Abstract
Time-domain models of dynamical systems are formulated in many applications in terms of differential-algebraic equations (DAEs). In the linear time-varying context, certain limitations of models of the form E(t)x′(t) + B(t)x(t) = q(t) have recently led to the properly stated formulation A(t)(D(t)x(t))′ + B(t)x(t) = q(t), which allows for explicit descriptions of problem solutions in regular DAEs with arbitrary index, and provides precise functional input-output characterizations of the system. In this context, the present paper addresses critical points of linear DAEs with properly stated leading term; such critical points describe different types of singularities in the system. Critical points are classified according to a taxonomy which reflects the phenomenon from which the singularity stems; this taxonomy is proved independent of projectors and also invariant under linear time-varying coordinate changes and refactorizations. Under certain working assumptions, the analysis of such critical problems can be carried out through a scalarly implicit decoupling, yielding a singular inherent ODE. Certain harmless problems for which this decoupling can be rewritten in explicit form are characterized. Some electrical circuit applications, including a linear time-varying analogue of Chua's circuit, are discussed for illustrative purposes.
1. Introduction
In different engineering and applied science contexts, dynamical system models often take the form of a differential-algebraic equation (DAE) Citation1-4. As discussed below, the dynamical behaviour of regular DAEs (see section 2.2) can be eventually described in terms of an ordinary differential equation (ODE): from this point of view, the main differences between ODEs and DAEs arise therefore from the modelling and computational perspectives. In non-regular contexts, new dynamical phenomena can be displayed and DAE models are essential to capture this singular behaviour; these singularities typically preclude the existence of a global state space (ODE) model of the system. In particular, in circuit applications DAEs are nowadays pervasive, and modern simulation programs such as SPICE or TITAN set up circuit equations in differential-algebraic form Citation5-15. DAEs are encountered very often also in mechanics, controls, power system theory, chemical processes, etc., sometimes under other names such as descriptor, generalized, constrained or semi-state systems Citation1,Citation3,Citation16.
In a linear time-varying setting, DAE models are often set up in the form
Therefore, recent attention has been driven to models taking the form
Note that the form Equation(2) comprises standard-form linear DAEs Equation(1) if there exists a C 1 basis for ker E(t) since, letting P(t) be a C 1 projector along ker E(t), we may rewrite Equation(1) as E(t)(P(t)x(t))′ + [F(t) − E(t)P′(t)]x(t) = q(t). Note that such a basis exists in particular if E(t) is C 1 with constant rank. Therefore, all the results apply to linear DAEs in the classical formulation Equation(1).
In this context, the present paper addresses singular or critical points of linear DAEs of the form Equation(2), the results being applicable to Equation(1) via the above-mentioned reformulation. Critical points may reflect different singular phenomena arising in the system, including impasse behaviour, bifurcations, voltage collapse in power system models, unboundedness of solutions, etc. (see Citation4 and the bibliography therein). But, beyond some particular structures, a complete analysis characterizing critical points in time-varying DAE models has not been developed so far; such a general analysis is allowed by the proper formulation Equation(2).
Roughly speaking, critical points of linear DAEs are sometimes defined in terms of the non-existence or non-uniqueness of solutions at those points. This is the case for instance in Citation25, where a definition of a critical point is provided for analytic problems relying upon the impossibility to continue a given solution. In the present paper we face critical points from a different point of view, described below.
Most approaches to the analysis of DAEs Citation1,Citation2,Citation4,Citation22,Citation26-29 are based on an iterative or recursive definition of an index, and end up with some kind of related (underlying/inherent/reduced, etc.) ODE. The index definition usually implies that this ODE is regular, and all initial value problems are uniquely solvable. But, sometimes, although the analysis procedure can be completed, it ends up with a non-solvable (or a non-uniquely solvable) ODE. In linear cases, non-solvable continuous problems are related to singularities and, in the analytic setting, can be tackled via Fuchs – Frobenius theory. These singularities arise typically in the last step of the analysis procedure, and drive the problem to the setting of singular ODEs.
But we look at the somewhat pathological behaviour not from the ODE perspective but from the DAE viewpoint. Hence, broadly speaking, critical points will be those where the DAE analysis procedure cannot be pursued beyond a certain step; formally, a point will be called critical if no neighbourhood of t * admits an index or, equivalently, there is no regularity interval including t *. Instances of these phenomena, beyond the above-mentioned last-step singularities, are rank-changing points of identically singular, non-analytic matrix functions A(t) in Equation(2) or E(t) in Equation(1). Note that previous approaches Citation4,Citation25,Citation29 do not detail to what extent these critical points can be handled. If the analysis procedure can be adapted in order to accommodate these critical points, then it would typically end up with a singular ODE, although some (informally called harmless) cases may result in a non-singular (hence solvable) linear ODE. In contrast, in the most involved cases there is simply no way to obtain such an ODE.
From this point of view, we show in this paper how to adapt the projector-based analysis of linear DAEs introduced in Citation22-24 in order to accommodate critical points. We work with mild smoothness assumptions, driving the results beyond the analytic setting of Citation25,Citation29. Our approach extends the results of Citation30 for index-1 DAEs in standard form, and is directly based on the work on regular points Citation19. Background in this direction is compiled in section 2.2.
Critical points will be classified in section 3 according to a taxonomy which reflects the phenomenon from which the singularity stems. To emphasize that critical points arise at a well-defined step we use the notions of a nice at level k DAE, and admissible up to level k projector sequences. As shown in Theorem 3.3 in section 3.1, the different types of critical points are independent of projectors and invariant with respect to rescaling and linear, time-dependent coordinate changes. In addition, Theorem 3.5 shows that for sufficiently smooth (C m−1) problems, all critical points fall in the types A and B defined in the above-mentioned taxonomy. Several examples in section 3.2 illustrate these notions.
Section 4 focuses on A-critical points; in section 4.1, we discuss working assumptions which, allowing to relax constant rank conditions, still make it possible to construct a chain of continuous matrix functions which particularizes to a tractability chain at regular points. The main result in this section is Theorem 4.3 in section 4.2, which proves that, under these working assumptions, the dynamical behaviour of the DAE can be unveiled through a decoupling based on a scalarly implicit inherent ODE. Again, this is made possible by the proper formulation Equation(2). Our working scenario allows for a uniform over singularities treatment, in problems which include non-isolated critical cases beyond the analytic setting of Citation25,Citation29. The above-mentioned harmless phenomenon is then discussed in section 4.3. Type-B critical points are in the scope of future work.
Finally, this framework is applied in section 5 to the analysis of critical points of the linear, time-varying analogue of Chua's circuit with current-controlled resistors introduced in Citation19.
2 Background
2.1 Modelling semi-state systems via DAEs with properly stated leading term
DAEs have been the focus of increasing interest in the last decades as models for dynamical systems. The differential-algebraic nature stems from the wish to capture explicitly algebraic (non-differential) constraints, or may arise in more subtle ways, as discussed below Citation1,Citation3,Citation4. A classical, rather general model in this context is the quasilinear one
In the last few years, the more general model
Note that linear time-varying systems of the form Equation(1) and Equation(2) arise often as linearizations of Equation(3) and Equation(5), respectively, along trajectories. In particular, the stability properties of periodic trajectories of non-linear DAEs of the form Equation(3) are examined in Citation32,Citation33 via an extension of Floquet theory to periodic linear DAEs in the context defined by Equation(1). In the same direction, an application to the study of non-linear oscillations in electrical circuits using models of the form Equation(2) and Equation(5) can be found in Citation34.
In the following paragraphs, the advantages of Equation(5) and Equation(2) over Equation(3) and Equation(1), respectively, are discussed via applications in circuit theory, adjoint formulations relevant in control engineering, and also in terms of input-output system characterizations; the latter, together with the discussion carried out in section 2.2, clarifies in particular the relation between the modelling approach underlying the proper formulation and the functional characterization of problem solutions.
Circuit theory
Semi-state or differential-algebraic models are nowadays pervasive in circuit simulation programs. On the one hand, a global state-space (ODE) description of a non-linear circuit may not exist in the presence of singularities. On the other hand, allowing certain redundancy between model variables is crucial in order to set up automatically circuit equations: even if a global state equation exists for a non-linear circuit, its formulation can be simply impossible to obtain via a computer program, in contrast to what happens with semi-state models such as those arising in Modified Nodal Analysis (MNA) used in SPICE or TITAN Citation5-8,Citation13-15,Citation31. In the DAE context, the formulation of a state equation can be seen as a reduction problem for semi-state systems Citation12; it is also important to distinguish between hypotheses allowing for the derivation of such a state-space equation and conditions arising in qualitative analyses of circuit dynamics Citation11.
MNA models take the form
Note that system Equation(6) has the properly stated form Equation(5). Only if the characteristics ψ, γ, and φ are smooth (what is not realistic in many real problems) it is possible to rewrite the system in the form Equation(3) via incremental capacitance, conductance, and inductance matrices. Such a rewriting presents additional drawbacks from both numerical and analytical points of view: see Citation20,Citation21 and the input-output discussion below, respectively.
The circuit models Equation(17) and Equation(30) below are LTV instances of Equation(6). As detailed later, critical points in these models will reflect certain pathological circuit configurations depicted at given time instants. Note finally that also distributed systems have been recently framed in the context of PDAEs: see Citation6,Citation13,Citation14 and the bibliography therein.
Adjoint formulations
Adjoint systems Citation35 play an important role both in ODE theory and also in optimal control and engineering applications (see Citation36 as a sample). The extension of adjoint formulations to the DAE context in the setting of Equation(1) finds, however, a fundamental drawback, given by the fact that the adjoint form −(E T(t)y)′ + F T(t)y = p(t) does not fall in the framework of Equation(1). In contrast, the adjoint of Equation(2) is −D T(t)(A T(t)y)′ + B T(t)y = p(t) and has the same structure as Equation(2). This symmetry provides additional advantages regarding fundamental matrix solutions, Lagrange identities, indices and solvability. Details and also optimal control applications can be found in Citation17,Citation37,Citation38 and references therein.
Input-output semi-state system formulations and inverse models
As discussed in this paragraph, the properly stated leading term in Equation(2) is also relevant in providing precise input-output functional descriptions of semi-state systems, and also in the formulation of inverse models: the latter in turn has applications within control theory Citation39.
In the form Equation(1), if E is the identity and F is continuous, a linear explicit ODE results, and C 0 excitations q are well-known to be mapped bijectively onto C 1 solutions x: the system behaves as an input-output smoothing operator. An initial condition x(t 0) = x 0 yields a bijection between the C 0 space of excitations q and the space {x ∈ C 1: x(t 0) = x 0}. An inverse ODE model results, mapping x onto q via x → q = x′ + Fx
Now, addressing DAEs in the classical form Equation(1) under an index-1 assumption, solutions x cannot be guaranteed to be C 1 if the excitation q is just C 0 (see e.g. Citation19,Citation30). Forcing q, E, F to be C 1 is not of great help, since the solution will not be C 2, and in a hypothetical inverse model mapping x ∈ C 1 to q = Ex′ + Fx, q will not be in general C 1. This shows that Ci spaces and the formulation Equation(1) are not the right ones to frame DAEs from an input-output functional perspective. In contrast, a continuous excitation q can be proved (see section 2.2) to yield a solution for the formulation Equation(2) under an index-1 assumption within the space As detailed in Citation19, a bijection and an inverse model x → q = A(Dx)′ + Bx can be then precisely defined. The extension to higher index cases follows from the explicit problem solution depicted in Equation(12), allowed by the specific form of Equation(2).
2.2 Regular points
We compile below a summary of the results of Citation19 needed for our present discussion. The analysis will be focused on DAEs of the form Equation(2); the DAE is said to be properly stated on a given interval if the coefficients A and D satisfy the transversality condition
Define
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Gi has constant rank ri on ;
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(N 0⊕ · · · ⊕N i−1) ∩ Ni = {0} on .
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(b′)QiQj = 0, for all 0 ≤ j < i and all .
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(c) ,
The procedure is then pursued by defining G i+1, etc., If both A and D are invertible on , then system Equation(2) is said to be regular with tractability index zero (on ); otherwise, the DAE Equation(2) is said to be regular with tractability index μ ∈ ℕ on if there exists an admissible projector sequence Q 0, …, Q μ−1, and r μ−1 < r μ = m. The latter identity can be rephrased as the non-singularity of G μ on .
In virtue of the proper statement of the DAE, solutions of a regular system with arbitrarily high index can be computed explicitly in the original setting of the problem, in contrast to other approaches Citation27,Citation29. Indeed, regular index μ DAE decouples into a system of the form Citation19,Citation23:
Now, a point is called regular if there exists an open interval , with , such that the DAE is regular on . It is shown in Citation19 that this notion yields a well-defined, maximal, projector-independent family of pairwise-disjoint regularity intervals, each one with a well-defined index μ i . We denote as the open set of regular points.
The analysis of critical points, where the above regularity notion fails and the solution description Equation(12) is not feasible, is the goal of the present work. We will restrict the analysis of critical points to sufficiently smooth problems, for which the conditions that may fail at every step are just (a) or (b): see Definition 3.2 and Theorem 3.5. The attention will be later focused on situations in which it is the constant rank condition (a) the one that fails, leading to the analysis of A-critical points carried out in section 4.
3 Critical points
Critical points have not been fully analysed in the context of standard-form linear DAEs Equation(1): previous works Citation4,Citation25,Citation29 do not detail how the techniques presented there can accommodate critical points in non-analytic settings. Within the tractability index framework, some results for index-1 systems of the form Equation(1) can be found in Citation30; however, the ability to handle problems with arbitrary index provided by the proper formulation Equation(2) opens a way for the general discussion of critical points presented below.
3.1 Classification
Definition 3.1
Assume that the DAE Equation(2) has continuous coefficients A(t), D(t), B(t). A point t *is said to be critical if there is no regularity interval comprising it.
We will suppose the DAE coefficients to be smooth enough to avoid critical points arising from the failing of condition (c) on page 297; sufficient conditions for this will be given in Theorem 3.5. Our interest is directed to critical points corresponding to failures of the used algebraic constant-rank and transversality conditions, as acknowledged in Definition 3.2. Theorem 3.3 will prove that these notions are actually independent of the (admissible up to level k − 1) choice of projectors.
A continuous matrix function , an interval, has a rank drop at , if each neighbourhood of t * contains points where the rank is different from rk G(t *). Then, t * is called a rank-change or rank-drop point of G.
Definition 3.2
Given the DAE Equation(2) with continuous coefficients, will be said to be a critical point of
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type 0 if G 0 has a rank drop at t *;
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type k‐A, k ≥ 1, if there exists a neighbourhood of t * where the DAE is nice up to level k − 1, but Gk has a rank drop at t * for some (hence any) admissible sequence Q 0, …, Q k−1;
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type k‐B, k ≥ 1, if there exists a neighbourhood of t * where the DAE is nice up to level k − 1 and Gk has constant rank for some (hence any) admissible sequence Q 0, …, Q k−1, but the intersection Nk (t *) ∩ {N 0(t *) ⊕ ⋯ ⊕N k−1(t *)} is non-trivial, for these (hence any other) projectors and Gk .
We will often speak of level-k critical points to refer either to type k‐A or type k‐B. A critical point is of type A or B if it has type k‐A or k‐B with arbitrary k; the terms A-critical and B-critical will also be used for critical points of types A and B, respectively.
A critical point of the DAE Equation(2) is isolated if there is a neighbourhood of t * such that all points different from t * are regular. By definition, an isolated critical point t * of type (k + 1) is the border of two regularity intervals, say and , , . The characteristic values r 0, …, rk apply to both intervals since there is a sequence Q 0, …, Qk that is admissible on . In case of a type (k + 1)-B critical point, G k+1 has uniform rank r k+1 on and . However, there may be different further characteristics μ−, , and μ+, on resp. . In case of a type (k + 1)-A critical point, there are possibly different characteristic values , , too.
Critical points of type 0, that is, rank drops in G 0, may be caused by rank drops in A or D, or in both, but also by failures of the transversality condition Equation(7) for ker A and im D. At those points, D −, R and P 0 are no longer continuous; however, they may have continuous extensions through the critical point. We will focus our interest on cases in which P 0 has a continuous extension.
The taxonomy of critical points presented in Definition 3.3 should be required to be independent of projectors and invariant under linear time-varying coordinate changes x(t) = K(t)y(t) and premultiplication by a continuous matrix function L(t). This is indeed the case, as proved below.
Theorem 3.3
The definitions of critical points of types k-A and k-B are independent of the (admissible) choice of projectors.
Additionally, if t* be a critical point of type k-A or k-B, 1 ≤ k ≤ m, or of type 0, for the DAE Equation(2), then t* is a critical point of the same type for the rescaled, transformed DAE
Proof
Concerning the first assertion, let Q 0, …, Q k−1, and , k ≥ 1, be admissible (up to level k − 1) projector sequences on a given subinterval , and denote as Gi , the corresponding matrices constructed from these sequences. The fact that the notion of a critical point of type k‐A is independent of the choice of the admissible sequence follows from the identity . The latter in turn is due to the identity , with a non-singular factor defined recursively for i ≤ k as , for certain Zij [Citation25, Th. 2.3]. Therefore, rank changes occur exactly at the same points for both matrices Gk and and the result for A-critical points is proved.
Regarding type k‐B critical points, we use the identity to check that . This leads to the identity for all i ≤ k−1, owing to the form of Zi , since and, conversely, The direct sum is supported here on the admissibility (up to level k − 1) of the projector sequences.
Additionally, from the identity and the expression defining Zk , we derive analogously and, conversely, We then get and, therefore, Type-B critical points are defined by the case in which this dimension is no longer zero and this occurs simultaneously for both projector sequences Q 0, …, Q k-1, and
The second assertion follows from the construction of the projectors [Qtilde]i = K −1 QiK Citation23 for Equation(14), which results in the identities [Gtilde]i = LGiK. The rank of Gi is therefore transferred to [Gtilde]i and type 0 and type-A singularities are hence invariant. Additionally, Ñi = ker [Gtilde]i = K −1 Ni , so that the loss of transversality in the Ni spaces defining type-B singularities is also transferred to Ñi .
From the proof of Theorem 3.3 above we immediately get the following result.
Corollary 3.4
With every critical point of type k we may associate a characteristic critical value in an invariant manner, namely:
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rk Gk (t *) for those of type A;
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dim (N 0 ⊕ ⋯ ⊕ N k−1) ∩ Nk (t *) for those of type B.
In sufficiently smooth cases, the only critical points are those of types A and B, as proved below. This means that the types introduced in Definition 3.2 capture the different critical behaviour that can be displayed in linear DAEs with sufficiently smooth coefficients.
Theorem 3.5
Assume that the coefficients A(t), D(t), B(t) in the DAE Equation(2) are C m−1. Then every critical point is of type k-A or k-B, with 1 ≤ k ≤ m, or of type 0.
Proof
Note that, if the DAE is algebraically nice at a given level k ≤ m − 1, then the smoothness requirement (iv) in [Citation19 Definition 5] can be satisfied. This is due to the fact that, in the matrix chain construction, supposed the DAE to be algebraically nice at level 0, we can take Q 0 in the class C m−1, so that G 1 = G 0 + BQ 0 is also C m−1. If neither type 1-A nor type 1-B singularities are met, then we may choose a preadmissible Q 1 in C m−1, so that Q 1 will actually be admissible. Then B 1 and so G 2 will be in the class C m−2.
If no critical points are displayed in subsequent levels, we can continue the sequence in an admissible manner up to G m−1, Q m−1 in C 1, so that continuous B m−1 and Gm can be constructed. Now, if Gm is singular and has constant rank (so that regular points with index m and critical points of type m‐A are also ruled out), then (N 0 ⊕ · · · ⊕ N m−1) ∩ Nm = ℝ m ∩ Nm = Nm ≠ {0} and a critical point of type m‐B is met.
This means that, for sufficiently smooth (meaning in this context C m−1) DAEs, there will be no difficulties in the particular smoothness requirement (c) on page 297. In this situation, a DAE will be nice at a given level if and only if it is algebraically nice at the same level, and if a preadmissible sequence up to a given level exists, then an admissible one exists at the same level. This is implicit in the classification of critical points depicted in .
3.2 Examples
Example 1
As indicated in the Introduction, critical points may well arise in the last step of the chain construction, leading to a singular inherent ODE. This is the case in the present example, where a type 1-A critical point is displayed in an index-1 context. Consider the DAE
Observe that here, for t > 1 and t < 1, the set S 0(t): = {z ∈ ℝ2 : B(t)z ∈ im G 0(t)} (which comprises the solutions of homogeneous index-1 DAEs) reads
The homogeneous inherent ODE Equation(16) with q 1 = 0, q 2 = 0 has the solutions u(t) = (t − 1)2 u(0). The solutions of the homogeneous DAE Equation(15) with q = 0 are then
Example 2
Consider the circuit displayed in , defined by a parallel connection of an independent voltage source, a capacitor and an inductor which are linear time-invariant, and a time-varying current-controlled current source (CCCS).
Modified Node Analysis (MNA) equations read for this circuit
Write in the sequel α(t) = γ(t) − 1, so that α(t) = 0 iff t = 0. Choose , and , so that results. The matrix G 1(t) has constant rank r 1 = 2, and the null space N 1 = ker G 1 is continuous. Compute N 0(t) ∩ N 1(t) = {z ∈ ℝ3 : z 1 = 0, z 2 = 0, α(t)z 3 = 0}, hence N 0(t) ∩ N 1(t) = {0}, for t ≠ 0, N 0(0) ∩ N 1(0) = N 0(0). Therefore, t * = 0 is a critical point of type 1-B.
On ℝ− and ℝ+ we may choose , yielding Q 1 = Q 1 P 0, , , , . It results that G 2(t) is non-singular except at the critical point t * = 0, the problem hence being regular with index 2 in ℝ− and ℝ+.
The systematic analysis of these B-critical points is the focus of future work.
Example 3
The last example in this section attempts to illustrate that critical points of type A do not necessarily yield a singular inherent ODE. This harmless phenomenon will be discussed in more detail in section 4.3. Consider the DAE
On regularity intervals the solution of the DAE is given by the expression
On intervals where α(t) vanishes identically, we simply have , D = [0 0], , G 0 = 0, Q 0 = I, r 0 = 0, G 1 = I, r 1 = 2, μ = 1, that is, the DAE Equation(18) is there regular with index one. Letting Q 1 = 0, P 1 = I, G 2 = G 1, we find DP 0 P 1D − = 0 and Equation(19), Equation(20) are valid also in this case.
If t *∈ (−∞, ∞) is such that α(t *) = 0 but in any neighbourhood of t * there are points t with α(t) ≠ 0, then t * is no longer regular but a type 0 critical point. In the particular case
If we consider Equation(18) with α (t) = t k or with α(t) = t 1/3 instead of Equation(21), formulas Equation(19), Equation(20) result again, and, furthermore, now the characteristics r 0 = 1, r 1 = 1, r 2 = 2, μ = 2 are equal for both (−∞, 0) and (0, ∞). Observe that, in all cases, it holds on regularity intervals that DP 0 P 1 D − = 0, so that there is a trivial smooth extension on the whole interval . Moreover, the function G 2 can be continuously extended on and the extension remains non-singular.
4 A-critical points
4.1 Working assumptions
Generally speaking, the existence of critical points in precludes the construction of a tractability chain defined on the whole . We figure out in this section working assumptions which make possible a ‘uniform over singularities’ treatment, that is, the construction of a globally defined matrix chain which yields a tractability chain at regular points. These assumptions will allow us to unravel the behaviour through a scalarly implicit inherent ODE, as shown in Theorem 4.3 in section 4.2.
The first assumption defines a setting which includes, in particular, problems with isolated critical points. The latter (and therefore Assumption 1) is satisfied by linear DAEs with analytic coefficients and a non-trivial regular set.
Assumption 1
The set of regular points is dense in .
We restrict further the attention to problems with almost uniform characteristic values, defined by the following working hypothesis. Note that it captures situations in which there are rank changes in some of the G i -matrices, as long as there exists a smooth extension of the kernels N i through the critical points (what is again the case for analytic problems Citation29) and these extensions satisfy a transversality property analogous to (b) – (b′). The latter excludes type-B critical points, as proved in Proposition 4.2.
Assumption 2
There exist continuous projector functions Q 0, …, Q m−1 on such that, for 0 ≤ i ≤ m − 1:
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Q i is onto ker G i for all ;
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Q i Q j = 0 for all , 0 ≤ j < i;
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DP 0 ⋯ P i D − is continuously differentiable in , and (DP 0 ⋯ P i D −)′,
D −(DP 0 ⋯ P i D −)′D have continuous extensions on .
Proposition 4.1
Under Assumptions 1 and 2, the DAE has the same characteristic values and, in particular, the same index μ in the whole .
Note that Q μ = … = Q m−1 = 0 and then G μ = … = G m in the whole . Since the projectors Q i realize the ‘regular’ matrix chain on and are well-defined and continuous on the whole interval, we will say that the DAE is almost uniformly regular with index μ on .
Proposition 4.2
Assumptions 1 and 2 rule out type B critical points on .
Proof
Fix . Assume that the DAE is nice up to level k − 1, for some k ≥ 1, and that rk G k is constant in some neighbourhood of t *, so that t * is not a type k-A critical point, and Nk = ker Gk has constant dimension around t *. We need to show that t * cannot be a type k-B critical point, namely, that Nk (t *) and N 0(t *) ⊕ · · · ⊕ N k−1(t *) intersect trivially. From item (ii) in Assumption 2, it follows that im Q k (t) is transversal to N 0 (t) ⊕ · · · ⊕ N k−1(t) for all t in some neighbourhood of t *; it then suffices to show that im Q k (t *) = N k (t *). On the one hand, both spaces have the same dimension due to the continuity of Q k and the constant dimension of N k ; on the other hand, from the vanishing of the continuous product G k Q k in the dense set , it follows that G k (t *) Q k (t *) = 0, so that im Q k (t *) ⫅ ker G k (t *) = N k (t *). This proves that type k-B critical points are precluded by Assumptions 1 and 2, for 1 ≤ k ≤ m.
We will hence call the matrix chain G i constructed under Assumptions 1 and 2 an A-critical chain. This provides a setting beyond just μ-A critical points; note that these last step critical points do not put in question the matrix chain construction (and the regular framework could be essentially used), in contrast to the ones in previous levels.
4.2 Decoupling
The importance of Assumptions 1 and 2 above rely on the fact that they allow one to extend a tractability chain through critical points, making it possible to describe the behaviour of the properly stated DAE through a scalarly implicit decoupling which generalizes Equation(12). This extends Theorem 3 of Citation30, which holds for index-1 problems in the classical form Equation(1).
Theorem 4.3
Let Assumptions 1 and 2 hold. Then,, , is a solution of the DAE Equation (2) if and only if it can be written as
Proof
Let be a solution of Equation(2) on some subinterval . Since the identity of the continuous matrix maps AR = A holds in the dense subset , it remains true in the whole . We may then write A = ADD − in Equation(2). Premultiplying by and using
If we multiply Equation(26) by Q μ−1, then by P 0 ⋯ P μ−2 if μ ≥ 2, and in turn by Q 0 P 1 ⋯ P μ−1 and P 0 ⋯ P k−1 Q k P k+1 ⋯ P μ−1, we get the relations depicted in Equation(24) for v 0 = Q 0 x, v i = P 0 ⋯ P i−1 Q i x, i = 1, …, μ−1.
The local invariance of the space im DP 0 ⋯ P μ−1 owes to the fact that α = (I−DP 0 ⋯ P μ−1 D −)u satisfies the homogeneous equation ω μ [α′ + (DP 0 ⋯ P μ−1 D −)′α] = 0 on . Since ω μ ≠ 0 on a dense set, it follows that α′ + (DP 0 ⋯ P μ−1 D −)′α = 0 on , and therefore a vanishing initial condition for α (in virtue of u = DP 0 ⋯ P μ−1 D − u) yields a trivial solution in the whole .
Conversely, we need to show that, provided that u, v 0, …, v μ−1 satisfy Equation(23) – Equation(24) in some subinterval , then x = D − u + v 0 + ⋯ + v μ−1 solves the DAE Equation(2) on . From the smoothness properties of u, v 0, …, v μ−1 it follows that , and additionally we know the relation A(Dx)′ + Bx−q = 0 to be satisfied on . Now, A(Dx)′ + Bx − q is continuous and, again by density of , the identity A(Dx)′ + Bx − q = 0 must hold in the whole , showing that x actually solves Equation(2).
Under Assumptions 1 and 2, the behaviour of a DAE with critical points is therefore addressed in terms of the scalarly implicit decoupling Equation(23) – Equation(24). This way, the analysis of a critical DAE is driven to the singular ODE setting; this is analogous to the approach carried out in the reduction approach of Rabier and Rheinboldt Citation29 and also in the framework of the strangeness index by Ilchmann and Mehrmann Citation25. Note that in those works the results hold only for analytic problems.
Note, however, that not necessarily ω μ vanishes at all critical points. This reflects that critical points, defined in terms of the failing of the regularity assumptions in the chain construction, do not exclude cases with unique solvability properties. A well-known example is Equation(29) below. The same phenomenon is acknowledged in the above-mentioned approaches Citation25,Citation29. A complete formal characterization of these ‘harmless’ critical points is currently an open problem. Proposition 4.4 below provides a result in this direction. An important consequence of this proposition is that this type of harmless critical points cannot follow from rank deficiencies in the leading matrix in almost uniformly index-1 problems such as those of Citation30. Obviously, a type μ-A critical point will always yield a zero in ω μ and therefore can never be harmless.
4.3 Harmless A-critical chains
As shown in section 4.2, the behaviour of critical DAEs, under Assumptions 1 and 2, can be unveiled through a scalarly implicit decoupling. The leading coefficient ω μ will actually vanish iff G μ is singular. Therefore we might speak of ‘harmless’ critical points t * in A-critical chains if G μ (t *) is non-singular. Obviously, μ-A critical points are never harmless. But this is not always the case for lower level A critical points. Consider the DAE
It therefore seems to be an interesting problem to check when type k-A critical points are lifted to the (k + 1)-level or, more generally, how rank deficiencies overlap or accumulate at different levels, for every critical point t * (see ). A result in this direction follows.
Proposition 4.4
Under Assumptions 1 and 2, a type (μ − 1)-A critical point t* leads to a singular G μ (t*).
Proof
Note that the identity G μ−1 Q μ−1 = 0 holds in the whole by continuity and due to the fact that it holds in the dense subset . Since G μ−1 undergoes a rank drop at t *, and Q μ−1 has constant rank by assumption, it follows that im Q μ−1(t *) is strictly contained in ker G μ−1(t *). Equivalently, there exists a non-trivial vector z ∈ ker G μ−1(t *) − im Q μ−1(t *). Because of this, it is
Corollary 4.5
A necessary condition for t* to be harmless is that G μ−1 has constant rank in some neighbourhood of t* .
5 Critical points of a linear time-varying Chua circuit
Following Citation19, we consider the linear time varying analogue of Chua's circuit Citation40 with current-controlled resistors depicted in . The framework presented in previous sections will make it possible to classify the critical points arising in the DAE model of this circuit; in particular, the harmless nature of certain type-0 critical points owing to the vanishing of the values of reactances will be discussed in an index-2 context.
Due to the current-control assumption and the eventual vanishing of R 1(t) and R 2(t), resistors' currents appear as variables in the Modified Nodal Analysis (MNA) model
We assume in the sequel that the resistor R 1 verifies R 1(t) > 0 for t < 0, and R 1(t) = 0 for t ≥ 0. This models a persistent short-circuit in the interval [0, ∞); note that the model Equation(30) is valid in the whole real line. Below we consider critical points due to the vanishing of R 2, C 1, C 2 or L in both subintervals.
R 1(t) > 0. In Citation19, it is shown that the simultaneous non-vanishing at a given t of R 1, R 2, C 1, C 2 and L defines t as a regular index-1 point. Assume that the non-vanishing of these values holds true in some open dense subset of ℝ− (where R 1 > 0), and let us analyse the effect of the vanishing of R 2, C 1, C 2 or L. To this end, write
Due to the asserted exclusion of harmless critical points in these almost uniform index-1 problems, the vanishing at a given t of any of the values C 1, C 2, L or R 2 is expected to yield a singularity in the scalarly implicit decoupling of the DAE. This is indeed the case, since the scalarly implicit inherent ODE Equation(23) can be checked to read
R 1(t) > 0. Let us now consider the behaviour in ℝ+ ∪ {0}, where R 1 does vanish. Assume that the conditions R 2 ≠ 0, 0 ≠ C 1 ≠ −C 2 ≠ 0, L ≠ 0 hold in some dense subset of ℝ+, so that the DAE has index-2 there, according to Citation19. Let us again consider the effect of the vanishing of some of these values at certain points. We assume that no more than one of the values of R 2, C 1, C 2 or L vanishes at a given t.
Looking at Equation(31), the vanishing of C 1, C 2 or L yields again a critical point of type 0. Take P 0 and Q 0 as in Equation(33), so that
Assuming C 1 + C 2 ≠ 0, we may take
Acknowledgements
Research supported by the DFG Forschungszentrum Mathematics for Key Technologies (MATHEON) in Berlin. The second author (corresponding author) acknowledges additional support from Vicerrectorado de Investigación, Universidad Politécnica de Madrid; his work is framed in Projects MTM2004-5316 and MTM2005-3894 of Ministerio de Educación y Ciencia, Spain.
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Appendix: The coefficients of Equation(24)
For k = 1, …, μ − 1, j = k + 2, …, μ − 1: