Asymptotic Stability of port-Hamiltonian Systems

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Asymptotic Stability of

port-Hamiltonian Systems

Marcus Waurick∗and Hans Zwart†

We characterise asymptotic stability of port-Hamiltonian systems by means

of matrix conditions using well-known resolvent criteria from C0-semigroup

theory. The idea of proof is based on a recent characterisation of exponential

stability established in [12], which was inspired by a structural observation

concerning port-Hamiltonian systems from [10]. We apply the result to study

the asymptotic stability of a network of vibrating strings.

Keywords port-Hamiltonian systems, stability, C0-semigroup, Inﬁnite-dimensional sys-

tems theory

MSC2020 93D23, 37K40, 47D06, 34G10

Contents

1. Introduction 2

2. Characterisation of asymptotic/semi-uniform stability 5

3. Exponential stability revisited 8

4. Example, network of vibrating strings 10

5. Vibrating strings and exponential stability 20

6. Conclusion 26

A. Well-posedness of Ordinary Diﬀerential Equations with measurable co-

eﬃcients 27

∗Institut für Angewandte Analysis, TU Bergakademie Freiberg, Germany

†Department of Applied Mathematics, University of Twente, Twente, The Netherlands and Depart-

ment of Mechanical Engineering, Technische Universiteit Eindhoven, 5600 MB Eindhoven, The

Netherlands

arXiv:2210.11775v2 [math.AP] 11 Dec 2023

1. Introduction

In this note we discuss the stability of port-Hamiltonian partial diﬀerential equations

(p.d.e.’s) of the form

∂x

∂t (ζ, t) =

k=0

∂k

∂ζk(H(ζ)x(ζ, t)) (1)

on the spatial interval [a, b]. Here, Pk∈Cn×nsatisfy P∗

k= (−1)k+1Pk(k∈ {0,··· , N})

and PNis invertible. The Hamiltonian density H: (a, b)→Cn×nis uniformly bounded,

H(ζ)∗=H(ζ)and there exists an m > 0such that H(ζ)≥mI for almost all ζ∈[a, b].

The above p.d.e. is completed with boundary conditions given as







(Hx)(b, t)

∂N−1(Hx)

∂ζN−1(b, t)

(Hx)(a, t)

∂N−1(Hx)

∂ζN−1(a, t)







= 0,(2)

where WBis an nN ×2nN matrix which we assume to be of full rank.

One possible approach to address properties of the above system is to invoke the theory

of strongly continuous semigroups of bounded linear operators, in that one reformulates

the above as an abstract ordinary diﬀerential equation in an (inﬁnite-dimensional) state

space X. For this, a suitable functional analytic setting is conveniently formulated in

the weighted Hilbert space X=L2,H((a, b); Cn), which coincides with L2((a, b); Cn)as

set and is endowed with the weighted inner product given by

⟨f, g⟩H=1

2Zb

g(ζ)∗H(ζ)f(ζ)dζ. (3)

Xis known as the energy space. In [7] necessary and suﬃcient conditions are given

when the operator associated with the above p.d.e. plus boundary conditions generates

a contraction semigroup. Since the notation will be used in our stability result, we repeat

this theorem. We deﬁne the nN ×nN matrix Qand the 2nN ×2nN matrix Rext as

Q= [Qij ], i, j ∈ {1,··· , N}with Qij =(0i+j > N

(−1)i−1Pki+j−1 = k(4)

and

Rext =1

√2Q−Q

InN InN .

Note that Rext is invertible since PNis invertible.

Theorem 1.1 ([7, 8, 9]).Consider the operator

A=PN

dζN+··· +P1

dζ +P0H(5)

with domain

dom(A) = {x∈L2,H((a, b); Cn)| Hx∈HN((a, b); Cn), WB







(Hx)(b)

dN−1Hx

dζN−1(b)

(Hx)(a)

dN−1Hx

dζN−1(a)







= 0},(6)

where WBis a full rank nN ×2nN matrix. Then the following conditions are equivalent:

(i) Agenerates a contraction semigroup X;

(ii) WBsatisﬁes

WBR−1

ext 0InN

InN 0WBR−1

ext∗≥0; (7)

(iii) For all x∈dom(A)there holds

⟨Ax, x⟩H+⟨x, Ax⟩H≤0.

Although the characterisation of contraction semigroups is given for the operator

associated to the p.d.e. (1) with boundary conditions (2), other properties have only

been studied for the case N= 1, see e.g. [9]. Recently, for that class the characterisation

of exponential stability was found. In order to state this result, we need to consider the

following parametrised ordinary diﬀerential equation

dζ (ζ) = iωP −1

1(H(ζ)−1+P0)v(ζ)for ζ∈(a, b)and ω∈R.(8)

Let Φωdenote its fundamental solution1, i.e.,

Φω: [a, b)→Cn×nwith Φω(a) = In,(9)

and satisﬁes (8). The characterisation of exponential stability now reads as follows.

Theorem 1.2 ([12]).Let (A, dom(A)) be given as in Theorem 1.1 for N= 1, and be

such that Agenerates a contraction semigroup on X. Furthermore, assume that

sup

ω∈R∥Φω∥∞= sup

ω∈R

sup

ζ∈(a,b)∥Φω(ζ)∥<∞.(10)

Then the following conditions are equivalent:

1In Appendix A we show that this exists and is uniquely determined, even when the right-hand side

of (8) is not continuous.

(i) (T(t))t≥0is exponentially stable;

(ii) Qω:=WBΦω(b)

Inis invertible and supω∈R∥Q−1

ω∥<∞.

Inspired by this theorem, we characterise the semi-uniform/asymptotic stability of the

contraction semigroup. For this, we invoke the celebrated Batty–Duyckaerts theorem,

[1]. This result can be formulated for the general class, and so we do not assume that

N= 1. We introduce the following extension of the diﬀerential equation, see also (8),

dζ (ζ) = Pλ(ζ)v(ζ)for ζ∈(a, b)and λ∈C,(11)

where vtakes its values in CnN and

Pλ(ζ) = 





0I0··· 0

0 0 I0.

.......0

0··· ··· 0I

λP −1

NH(ζ)−1−P−1

NP0−P−1

NP1··· ··· −P−1

NPN−1







(12)

Let Ψλdenote its fundamental solution2, i.e.,

Ψλ: [a, b)→CnN×nN with Ψλ(a) = InN ,(13)

and satisﬁes (11). Note that for N= 1 we have that Ψiω = Φω, see (8), (9).

Theorem 1.3. Assume that (A, dom(A)) as given in (5) and (6) generates a contraction

semigroup (T(t))t≥0on X. Then the following conditions are equivalent:

(i) (T(t))t≥0is semi-uniformly stable (with decay rate f: [0,∞)→[0,∞)), i.e.,

f(t)→0as t→ ∞ and for all x∈dom(A)

∥T(t)x∥H≤f(t)∥x∥dom(A)t≥0.

(ii) (T(t))t≥0is asymptotically stable, i.e, for all x∈Xthere holds T(t)x→0as

t→ ∞;

(iii) The operator Awith domain dom(A)has no eigenvalues on the imaginary axis;

(iv) The square matrix Q(iω):=WBΨiω(b)

InN is invertible for all ω∈R.

(v) For all ω∈Rthe set

{va∈CnN | Q(iω)va= 0 and v∗

a[Ψiω (b)∗InN ]Q0

0−QhΨiω (b)

InN iva= 0}

contains only the zero element.

2In Appendix A we show that this exist, even when the right-hand side of (11) is not continuous

The proof of this theorem will be given in the following section. Furthermore, we

also adopt the insights drawn from this equivalence to shed some more light on the

characterisation of exponential stability from [12]. We exemplify our results in the

sections 4 and 5. We provide a conclusion afterwards. This manuscript is supplemented

with an appendix gathering known material for non-autonomous o.d.e.’s with bounded

and measurable coeﬃcients.

2. Characterisation of asymptotic/semi-uniform

stability

We recall the celebrated Batty–Duyckaerts result on semi-uniform stability.

Theorem 2.1 (Batty–Duyckaerts, [1]; see, [2, Theorem 3.4]).Let (T(t))t≥0be a bounded

C0-semigroup on a Banach space X, with generator A,µ∈ρ(A). Then the following

conditions are equivalent:

(i) σ(A)∩iR=∅;

(ii) limt→∞ ∥T(t)(µ− A)−1∥= 0;

(iii) there exists a function f: [0,∞)→[0,∞)with f(t)→0as t→ ∞ and such that

∀x0∈dom(A): ∥T(t)x∥ ≤ f(t)∥x0∥dom(A),

i.e., Tis semi-uniformly stable (with decay rate f).

In order to apply the Batty–Duyckaerts result, we provide some (standard) observa-

tions.

Theorem 2.2. The operator Awith domain dom(A)as given by (5) and (6) has compact

resolvent on the energy space X.

Proof. The assertion for H=Infollows upon using the Arzelà–Ascoli Theorem since

dom(A)embeds continuously into HN((a, b); Cn). The general case then follows since

Xis isomorphic to L2((a, b); Cn).

Lemma 2.3. Let Bbe a closed, densely deﬁned, linear operator on a Hilbert space X,

with ρ(B)̸=∅. If dom(B)→Xis compact, then

σ(B) = {λ∈C|0<dim(ker(λI − B)) <∞} =:σp(B).

Proof. For the proof we invoke standard theory of (unbounded) Fredholm operators,

see e.g., [5, Theorem 3.6] for an overview and suitable references. Let µ∈ρ(B)and

λ∈σ(B). Since µI − B is continuously invertible, µI − B is a Fredholm operator. The

compactness of dom(B)→Xyields that (µI − B)−1is compact and, hence, (λ−µ)I

is a relatively compact perturbation of µI − B and so λI −µI + (µI − B) = λI − B is

Fredholm and ind(µI − B) = ind(λI − B). In particular, dim(ker(λI − B)) <∞. Since

λ∈σ(B),λI − B fails to be continuously invertible, which can, thus, only happen if

ker(λI − B)̸={0}; whence dim(ker(λI − B)) >0.

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AsymptoticStabilityofport-HamiltonianSystemsMarcusWaurick∗andHansZwart†Wecharacteriseasymptoticstabilityofport-Hamiltoniansystemsbymeansofmatrixconditionsusingwell-knownresolventcriteriafromC0-semigrouptheory.Theideaofproofisbasedonarecentcharacterisationofexponentialstabilityestablishedin[12],which...

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