On Dirac structure of innite-dimensional stochastic port-Hamiltonian systems Fran cois Lamolineab Anthony Hastirb aUniversity of Luxembourg Luxembourg Centre for Systems Biomedicine Avenue du Swing 6 L-4367 Belvaux Luxembourg

2025-04-27 0 0 454.15KB 8 页 10玖币

On Dirac structure of inﬁnite-dimensional stochastic port-Hamiltonian systems

Fran¸cois Lamolinea,b, Anthony Hastirb

aUniversity of Luxembourg, Luxembourg Centre for Systems Biomedicine, Avenue du Swing 6, L-4367 Belvaux, Luxembourg

bUniversity of Namur, Department of Mathematics and Namur Institute for Complex Systems (naXys), Rue de Bruxelles 61, B-5000

Namur, Belgium

Abstract

Stochastic inﬁnite-dimensional port-Hamiltonian systems (SPHSs) with multiplicative Gaussian white noise are consid-

ered. In this article we extend the notion of Dirac structure for deterministic distributed parameter port-Hamiltonian

systems to a stochastic ones by adding some additional stochastic ports. Using the Stratonovich formalism of the

stochastic integral, the proposed extended interconnection of ports for SPHSs is proved to still form a Dirac structure.

This constitutes our main contribution. We then deduce that the interconnection between (stochastic) Dirac structures

is again a (stochastic) Dirac structure under some assumptions. These interconnection results are applied on a system

composed of a stochastic vibrating string actuated at the boundary by a mass-spring system with external input and

output. This work is motivated by the problem of boundary control of SPHSs and will serve as a foundation to the

development of stabilizing methods.

Keywords: Inﬁnite-dimensional systems – Stochastic partial diﬀerential equations – Dirac structures – Boundary

control

1. Introduction

Linear distributed port-Hamiltonian systems constitute

a powerful class of systems for the modelling, the anal-

ysis, and the control of distributed parameter systems.

It enables us to model many physical systems such as

beam equations, transport equations or wave equations,

see for instance Jacob and Zwart (2012). A comprehensive

overview of the literature on this class of systems can be

found in Rashad Hashem et al. (2020). In order to cover an

even larger set of systems that admits a port-Hamiltonian

representation, some authors have deﬁned the notion of

dissipative or irreversible port-Hamiltonian systems, see

e.g. Mora et al. (2021a,b); Caballeria et al. (2021) and

Ramirez et al. (2022). Port-Hamiltonian systems (PHSs)

are characterized by a Dirac structure and an Hamilto-

nian. Dirac structures consist of the power-preserving in-

terconnection of diﬀerent ports elements and were ﬁrst in-

troduced in the context of port-Hamiltonian systems in

van der Schaft and Maschke (2002). It was then extended

for higher-order PHSs in Le Gorrec et al. (2005) and Vil-

legas (2007). A fundamental property of Dirac structures

is that the composition of Dirac structures still forms a

Dirac structure, provided some assumptions. This induces

the main aspect of the port-Hamiltonian modelling, which

is that the power-conserving interconnection of PHSs is

still a port-Hamiltonian system.

Email addresses: francois.lamoline@uni.lu (Fran¸cois

Lamoline), anthony.hastir@unamur.be (Anthony Hastir)

Stochastic models are powerful to take into account

neglected random eﬀects that may occur when working

with real plants. Especially, random forcing, parameter

uncertainty or even boundary noise can impact the be-

havior of dynamical systems. In particular, PHSs inter-

act with their environment through external ports, which

can be a cause of randomness in many diﬀerent ways

as explained in Lamoline (2021). The stochastic exten-

sion of port-Hamiltonian systems was ﬁrst proposed in

L´azaro-Cam´ı and Ortega (2008) for Poisson manifolds.

On ﬁnite-dimensional spaces the class of nonlinear time-

varying stochastic port-Hamiltonian systems (SPHSs) was

presented in Satoh and Fujimoto (2013). A stochastic ex-

tension of distributed port-Hamiltonian systems was ﬁrst

developed in Lamoline (2019) and in Lamoline and Winkin

(2020). In addition the passivity property of SPHSs was

investigated in Lamoline and Winkin (2017) and Lamoline

(2021). However, in these works only a state space repre-

sentation described by a stochastic diﬀerential equation

(SDE) is given. As far as known, few eﬀorts have been

done for describing the underlying geometric structure of

SPHSs. One can cite Cordoni et al. (2019), where ﬁnite-

dimensional stochastic port-Hamiltonian systems are mod-

elled using the Stratonovich and Ito formalisms.

In this paper the notion of Dirac structure with stochas-

tic port-variables is explored. Our central idea consists

in extending the original Dirac structure of deterministic

ﬁrst-order PHSs by adding further noise ports to the port-

based structure. These speciﬁc noise ports are devoted

to represent the interaction of the dynamical system with

Preprint submitted to European Journal of Control October 13, 2022

arXiv:2210.06358v1 [math.OC] 12 Oct 2022

its random environment. In order to preserve the power-

preserving interconnection we consider the Stratonovich

formulation of the stochastic integral, see for instance

Duan and Wang (2014). Interested readers may also be

referred to Ruth F. Curtain (1978) and Da Prato and

Zabczyk (2014) for further details on inﬁnite-dimensional

SDEs.

The content of this article is as follows. In Section 2 we

introduce the basic concepts on Dirac structures together

with the class of deterministic port-Hamiltonian systems.

In Section 3 a port-based representation for SPHSs is pre-

sented and it is shown to form a Dirac structure, which

is the main contribution of the paper. Section 4 is ded-

icated to the illustration of our central result, by show-

ing that some interconnection between the newly deﬁned

Dirac structure and another arbitrary Dirac structure that

shares common ports is still a Dirac structure. A stochas-

tic damped vibrating string actuated by a mass-spring sys-

tem at the boundary is then presented as an example. We

conclude and discuss some future works in 5.

2. Background on Dirac structure

In this section we introduce some notions on distributed

port-Hamiltonian systems, Tellegen structures and Dirac

structures. Let us ﬁrst recall the deﬁnitions of Tellegen

and Dirac structures for linear distributed PHSs, see e.g.

van der Schaft and Maschke (2002), Le Gorrec et al. (2005)

and Kurula et al. (2010). Let Eand Fbe two Hilbert

spaces endowed with the inner products h·,·iEand h·,·iF,

respectively. The spaces Eand Fdenote the eﬀort and

the ﬂow spaces, respectively. We deﬁne the bond space

B:= F × E equipped with the following inner product

f1

e1,f2

e2B

=hf1, f2iF+he1, e2iE(1)

for all (f1, e1),(f2, e2)∈ B.

To deﬁne Tellegen or Dirac structures, the bond space is

endowed with the bilinear symmetric pairing given by

f1

e1,f2

e2+

=hf1, j−1e2iF+he1, jf2iE,(2)

with j:F → E being an invertible linear mapping. The

bilinear pairing h·,·i+represents the power.

Let Vbe a linear subspace of B. The orthogonal subspace

of Vwith respect to the bilinear pairing h·,·i+is deﬁned

V⊥:= {b∈ B :hb, vi+= 0,for all v∈ V}.(3)

These tools enable us to deﬁne Tellegen and Dirac struc-

tures, see (Kurula et al., 2010, Deﬁnition 2.1).

Deﬁnition 2.1. A linear subspace Dof the bond space

B:= F×E is called a Tellegen structure if D ⊂ D⊥, where

the orthogonal complement is understood with respect to

the bilinear pairing h·,·i+, see (2).

Deﬁnition 2.2. A linear subspace Dof the bond space B

is said to be a Dirac structure if

D⊥=D.(4)

Note that the condition (4) implies that the power of any

element of the Dirac structure is equal to zero, i.e.,

f

e,f

e+

= 2hf, j−1eiF= 0,

for any (f, e)∈ D, where the relation hf, j−1eiF=

hjf, eiEhas been used. The underlying structure of port-

Hamiltonian systems forms a Dirac structure, which links

the port-variables in a way that the total power is equal to

zero. A distributed port-Hamiltonian system is described

by the following partial diﬀerential equation

∂ε

∂t (ζ, t) = P1

∂

∂ζ (H(ζ)ε(ζ, t)) + P0H(ζ)ε(ζ, t),(5)

where ε(ζ, t)∈Rnfor ζ∈[a, b] and t≥0. In addi-

tion, P1=PT

1∈Rn×nis invertible, P0=−PT

0∈Rn×n,

and H ∈ L∞([a, b]; Rn×n) is symmetric and satisﬁes mI ≤

H(ζ) for all ζ∈[a, b] and some constant m > 0. The state

space X:= L2([a, b]; Rn) is endowed with the energy inner

product hε1, ε2iX=hε1,Hε2iL2=Rb

aε1(ζ)TH(ζ)ε2(ζ)dζ,

for all ε1, ε2∈X. The energy associated to (5) is given by

E(t) = 1

2kε(t)k2

The boundary ports denoted by f∂and e∂are given by

f∂(t)

e∂(t)=1

√2P1−P1

I I (Hε(t))(b)

(Hε(t))(a)=: R0(Hε(t))(b)

(Hε(t))(a)

(6)

and represent a linear combination of the restriction at the

boundary variables. Note that the notation (Hε(t))(a) :=

H(a)ε(a, t) has been used. We complete the PDE (5) with

the following homogeneous boundary conditions

0 = WBf∂(t)

e∂(t),(7)

where WB∈Rn×2n. It can be easily seen that the PDE

(5) together with the boundary conditions (7) admits the

abstract representation ˙ε(t) = Aε(t), ε(0) = ε0∈ X where

the linear (unbounded) operator Ais deﬁned by

Aε:= P1

dζ (Hε) + P0Hε(8)

for εon the domain

D(A) = ε∈X:Hε∈H1([a, b]; Rn), WBf∂

e∂= 0.

(9)

Before introducing the concept of Dirac structure for (5)

and (7), let us recall the following two results from Jacob

and Zwart (2012).

Lemma 2.1. Consider the operator Adeﬁned by (8) with

domain (9). Then the following result holds:

hAε, εiX+hε, AεiX= 2fT

∂e∂.(10)

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OnDiracstructureofinnite-dimensionalstochasticport-HamiltoniansystemsFrancoisLamolinea,b,AnthonyHastirbaUniversityofLuxembourg,LuxembourgCentreforSystemsBiomedicine,AvenueduSwing6,L-4367Belvaux,LuxembourgbUniversityofNamur,DepartmentofMathematicsandNamurInstituteforComplexSystems(naXys),RuedeBruxe...

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On Dirac structure of innite-dimensional stochastic port-Hamiltonian systems Fran cois Lamolineab Anthony Hastirb aUniversity of Luxembourg Luxembourg Centre for Systems Biomedicine Avenue du Swing 6 L-4367 Belvaux Luxembourg

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