ + x , with x ∈ K.

Remark 3 (Relationship between norm and seminorm).
A seminorm on R n with kernel K induces a norm on K ⊥ by restriction.Vice-versa, given a subspace K of R n , a norm • on K ⊥ , denoted by • ⊥ , induces a seminorm on R n with kernel K by projection: Definition 6 (Generalized p ergodicity coefficient [28]).
Given p ∈ [1, ∞] and a vector subspace K ⊂ R m , the generalized p ergodicity coefficient The ergodicity coefficient ( 6) is the norm of the operator defined on the real (normed) linear space

A. Projection and Distance Seminorms
In the following we provide the definition of projection and distance seminorms.These two seminorms will play a fundamental role in the duality result.
Definition 8 (Projection and distance seminorms).Let K ⊂ R n be a vector space and Π ⊥ ∈ R n×n be the orthogonal projection matrix onto K ⊥ .For each p ∈ [1, ∞], define the p -projection seminorm with respect to K by and the p -distance seminorm with respect to K by Note that the optimization problem ( 8) is well posed since the norm function is convex.7) and (8).Next, we compute

This completes the proof of statement (ii).
It is not true in general that |||x||| K proj,p ≤ x p .Example 10 (Seminorms for consensus and stationary distribution).When K = span{1 n } and p ∈ {1, 2, ∞}, explicit formulas for the p -projection and distance seminorms are either easily derivable or available in the literature [4], [14], [23].For each x ∈ R n , with the shorthand Next, sort the entries of x according to Figure 1 illustrates the unit disks for these seminorms on R 3 .
Example 11 (Total variation distance).The total variation [21, Section 4.1] is a metric on the simplex ∆ n defined by Given any two vectors x, y ∈ ∆ n , a simple derivation shows where is the 1 -projection seminorm with respect to the kernel K = span{1 n }.

B. Duality
In this section we establish a useful duality relationship between projection and distance seminorms.We start with the notion of dual seminorm.
Proof.Let y, z ∈ V , and let a ∈ R. Since x = 0 n satisfies |||x||| K ≤ 1 and x ∈ K ⊥ , |||y||| K ≥ y, 0 n = 0, establishing the non-negativity of |||•||| K .To prove homogeneity, Finally, to prove sub-additivity, When V = R n , we make the usual identification (R n ) = R n .In this case, the kernel of the dual seminorm is identical to the kernel of the primal seminorm.
Next, we present an important generalization to arbitrary p / q norms of the Markov contraction inequality from [14, Lemma 2.3].
Proof.For each u ∈ K satisfying |||y||| The result follows from minimizing with respect to u.

Remark 15 (Markov contraction and Hölder's inequalities).
For the inner product of vectors perpendicular to a subspace, the Markov contraction inequality provides a tighter bound than the Hölder's inequality x T y ≤ x p y q .In fact, as a consequence of Lemma 9(ii), Next, we recall that, for unconstrained vectors, the Hölder's inequality provides a tight bound in the sense that, for all x ∈ R n , there exists y ∈ R n such that x T y = x p y q .We now show this tightness result also for the Markov contraction inequality, thereby establishing the duality relationship between projection and the distance seminorms.
To prove the opposite inequality, choose any y ∈ R n such that |||y||| K proj,q ≤ 1 and y ∈ K ⊥ .Then ||y|| q = |||y||| K proj,q ≤ 1, so by Lemma 14, To prove equality (10) we notice, as in the previous first case, that if x ∈ K, then |||x||| K proj,q = 0, while y ∈ K ⊥ implies that y T x = 0, so both sides of (10) are zero.Otherwise, in the second case, if x ∈ K, by Lemma 39, there exists .
To prove the opposite inequality, choose any y ∈ R n such that |||y||| K dist,p ≤ 1 and y ∈ K ⊥ .Lemma 14 implies This concludes the proof.

A. Induced Matrix Seminorms
In the following we list some basic properties related to induced matrix seminorms.

Lemma 17 (Properties of induced matrix seminorms). Let |||•|||
K be a seminorm on R n with kernel K.For any Proof.Property (i) was proven in [19].To prove property (ii), assume x * ∈ K ⊥ to be a vector with To prove property (iii), decompose any vector x ∈ R n as x = x ⊥ + x , with x ⊥ ∈ K ⊥ and x ∈ K, and notice that where the second equality is based on Lemma 2 and exploits the fact that AK ⊆ K. To prove property (iv) we notice that, by adopting the same decomposition as before where the first equality is based on Lemma 2 and exploits the fact that AK ⊆ K, while the inequality derives from Definition 4. Property (v) can be found in [19] and can be proved by following arguments similar to property (ii).
Based on Theorem 16 we are now in the position to provide one of the main results of this manuscript.
For a matrix A ∈ R k×n , and for p, q ∈ [1, ∞], with p −1 + q −1 = 1 it holds that where • p : R n → R ≥0 and • q : R n → R ≥0 are dual norms.
The following theorem represents a generalization of the duality relationship between induced matrix norms (11) to seminorms.
Theorem 18 (Duality of induced matrix seminorms).Let p, q ∈ [1, ∞] such that p −1 + q −1 = 1.For any matrix A ∈ R n×n , and any vector space K ⊆ R n , Additionally, if AK ⊆ K, then Proof.Eqn. ( 12) is a direct consequence of Theorem 16: = max To prove (13) note that where the second-last equality follows from the fact that In the following we provide some explicit expressions for the distance seminorm of row-stochastic matrices and the projection seminorm of column stochastic matrices for the case in which the kernel of the seminorms is the consensus subspace.The explicit expressions can be derived by the ones available in the literature for ergodicity coefficients [16], [30] and by the duality result from Theorem 18.
The second equality in (15) follows from Lemma 7, since and thus A T Π n A b 2 Π n .This way we have proved that Formulas ( 17) − (19) are derived by duality.
Finally, we include a comparative analysis for induced seminorms and the notion of optimal deflation given by [23].

B. Induced Matrix Log Seminorms
We now present a duality result for induced matrix log seminorms which is parallel to the one in Theorem 18.
Theorem 23 (Explicit formulas for distance logarithmic seminorms).Consider the consensus distance and projection seminorms.Let L ∈ R n×n be the Laplacian matrix corresponding to an adjacency matrix A ∈ R n×n without self-loops, and let d out = A1 n .For each i ∈ {1, 2, . . ., n}, sort the off-diagonal entries of Ae j according to Observe that S h is row-stochastic for every h > 0, and its entries are For each j ∈ {1, 2, . . ., n}, sort the entries of S h e j as Assume h is so small that (S h ) (1),j = (S h ) j,j .Then by (14), a (i),j .
Substituting into (5) yields the formula for µ dist,1 (−L), since the order of the off-diagonal elements of Ae j is identical to the order of the off-diagonal elements of S h e j for all h > 0.

Corollary 24 (Explicit formulas for projection logarithmic seminorms). Consider the distance and the projection seminorms with ker(|||•|||
R n×n be the Laplacian matrix corresponding to an adjacency matrix A ∈ R n×n without self-loops, and let d out = A1 n .For each j ∈ {1, 2, . . ., n}, sort the off-diagonal entries of e T i A according to a i, (1)

V. SEMICONTRACTING DYNAMICAL SYSTEMS
We exploit now the duality result of induced matrix seminorms and induced matrix logarithmic seminorms for the study of strong semicontractivity of dynamical systems.We also provide some theoretical results that formalize semicontractivity conditions for linear and nonlinear dynamical systems both in discrete and continuous time.
Given a vector subspace K ⊂ R n and a vector field f : R n → R n , the perpendicular vector field f ⊥ : R n → K ⊥ and the parallel vector field f : R n → K are denoted for all Lemma 26 (Differential characterization of invariance).Given a continuously differentiable map f :

A. Discrete Time Semicontraction
Let us consider the discrete time, time varying, nonlinear dynamics with k ∈ Z ≥0 , x ∈ R n .We assume f to be continuously differentiable in the second argument.In the following we give a generalized definition of strongly semicontracting discrete time system with respect to the one in [17].The generalization applies to systems with arbitrary contraction step.

Definition 27 (Semicontracting discrete time systems). Let |||•|||
K be a seminorm on R n with kernel K.If there exists m ∈ N, ρ < 1 and a domain C ⊆ R n for which the timevarying vector field f : for all k ∈ Z ≥0 and x ∈ C, then the vector field is strongly semicontracting on C with rate m √ ρ.
Lemma 28 provides sufficient conditions for two fundamental discrete-time systems to be strongly semicontracting.
Lemma 28 (Strong semicontractivity of discrete-time affine systems).Given a subspace K ⊂ R n and p, q ∈ [1, ∞] with p −1 + q −1 = 1, consider a sequence of matrices {A(k)} k∈Z ≥0 ⊂ R n×n satisfying: (i) Then the system is strongly semicontracting with rate ρ in the distance q seminorm with kernel K. Moreover is strongly semicontracting with rate ρ in the projection p seminorm with kernel K.Moreover, for any x(0), y(0) satisfying x(0) − y(0 Proof.The proof of part (i) follows from equation ( 13) in Theorem 18, and from the conditional submultiplicative property iii) Lemma 17: The proof of part (ii) follows from Lemma 17 part (i) since x(k) − y(k) ∈ K ⊥ , ∀k ∈ Z ≥0 , as a consequence of the invariance assumption (invariance) and therefore For example, when the subspace K is the consensus subspace, the matrices {A(k)} ∞ k=0 are row-stochastic and the term b(k) ≡ 0 n ∀k ∈ Z ≥0 , the systems ( 23) and ( 24) are the standard averaging (1a) and flow systems (1b) in the Introduction and the bounds ( 25) and ( 26) are precisely the bounds (3) and ( 5) stated in the Introduction.
The following theorem focuses on strong semicontractivity of discrete-time dynamical systems that enjoy the invariance property of the kernel of the seminorm.
Theorem 29 (Discrete time semicontracting dynamics with invariance property).Consider a system as in (21).Let K ⊂ R n be an f -invariant subspace, and suppose that f is strongly semicontracting with rate ρ < 1, with respect to a seminorm |||•||| K on R n with kernel K.Then, (i) the system admits the cascade decomposition (ii) the perpendicular dynamics (28) are strongly contracting on K ⊥ with rate ρ, with respect to • ⊥ : K ⊥ → R ≥0 ; and (iii) for any two trajectories x(k), y(k) of (21), Proof.Regarding part (i), the cascade decomposition ( 27)-( 28) follows from the observation that where the second equality is due to the f -invariance of K. Part (ii) follows from where the second equality follows from the fact that for a generic matrix A, The following theorem focuses on strong semicontractivity of discrete-time dynamical systems that enjoy the invariance property of the orthogonal complement of the kernel of the seminorm.
Theorem 30 (Discrete time semicontracting dynamics with conservation property).Consider a system as in (21).
strongly semicontracting with rate ρ < 1 with respect to a seminorm |||•||| K on R n with kernel K.Then, (i) the system admits the cascade decomposition (ii) for each x ∈ K, the vector field is strongly contracting with rate ρ, with respect to for all k ∈ Z ≥0 .
Proof.Regarding part (i), the cascade decomposition ( 29)- (30) follows from the observation that where the second equality is due to the f -invariance of K ⊥ .To prove (ii), fix x ∈ K, and pick any x ⊥ , y ⊥ ∈ K ⊥ .Then where the first inequality is due to the subadditivity property and the second one follows from point (ii) and the invariance of K ⊥ .

B. Continuous Time Semicontraction
Let us consider the continuous time, time varying, nonlinear dynamics with t ∈ R ≥0 , x ∈ R n .We assume f to be continuously differentiable in the second argument.
Definition 31 (Semicontracting continuous time systems).Let |||•||| K be a seminorm on R n with kernel K.The time-varying Lemma 32 provides sufficient conditions for two fundamental continuous time dynamical systems to be strongly infinitesimally semicontracting.
Lemma 32 (Strong semicontractivity of continuous-time affine systems).Given a subspace K ⊂ R n and p, q ∈ [1, ∞] with p −1 + q −1 = 1, consider a sequence of matrices {A(t)} t∈R ≥0 ⊂ R n×n satisfying: (i) The system is strongly infinitesimally semicontracting with rate c in the distance p seminorm with kernel K, moreover is strongly infinitesimally semicontracting with rate c in the projection q seminorm with kernel K, moreover, for any x(0), y(0) satisfying x(0) − y(0 Proof.The proof of part (i) follows from Theorem 13, part i) in [17].To prove part (ii) we follow a similar reasoning as in Theorem 11 from [17].In fact, for all x(0), y(0) such that x(0) − y(0) ∈ K ⊥ , since the solutions t → x(t) of ( 32) are differentiable, by defining z(t) x(t) − y(t), for small h, one can write since z(t) ∈ K ⊥ and A T K ⊥ ⊆ K ⊥ by hypothesis.Therefore, by Lemma 2 and Lemma 17 part (i) Taking the limit as h → 0 + , one gets Eq. ( 33) follows from the fact that µ dist,p (A(t)) = µ proj,q (A T (t)) ≤ −c for all t.
The following theorem focuses on strong infinitesimal semicontractivity of continuous-time dynamical systems that enjoy the invariance property of the kernel of the seminorm.This theorem extends Theorem 13 from [17] through the formulation of a cascade decomposition and by establishing a strong contractivity property on the orthogonal complement to the seminorm kernel.
Theorem 33 (Continuous time semicontracting dynamics with invariance property, partially from [17]).Consider a system as in (31).Let K ⊂ R n be an f -invariant subspace and suppose that f is strongly infinitesimally semicontracting with rate c > 0, with respect to a seminorm |||•||| K in R n with kernel K.Then, (i) the system admits the cascade decomposition (ii) the perpendicular dynamics (35) are strongly infinitesimally contracting on K ⊥ with rate c, with respect to • ⊥ : K ⊥ → R ≥0 ; (iii) for any two trajectories x(t), y(t) of (31),
Proof.Regarding part (i), the cascade decomposition is obtained by following the same reasoning as in Theorem 29.Part (ii) follows from where the first equality follows from the fact that for a generic matrix A, µ The following theorem focuses on strong semicontractivity of continuous-time dynamical systems that enjoy the invariance property of the orthogonal complement of the kernel of the seminorm.
Theorem 34 (Continuous time semicontracting dynamics with conservation property).Consider a system as in (31).Let K ⊂ R n be such that K ⊥ is an f -invariant subspace.Let f : R ≥0 × R n → R n be strongly infinitesimally semicontracting with rate c > 0 with respect to a seminorm |||•||| K on R n with kernel K.
Then, (i) the system admits the cascade decomposition (ii) for each x ∈ K, the vector field x ⊥ → f ⊥ (t, x + x ⊥ ) is strongly infinitesimally contracting with rate c, with respect to with constant ∈ R with respect to some metric d K on K, then for any two trajectories x(t), y(t) of (31), satisfying x(0) − y(0 Proof.The proof follows the same arguments as Theorem 30 for discrete time systems.

VI. GRAPH THEORETICAL CONDITIONS FOR SEMICONTRACTIVITY
We now provide graph theoretical conditions for the systems ( 1) and ( 2) to be semicontracting with respect to p distance and projection seminorms, for p ∈ {1, 2, ∞}.For the discrete time case, the following conditions are topological abstractions of algebraic conditions in [16], [23].Lemma 36 is novel.
Lemma 35 (Topological conditions for discrete-time averaging systems).The averaging system (1a) x(k + 1) = Ax(k) with A row stochastic is strongly semicontracting in the (i) Proof.Condition (i) ensures, in particular, that there exists m ∈ N such that A m has at least n 2 + 1 nonzero entries in each column so the expression in (14) takes value less than one.Consequently, the system is strongly semicontracting according to condition (22) in Definition 27.
Condition (ii) directly follows from Lemma 21 and Theorem 8 in [23].Finally, according to Corollary 4.5 in [5], condition (iii) ensures that there exists m ∈ N such that A m (has a column with all nonzero entries and hence) is scrambling.Consequently, according to Corollary 3.9 in [16] and condition (22) in Definition 27 the system is strongly semicontracting.
Lemma 36 (Topological conditions for continuous-time averaging).The averaging system (2) ẋ = −Lx with L the Laplacian of a graph with adjacency matrix A and without self-loops, is strongly infinitesimally semicontracting in the (i) min{a ik , a jk } > 0 ∀i = j that for nonnegative adjacency matrices is true if and only if: (i, j) is an edge or, (j, i) is an edge or (i, k) and (j, k) are an edge for some third node k.

VII. CONCLUSIONS
We have studied seminorms on vector spaces and induced matrix seminorms for discrete-and continuous-time dynamical systems.We have shown how the natural distance and projection seminorms are dual and how the long-studied p ergodic coefficients of a row-stochastic matrix are precisely induced matrix seminorms.We have provided a comprehensive treatment of semicontraction for discrete-and continuous-time systems with invariance or conservation properties.Future research directions include the application of semicontraction theory to systems with symmetries, such as robotic vehicles (SE(3) symmetry) and coupled oscillators (torus symmetry), as well as systems with invariance properties, such as population games and evolutionary dynamics (whose state space is the simplex).A long-term elusive task is the definition of an ergodic coefficient that is strictly less than unity for rowstochastic matrices satisfying weak connectivity properties.

APPENDIX I SEMINORM COEFFICIENTS
Here we recall some useful properties of standard p-norms.In the following, for a differentiable function f : R n → R, we denote by ∇(f ) its gradient.
Based on Lemma 37, we establish a novel and useful characterization of the distance and projection seminorms.
Lemma 38 (Coefficients for distance seminorms).Let p, q ∈ [1, ∞] be such that p −1 + q −1 = 1 and let K ⊂ R n be a vector subspace.There exists a distance coefficient map ψ p : R n → K ⊥ such that, for all x ∈ R n , (i) Proof.Let V ∈ R n×k be a a matrix whose columns are a basis for K, so that we can write where G p is the subdifferential G p = ∂|| • || p ⊂ R n .Consequently, there exists a vector ψ p (x) ∈ G p (x − V α * ) such that ψ p (x) ∈ ker(V T ) = K ⊥ .Note that |||ψ p (x)||| K proj,q = ψ p (x) q , that ψ p (x) T x = ψ p (x) T (x − V α * ), and that |||x||| K dist,p = x − V α * p , so we need only to show for each p ∈ [1, ∞] that ψ p (x) q = 1 and that ψ p (x) T (x − V α * ) = x − V α * p .Case p = 1: Using the standard formula for the subgradient of the absolute value function [3], ψ 1 (x) ∈ G 1 (x − V α * ) implies that Case p ∈ (1, ∞): If p ∈ (1, ∞), then • p is differentiable, so G p (z) = ∇ z p for all z ∈ R n , and thus ψ p (x) = ∇||x − V α * || p (where the gradient is taken with respect to x − V α * ).If x / ∈ K ⊥ , then x − V α * = 0 n , so ||ψ p (x)|| q = 1 due to Lemma 37. A further consequence of this lemma is that Case p = ∞: Let I ⊆ {1, 2, . . ., n} be the set of indices such that x−V α * ∞ = |x−V α * | i .Using a standard formula for the subdifferential of a pointwise maximum [3] where conv denotes the convex hull, and the subdifferential of each absolute value is with respect to its argument.Therefore, there exist g i ∈ ∂|x − V α * | i for each i ∈ I, as well as convex weights λ i , such that For each g i , we have [g i ] j = 0 for j = i, since |z| i only depends on z i for any z ∈ R n .Furthermore, if x / ∈ K, then x − V α * = 0 n , so |x − V α * | i > 0 for all i ∈ I, which implies that [g i ] i = sgn(x − V α * ) i .Together, these two observations imply that Lemma 39 (Coefficients for projection seminorms).Let p, q ∈ [1, ∞] be such that p −1 + q −1 = 1 and K ⊂ R n be a vector subspace.There exists a projection coefficient map ζ p : R n → K ⊥ such that, for all x ∈ R n , (i) ζ p (x) = 0 n if x ∈ K and |||ζ p (x)||| K dist,q ≤ 1 otherwise, and (ii) |||x||| K proj,p = ζ p (x) T x.Proof.Let x ∈ R n and define x ⊥ = Π ⊥ x.
Here a conjecture on the equivalence between p distance seminorm and p-optimal deflation as in Definition 20.For each p ∈ [1, ∞] and row-stochastic matrix A ∈ R n×n , Here some reasons in support of this conjecture.(i) Expressions given in [22] 4 to the projection seminorm, with kernel K = span{1 n }, it would lead to the orthogonality constraint with respect to K and consequently to the equivalence between |A| p and A T K proj,q .
1 distance consensus seminorm if A is doubly stochastic and G(A) is strongly connected and aperiodic; (ii) 2 distance consensus seminorm if A is doubly stochastic and G(A) is weakly connected with self loops at each node; (iii) ∞ distance consensus seminorm if G(A) has self loops at each node and a globally reachable node.
1 distance consensus seminorm if A is doubly stochastic and every node has at least n 2 in-neighbors, (ii) 2 distance consensus seminorm if A is doubly stochastic and G(A) is weakly connected, (iii) ∞ distance consensus seminorm if every two nodes are either (weakly) adjacent or have a common out-neighbor.< 1, ∀i that for A doubly stochastic is fulfilled if and only if each node has at least n 2 in-neighbors.To prove (ii) note that for A doubly stochastic LΠ n = Π n L and hencethe formula for µ dist,2 (−L) in Theorem 24 reads as for p ∈ {1, 2, ∞} of x ∈ R n and of A ∈ R n×n row stochastic, of |x| p and |A| p coincide with the ones of |||x||| If the envelope theorem [35, Theorem 1.F.1] could be applied