Abstract
Linear time-varying differential-algebraic equations with symmetries are studied. The structures that we address are self-adjoint and skew-adjoint systems. Local and global canonical forms under congruence are presented and used to classify the geometric properties of the flow associated with the differential equation as symplectic or generalized orthogonal flow. As applications, the results are applied to the analysis of dissipative Hamiltonian systems arising from circuit simulation and incompressible flow.
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1 Introduction
We study regular linear variable coefficient differential-algebraic equations (DAEs)
which additionally possess certain symmetries, in particular self-adjoint and skew-adjoint structures. Here \(\mathbb {I}\subseteq \mathbb {R}\) is a compact non-trivial time interval and \(C^{k}(\mathbb {I},\mathbb {R}^{n,n})\) with \(k\in {\mathbb {N}}_{0}\cup \{\infty \}\) denotes the set of k times continuously differentiable functions from \(\mathbb {I}\) into the set of real n × n matrices \(\mathbb {R}^{n,n}\). In the case k = 0 we drop the superscript. A function is said to be sufficiently smooth if k is sufficiently large such that all needed derivatives exist.
The discussed classes of DAEs with symmetries are defined as follows.
Definition 1.1
The DAE (1) and its associated pair (E,A) of matrix functions are called self-adjoint if
as equality of functions.
Definition 1.2
The DAE (1) and its associated pair (E,A) of matrix functions are called skew-adjoint if
as equality of functions.
Systems with these types of symmetries arise in the modeling of physical systems, e.g., by a (generalized) Hamiltonian formalism or in optimal control problems leading to a self-adjoint structure or by so-called dissipative Hamiltonian systems leading to a skew-adjoint structure.
Our main motivation to study differential-algebraic equations with self-adjoint and skew-adjoint pairs arises from multi-physics, multi-scale models that are coupling different physical domains that may include mechanical, mechatronic, fluidic, thermic, hydraulic, pneumatic, elastic, plastic, or electric components, see e.g. [1, 2, 11, 19, 38, 39].
An important class of problems where multi-physics and multi-scale modeling arises is the human cariovascular system, where model hierarchies of detailed models for the blood flow in large vessels, modeled via the Navier–Stokes equations, and reduced or surrogate models for the capillary vessels, modeled via electrical network equations, are coupled together to improve computational efficiency, while at the same time achieving a desired simulation accuracy, see [13, 14, 32, 33, 35]. Due to the physical background, after space discretization and linearization along a solution, as well as ignoring dissipation terms, all the components arising in this application can be expressed as DAE systems with symmetries.
Motivated by classical bond graph theory, see [31, 42], and to deal with general multi-physics and multi-scale coupling, in recent years the framework of port-Hamiltonian systems has become an important modeling paradigm, see [3, 15, 20, 28, 30, 43, 44] or the recent survey [29], that encodes underlying physical principles directly into the algebraic structure of the coefficient matrices and in the geometric structure associated with the flow of the dynamical system. This leads to a remarkably flexible modeling approach, which has also been extended to include algebraic constraints, so that the resulting model is a port-Hamiltonian differential-algebraic equation (pHDAE), [3, 28, 45]. Such systems allow for automated modeling in a modularized network based fashion, and they are ideal for building model hierarchies of very fine models for numerical simulation and reduced or surrogate models for control and optimization. This makes them particularly suited also for large networks, such as power, gas, or district heating networks where such model hierarchies are used to adapt the simulation and optimization techniques to user needs, [9, 18, 28].
Since in this paper we will mainly discuss linear time-varying DAE systems with symmetries, we introduce the structure of pHDAEs for this case as in [3], see [28] for the more general nonlinear framework. Linear time-varying DAE systems arise when general nonlinear DAE systems are linearized along non-stationary solutions [6] and the study of this class is essential for the understanding of the general nonlinear DAEs, see [22]. For nonlinear pHDAE with non-quadratic Hamiltonian systems also a local approximation of the Hamiltonian by a quadratic Hamiltonian is typical.
Linear time-varying pHDAE systems with quadratic Hamiltonian have the form
with state x, input u, output y and coefficients \(E\in C^{1}(\mathbb {I},\mathbb {R}^{n,n})\), \(J,R,K \in C(\mathbb {I},\mathbb {R}^{n,n})\), \(G,P\in C(\mathbb {I},\mathbb {R}^{n, m})\), \(S,N\in C(\mathbb {I},\mathbb {R}^{m, m})\), S = ST,N = −NT. As energy function we employ a quadratic Hamiltonian
where E(t) is the local Hessian of the Hamiltonian when linearizing around a nonstationary solution and the pair of coefficients (E,J − EK) is skew-adjoint, while the matrix function associated with dissipation of energy
is pointwise positive semidefinite. Furthermore, typically one also has that E is pointwise positive semidefinite as well. If the system does not have an output equation and the input is considered as an inhomogeneity then this is called a dissipative Hamiltonian DAE (dHDAE).
The underlying skew-adjoint structure arises if the dissipation term R is neglected, i.e., if \(\mathcal {W}=0\) in (4). Typically the problems with dissipation can be considered as a perturbed symmetry structure and a dissipative term can be treated separately as a by-product in simulation methods (Fig. 1).
To illustrate applications for dHDAEs and DAEs with symmetries consider the following simple examples, for further applications see [3, 28, 45, 46] or the recent survey [29].
Example 1.3
Consider the pHDAE formulation of an electrical circuit from [28]. Denoting by Vi the voltages and by Ii the currents, where L > 0 models an inductor, C1,C2 > 0 capacitors, RG,RL,RR > 0 resistances, and EG a controlled voltage source, one obtains a pHDAE
with \(x=\begin {bmatrix} I &V_{1}&V_{2}&I_{G}&I_{R} \end {bmatrix}^{T}\), u = EG, y = IG, E = diag(L,C1,C2,0,0),
The quadratic Hamiltonian, describing the energy stored in the inductor and the two capacitors, is given by
If the generator is shut down (i.e. EG = 0), then the system approaches an equilibrium solution for which \(\frac {d}{dt} {\mathscr{H}}(x)=0\), so that I = IG = IR = 0 and thus x = 0 in the equilibrium. Without the damping by the resistances (i.e. setting RL = RG = RR = 0) this is a skew-adjoint DAE.
Example 1.4
A classical example of a partial differential equation which, after proper space discretization leads to a pHDAE, see e.g. [12, 34], are the incompressible or nearly incompressible Navier–Stokes equations describing the flow of a Newtonian fluid in a domain Ω,
together with suitable initial and boundary conditions, see e.g. [41]. When one linearizes around a prescribed vector field \(v_{\infty }\), then one obtains the linearized Navier–Stokes equations, and if \(v_{\infty }\) is constant in space and the term \((v_{\infty }\cdot \nabla ) v\) is neglected then one obtains the Stokes equation.
Performing a finite element discretization in space, see for instance [26], a Galerkin projection leads to a dHDAE of the form
where M = MT > 0 is the mass matrix, \(A_{S}=-{A_{S}^{T}}\), \(A_{H}={A_{H}^{T}}\geq 0\) are skew-symmetric and symmetric part of the discretized and linearized convection-diffusion operator, B is the discretized gradient operator, BT is the discretized divergence operator, which we assume to be normalized so that it is of full row rank, and C = CT > 0 is a stabilization term of small norm that is needed for some finite element spaces, see e.g. [4, 34, 36, 37]. Here v and p denote the discretized velocity and pressure, respectively, and f is a forcing term. Without the damping (i.e. for AH = 0 and C = 0) we again have a skew-adjoint DAE.
The class of problems with self-adjoint structure arises most prominently in the context of constrained generalized Hamiltonian systems and in optimal control problems, where the operators associated with the optimality conditions have this structure.
Example 1.5
In [23] the optimality conditions were derived for the linear-quadratic optimal control problem of minimizing a cost functional
subject to the DAE constraint
with \(E,A,W\in C(\mathbb {I},\mathbb {R}^{n,n})\), \(B \in C(\mathbb {I},\mathbb {R}^{n,m})\), \(S \in C(\mathbb {I},\mathbb {R}^{n,m})\), \(R \in C(\mathbb {I},\mathbb {R}^{m,m})\), \(f \in C(\mathbb {I},\mathbb {R}^{n})\) and \(M_{e} \in \mathbb {R}^{n,n}\), where R = RT, W = WT and \(M_{f}={M_{f}^{T}}\).
After some appropriate reformulation (via some index reduction process) and under some smoothness conditions, the optimality condition is given by a boundary value problem
with boundary conditions x(t0) = x0, E(tf)Tλ(tf) − Mfx(tf) = 0. The associated pair of coefficient functions obviously is a a self-adjoint pair, see [25].
Example 1.6
Linear multibody systems with linear holonomic constraints, see [16, 27], take the form
where \(p,q \in C^{1}(\mathbb {I}, \mathbb {R}^{n})\), \(W,M\in \mathbb {R}^{n,n}\) with W = WT, M = MT, and \(G\in \mathbb {R}^{m,n}\). If the mass matrix is positive definite, i.e. M > 0, then we can multiply the second equation by − M and the constraint by − 1 to obtain, after switching the first and second equation,
which has the structure of a self-adjoint DAE.
If W > 0 then we can also multiply the second equation of the constrained Hamiltonian system by W to obtain
which now has the structure of a skew-adjoint DAE.
Remark 1.7
It should be noted that if in a self-adjoint system \(E \dot {x}=A x+f\), both E and A are constant in time and invertible, then by multiplication with E− 1 and a change of variables z = Ax we get a system \( A^{-1} \dot {z} = E^{-1} z+ E^{-1} f\) that is skew-adjoint. A similar construction has also been proposed for dissipative Hamiltonian systems in [10].
Having illustrated the importance of DAEs with symmetries, in this paper we present canonical forms for DAEs (1) with self-adjoint and skew-adjoint structure and show the consequences for the resulting flows.
The paper is organized as follows. In Section 2 we recall some results for general DAEs. In Section 3 we discuss local and global canonical forms for self-adjoint and in Section 4 for skew-adjoint DAEs.
2 Preliminaries
Linear time-varying DAE systems have been extensively discussed, see [22] for a detailed analytical and numerical treatment. In this section we recall some basic concepts from the general theory of DAEs. Our first concept is that of regularity for DAEs that is concerned with the existence of solutions at least for sufficiently smooth inhomogeneity (surjectivity) and uniqueness of the solution in the cases where the initial condition x(t0) = x0 allows for a solution (injectivity).
Definition 2.1
The pair (E,A) and the corresponding DAE (1) are called regular if
-
1.
the DAE (1) is solvable for every sufficiently smooth f,
-
2.
the solution is unique for every \(t_{0}\in \mathbb {I}\) and every \(x_{0}\in \mathbb {R}^{n}\) allowing for a solution of the DAE with x(t0) = x0,
-
3.
the solution depends smoothly on f, t0, and x0.
The most important technique for the analysis of general linear DAEs is the construction of suitable local and global canonical forms under global equivalence. Since we will refer to these results and some techniques from their derivation, we include here the necessary material from the general (square) case.
We start with (global) equivalence which refers to time-dependent scaling of the DAE and changes of basis.
Definition 2.2
Two pairs (Ei,Ai), \(E_{i},A_{i}\in C(\mathbb {I},\mathbb {R}^{n,n})\), i = 1,2, of matrix functions are called (globally) equivalent if there exist pointwise nonsingular matrix functions \(P\in C(\mathbb {I},\mathbb {R}^{n,n})\) and \(Q\in C^{1}(\mathbb {I},\mathbb {R}^{n,n})\) such that
as equality of functions. We write \((E_{1},A_{1}) \sim (E_{2},A_{2})\).
It is easy to see that the relation defined in Definition 2.2 indeed is an equivalence relation, [22].
The derivation of canonical forms then relies on the following property of matrix functions on intervals, see [7, 22].
Theorem 2.3
Let \(E\in C^{k}(\mathbb {I},\mathbb {R}^{m,n})\), \(k\in {\mathbb {N}}_{0} \cup \{\infty \}\), with rankE(t) = r for all \(t\in \mathbb {I}\). Then there exist pointwise orthogonal functions \(U\in C^{k}(\mathbb {I},\mathbb {R}^{m,m})\) and \(V\in C^{k}(\mathbb {I},\mathbb {R}^{n,n})\) such that
with pointwise nonsingular \({{\varSigma }}\in C^{k}(\mathbb {I},\mathbb {R}^{r,r})\).
We then have the following result on a local canonical form, i.e., a canonical form that requires the restriction to certain subintervals, see [22, Sections 3.1 and 3.3]. In the following, non-specified blocks of matrices or matrix functions are denoted by ∗.
Theorem 2.4
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth. Then there exist pairwise disjoint open intervals \(\mathbb {I}_{j}\), \(j\in {\mathbb {N}}\), with
such that on every \(\mathbb {I}_{j}\) one has
where G is structurally nilpotent according to
Furthermore, the size d of the differential part and the size a of the algebraic part are the same for every interval.
A restriction to subintervals is necessary to obtain the structural nilpotent form of G, which requires constant rank assumptions. This phenomenon occurs naturally for example in robotics when position constraints have a singularity due to two robot arms forming a straight line, see e.g. [8], and no structural nilpotent G exists globally.
A global canonical form, i.e., a canonical form that does not require the restriction to certain subintervals, was given in [5]. We state this result here in a version for real-valued problems omitting the last step of the proof which would require complex-valued transformations. We include the proof for later reference.
Theorem 2.5
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth. Then we have
where
is uniquely solvable for every sufficiently smooth f2 without specifying initial conditions.
Proof
If the homogeneous equation
has only the trivial solution, then the first block in (7) is missing (i.e., d = 0) and the claim holds trivially by assumption. In any case, the solution space is finite dimensional, since otherwise we could not select a unique solution by prescribing initial conditions.
Let therefore d≠ 0 and let {ϕ1,…,ϕd} be a basis of the solution space. Setting Φ = [ϕ1 ⋯ ϕd], we have
since, if we had rankΦ(t) < d for some \(t_{0}\in \mathbb {I}\), then there would exist coefficients \(\alpha _{1},\ldots ,\alpha _{d}\in \mathbb {R}\), not all being zero, with
and α1ϕ1 + ⋯ + αdϕd would be a nontrivial solution of the homogeneous initial value problem.
Hence, by Theorem 2.3 there exists a smooth, pointwise nonsingular matrix function U with
Defining
yields a pointwise nonsingular matrix function \(Q=[{{\varPhi }}~~{{\varPhi }}^{\prime }]\). Since \(E\dot {{\varPhi }}=A{{\varPhi }}\), we obtain
In this relation, E1 has full column rank d. To see this, suppose that \(\text {rank} E_{1}(\hat {t})<d\) for some \(\hat {t}\in \mathbb {I}\). Then there would exist a vector w≠ 0 with
Defining in this situation
we would obtain a smooth inhomogeneity f. The function x given by
would then solve
on \(\mathbb {I}\setminus \{\hat {t}\}\) in contradiction to the assumption of unique solvability, which includes by definition that solutions are defined on the entire interval \(\mathbb {I}\).
Hence, since E1 has full column rank, there exists a smooth, pointwise nonsingular matrix function P with
and thus
The equation
only admits the trivial solution. To see this, suppose that x2≠ 0 is a nontrivial solution and x1 a solution of the ODE
Then we obtain
Since x2≠ 0, transforming back gives
or \(E(t)\dot {x}(t)=A(t)x(t)\) with
But this contradicts the construction of ϕ1,…,ϕd, and hence (7) holds. □
In the presence of symmetries, we of course want to maintain these properties, which requires to restrict the allowed equivalence transformations. We will make use of the following notions and properties.
Definition 2.6
Two pairs (Ei,Ai), \(E_{i},A_{i}\in C(\mathbb {I},\mathbb {R}^{n,n})\), i = 1,2, of matrix functions are called congruent if there exist a pointwise nonsingular matrix function \(Q\in C^{1}(\mathbb {I},\mathbb {R}^{n,n})\) such that
as equality of functions. We write (E1,A1) ≡ (E2,A2).
Again, it is easy to see that the relation defined in Definition 2.6 indeed is an equivalence relation, see [25].
The following result then modifies Theorem 2.3 provided some symmetry property holds.
Theorem 2.7
Let \(E\in C^{k}(\mathbb {I},\mathbb {R}^{n,n})\), \(k\in {\mathbb {N}}_{0}\cup \{\infty \}\), with rankE(t) = r for all \(t\in \mathbb {I}\) and let kernelE(t)T = kernelE(t) for all \(t\in \mathbb {I}\). Then there exist a pointwise orthogonal function \(Q\in C^{k}(\mathbb {I},\mathbb {R}^{n,n})\) such that
with pointwise nonsingular \({{\varSigma }}\in C^{k}(\mathbb {I},\mathbb {R}^{r,r})\).
Proof
With Q = V from Theorem 2.3, the pointwise property kernelET = kernelE allows to choose U = QT. □
In the next two sections we will employ these preliminary results to derive canonical forms for self-adjoint and skew-adjoint DAEs.
3 Canonical Forms for Self-Adjoint Pairs of Matrix Functions
In this section we study canonical forms under congruence for self-adjoint DAEs. We survey, modify, and extend previous results from [25]. We first recall that congruence preserves the self-adjoint structure.
Lemma 3.1
Consider two pairs of matrix functions (E,A) and \((\widetilde {E},\widetilde {A})\) that are congruent and let (E,A) be self-adjoint. Then \((\widetilde {E},\widetilde {A})\) is self-adjoint as well.
For self-adjoint pairs the following local canonical form under pointwise orthogonal congruence transformations is due to [25] stated here for the special case of a regular pair of matrix functions.
Theorem 3.2
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth and let (E,A) be skew-adjoint. Then there exist pairwise disjoint open intervals \(\mathbb {I}_{j}\), \(j\in {\mathbb {N}}\), with (6) such that on every \(\mathbb {I}_{j}\) there exists a pointwise orthogonal \(Q\in C(\mathbb {I},\mathbb {R}^{n,n})\) with
![figure a](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figa_HTML.png)
where
![](http://media.springernature.com/lw337/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figb_HTML.png)
and Δ,Σ22,Γ1,…,Γw are pointwise nonsingular. Furthermore,
Theorem 3.2 can be further refined by allowing for a restricted class of non-orthogonal transformations, see again [25].
Theorem 3.3
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth and let (E,A) be skew-adjoint. Then there exist pairwise disjoint open intervals \(\mathbb {I}_{j}\), \(j\in {\mathbb {N}}\), with (6) such that on every \(\mathbb {I}_{j}\) there exists a pointwise nonsingular \(Q\in C(\mathbb {I},\mathbb {R}^{n,n})\) with
![](http://media.springernature.com/lw489/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figc_HTML.png)
where
![](http://media.springernature.com/lw326/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figd_HTML.png)
and Σ22 pointwise nonsingular. Furthermore,
By successively resolving the algebraic equations in the fourth, third and first row of the DAE with solution \([{x}_{1}^{T} ~~ {x}_{2}^{T} ~~ {x}_{3}^{T} ~~ {x}_{4}^{T}]^{T}\) and inhomogeneity \([{f}_{1}^{T} ~~ {f}_{2}^{T} ~~ {f}_{3}^{T} ~~ {f}_{4}^{T}]^{T}\) associated with (8), we can directly solve for x1 in terms of linear combinations of derivatives of f4, for x3 in terms of x1 and f3, and for x4 in terms of linear combinations of derivatives of f1 and all other components. The only dynamic behavior related to (8) is described by the second block row. Inserting x1 obtained from the last block row and calling the updated inhomogeneity \(\widetilde {f}_{2}\), the associated ODE reads
The matrix function M = J− 1C satisfies MTJ − JM = 0 and lies therefore pointwise in the Lie algebra of Hamiltonian matrices, see e.g. [17]. Thus, the flow corresponding to (9) defined by
satisfies \({{{\varPhi }}_{2}^{T}} J {{\varPhi }}_{2} =J\), see e.g. [16], and is therefore symplectic.
In [25], also a global canonical form for self-adjoint DAEs and the associated pairs (E,A) was derived. The following modified result differs in the assumptions, and, moreover, the resulting canonical form is more refined.
Theorem 3.4
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth and let (E,A) be self-adjoint. Then we have
where
is uniquely solvable for every sufficiently smooth f3 without specifying initial conditions. Furthermore,
Proof
According to Theorem 3.3, the size d of the differential part is given by d = 2p. This implies that the solution space of the homogeneous equation
is of dimension 2p. If p = 0, then the first two blocks are missing and the claim holds trivially by assumption.
Let therefore p≠ 0 and let {ϕ1,…,ϕ2p} be a basis of the solution space. Setting Φ = [ϕ1 ⋯ ϕ2p], we have
as in the general case of Theorem 2.5.
Hence, by Theorem 2.3 there exists a smooth, pointwise nonsingular matrix function U with
Defining
with a = n − 2p yields a pointwise nonsingular matrix function \(Q=[{{\varPhi }}~~{{\varPhi }}^{\prime }]\). Since \(E\dot {{\varPhi }}=A{{\varPhi }}\), we obtain
with
To simplify the notation, we omit now and later at similar instances the tildes thus re-using the same notation for possibly different quantities and write
As in the general case, we can conclude that
Since self-adjointness is conserved, we additionally have
In particular, E11 is constant and skew-symmetric. Hence, see [25], there exists an orthogonal symplectic matrix \(U\in \mathbb {R}^{2p,2p}\) with
where \(\widetilde {E}_{12}\in \mathbb {R}^{p,p}\).
Omitting again the tildes, we write
Conservation of self-adjointness and full rank of the leading block yields
Hence, there exists a smooth, pointwise nonsingular matrix function V with
leading to
Omitting the tildes again, we write
Conservation of self-adjointness yields
Finally, after a congruence transformation with,
we arrive at
Omitting again the tildes then yields
which is just (10), where (11) follows along the same lines as in the general case and (12) follows by the conservation of self-adjointness. □
Theorem 3.4 shows that the symmetry defined by the self-adjoint structure under some weak assumptions leads to the geometric property that the flow associated with the dynamic part of the DAE transformed to global canonical form is symplectic, i.e. this property of classical Hamiltonian systems is lifted to the DAE case via the self-adjoint structure. In the next section we show that an analogous property holds for skew-adjoint DAEs.
4 Canonical Forms for Skew-Adjoint Pairs of Matrix Functions
While the self-adjoint structure had been analyzed for DAEs already in [25], the skew-adjoint structure and its importance for energy based modeling has only recently been observed in [3, 40], where also local canonical forms under constant rank assumptions have been studied. In this section we show that also global canonical forms under congruence can be derived for skew-adjoint DAEs. These then again imply the geometric properties of an underlying flow.
Lemma 4.1
Consider two pairs of matrix functions (E,A) and \((\widetilde {E},\widetilde {A})\) that are congruent and let (E,A) be skew-adjoint. Then \((\widetilde {E},\widetilde {A})\) is skew-adjoint as well.
Proof
The proof is analogous to that of Lemma 3.1 presented in [25]. □
The following result on a local canonical form under pointwise orthogonal congruence transformations is due to [40].
Theorem 4.2
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth and let (E,A) be skew-adjoint. Then there exist pairwise disjoint open intervals \(\mathbb {I}_{j}\), \(j\in {\mathbb {N}}\), with (6) such that on every \(\mathbb {I}_{j}\) there exists a pointwise orthogonal \(Q\in C(\mathbb {I},\mathbb {R}^{n,n})\) with
![figure e](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Fige_HTML.png)
where
![](http://media.springernature.com/lw337/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figf_HTML.png)
and Δ,Σ22,Γ1,…,Γw are pointwise nonsingular. Furthermore,
Theorem 3.2 can be further refined by allowing for a restricted class of non-orthogonal transformations, which yields the following local canonical form which, using a recent result of [21], is more refined than that in [40].
Theorem 4.3
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth and let (E,A) be skew-adjoint. Then there exist pairwise disjoint open intervals \(\mathbb {I}_{j}\), \(j\in {\mathbb {N}}\), with (6) such that on every \(\mathbb {I}_{j}\) there exists a pointwise nonsingular \(Q\in C(\mathbb {I},\mathbb {R}^{n,n})\) with
![](http://media.springernature.com/lw496/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figg_HTML.png)
where
![](http://media.springernature.com/lw326/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figh_HTML.png)
and Σ22 pointwise nonsingular. Furthermore,
Proof
Compared with the result in [40], to obtain the stated local canonical form we need a smooth version of Sylvester’s law of inertia, which is proved in [21]. In particular, this result shows the existence of a smooth transformation W with WT ΔW = S, where Δ is pointwise nonsingular and symmetric. □
By successively resolving the algebraic equations in the fourth, third and first row of the DAE with solution \([{x}_{1}^{T} ~~ {x}_{2}^{T} ~~ {x}_{3}^{T} ~~ {x}_{4}^{T}]^{T}\) and inhomogeneity \([{f}_{1}^{T} ~~ {f}_{2}^{T} ~~ {f}_{3}^{T} ~~ {f}_{4}^{T}]^{T}\) associated with (8), we can directly solve for x1 in terms of linear combinations of derivatives of f4, for x3 in terms of x1 and f3, and for x4 in terms of linear combinations of derivatives of f1 and all other components. The only dynamic behavior related to (8) is described by the second block row. Inserting x1 obtained from the last block row and calling the updated inhomogeneity \(\widetilde {f}_{2}\), the associated ODE reads
The matrix function M = S− 1J has the property that SM is skew-symmetric, i.e., that (SM)T = −(SM) or MTS + SM = 0, and lies therefore pointwise in the Lie algebra belonging to the quadratic Lie group
the so-called generalized orthogonal group with inertia(p,q,0), see e.g. [17], implying that the flow belonging to (14) lies in O(p,q). Observe the special case p = 0 or q = 0 where the flow is then orthogonal.
We also obtain a global canonical form.
Theorem 4.4
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth and let (E,A) be skew-adjoint. Then we have
where
is uniquely solvable for every sufficiently smooth f3 without specifying initial conditions. Furthermore,
Proof
According to Theorem 4.3, the size d of the differential part is given by d = p + q. This implies that the solution space of the homogeneous equation
is of dimension p + q. If p + q = 0, then the first two blocks are missing and the claim holds trivially by assumption.
Let therefore p + q≠ 0 and let {ϕ1,…,ϕp+q} be a basis of the solution space. Setting Φ = [ϕ1 ⋯ ϕp+q], we have
as in the general case of Theorem 2.5.
Hence, by Theorem 2.3 there exists a smooth, pointwise nonsingular matrix function U with
Defining
with a = n − (p + q) yields a pointwise nonsingular matrix function \(Q=[{{\varPhi }}~~{{\varPhi }}^{\prime }]\). Since \(E\dot {{\varPhi }}=A{{\varPhi }}\), we obtain
with
As in the proof of Theorem 3.4, we omit now and later at similar instances the tildes thus re-using the same notation for possibly different quantities and write
As in the general case, we can conclude that
Since skew-adjointness is conserved, we additionally have
In particular, E11 is constant and symmetric. Moreover, in the following we will show that E11 is nonsingular.
Due to Sylvester’s law of inertia, there is a nonsingular matrix \(U\in \mathbb {R}^{d,d}\), d = p + q + r, with
where S = diag(Ip,−Iq).
Omitting again the tildes, we write
Conservation of skew-adjointness and full rank of the leading block yields
Hence, there exists a smooth, pointwise nonsingular matrix function V with
leading to
Omitting again the tildes, we write
Conservation of self-adjointness yields
and
Finally, after a congruence transformation with,
we arrive at
The corresponding homogeneous DAE \(\widetilde {E}\dot {\widetilde {x}}=\widetilde {A}{\widetilde {x}}\) has the form
If we take \({\widetilde {x}}_{1}\) and \({\widetilde {x}}_{3}\) as solutions of the first two equations, the fourth equation must determine \({\widetilde {x}}_{4}\), possibly imposing a suitable initial condition. Finally, the third equation then fixes \({\widetilde {x}}_{2}\) using a suitable initial condition. Hence the dimension of the solution space is at least p + q + 2r. But the dimension was assumed to be p + q implying that r = 0.
Omitting the tildes again, skipping the second and third block row and column, which are of zero dimension, and renumbering the indices gives
which is just (15), where (16) follows along the same lines as in the general case and (17) follows by the conservation of skew-adjointness. □
The presented canonical forms have some direct consequences in the case that E is pointwise positive semidefinite.
Corollary 4.5
Let (E,A) be regular with \(E,A\in C(\mathbb {I},\mathbb {R}^{n,n})\) sufficiently smooth and let (E,A) be skew-adjoint with E pointwise positive semidefinite. Then in the canonical form (13) E41 = 0 and the flow associated with the dynamical part of the system is orthogonal.
Proof
Since QTEQ is positive semidefinite for positive semidefinite E, it follows that the blocks in positions (1,3), (3,1), (4,1) and (1,4) of (13) are zero and that S is pointwise positive definite. Thus q = 0 and the flow corresponding to (14) is orthogonal. □
Remark 4.6
Corollary 4.5 immediately implies also an upper bound on the so-called differentiation index of the DAE, see [22], which is at most two. This follows directly from (13), since with E41 = 0 at most one differentiation of the inhomogeneity is needed.
Remark 4.7
The construction of the local and global canonical forms is possible, but not really practical computationally. As in the procedure for general DAE systems it is more practical to use derivative arrays to filter out a subsystem that consists of separated algebraic and differential equations, where the consistency of the initial value can be checked and where it can be assured that the constraints are preserved in the numerical procedure, see [22]. For skew- and self-adjoint systems this is possible as well, see the recent paper [24].
To illustrate the canonical forms we discuss two examples.
Example 4.8
Consider the circuit in Example 1.3. In this case the construction of the local and global form is extremely simple. Reordering the rows and columns in the order second, third, first, fourth, and fifth, we obtain the skew-adjoint system in canonical form
![figure i](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10013-022-00596-x/MediaObjects/10013_2022_596_Figi_HTML.png)
If u is given and the resistive terms are neglected, i.e. RL,RG,RR = 0, then this is skew-adjoint DAE, in which the last two equations can be solved as V2 = 0 and V1 = −u and from the first two equations we get \(I_{G}=-C_{1} \dot {V}_{1}= C_{1} \dot {u}\) and \(I_{R}=-C_{2} \dot {V}_{2}=0\) and the dynamic equation is
which has the orthogonal flow I = 1.
Example 4.9
Consider the linear time-varying DAE (5) in Example 1.4 with AH = 0 and C = 0. This corresponds to the situation that dissipation due to friction is negligible and that no artificial stabilization term in the space discretization is used. Performing a full rank decomposition
with B1 nonsingular (which corresponds to the partitioning of the velocity into the parts that have divergence zero and nonzero), and applying a congruence transformation with
yields a transformed system
The third equation yields v1 = 0 and the first equation gives
while the underlying dynamics of the system is described by the skew-adjoint DAE
with constant \(M_{22}=M_{22}^{T}>0\) and pointwise skew-symmetric J22. After a change of basis with the positive definite square root \(M_{22}^{1/2}\) of M22 according to \(\tilde {v}_{2}= M_{22}^{1/2}v_{2}\) and scaling the equation by its inverse \(M_{22}^{-1/2}\), we obtain an ODE system
with pointwise skew-symmetric \(\tilde {J}_{22}\), which has an orthogonal flow.
5 Conclusions
We have derived local and global canonical forms under congruence transformations for self-adjoint and skew-adjoint systems of linear variable coefficient differential-algebraic equations. The associated flows for the dynamical part of the system are shown to be symplectic or in the generalized orthogonal groups. The results are illustrated at the hand of examples from electrical network and flow simulation.
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Acknowledgements
Partially supported by the Research In Pairs program of Mathematisches Forschungsinstitut Oberwolfach, whose hospitality is gratefully acknowledged.
Volker Mehrmann was partially supported by the Deutsche Forschungsgemeinschaft through Project A2 of CRC 910 Control of self-organizing nonlinear systems: Theoretical methods and concepts of application.
Volker Mehrmann was partially supported by Deutsche Forschungsgemeinschaft through the Excellence Cluster Math+ in Berlin and Priority Program 1984 ‘Hybride und multimodale Energiesysteme: Systemtheoretische Methoden für die Transformation und den Betrieb komplexer Netze’.
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Kunkel, P., Mehrmann, V. Local and Global Canonical Forms for Differential-Algebraic Equations with Symmetries. Vietnam J. Math. 51, 177–198 (2023). https://doi.org/10.1007/s10013-022-00596-x
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DOI: https://doi.org/10.1007/s10013-022-00596-x
Keywords
- Differential-algebraic equation
- Self-adjoint system
- Skew-adjoint system
- Dissipative Hamiltonian system
- Canonical form under congruence