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On the stability of port-Hamiltonian descriptor systems thanks: H.G. acknowledges the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 "Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

Hannes Gernandt Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10587 Berlin, Germany (gernandt@math.tu-berlin.de).    Frédéric E. Haller Universität Hamburg, Bundesstraße 55, 20146 Hamburg (frederic.haller@uni-hamburg.de).
Abstract

We characterize stable differential-algebraic equations (DAEs) using a generalized Lyapunov inequality. The solution of this inequality is then used to rewrite stable DAEs as dissipative Hamiltonian (dH) DAEs on the subspace where the solutions evolve. Conversely, we give sufficient conditions guaranteeing stability of dH DAEs. Further, for stabilizable descriptor systems we construct solutions of generalized algebraic Bernoulli equations which can then be used to rewrite these systems as pH descriptor systems. Furthermore, we show how to describe the stable and stabilizable systems using Dirac and Lagrange structures.

Keywords: Descriptor systems, port-Hamiltonian systems, stability, differential-algebraic equations, linear matrix inequalities, Dirac structure.

1 Introduction

In this note, we consider generalized port-Hamiltonian systems (pH systems) of the form

ddtEx(t)=(JR)Qx(t)+(BP)u(t),y(t)=(B+P)TQx(t)+(SN)u(t),\displaystyle\begin{split}\tfrac{\rm d}{{\rm d}t}Ex(t)&=(J-R)Qx(t)+(B-P)u(t),\\ y(t)&=(B+P)^{T}Qx(t)+(S-N)u(t),\end{split} (1)

where E,J,R,Qn×nE,J,R,Q\in\mathbb{R}^{n\times n}, B,Pn×kB,P\in\mathbb{R}^{n\times k} and S,Nk×kS,N\in\mathbb{R}^{k\times k} and

[RPPTS]0,[JBBTN]=[JBBTN]T,QTE=ETQ,\displaystyle\left[\begin{smallmatrix}R&P\\ P^{T}&S\end{smallmatrix}\right]\geq 0,\ \left[\begin{smallmatrix}J&B\\ -B^{T}&N\end{smallmatrix}\right]=-\left[\begin{smallmatrix}J&B\\ -B^{T}&N\end{smallmatrix}\right]^{T},\ Q^{T}E=E^{T}Q, (2)

see e.g. [MM19]. In comparison to classical port-Hamiltonian systems where the coefficient EE is the identity, here it might be singular.

If in (1) neither inputs nor outputs are present then the system reduces to the differential-algebraic equation (DAE)

ddtEx(t)=(JR)Qx(t)\displaystyle\tfrac{\rm d}{{\rm d}t}Ex(t)=(J-R)Qx(t) (3)

where the conditions (2) simplify to

R0,J=JT,QTE=ETQ.R\geq 0,\quad J=-J^{T},\quad Q^{T}E=E^{T}Q. (4)

If we further assume QTE0Q^{T}E\geq 0, these equations are called dissipative Hamiltonian DAEs (dH DAEs), see [MMW21].

The first aim of this paper is to show that stable DAEs

ddtEx(t)=Ax(t)\displaystyle\tfrac{{\rm d}}{{\rm d}t}Ex(t)=Ax(t) (5)

can be rewritten as dH DAEs (3) where the matrix QQ is given by the solution of a generalized Lyapunov inequality. For this rewriting we have to restrict the coefficients of (5) to the smallest subspace where the solutions evolve, also called the system space, see [RRV15]. Furthermore, it is shown that such a reformulation as pH systems is also possible for stabilizable systems

ddtEx(t)=Ax(t)+Bu(t)\tfrac{{\rm d}}{{\rm d}t}Ex(t)=Ax(t)+Bu(t)

by introducing a suitable output and using the solutions of generalized algebraic Bernoulli equations.

The second aim of the paper is to study the stability of DAEs (5) which are given in the geometric pH framework from [vdSM18]. This more general framework contains dH DAEs (3) which was recently shown in [GHR21]. Here it is shown how systems of the form (1) can be embedded into the more general geometric pH framework from [vdSM18].

The outline of the paper is as follows. After recalling some terminologies of matrix pencils in Section 2, we study the behavioral stability of DAEs in Section 3. We characterize the stability of DAEs using a generalized Lyapunov inequality and use their solutions to rewrite stable DAEs as dH DAEs. Further, we prove that dH DAEs which fulfill kerQkerE\ker Q\subseteq\ker E are stable. In Section 4 we rewrite behaviorally stabilizable DAEs as pH systems (1). Finally, in Section 5 we embedded dH DAEs (3) and pH systems (1) into the geometric framework from [vdSM18].

2 Preliminaries

Matrix pencils sEAsE-A are linear matrix polynomials with coefficients in E,An×nE,A\in\mathbb{R}^{n\times n}. We briefly write sEA[s]n×nsE-A\in\mathbb{R}[s]^{n\times n} and this pencil is called regular if det(sEA)[s]\det(sE-A)\in\mathbb{R}[s] is non-zero. The spectrum σ(E,A)\sigma(E,A) of a matrix pencil sEAsE-A is the set of all complex numbers λ\lambda for which λEA\lambda E-A is not invertible. An eigenvalue is called simple if ker(λEA)\ker(\lambda E-A) has dimension one and semi-simple if the dimension of this subspace coincides with the multiplicity of λ\lambda as a root of det(sEA)\det(sE-A).

Recall that every pencil sEA[s]n×msE-A\in\mathbb{R}[s]^{n\times m} can be transformed to quasi Kronecker form, see e.g. [BT12], i.e., there exists invertible Sn×nS\in\mathbb{R}^{n\times n} and Tm×mT\in\mathbb{R}^{m\times m} such that S(sEA)TS(sE-A)T is block diagonal with the following four types of blocks

sIn0A0\displaystyle sI_{n_{0}}-A_{0} [s]n0×n0,\displaystyle\in\mathbb{R}[s]^{n_{0}\times n_{0}}, (6a)
sNαiIαi\displaystyle sN_{\alpha_{i}}-I_{\alpha_{i}} =[1ss1][s]αi×αi,\displaystyle=\left[\begin{smallmatrix}-1&s&&\\ &\ddots&\ddots&\\ &&\ddots&~{}~{}s\\ &&&-1\end{smallmatrix}\right]\in\mathbb{R}[s]^{\alpha_{i}\times\alpha_{i}}, (6b)
sKβiLβi\displaystyle sK_{\beta_{i}}-L_{\beta_{i}} =[s1s1][s](βi1)×βi,\displaystyle=\left[\begin{smallmatrix}s&-1&&\\ &\ddots&\ddots&\\ &&s&-1\end{smallmatrix}\right]\in\mathbb{R}[s]^{(\beta_{i}-1)\times\beta_{i}}, (6c)
sKγiTLγiT\displaystyle\quad sK_{\gamma_{i}}^{T}-L_{\gamma_{i}}^{T} [s]γi×(γi1).\displaystyle\in\mathbb{R}[s]^{\gamma_{i}\times(\gamma_{i}-1)}. (6d)

The above indices are collected in multi-indices α=(α1,,αα)α\alpha=(\alpha_{1},\ldots,\alpha_{\ell_{\alpha}})\in\mathbb{N}^{\ell_{\alpha}}, β=(β1,,ββ)β\beta=(\beta_{1},\ldots,\beta_{\ell_{\beta}})\in\mathbb{N}^{\ell_{\beta}}, γ=(γ1,,γγ)γ\gamma=(\gamma_{1},\ldots,\gamma_{\ell_{\gamma}})\in\mathbb{N}^{\ell_{\gamma}}. For a multi-index δδ\delta\in\mathbb{N}^{\ell_{\delta}} recall |δ|=i=1δδi|\delta|=\sum_{i=1}^{\ell_{\delta}}\delta_{i}. Note that for βi=1\beta_{i}=1 and γi=1\gamma_{i}=1 one adds a zero column and a zero row to the pencil, respectively. The largest αi\alpha_{i} is called the index of the DAE. Furthermore, we write +{z|Rez>0}\mathbb{C}_{+}\coloneqq\{z\in\mathbb{C}~{}|~{}\operatorname{Re}z>0\}, {z|Rez<0}\mathbb{C}_{-}\coloneqq\{z\in\mathbb{C}~{}|~{}\operatorname{Re}z<0\} and ¯+\overline{\mathbb{C}_{-}}\coloneqq\mathbb{C}\setminus\mathbb{C}_{+}.

3 Behavioral stable and dH DAEs

In this section, we focus on DAEs

ddtEx(t)=Ax(t),\displaystyle\tfrac{{\rm d}}{{\rm d}t}Ex(t)=Ax(t), (7)

with E,An×nE,A\in\mathbb{R}^{n\times n} and denote the set of all such pairs [E,A][E,A] by Σn\Sigma_{n}. Here we follow the behavioral approach from [PW97] and introduce the behavior of [E,A][E,A] by

𝔅[E,A]{xC(,n)|ddtEx=Ax},\displaystyle\mathfrak{B}_{[E,A]}^{\infty}\coloneqq\{x\in C^{\infty}(\mathbb{R},\mathbb{R}^{n})\,\big{|}\tfrac{{\rm d}}{{\rm d}t}Ex=Ax\},

A definition with less smoothness assumptions can be found in [BR13].

We say that [E,A]Σn[E,A]\in\Sigma_{n} is stable if

x𝔅[E,A]M0:supt0x(t)M.\forall x\in\mathfrak{B}_{[E,A]}^{\infty}\,\,\exists M\geq 0:\quad\sup_{t\geq 0}\|x(t)\|\leq M.

The aim is to show that stable DAEs [E,A][E,A] can be rewritten as a dH DAE (3), i.e.,

ddtEx(t)=(JR)Qx(t),J=JT,R0,QTE0.\displaystyle\begin{split}&\tfrac{\rm d}{{\rm d}t}Ex(t)=(J-R)Qx(t),\\ &J=-J^{T},\quad R\geq 0,\quad Q^{T}E\geq 0.\end{split} (8)

To this end, we introduce the system space as follows

𝒱sys[E,A]\displaystyle\mathcal{V}_{\rm sys}^{[E,A]} {x(0)|x𝔅[E,A]}n,\displaystyle\coloneqq\{x(0)~{}|~{}x\in\mathfrak{B}_{[E,A]}^{\infty}\}\subseteq\mathbb{R}^{n},

see also [RRV15] for an equivalent definition. To determine the system space first observe that the behavior and hence the system space transforms as follows

𝔅[SET,SAT]\displaystyle\mathfrak{B}_{[SET,SAT]}^{\infty} =T1𝔅[E,A],𝒱sys[SET,SAT]\displaystyle=T^{-1}\mathfrak{B}_{[E,A]}^{\infty},\quad\mathcal{V}_{\rm sys}^{[SET,SAT]} =T1𝒱sys[E,A]\displaystyle=T^{-1}\mathcal{V}_{\rm sys}^{[E,A]} (9)

for invertible S,Tn×nS,T\in\mathbb{R}^{n\times n}. This combined with the quasi Kronecker form (6) is the following representation of the system space.

Lemma 3.1.

Let [E,A]Σn[E,A]\in\Sigma_{n} with invertible S,Tn×nS,T\in\mathbb{R}^{n\times n} such that S(sEA)TS(sE-A)T is in quasi Kronecker form (6). Then

𝒱sys[E,A]=T(n0×{0}|α|×|β|×{0}|γ|γ).\displaystyle\mathcal{V}_{\rm sys}^{[E,A]}=T(\mathbb{R}^{n_{0}}\times\{0\}^{|\alpha|}\times\mathbb{R}^{|\beta|}\times\{0\}^{|\gamma|-\ell_{\gamma}}). (10)

Below, the stability is characterized either in terms of the eigenvalues or a Lyapunov inequality on the system space. To this end, for a subspace n\mathcal{L}\subseteq\mathbb{R}^{n} and matrices M,Nn×nM,N\in\mathbb{R}^{n\times n} we say that

M=N:Mx=Nxx.M=_{\mathcal{L}}N\quad:\Leftrightarrow\quad Mx=Nx\quad\forall~{}x\in\mathcal{L}.

If we further assume that MM and NN are symmetric, we say that

MN\displaystyle M\geq_{\mathcal{L}}N\quad :xTMxxTNxx,\displaystyle:\Leftrightarrow\quad x^{T}Mx\geq x^{T}Nx\quad\forall~{}x\in\mathcal{L},
M>N\displaystyle M>_{\mathcal{L}}N\quad :xTMx>xTNxx{0}.\displaystyle:\Leftrightarrow\quad x^{T}Mx>x^{T}Nx\quad\forall~{}x\in\mathcal{L}\setminus\{0\}.
Proposition 3.2.

Let [E,A]Σn[E,A]\in\Sigma_{n} then the following statements are equivalent.

  • (a)

    [E,A][E,A] is stable.

  • (b)

    sEAsE-A is regular and there exists a symmetric Xn×nX\in\mathbb{R}^{n\times n} with X>E𝒱sys[E,A]0X>_{E\mathcal{V}_{\rm sys}^{[E,A]}}0 and X(E𝒱sys[E,A])=E𝒱sys[E,A]X(E\mathcal{V}_{\rm sys}^{[E,A]})=E\mathcal{V}_{\rm sys}^{[E,A]} such that

    ATXE+ETXA𝒱sys[E,A]0.\displaystyle A^{T}XE+E^{T}XA\leq_{\mathcal{V}_{\rm sys}^{[E,A]}}0. (11)
  • (c)

    The pencil sEAsE-A is regular, σ(E,A)¯\sigma(E,A)\subseteq\overline{\mathbb{C}_{-}} and the eigenvalues on the imaginary axis are semi-simple.

Proof.

Since the above conditions are invariant under multiplication from left and right with invertible matrices, we can assume that sEAsE-A is already given in Kronecker form (6). If [E,A][E,A] is stable then β\beta cannot be present in the Kronecker form. Indeed if β11\beta_{1}\geq 1 then for all fC(,)f\in C^{\infty}(\mathbb{R},\mathbb{R}), there exists a solution x~\tilde{x} to

ddtKβ1x~(t)=Lβ1x~(t),x~β1=f,\tfrac{{\rm d}}{{\rm d}t}K_{\beta_{1}}\tilde{x}(t)=L_{\beta_{1}}\tilde{x}(t),\quad\tilde{x}_{\beta_{1}}=f,

which contradicts stability. Since sEAsE-A is square also γ\gamma is not present and therefore sEAsE-A is regular. Furthermore, the stability of [In0,A0][I_{n_{0}},A_{0}] shows σ(E,A)¯\sigma(E,A)\subseteq\overline{\mathbb{C}_{-}} and that all eigenvalues on the imaginary axis are semi-simple. This proves (c). Clearly, (c) implies (a). To prove the equivalence of (b) and (c), we assume that sEAsE-A is regular. Then Lemma 3.1 implies that

𝒱sys[E,A]=n0×{0}|α|,E𝒱sys[E,A]=𝒱sys[E,A].\displaystyle\mathcal{V}_{\rm sys}^{[E,A]}=\mathbb{R}^{n_{0}}\times\{0\}^{|\alpha|},\quad E\mathcal{V}_{\rm sys}^{[E,A]}=\mathcal{V}_{\rm sys}^{[E,A]}.

Furthermore, (11) holds if and only if

[A0T00I|α|]X[In000N]+[In000NT]X[A000I|α|]n0×{0}|α|0\left[\begin{smallmatrix}A_{0}^{T}&0\\ 0&I_{|\alpha|}\end{smallmatrix}\right]X\left[\begin{smallmatrix}I_{n_{0}}&0\\ 0&N\end{smallmatrix}\right]+\left[\begin{smallmatrix}I_{n_{0}}&0\\ 0&N^{T}\end{smallmatrix}\right]X\left[\begin{smallmatrix}A_{0}&0\\ 0&I_{|\alpha|}\end{smallmatrix}\right]\leq_{\mathbb{R}^{n_{0}}\times\{0\}^{|\alpha|}}0

which is, by considering the upper-left block entries, equivalent to the existence of some X1n0×n0X_{1}\in\mathbb{R}^{n_{0}\times n_{0}} with X1>0X_{1}>0 such that

A0TX1+X1A00\displaystyle A_{0}^{T}X_{1}+X_{1}A_{0}\leq 0 (12)

and hence to σ(A0)¯\sigma(A_{0})\subseteq\overline{\mathbb{C}_{-}} and semi-simple eigenvalues on the imaginary axis. If X1>0X_{1}>0 is a solution of (12) then X[X1000]X\coloneqq\left[\begin{smallmatrix}X_{1}&0\\ 0&0\end{smallmatrix}\right] fulfills XE𝒱sys[E,A]=E𝒱sys[E,A]XE\mathcal{V}_{\rm sys}^{[E,A]}=E\mathcal{V}_{\rm sys}^{[E,A]} and XE𝒱sys[E,A]>0X_{E\mathcal{V}_{\rm sys}^{[E,A]}}>0.

Remark 3.3.

Oftentimes asymptotically stable DAEs are of interest which are defined by

x𝔅[E,A]:limtx(t)=0.\forall x\in\mathfrak{B}_{[E,A]}^{\infty}:\lim\limits_{t\rightarrow\infty}\|x(t)\|=0.

Analogously to Proposition 3.2, such DAEs can be characterized by a strict inequality in (11) or by σ(E,A)\sigma(E,A)\subseteq\mathbb{C}_{-}. Related characterizations with Lyapunov equations were previously given e.g. in [Sty02] and recently in [AAM21, Theorem 4].

The proposition below provides a dH formulation (8) of stable DAEs [E,A]Σn[E,A]\in\Sigma_{n} on the system space. Here we use the pseudo-inverse QQ^{\dagger} of a matrix QQ.

Proposition 3.4.

Let [E,A]Σn[E,A]\in\Sigma_{n} be stable and let Xn×nX\in\mathbb{R}^{n\times n} be a solution of (11). Define QXEQ\coloneqq XE and

J12(AQ(AQ)T),R12(AQ+(AQ)T).J\coloneqq\tfrac{1}{2}(AQ^{\dagger}-(AQ^{\dagger})^{T}),\ \ R\coloneqq-\tfrac{1}{2}(AQ^{\dagger}+(AQ^{\dagger})^{T}).

Then [E,(JR)Q][E,(J-R)Q] is a dH DAE on 𝒱sys[E,A]\mathcal{V}_{\rm sys}^{[E,A]} in the sense that

J=E𝒱sys[E,A]JT,A=𝒱sys[E,A](JR)Q,RE𝒱sys[E,A]0,QTE>𝒱sys[E,A]0.\begin{split}J&=_{E\mathcal{V}_{\rm sys}^{[E,A]}}-J^{T},\\ A&=_{\mathcal{V}_{\rm sys}^{[E,A]}}(J-R)Q,\end{split}\quad\quad\begin{split}R&\geq_{E\mathcal{V}_{\rm sys}^{[E,A]}}0,\\ Q^{T}E&>_{\mathcal{V}_{\rm sys}^{[E,A]}}0.\end{split}

Proof.

Note that sEAsE-A is regular by Proposition 3.2. Let Xn×nX\in\mathbb{R}^{n\times n} be a solution to (11) as given by Proposition 3.2. Since XE𝒱sys[E,A]=E𝒱sys[E,A]=𝒱sys[E,A]XE\mathcal{V}_{\rm sys}^{[E,A]}=E\mathcal{V}_{\rm sys}^{[E,A]}=\mathcal{V}_{\rm sys}^{[E,A]} by regularity of sEAsE-A and (10),

QQ=𝒱sys[E,A]EE=𝒱sys[E,A]I=𝒱sys[E,A]EE=𝒱sys[E,A]QQQ^{\dagger}Q=_{\mathcal{V}_{\rm sys}^{[E,A]}}E^{\dagger}E=_{\mathcal{V}_{\rm sys}^{[E,A]}}I=_{\mathcal{V}_{\rm sys}^{[E,A]}}EE^{\dagger}=_{\mathcal{V}_{\rm sys}^{[E,A]}}QQ^{\dagger}

and thus (JR)Q=AQQ=𝒱sys[E,A]A(J-R)Q=AQ^{\dagger}Q=_{\mathcal{V}_{\rm sys}^{[E,A]}}A. Since X>E𝒱sys[E,A]0X>_{E\mathcal{V}_{\rm sys}^{[E,A]}}0, we have for all x𝒱sys[E,A]{0}x\in\mathcal{V}_{\rm sys}^{[E,A]}\setminus\{0\}

xTQTEx=xTETXEx>0.x^{T}Q^{T}Ex=x^{T}E^{T}XEx>0.

With E𝒱sys[E,A]=Q𝒱sys[E,A]E\mathcal{V}_{\rm sys}^{[E,A]}=Q\mathcal{V}_{\rm sys}^{[E,A]} we find that RE𝒱sys[E,A]0R\geq_{E\mathcal{V}_{\rm sys}^{[E,A]}}0 is equivalent to

0𝒱sys[E,A]QTRQ\displaystyle 0\leq_{\mathcal{V}_{\rm sys}^{[E,A]}}Q^{T}RQ =QT(AQ+(Q)TAT)Q\displaystyle=-Q^{T}(AQ^{\dagger}+(Q^{\dagger})^{T}A^{T})Q
=QTAATQ\displaystyle=-Q^{T}A-A^{T}Q
=𝒱sys[E,A]ETXAATXE,\displaystyle=_{\mathcal{V}_{\rm sys}^{[E,A]}}-E^{T}XA-A^{T}XE,

which holds by (11). Moreover, J=E𝒱sys[E,A]JTJ=_{E\mathcal{V}_{\rm sys}^{[E,A]}}J^{T} holds trivially.

Remark 3.5.

If [E,A]Σn[E,A]\in\Sigma_{n} is stable with index at most one then we can redefine QQ in Proposition 3.4 in such a way that the relations in (b) hold on n\mathbb{R}^{n}. Here we assume for simplicity that EE and AA are already given in quasi Kronecker form. Then E=[In0000]E=\left[\begin{smallmatrix}I_{n_{0}}&0\\ 0&0\end{smallmatrix}\right] and A=[A000Inn0]A=\left[\begin{smallmatrix}A_{0}&0\\ 0&I_{n-n_{0}}\end{smallmatrix}\right]. Let X0>0X_{0}>0 satisfy A0TX0+X0A00A_{0}^{T}X_{0}+X_{0}A_{0}\leq 0. Then we define Q^[X000Inn0]\hat{Q}\coloneqq\left[\begin{smallmatrix}X_{0}&0\\ 0&-I_{n-n_{0}}\end{smallmatrix}\right] which is invertible and with J12(AQ^1(AQ^1)T)J\coloneqq\tfrac{1}{2}(A\hat{Q}^{-1}-(A\hat{Q}^{-1})^{T}) and R12(AQ^1+(AQ^1)T)R\coloneqq-\tfrac{1}{2}(A\hat{Q}^{-1}+(A\hat{Q}^{-1})^{T}) we have Q^TE0\hat{Q}^{T}E\geq 0 and A=(JR)Q^A=(J-R)\hat{Q}. If sEAsE-A has index greater than one then this extension of QQ is still possible but leads to Q^TE0\hat{Q}^{T}E\ngeq 0.

The following example from [MMW18] shows that not every dH DAE given by (8) is stable. Consider

sEJQ=[s10s],J[0110],Q[0001],sE-JQ=\left[\begin{smallmatrix}s&1\\ 0&s\end{smallmatrix}\right],\quad J\coloneqq\left[\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right],\quad Q\coloneqq\left[\begin{smallmatrix}0&0\\ 0&1\end{smallmatrix}\right],

which is port-Hamiltonian and has a Jordan block of size 2 at zero and is therefore unstable. The above example is an ordinary differential equation and xxTQTEx=xTQxx\mapsto x^{T}Q^{T}Ex=x^{T}Qx is not a Lyapunov function.

It will be shown below, that an additional assumption which guarantees the stability of dH DAEs is

kerQkerE.\displaystyle\ker Q\subseteq\ker E. (13)

An interpretation of this condition is given later in the geometric formulation of DAEs in Section 5 where it ensures that the only trajectories of a system with vanishing effort are the purely algebraic solutions.

Proposition 3.6.

Let [E,(JR)Q][E,(J-R)Q] be a dH DAE (8) with kerQkerE\ker Q\subseteq\ker E. Then the following holds:

  • (a)

    If sEQsE-Q is regular then QQ is invertible.

  • (b)

    Let QQ be invertible. Then the dH DAE (8) is stable if and only if kerJkerR(QkerE)={0}\ker J\cap\ker R\cap(Q\ker E)=\{0\}.

Proof.

If sEQsE-Q is regular then kerEkerQ={0}\ker E\cap\ker Q=\{0\}. This together with kerQkerE\ker Q\subseteq\ker E implies kerQ={0}\ker Q=\{0\} and hence QQ is invertible. This proves (a).

Next, observe that

ker(JR)=kerJkerR.\displaystyle\ker(J-R)=\ker J\cap\ker R. (14)

If xker(JR)x\in\ker(J-R) then xT(JR)x=0x^{T}(J-R)x=0 and this implies 0=xT(JTR)x=xT(J+R)x0=x^{T}(J^{T}-R)x=-x^{T}(J+R)x. Hence xkerRx\in\ker R and therefore xkerJx\in\ker J. The reverse inclusion is trivial.

To characterize the stability of (8), we use Proposition 3.2 (c). If (8) is stable then sE(JR)QsE-(J-R)Q is regular. Hence, using the invertibility of QQ and (14) we find

{0}=kerEker(JR)Q=(QkerE)kerJkerR.\displaystyle\{0\}=\ker E\cap\ker(J-R)Q=(Q\ker E)\cap\ker J\cap\ker R. (15)

Conversely, assume that (15) holds. The pencil sEQ1(JR)sEQ^{-1}-(J-R) is semi-dissipative Hamiltonian and according to [MMW21, Thm. 2] in the quasi-Kronecker form (6) one has βi,γi1\beta_{i},\gamma_{i}\leq 1 and the eigenvalue conditions in Proposition 3.2 (c) are satisfied for the regular part (6a). Since

{0}=kerEker(JR)Q=kerEQ1ker(JR),\{0\}=\ker E\cap\ker(J-R)Q=\ker EQ^{-1}\cap\ker(J-R),

the multi-index β\beta cannot be present in the quasi-Kronecker form of sEQ1(JR)sEQ^{-1}-(J-R). Hence γ\gamma is not present implying that sE(JR)QsE-(J-R)Q is regular and therefore stable.

Remark 3.7.

Similar considerations can be done for systems which are stable backwards in time, i.e., [E,A][-E,A] is stable. A characterization similar to Proposition 3.2 is straightforward and as a consequence, every DAE which has semi-simple eigenvalues on the imaginary axis can be rewritten on the system space 𝒱sys[E,A]\mathcal{V}_{\rm sys}^{[E,A]} in the form (8) but with QTE=ETQQ^{T}E=E^{T}Q instead of QTE0Q^{T}E\geq 0.

4 Stabilizable systems as pH systems

In this section, we consider descriptor systems

ddtEx(t)=Ax(t)+Bu(t),\tfrac{{\rm d}}{{\rm d}t}Ex(t)=Ax(t)+Bu(t),

with E,An×nE,A\in\mathbb{R}^{n\times n}, Bn×kB\in\mathbb{R}^{n\times k} for n,kn,k\in\mathbb{N} and we write [E,A,B]Σn,k[E,A,B]\in\Sigma_{n,k}. The behavior and system space of [E,A,B][E,A,B] are given by

𝔅[E,A,B]{(x,u)C(,n)×C(,k)|ddtEx=Ax+Bu},𝒱sys[E,A,B]{(x(0),u(0))|(x,u)𝔅[E,A,B]}n+k.\mathfrak{B}_{[E,A,B]}^{\infty}\coloneqq\{(x,u)\in C^{\infty}(\mathbb{R},\mathbb{R}^{n})\times C^{\infty}(\mathbb{R},\mathbb{R}^{k})\,\big{|}\\ \tfrac{{\rm d}}{{\rm d}t}Ex=Ax+Bu\},\\ \mathcal{V}_{\rm sys}^{[E,A,B]}\coloneqq\{(x(0),u(0))~{}|~{}(x,u)\in\mathfrak{B}_{[E,A,B]}^{\infty}\}\subseteq\mathbb{R}^{n+k}.

In the following it is shown that systems which can be stabilized in behavioral sense can be rewritten as pH systems (1) on the system space after introducing a suitable output.

We say that the system [E,A,B]Σn,k[E,A,B]\in\Sigma_{n,k} is behaviorally stabilizable if

x0𝒱sys[E,A]M0(x,u)𝔅[E,A,B]:x(0)=x0,supt0x(t)M.\displaystyle\begin{split}\forall x_{0}\in\mathcal{V}_{\rm sys}^{[E,A]}\,\,~{}\exists M\geq 0~{}\exists(x,u)\in\mathfrak{B}_{[E,A,B]}^{\infty}:\\ x(0)=x_{0},\quad\sup\limits_{t\geq 0}\|x(t)\|\leq M.\end{split} (16)

Furthermore, the class of behavioral stabilizable systems [E,A,B]Σn,k[E,A,B]\in\Sigma_{n,k} for which sEAsE-A is regular and only has semi-simple eigenvalues on the imaginary axis are denoted by Σn,ks\Sigma_{n,k}^{s}.

If [E,A,B]Σn,ks[E,A,B]\in\Sigma_{n,k}^{s} then applying a Jordan decomposition on the first block entry in the quasi Kronecker form (6) we see that there exists some invertible S,Tn×nS,T\in\mathbb{R}^{n\times n}, n1,n2n_{1},n_{2}\in\mathbb{N} with n1+n2=n0n_{1}+n_{2}=n_{0} such that

S(sEA)T=[sIn1A1000sIn2A2000sNαI|α|],x=(x1x2x3)SB=[B1B2Bα],σ(A1)+,[In2,A2]stable.\displaystyle\begin{split}&S(sE-A)T=\left[\begin{smallmatrix}sI_{n_{1}}-A_{1}&0&0\\ 0&sI_{n_{2}}-A_{2}&0\\ 0&0&sN_{\alpha}-I_{|\alpha|}\end{smallmatrix}\right],\quad x=\left(\begin{smallmatrix}x_{1}\\ x_{2}\\ x_{3}\end{smallmatrix}\right)\\ &SB=\left[\begin{smallmatrix}B_{1}\\ B_{2}\\ B_{\alpha}\end{smallmatrix}\right],\quad\sigma(A_{1})\subseteq\mathbb{C}_{+},\quad[I_{n_{2}},A_{2}]~{}\text{stable.}\end{split} (17)

Furthermore, [In1,A1,B1][I_{n_{1}},A_{1},B_{1}] is controllable.

Lemma 4.1.

If [E,A,B]Σn,ks[E,A,B]\in\Sigma_{n,k}^{s} fulfills (17) with S=T=InS=T=I_{n} then a stabilizing feedback is given by u(t)=B1TP1x1(t)u(t)=-B_{1}^{T}P_{1}x_{1}(t) and P1>0P_{1}>0 is the unique solution of

A1TP1+P1A1=P1B1B1TP1.\displaystyle A_{1}^{T}P_{1}+P_{1}A_{1}=P_{1}B_{1}B_{1}^{T}P_{1}. (18)

Proof.

Clearly, (18) has a unique positive definite solution if and only if

(A1T)TP11+P11(A1T)=B1B1T(-A_{1}^{T})^{T}P_{1}^{-1}+P_{1}^{-1}(-A_{1}^{T})=-B_{1}B_{1}^{T}

has a unique positive definite solution. Since [In1,A1,B1][I_{n_{1}},A_{1},B_{1}] is controllable and σ(A1T)\sigma(-A_{1}^{T})\subset\mathbb{C}_{-} such a solution exists by [TSH01, Thm. 3.28]. It remains to show that u(t)=B1TP1x1(t)u(t)=-B_{1}^{T}P_{1}x_{1}(t) is a stabilizing feedback, i.e., that (16) holds. Note that [In1,A1B1B1TP1,B1][I_{n_{1}},A_{1}-B_{1}B_{1}^{T}P_{1},B_{1}] is also controllable and that (18) is also equivalent to

(A1TP1TB1B1T)TP11+P11(A1TP1TB1B1T)=B1B1T.(A_{1}^{T}-P_{1}^{T}B_{1}B_{1}^{T})^{T}P_{1}^{-1}+P_{1}^{-1}(A_{1}^{T}-P_{1}^{T}B_{1}B_{1}^{T})=-B_{1}B_{1}^{T}.

Now invoking [TSH01, Thm. 3.28] again shows σ(A1B1B1TP1)\sigma(A_{1}-B_{1}B_{1}^{T}P_{1})\subseteq\mathbb{C}_{-}. Moreover, with

x1(t)=e(A1B1B1TP1)tx1(0),t0,x_{1}(t)=e^{(A_{1}-B_{1}B_{1}^{T}P_{1})t}x_{1}(0),\quad\forall t\geq 0,

there exist M,β>0M,\beta>0 such that for all k0k\geq 0 which are smaller than the index of [E,A][E,A]

x1(k)(t)Meβt,t0.\displaystyle\|x_{1}^{(k)}(t)\|\leq Me^{-\beta t},\quad\forall t\geq 0.

Therefore, we have for some M^>0\hat{M}>0 such that for all k0k\geq 0 which are smaller then the index of [E,A][E,A]

u(k)(t)M^eβt,t0.\displaystyle\|u^{(k)}(t)\|\leq\hat{M}e^{-\beta t},\quad\forall t\geq 0. (19)

Since [In2,A2][I_{n_{2}},A_{2}] is stable, the variation of constants formula implies that the solution x2x_{2} of x˙2=A2x2(t)+B2u(t)\dot{x}_{2}=A_{2}x_{2}(t)+B_{2}u(t) are bounded. Next, we only consider the first block-entry (if any) of sNαI|α|sN_{\alpha}-I_{|\alpha|}, sNα1Iα1sN_{\alpha_{1}}-I_{\alpha_{1}}, since the others are treated analogously. With Bα=[Bα1TBα(α)T]TB_{\alpha}=\left[\begin{smallmatrix}B_{\alpha_{1}}^{T}&\cdots&B^{T}_{\alpha_{\ell(\alpha)}}\end{smallmatrix}\right]^{T} the solution xα1(t)=(xα1,1(t),,xα1,α1(t))Tx_{\alpha_{1}}(t)=(x_{{\alpha_{1}},1}(t),\ldots,x_{{\alpha_{1}},{\alpha_{1}}}(t))^{T} of [Nα1,Iα1][N_{\alpha_{1}},I_{\alpha_{1}}] fulfills

(x˙α1,2x˙α1,α10)=ddtNα1(xα1,1xα1,α11xα1,nα1)=(xα1,1xα1,α11xα1,nα1)+Bα1u\left(\begin{smallmatrix}\dot{x}_{{\alpha_{1}},2}\\ \vdots\\ \dot{x}_{{\alpha_{1}},{\alpha_{1}}}\\ 0\end{smallmatrix}\right)=\tfrac{{\rm d}}{{\rm d}t}N_{\alpha_{1}}\left(\begin{smallmatrix}x_{{\alpha_{1}},1}\\ \vdots\\ x_{{\alpha_{1}},{\alpha_{1}}-1}\\ x_{{\alpha_{1}},n_{\alpha_{1}}}\end{smallmatrix}\right)=\left(\begin{smallmatrix}x_{{\alpha_{1}},1}\\ \vdots\\ x_{{\alpha_{1}},{\alpha_{1}}-1}\\ x_{{\alpha_{1}},n_{\alpha_{1}}}\end{smallmatrix}\right)+B_{\alpha_{1}}u

and inspecting the last row leads to

xα1,α1(t)=α1u(t)Bα1M^eβt,\|x_{{\alpha_{1}},{\alpha_{1}}}(t)\|=\|{\alpha_{1}}u(t)\|\leq\|B_{\alpha_{1}}\|\hat{M}e^{-\beta t},

which tends to zero as tt\rightarrow\infty. Similarly, the penultimate row leads to

xα1,α11(t)\displaystyle x_{{\alpha_{1}},{\alpha_{1}}-1}(t) =x˙α1,α1(t)eα11TBα1u(t)\displaystyle=\dot{x}_{{\alpha_{1}},{\alpha_{1}}}(t)-e_{{\alpha_{1}}-1}^{T}B_{\alpha_{1}}u(t)
=Bα1u˙(t)eα11TBα1u(t),\displaystyle=B_{\alpha_{1}}\dot{u}(t)-e_{{\alpha_{1}}-1}^{T}B_{\alpha_{1}}u(t),

which is again exponentially bounded by (19). Repeating the last step with the remaining rows starting with the α12{\alpha_{1}}-2th row shows that xα1(t)0\|x_{\alpha_{1}}(t)\|\rightarrow 0 as tt\rightarrow\infty which completes the proof that uu stabilizes the solution of [E,A,B][E,A,B].

Below, we obtain another characterization of stabilizability using solutions to certain matrix equalities on the system space which allows us to reformulate the system in a port-Hamiltonian way. Similar equations for stabilization of DAEs have been studied under the name generalized algebraic Bernoulli equation in [BBQO07].

Lemma 4.2.

Let [E,A,B]Σn,ks[E,A,B]\in\Sigma_{n,k}^{s}. Then there exists some X1,X2E𝒱sys[E,A]0X_{1},X_{2}\geq_{E\mathcal{V}_{\rm sys}^{[E,A]}}0 such that the following holds.

ATX1E+ETX1A=𝒱sys[E,A]ETX1BBTX1E,\displaystyle A^{T}X_{1}E+E^{T}X_{1}A=_{\mathcal{V}_{\rm sys}^{[E,A]}}E^{T}X_{1}BB^{T}X_{1}E, (20)
ATX2E+ETX2A𝒱sys[E,A]0,\displaystyle A^{T}X_{2}E+E^{T}X_{2}A\leq_{\mathcal{V}_{\rm sys}^{[E,A]}}0, (21)

with (X2±X1)E𝒱sys[E,A]=E𝒱sys[E,A](X_{2}\pm X_{1})E\mathcal{V}_{\rm sys}^{[E,A]}=E\mathcal{V}_{\rm sys}^{[E,A]} and X1+X2>E𝒱sys[E,A]0X_{1}+X_{2}>_{E\mathcal{V}_{\rm sys}^{[E,A]}}0.

Proof.

All conditions are invariant under transformations of the form [E,A,B][SET,SAT,SB][E,A,B]\rightarrow[SET,SAT,SB] for all invertible S,Tn×nS,T\in\mathbb{R}^{n\times n}. Hence we can assume without restriction that [E,A,B][E,A,B] is already given in the block diagonal form on the right-hand side of (17). Introduce E~=diag(In2,N)\tilde{E}=\operatorname{diag}(I_{n_{2}},N), A~=diag(A2,In3)\tilde{A}=\operatorname{diag}(A_{2},I_{n_{3}}) and then we set X1diag(P1,0n2+n3)X_{1}\coloneqq\operatorname{diag}(P_{1},0_{n_{2}+n_{3}}), where P1>0P_{1}>0 is a solution of (18) and X2diag(0n1,P21,0n3)X_{2}\coloneqq\operatorname{diag}(0_{n_{1}},P_{2}^{-1},0_{n_{3}}) where P2>0P_{2}>0 is a solution of the Lyapunov inequality A2TP2+P2A20A_{2}^{T}P_{2}+P_{2}A_{2}\leq 0. Clearly, X1,X2E𝒱sys[E,A]0X_{1},X_{2}\geq_{E\mathcal{V}_{\rm sys}^{[E,A]}}0 and they satisfy (20) and (21), respectively. Furthermore, E𝒱sys[E,A]=n1+n2×{0}n3E\mathcal{V}_{\rm sys}^{[E,A]}=\mathbb{R}^{n_{1}+n_{2}}\times\{0\}^{n_{3}} and hence X2±X1=diag(±P1,P21,0n3)X_{2}\pm X_{1}=\operatorname{diag}(\pm P_{1},P_{2}^{-1},0_{n_{3}}) map E𝒱sys[E,A]E\mathcal{V}_{\rm sys}^{[E,A]} into itself with X1+X2>E𝒱sys[E,A]0X_{1}+X_{2}>_{E\mathcal{V}_{\rm sys}^{[E,A]}}0.

Based on this result, we show how to interpret a stabilizable system as a port-Hamiltonian system (1) which can be viewed as an analogue to Proposition 3.4.

Proposition 4.3.

Let [E,A,B]Σn,ks[E,A,B]\in\Sigma_{n,k}^{s}. Then there exist X1,X2E𝒱sys[E,A]0X_{1},X_{2}\geq_{E\mathcal{V}_{\rm sys}^{[E,A]}}0 such that with the choices of

Q\displaystyle Q (X2X1)E,\displaystyle\coloneqq(X_{2}-X_{1})E, (22)
J\displaystyle J 12(AQ(AQ)T),\displaystyle\coloneqq\tfrac{1}{2}(AQ^{\dagger}-(AQ^{\dagger})^{T}),
R\displaystyle R 12(AQ+(AQ)T),\displaystyle\coloneqq-\tfrac{1}{2}(AQ^{\dagger}+(AQ^{\dagger})^{T}),

the system

ddtEx(t)\displaystyle\tfrac{\rm d}{{\rm d}t}Ex(t) =(JR)Qx(t)+Bu(t),\displaystyle=(J-R)Qx(t)+Bu(t), (23)
y(t)\displaystyle y(t) =BTQx(t)+u(t),\displaystyle=B^{T}Qx(t)+u(t),

is port-Hamiltonian on 𝒱sys[E,A]×k\mathcal{V}_{\rm sys}^{[E,A]}\times\mathbb{R}^{k} in the sense that

[JBBT0]\displaystyle\left[\begin{smallmatrix}J&B\\ -B^{T}&0\end{smallmatrix}\right] =E𝒱sys[E,A]×k[JBBT0]T,\displaystyle=_{E\mathcal{V}_{\rm sys}^{[E,A]}\times\mathbb{R}^{k}}-\left[\begin{smallmatrix}J&B\\ -B^{T}&0\end{smallmatrix}\right]^{T}, (24)
[R00I]\displaystyle\left[\begin{smallmatrix}R&0\\ 0&I\end{smallmatrix}\right] E𝒱sys[E,A]×k0,\displaystyle\geq_{E\mathcal{V}_{\rm sys}^{[E,A]}\times\mathbb{R}^{k}}0,
ETQ\displaystyle E^{T}Q =𝒱sys[E,A]QTE,A=𝒱sys[E,A](JR)Q.\displaystyle=_{\mathcal{V}_{\rm sys}^{[E,A]}}Q^{T}E,\quad A=_{\mathcal{V}_{\rm sys}^{[E,A]}}(J-R)Q.

Proof.

The proof is completely analogous to the proof of Proposition 3.4 except that we use (20) and (21) instead of (11) to prove the inequality in (24). To be more precise, let X1,X2X_{1},X_{2} be the solutions of (20) and (21), respectively, given by Lemma 4.2. Then with (X1X2)E𝒱sys[E,A]=E𝒱sys[E,A]=𝒱sys[E,A](X_{1}-X_{2})E\mathcal{V}_{\rm sys}^{[E,A]}=E\mathcal{V}_{\rm sys}^{[E,A]}=\mathcal{V}_{\rm sys}^{[E,A]} we have

RE𝒱sys[E,A]02QTRQ𝒱sys[E,A]0R\geq_{E\mathcal{V}_{\rm sys}^{[E,A]}}0\Leftrightarrow 2Q^{T}RQ\geq_{\mathcal{V}_{\rm sys}^{[E,A]}}0

and

2QTRQ\displaystyle 2Q^{T}RQ =𝒱sys[E,A]QTAATQ\displaystyle=_{\mathcal{V}_{\rm sys}^{[E,A]}}-Q^{T}A-A^{T}Q
=ET(X2X1)AAT(X2X1)E\displaystyle=-E^{T}(X_{2}-X_{1})A-A^{T}(X_{2}-X_{1})E
𝒱sys[E,A]ETX1BBTX1E𝒱sys[E,A]0.\displaystyle\geq_{\mathcal{V}_{\rm sys}^{[E,A]}}E^{T}X_{1}BB^{T}X_{1}E\geq_{\mathcal{V}_{\rm sys}^{[E,A]}}0.

Remark 4.4.

In the context of Proposition 4.3, one can show that the solutions (x,u)(x,u) of [E,ABBT(X2X1)E][E,A-BB^{T}(X_{2}-X_{1})E] correspond to the solutions (x,u,y)(x,u,y) of (23) when imposing y=0y=0. This restriction corresponds to an interconnection (cf. [CvdSBn07]) with respect to the Dirac structure ran[0I]\operatorname{ran}\left[\begin{smallmatrix}0\\ I\end{smallmatrix}\right] which are key objects in the geometric formulation of pH systems (cf. Section 5). Furthermore, if sEAsE-A has index one then N=0N=0 in (17) and E𝒱sys[E,A]×kE\mathcal{V}_{\rm sys}^{[E,A]}\times\mathbb{R}^{k} coincides with [E00Ik]𝒱sys[E,A,B]\left[\begin{smallmatrix}E&0\\ 0&I_{k}\end{smallmatrix}\right]\mathcal{V}_{\rm sys}^{[E,A,B]}. If EE is invertible then 𝒱sys[E,A,B]=n+k\mathcal{V}_{\rm sys}^{[E,A,B]}=\mathbb{R}^{n+k}, E𝒱sys[E,A]=nE\mathcal{V}_{\rm sys}^{[E,A]}=\mathbb{R}^{n} and Q=X1+X2>0Q=X_{1}+X_{2}>0.

Remark 4.5.

In Proposition 4.3 we defined a suitable output to obtain a pH system. More generally, in [GS18, Thm. 3.6] it was shown that descriptor systems [E,A,B][E,A,B] with output y(t)=Cx(t)+Du(t)y(t)=Cx(t)+Du(t) can be rewritten as a pH system of the form (1) if there exists an invertible solution XX of a linear matrix inequality (typically referred to as Kalman-Yakubovich-Popov inequality). Moreover, one has X=QX=Q in (1). Such invertible solutions can be obtained by restricting the system (23) to the space 𝒱sys[E,A]×k\mathcal{V}_{\rm sys}^{[E,A]}\times\mathbb{R}^{k}. In this case QQ in (22) can be replaced by the invertible Q^(X2X1)E|𝒱sys[E,A]\hat{Q}\coloneqq(X_{2}-X_{1})E|_{\mathcal{V}_{\rm sys}^{[E,A]}}.

5 Stability of geometric dH DAEs

In this section we study the stability of DAEs which are given by the geometrical formulation used in [vdSM18].

In comparison to (1), the equations here are given implicitly. To this end, let \mathcal{L} and 𝒟\mathcal{D} be subspaces of n×n\mathbb{R}^{n}\times\mathbb{R}^{n} with range representations =ran[L1L2]\mathcal{L}=\operatorname{ran}\left[\begin{smallmatrix}L_{1}\\ L_{2}\end{smallmatrix}\right] and 𝒟=ran[D1D2]\mathcal{D}=\operatorname{ran}\left[\begin{smallmatrix}D_{1}\\ D_{2}\end{smallmatrix}\right] for some L1,L2,D1,D2n×nL_{1},L_{2},D_{1},D_{2}\in\mathbb{R}^{n\times n}. Then \mathcal{L} is Lagrangian and 𝒟\mathcal{D} is maximally dissipative if

L1TL2L2TL1\displaystyle L_{1}^{T}L_{2}-L_{2}^{T}L_{1} =0,\displaystyle=0, dim=n,and\displaystyle\dim\mathcal{L}=n,\quad\text{and} (25)
D2TD1+D1TD2\displaystyle D_{2}^{T}D_{1}+D_{1}^{T}D_{2} 0,\displaystyle\leq 0, dim𝒟=n.\displaystyle\dim\mathcal{D}=n. (26)

Furthermore, the matrices L1,L2,D1,D2L_{1},L_{2},D_{1},D_{2} can be chosen in such a way that L2=L2TL_{2}=L_{2}^{T} and D2+D2T0D_{2}+D_{2}^{T}\leq 0 holds. These 𝒟\mathcal{D} and \mathcal{L} can be used to implicitly define a DAE by demanding that the system trajectories z,e:nz,e:\mathbb{R}\rightarrow\mathbb{R}^{n} satisfy

(e(t),ddtz(t))𝒟,(z(t),e(t)).\displaystyle(e(t),-\tfrac{d}{dt}z(t))\in\mathcal{D},\quad(z(t),e(t))\in\mathcal{L}. (27)

These systems were called generalized pH DAE systems in [vdSM18] and zz and ee are called the state and effort, respectively. Therein, 𝒟\mathcal{D} was assumed to be a so-called Dirac structure, i.e., (26) holds with equality. In [GHR21], 𝒟\mathcal{D} in (27) is allowed to be dissipative and we include the port and the resistive variables already in the state for simplicity.

The DAE is then explicitly given by the range representation of the following subspace

𝒟\displaystyle\mathcal{D}\mathcal{L} {(x,z)|en:(x,e),(e,z)𝒟}\displaystyle\coloneqq\{(x,z)~{}|~{}\exists\,e\in\mathbb{R}^{n}:~{}(x,e)\in\mathcal{L},(e,z)\in\mathcal{D}\}
=ran[EA]\displaystyle~{}=\operatorname{ran}\begin{bmatrix}E\\ A\end{bmatrix} (28)

for some E,An×nE,A\in\mathbb{R}^{n\times n}. Moreover, the functions which fulfill ddtEx(t)=Ax(t)\tfrac{d}{dt}Ex(t)=Ax(t) and (27) with z(t)=Ex(t)z(t)=Ex(t) coincide, see [GHR21, Section 4].

To ensure stability of [E,A]Σn[E,A]\in\Sigma_{n} given by (28), we need the additional assumption that the Lagrangian subspace =[L1L2]\mathcal{L}=\left[\begin{smallmatrix}L_{1}\\ L_{2}\end{smallmatrix}\right] is nonnegative, i.e., L1TL20L_{1}^{T}L_{2}\geq 0 and, using a suitable CS-decomposition, see [MMW18, Prop. 3.1], we can choose L1L_{1} and L2L_{2} in such a way that L10L_{1}\geq 0.

As a main result of this section we provide sufficient conditions for the stability of DAEs given by (28). Here we use the orthogonal projector PP_{\mathcal{M}} onto a subspace n\mathcal{M}\subseteq\mathbb{R}^{n}. The result is an immediate consequence of [GHR21, Prop. 5.1, Prop. 6.3].

Proposition 5.1.

Let 𝒟=ran[D1D2]\mathcal{D}=\operatorname{ran}\left[\begin{smallmatrix}D_{1}\\ D_{2}\end{smallmatrix}\right] be maximally dissipative and =ran[L1L2]\mathcal{L}=\operatorname{ran}\left[\begin{smallmatrix}L_{1}\\ L_{2}\end{smallmatrix}\right] nonnegative Lagrangian with L1,L1TL20L_{1},L_{1}^{T}L_{2}\geq 0 and D2+D2T0D_{2}+D_{2}^{T}\leq 0. Further, let sEAsE-A be the pencil given by (28). Then with 𝒳ranD1ranL2\mathcal{X}\coloneqq\operatorname{ran}D_{1}\cap\operatorname{ran}L_{2} the following holds.

  • (a)

    sEAsE-A is regular if and only if

    kerP𝒳L1|𝒳kerP𝒳D2|𝒳={0},\displaystyle\ker P_{\mathcal{X}}L_{1}|_{\mathcal{X}}\cap\ker P_{\mathcal{X}}D_{2}|_{\mathcal{X}}=\{0\},
    D2(kerD1)L1(kerL2)={0}.\displaystyle D_{2}(\ker D_{1})\cap L_{1}(\ker L_{2})=\{0\}.
  • (b)

    [E,A][E,A] is stable if additionally L1(kerL2)={0}L_{1}(\ker L_{2})=\{0\} holds.

DH DAEs sE(JR)QsE-(J-R)Q can be written in the form (28) using 𝒟=gr(JR){(x,(JR)x)|xn}\mathcal{D}={\rm gr\,}(J-R)\coloneqq\{(x,(J-R)x)~{}|~{}x\in\mathbb{R}^{n}\} and =ran[EQ]\mathcal{L}=\operatorname{ran}\left[\begin{smallmatrix}E\\ Q\end{smallmatrix}\right]. Here, D1=InD_{1}=I_{n} and D2=JRD_{2}=J-R and hence the second condition in Proposition 5.1 (a) trivially holds.

Corollary 5.2.

Let [E,(JR)Q]Σn[E,(J-R)Q]\in\Sigma_{n} be a dH DAE (3) then it is regular if and only if kerEker(QTJQ)ker(QTRQ)={0}\ker E\cap\ker(Q^{T}JQ)\cap\ker(Q^{T}RQ)=\{0\}. Furthermore, [E,(JR)Q][E,(J-R)Q] is stable if additionally kerQkerE\ker Q\subseteq\ker E.

Proof.

Since stability and regularity are invariant under pencil equivalence we can use [MMW18, Prop. 3.1] and assume without restriction that E0E\geq 0, Q=QT0Q=Q^{T}\geq 0. If sE(JR)QsE-(J-R)Q is regular then kerEkerQ={0}\ker E\cap\ker Q=\{0\} and hence [MMW18, Prop. 3.1] implies that =ran[EQ]\mathcal{L}=\operatorname{ran}\left[\begin{smallmatrix}E\\ Q\end{smallmatrix}\right] is Lagrangian and without restriction E0E\geq 0, QTE0Q^{T}E\geq 0, Q=QT=Q2Q=Q^{T}=Q^{2} holds. By Proposition 5.1 with 𝒳=ranQ=(kerQT)\mathcal{X}=\operatorname{ran}Q=(\ker Q^{T})^{\perp} the regularity is equivalent to

Q(kerQTEQker(QTJQ)kerQTRQ)\displaystyle~{}~{}~{}~{}Q(\ker Q^{T}EQ\cap\ker(Q^{T}JQ)\cap\ker Q^{T}RQ) (29)
={Qx:QTEQx=QT(JR)Qx=0}\displaystyle=\{Qx:Q^{T}EQx=Q^{T}(J-R)Qx=0\}
=kerPranQE|ranQkerPranQ(JR)|ranQ\displaystyle=\ker P_{\operatorname{ran}Q}E|_{\operatorname{ran}Q}\cap\ker P_{\operatorname{ran}Q}(J-R)|_{\operatorname{ran}Q}
={0}.\displaystyle=\{0\}.

The assumptions on EE and QQ imply

kerQkerE=kerQE=kerEQ=kerQEQ\ker Q\dotplus\ker E=\ker QE=\ker EQ=\ker QEQ

and therefore (29) is equivalent to

Q(kerQEQkerQ(JR)Q)={0}\displaystyle~{}~{}~{}~{}~{}~{}~{}Q(\ker QEQ\cap\ker Q(J-R)Q)=\{0\}
(kerEkerQ)kerQ(JR)QkerQ\displaystyle\Longleftrightarrow(\ker E\dotplus\ker Q)\cap\ker Q(J-R)Q\subseteq\ker Q
kerEkerQ(JR)Q={0}.\displaystyle\Longleftrightarrow\ker E\cap\ker Q(J-R)Q=\{0\}.

Furthermore, (14) implies

kerQ(JR)Q=ker(QJQ)ker(QRQ)\ker Q(J-R)Q=\ker(QJQ)\cap\ker(QRQ)

which proves kerEkerQTJQkerQTRQ={0}\ker E\cap\ker Q^{T}JQ\cap\ker Q^{T}RQ=\{0\}. Conversely, if this intersection is trivial then kerEkerQ={0}\ker E\cap\ker Q=\{0\} and hence by [MMW18, Prop. 3.1] the subspace =ran[EQ]\mathcal{L}=\operatorname{ran}\left[\begin{smallmatrix}E\\ Q\end{smallmatrix}\right] fulfills dim=n\dim\mathcal{L}=n and is therefore Lagrangian. Hence (29) holds and sE(JR)QsE-(J-R)Q is regular by Proposition 5.1. Conversely, if kerEker(QT(JR)Q)={0}\ker E\cap\ker(Q^{T}(J-R)Q)=\{0\} holds then kerQkerE={0}\ker Q\cap\ker E=\{0\} and hence =ran[EQ]\mathcal{L}=\operatorname{ran}\left[\begin{smallmatrix}E\\ Q\end{smallmatrix}\right] is Lagrangian. Hence Proposition 5.1 implies that sE(JR)QsE-(J-R)Q is regular. To prove stability we apply Proposition 5.1 (b) and observe that EkerQ={0}E\ker Q=\{0\} is equivalent to (13), i.e., kerQkerE\ker Q\subseteq\ker E.

The above characterization of regularity was also obtained recently in [FMP+21]. However, Proposition 5.1 characterizes the regularity and the stability for a larger class of DAEs. If QQ is invertible then the assumptions in Proposition 5.1 and Proposition 3.6 (b) coincide.

In the remainder we consider pH descriptor systems (1) and formulate them implicitly similar to (27). Define

𝒟grD=gr[JRBP(B+P)TSN],ran[E00IkQ00Ik].\mathcal{D}\coloneqq{\rm gr\,}D={\rm gr\,}\left[\begin{smallmatrix}J-R&B-P\\ (B+P)^{T}&S-N\end{smallmatrix}\right],\quad\mathcal{L}\coloneqq\operatorname{ran}\left[\begin{smallmatrix}E&0\\ 0&I_{k}\\ Q&0\\ 0&I_{k}\end{smallmatrix}\right].

It follows from (2) that D+DT0D+D^{T}\leq 0 and, hence, 𝒟\mathcal{D} fulfills (26). Moreover, \mathcal{L} is Lagrangian if and only if QTE=ETQQ^{T}E=E^{T}Q and sEQsE-Q is regular, see [GHR21, Corollary 5.1].

The system (1) is then implicitly given by

(z(t),u(t),ddtz(t),y(t))𝒟=ran[E^A^],(z(t),u(t),-\tfrac{{\rm d}}{{\rm d}t}z(t),-y(t))\in\mathcal{D}\mathcal{L}=\operatorname{ran}\left[\begin{smallmatrix}\hat{E}\\ \hat{A}\end{smallmatrix}\right],

with E^[E00Ik]\hat{E}\coloneqq\left[\begin{smallmatrix}E&0\\ 0&I_{k}\end{smallmatrix}\right], A^D[Q00Ik]\hat{A}\coloneqq D\left[\begin{smallmatrix}Q&0\\ 0&I_{k}\end{smallmatrix}\right]. Hence, by Proposition 4.3 also stabilizable descriptor systems lie in the geometric framework of [vdSM18].

6 Conclusion

We studied the stability of dH DAEs (3) and DAEs given by the implicit pH formulation (28). In both cases we obtained that regularity together with the assumption kerQkerE\ker Q\subseteq\ker E guarantees the stability of theses DAEs. Furthermore, we showed in Proposition 3.4 that stable DAEs can always be reformulated as dH DAEs if we restrict the coefficients of the underlying equations to the system space. This restriction is not necessary if we consider index one DAEs, see Remark 3.5, and the matrix QQ in (3) can be obtained from the solution of the generalized Lyapunov inequality (11). Similarly, we showed in Proposition 4.3 that stabiliazble systems can be reformulated as pH systems (1) using the solutions of generalized algebraic Bernoulli equations (20) and (21). Finally, it was shown how dH DAEs (3) and pH systems (1) can be embedded into the geometric pH framework from [vdSM18].

Acknowledgements

The authors are indebted to Timo Reis for his valuable remarks and suggestions on an earlier draft of this manuscript. Furthermore, the authors would like to thank Volker Mehrmann for providing valuable references.

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