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Periodic solutions of semi-explicit differential-algebraic equations with time-dependent constraints
Boundary Value Problems volume 2014, Article number: 179 (2014)
Abstract
In this paper we investigate the properties of the set of T-periodic solutions of semi-explicit parametrized differential-algebraic equations with non-autonomous constraints of a particular type. We provide simple, degree-theoretic conditions for the existence of branches of T-periodic solutions of the considered equations. Our approach is based on topological arguments as regards differential equations on implicitly defined manifolds, combined with elementary facts of matrix analysis.
MSC: 34A09, 34C25, 34C40.
1 Introduction
Several mathematical models arising from physical and engineering problems can be described in terms of differential-algebraic equations (DAEs). Because of this, in recent years, there has been a lot of interest on these equations from both the point of view of pure and of applied mathematicians. Beside the more genuinely modelistic or numerical approaches, there are many books and papers that treat DAEs from an analytical perspective. Of all those, in order to avoid an impossibly long and necessarily incomplete list, we only mention [1]–[3] and references therein.
A relevant case is represented by first-order semi-explicit DAEs in Hessenberg form (see, e.g., [1]) that is,
where is a continuous map, and is sufficiently smooth. If we assume that the partial derivative, , of with respect to the third variable y is invertible, then (1.1) is said to be of index 1.
In this paper we are concerned with a parametrized special case of (1.1). In fact, we assume that the constraint has the form
where is , and the square-matrix-valued maps and are continuous. Here denotes the group of orthogonal matrices and the group of invertible ones.
Namely, we consider parametrized DAEs of the following form:
with f as in (1.1), and we assume that is invertible for all and (for technical reasons) that A is of class .
We also treat, in parallel, the following parametrized second-order DAEs:
In this case we assume the matrix-valued maps A and B to be of class and , respectively. The latter type of equations, in particular, may be used to represent some nontrivial physical systems as, for instance, constrained systems (see e.g.[2]).
We will assume throughout the paper that the matrix-valued function A satisfies the following property:
This assumption might seem unnatural, but it is not so. To understand why, consider the case when . In that case, if is a fixed reference frame in and is a moving frame, our assumption is equivalent to imposing the requirement that the angular velocity of is the zero vector. This is, in fact, an immediate consequence of the definition of angular velocity. An entirely similar statement holds for .
Furthermore, in this paper we will always assume that, for some given , the map f is T-periodic in the first variable and that A and B are periodic of the same period T. Following the approach of [4]–[9], we study qualitative properties of the set of T-periodic solutions of (1.2) and (1.3). Roughly speaking, we show the existence of an unbounded connected component of ‘nontrivial’ T-periodic solutions of (1.2) or (1.3) emanating from the set of the ‘trivial’ ones. In this setting, a trivial solution is a solution which is constant with respect to the moving frame defined by the time-dependent change of variable . However, precise statements will be given in Section 3.1 for first-order equations and in Section 3.2 for second-order ones. We also show, through some examples and remarks, how our constructions can be extended to include several equations of different forms.
Our continuation results are in the spirit of analogous ones by Furi and Pera for parametrized first- and second-order equations on differentiable manifolds (for more details see the survey [10]) and could be considered, in some sense, as consequences of recent results obtained by the last two authors in [6]–[9]. However, we wish to point out the following facts. First of all, while the continuation results on differentiable manifolds by Furi and Pera require the knowledge of the degree (often called characteristic or rotation) of suitable tangent vector fields, here (as in [6]–[9]) we give conditions only in terms of the well-known Brouwer degree, which is also easier to compute explicitly. On the other hand, in the present paper we tackle the case of time-dependent constraints (even if of a peculiar form). In other words, our results can be regarded as concerning ODEs on particular T-periodically moving manifolds defined implicitly. As already pointed out, we slightly modify and adapt to the present context the concept of ‘trivial’ and ‘nontrivial’ T-periodic solution. As far as we know, the techniques of Furi and Pera have never been applied to moving manifolds, and this novelty is our main original contribution to the subject.
This paper is organized as follows. In Section 2 we collect the preliminaries needed to approach the DAEs in (1.2) and (1.3). In Section 3 we give our main results and we get topological information on the set of T-periodic pairs to the considered equations; examples of applications of our methods are provided. Finally, in the Appendix, we give the proofs of the technical results of matrix analysis used throughout the paper.
2 Notation and preliminary results
Throughout the paper, will denote the Banach space of all the T-periodic continuous maps with the usual supremum norm, and will be the Banach space of all the T-periodic maps with the norm.
We will make use of the (extended) theory of the Brouwer degree for maps between open sets of . Namely, we say that a triple , with and F a proper map defined in some neighborhood of the open set , is admissible if is compact. For any admissible triple , the Brouwer degree of B in U respect to z is an integer that, roughly speaking, counts algebraically the elements of which lie in U. See e.g.[11] for a broader definition in the more general case of maps between oriented manifolds, or [12] for a quick introduction. Since in this paper the target point z will always be the origin, for the sake of simplicity, we will always omit it and write instead of .
2.1 First-order DAEs
Let us consider semi-explicit DAEs, depending on a parameter , of the following forms:
and
where we assume that and are continuous maps, is T-periodic in the first variable, and is and such that is invertible for all . Notice that, consequently, is a closed submanifold of . Furthermore observe that, even if (2.2) can be considered as a particular case of (2.1) (i.e. with identically), for our purposes the two equations need to be treated separately.
Given , by a solution of (2.1) we mean a pair of functions x and y defined on an interval I with the property that the following equalities hold for all : and . The notion of solution of (2.2) is analogous. Notice that one might wish to ask only the continuity of y. In fact, if x is , the assumptions on together with the implicit function theorem imply that y is .
In this section we recall two results from [6], [7] and [9] (see also [4], [5] for more general results) about the sets of T-pairs of (2.1) and of (2.2), namely, of those pairs with a T-periodic solution of (2.1) and of (2.2), respectively. Recall that a T-pair of (2.1) or of (2.2) is said to be trivial if and is constant. Observe that any T-pair of (2.2) of the form is necessarily trivial, because all solutions of (2.2) corresponding to are constant (because is nonsingular). However, the same statement is not true for (2.1) as shown by the following trivial example with and :
For the sake of simplicity we make some conventions. We will regard every space as its image in the following diagram of natural inclusions:
In particular, we will identify with its image in under the embedding which associates to any the map constantly equal to . Moreover, we will regard as the slice and, analogously, as . We point out that the images of the above inclusions are closed.
For simplicity, given , we will denote by the set consisting of all pairs such that .
The following is a consequence of Theorem 5.1 in [9].
Theorem 2.1
Let, , be as above. Defineby
Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (2.1) whose closure inmeetsand cannot be both bounded and contained in Ω.
Sketch of the proof
Taking in [9], Theorem 5.1] we see that there exists a connected set G of nontrivial T-pairs for (2.1) whose closure in meets , which, according to our identification in (2.3), means and is not contained in any compact subset of Ω. Let Γ be the connected component of the set of nontrivial T-pairs for (2.1) that contains G. Observe that since is a complete metric space, the theorem of Ascoli-Arzelà implies that any bounded set of T-pairs for (2.1) is actually relatively compact. Hence, if Γ is bounded then it is compact (being closed), but G cannot be contained in a compact subset of Ω. This implies that Γ cannot be both compact and contained in Ω. □
The same argument of the above proof shows that the following is a consequence of Theorem 2.2 in [6].
Theorem 2.2
Letandbe as above. Defineby
Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (2.2) whose closure inmeetsand cannot be both bounded and contained in Ω.
2.2 Second-order DAEs
Consider the following second-order parametrized DAEs:
and
where we assume that and are continuous maps, is T-periodic in the first variable, and is and such that is invertible for all .
Given , by a solution of (2.4) we mean a pair of functions x and y defined on an interval I with the property that the following equalities hold for all : and . Notice that, as in the first-order case, it is equivalent to ask only the continuity of y.
The structure of the set of solution pairs of (2.4) and of (2.5) has been studied in [8]. As in Section 2.1, we recall that by a T-pair of (2.4) and of (2.5) we mean a pair with a T-periodic solution of (2.4) and of (2.5), respectively. Again, a T-pair of (2.4) or of (2.5) is said to be trivial if and is constant. Unlike in the first-order case, it is not necessarily true that any T-pair of (2.5) of the form is trivial. In fact, if were a closed geodesics of the manifold with appropriate initial velocity then would be a nontrivial T-pair of (2.5). Similarly, it is not necessarily true that any T-pair of (2.4) of the form is trivial.
As in Section 2.1, for simplicity we will regard every space as its image in the following diagram of natural inclusions:
with the obvious analogous identifications.
Again, given , we will denote by the set consisting of all pairs such that .
The next results are straightforward consequences of Corollary 5.2 and Corollary 5.3 in [8], respectively (the arguments are similar to the proof of Theorem 2.1).
Theorem 2.3
Let, , be as above. Defineby
where. Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (2.4) whose closure inmeetsand cannot be both bounded and contained in Ω.
Theorem 2.4
Letandbe as above. Defineby
where. Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (2.5) whose closure inmeetsand cannot be both bounded and contained in Ω.
Remark 2.5
Common manipulations employed for order reduction of differential equations, when applied to (2.4) or (2.5) might not work for deducing Theorems 2.3 or 2.4 directly from Theorems 2.1 or 2.2. In fact, those procedures usually lead to equations whose form is not suited for our first-order results. Thus, despite the similar structure of (2.4) and (2.5) to (2.1) and (2.2), respectively, the former equations seem to need a specific study.
3 Coordinate transformation and main results
3.1 First-order DAEs
We first investigate parametrized DAEs of the following form:
where, as in the introduction, the map is continuous and T-periodic in the first variable, is and such that is invertible for all , and and are T-periodic continuous (square-)matrix-valued maps. We will assume that A is of class . The constraint that appears in (3.1) forces the motion to occur on a manifold that moves integrally to a reference frame defined by the time-dependent change of variable . As before, we consider the set of T-pairs of (3.1), namely, of those pairs with a T-periodic solution of (3.1). A T-pair of (3.1) is trivial if and is constant. In other words, is trivial if is constant with respect to the moving reference above, i.e., to the moving manifold that constitutes the constraint. Thus, loosely speaking, we could say that in this case is constant with respect to the ‘moving constraint’. Observe that when is trivial, then must be constantly equal to and similarly , so that for all t we have and . The former fact has an interesting consequence for x. Since , must be constant. Thus, when is trivial, we have , i.e., is invariant for for all t’s.
Let us apply, for all t, a change of coordinates in :
Let us rewrite the first of these two equations as . Differentiating with respect to t we get
Observe, in fact, that the operations of differentiation and transposition commute; that is,
From (3.3) we get . Thus, (3.1) can be rewritten in the new coordinates as follows:
where is defined by
If we assume that the matrix is constant, then we can obtain continuation results for T-pairs of (3.1) as consequences of the results in the previous section.
In the following we will adopt the same notation as in Section 2.1.
Theorem 3.1
Let f, g, A, and B be as above. Assume thatis constant and defineby. Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (3.1) whose closure inmeetsand cannot be both bounded and contained in Ω.
Proof
Consider the transformation (3.2). As discussed above, in the new coordinates ξ, η, (3.1) becomes (3.4), which we write as
where F is defined as in (3.5). Consider also the homeomorphism given by with ξ and η given by (3.2). Clearly ℌ establishes a homeomorphism between the space X of T-pairs of (3.1) and the space of T-pairs of (3.6), which preserves triviality. In the sense that ℌ takes trivial T-pairs of (3.1) to trivial ones of (3.6) and, vice versa, makes trivial T-pairs of (3.6) correspond to trivial ones of (3.1).
Let . Applying Theorem 2.1 we get the existence of a connected component, let us say ϒ, of nontrivial T-pairs for (3.6) whose closure in meets and cannot be both bounded and contained in . One sees immediately that has the required properties. □
In the following consequence of Theorem 3.1 we further assume that M is nonsingular and use the properties of the Brouwer degree to get a continuation result with the sole assumption that is a nonempty and compact subset of .
Corollary 3.2
Let f, g, A, and B be as above. Assume thatis constant and nonsingular. Letbe open. Assume that the setis nonempty and compact. Then there exists a connected component Γ of nontrivial T-pairs for (3.1) whose closure inmeetsand cannot be both bounded and contained in Ω.
Proof
Let ℱ be as in the assertion of Theorem 3.1. Since the first component of ℱ is nonsingular, the reduction property of the Brouwer degree implies
Observe now that since is never singular,
which is finite and nonzero. □
In the next result we assume and apply Theorem 2.2.
Theorem 3.3
Let f, g, A, and B be as above. Assume thatis identically zero and defineby
Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (3.1) whose closure inmeetsand cannot be both bounded and contained in Ω.
Proof
It follows from Theorem 2.2 with the same proof as of Theorem 3.1. □
Example 3.4
Take and . Let be any continuous mapping 2π-periodic in the first variable. Consider
where . It is readily verified that
where
Thus, the constraint can be regarded as the surface obeying the equation , in the space , revolving around the q axis (a full rotation takes time 2π). With the transformation (3.2) the above DAE becomes
where
Let . Since M is nonsingular and , Corollary 3.2 yields an unbounded connected component of nontrivial 2π-pairs for (3.7) that meets (regarded as a 2π-pair).
Remark 3.5
Notice that a similar coordinate transformation applies also to a slightly different situation. Consider the following DAE:
where A, B, f, and g are as in (3.1) and H is a matrix that commutes with A. Suppose, as above, that is constant (not necessarily invertible) and apply the transformation as indicated above. Equation (3.8) becomes
with F as in (3.4), so that the results of the previous section are applicable to (3.9).
Example 3.6
Consider the following DAE:
where , , are continuous mappings 2π-periodic in the first variable. If we put , (3.10) is of the form (3.8) with
and and defined by
respectively. Clearly, as in Remark 3.5, (3.10) becomes
where
Equation (3.11) is of the form considered in Corollary 3.2.
In our next example we consider periodic perturbations of a class of semi-linear DAEs (semi-linear DAEs find practical applications in robotics and electrical circuit modeling see e.g.[1], [13]). We will restrict ourselves to the case when the equation has a particular ‘separated variables’ form, that is,
, and are continuous maps, and is a constant matrix (here denotes the set of real matrices). Further, we assume that F and C are T-periodic, given.
Example 3.7
Consider (3.12) with and
and
The orthogonal matrices
realize a singular value decomposition for E. In particular, we have
Then, setting with and multiplying (3.12) by on the left, we can rewrite (3.12) as
that is,
where we have set . This equation can be rewritten as follows:
or, in our case, as
where we have put and . The above DAE is of the form (3.1) considered in Theorem 3.3. Observe that the map ω considered there is given by
The example considered above is a particular case of a more general procedure that we now roughly sketch. Take E, F, C, and S be as in (3.12), and let . Assume that , and that
Let P, Q be orthogonal matrices realizing a singular value decomposition for E. Multiply (3.12) by on the left, and put with . We get, as in Example 3.7,
Since P and Q realize a singular value decomposition of E, and since E, F, and C satisfy (3.13a) and (3.13b), an inspection of the proof of [14], Lemma 5.5] (see also [15]) shows us that for all t,
Set and . Then we can rewrite (3.14) as
or, equivalently
and, if is invertible for all t,
which is of type (3.1) with if also is invertible for all t.
3.2 Second-order DAEs
Let us now focus on parametrized second-order DAEs and proceed as in the first-order case. Consider
where is continuous and T-periodic in the first variable, is and such that is invertible for all , and the T-periodic matrix-valued maps and are of class and , respectively.
As in the first-order case, we consider the set of T-pairs of (3.16), namely, of those pairs with a T-periodic solution of (3.16). A T-pair of (3.16) is trivial if and is constant. In other words, is trivial if is constant with respect to the moving constraint. Again, if is trivial, then must be constantly equal to . Also, , so that the periodicity condition implies that must be constant as well. Thus, for all t, is invariant for .
Let us consider the following change of coordinates for all t:
We can rewrite the first of these equations as and, taking the derivative, we get
Let us multiply by A on the left the second of these equations. Reordering (and omitting the explicit dependence on t) we get
Moreover, since ,
Thus we can rewrite our DAE, in the new coordinates, as follows:
where , defined by
is clearly continuous and T-periodic.
Now, by Proposition A.3 (see the Appendix), we find that if is constant (and nonsingular), then is constant (and nonsingular) as well, as it is equal to . Thus, as for first-order equations, provided that is constant, this DAE can be treated with the methods of the previous section.
It is also worth noticing that , which appears in the expression of F, can also be conveniently expressed as . This trivial fact is readily established by differentiating the relation .
Proceeding as in the previous subsection, and using Theorems 2.3 and 2.4 in place of Theorems 2.1 and 2.2, we get the following results, remarkably similar to Theorems 3.1 and 3.3, and Corollary 3.2.
Theorem 3.8
Let f, g, A, and B be as above. Assume thatis constant and defineby. Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (3.16) whose closure inmeetsand cannot be both bounded and contained in Ω.
Corollary 3.9
Let f, g, A, and B be as above. Assume thatis constant and nonsingular. Letbe open. Assume that the setis nonempty and compact. Then there exists a connected component Γ of nontrivial T-pairs for (3.16) whose closure inmeetsand cannot be both bounded and contained in Ω.
Theorem 3.10
Let f, g, A, and B be as above. Assume thatis identically zero and defineby
Letbe open and assume thatis well defined and nonzero. Then there exists a connected component Γ of nontrivial T-pairs for (3.16) whose closure inmeetsand cannot be both bounded and contained in Ω.
In the next example we consider the same time-dependent constraint as in Example 3.4, but in the case of second-order DAEs.
Example 3.11
Let be any continuous mapping 2π-periodic in the first variable. Consider
where . Let A and g be as in Example 3.4. Applying the coordinate transformation as described above we rewrite our DAE as follows:
Here
so that . Let . Since M is nonsingular and , Corollary 3.9 yields an unbounded connected component of 2π-periodic pairs emanating from (regarded as a 2π-pair).
Remark 3.12
As in the first-order case, our coordinate transformation applies also to a slightly different situation. Consider the following DAE:
where A, B, f, and g are as in (3.16) and , , are matrices that commute with A. Suppose, as above, that is constant (not necessarily invertible) and apply the transformation as indicated above. Equation (3.8) becomes
with F as in (3.17), so that the results of Section 2.2 are applicable to (3.19).
Remark 3.13
Let us consider the following second-order DAE
where f, A, and B are as in (3.16) and is and T-periodic. We also assume that C has the same property as A, that is, is constant. Expanding the derivative on the left-hand side of the first equation in (3.20) and using the fact that , for all , we rewrite (3.20) as follows:
where, to keep the notation concise, the explicit dependence on t of A, B, and C is omitted. Proposition A.3 shows that is constant and, by Remark A.4(3), it follows that is constant as well being equal to . Hence, (3.21) is of the form (3.18). Notice that if we assume that A commutes with , then it commutes with as well. In conclusion, if commutes with for all , Remark 3.12 applies with and .
Appendix: some lemmas of Matrix Analysis
This section gathers, for reference purposes, a few simple facts - possibly well known - concerning time-dependent matrices.
Lemma A.1
Letbe asquare-matrix-valued function. Suppose that the mapis constant. Then
Proof
For the sake of simplicity, we drop the explicit indication of the dependence of A on t.
Let us put . Then is also constant. Taking the derivative with respect to t of both these relations, we get
Hence,
which implies (A.1).
From (A.3) and (A.1) it follows that
whence the assertion. □
Observe that under the hypothesis of Lemma A.1, since
(A.1) implies the symmetry of .
Lemma A.2
Letbe asquare-matrix-valued function. Assume thatis orthogonal for all t, then.
Proof
Differentiating the relation we obtain . Multiplying this relation on the left by and on the right by , we get
Since and , transposing yields
as desired. □
Equation (A.2) and Lemma A.2 together yield the following fact.
Proposition A.3
Letbe asquare-matrix-valued function. Assume thatis orthogonal for all t and that the mapis constant. Thenis constantly equal to. In particular, ifis constant and nonsingular then so is.
Remark A.4
Replacing A with , it is easy to verify that results analogous to Lemma A.1, Lemma A.2 and Proposition A.3 hold if we assume the constancy of instead of that of . Namely, if is a square-matrix-valued function such that is orthogonal for all t and the map is a constant, then
-
(1)
and ;
-
(2)
;
-
(3)
.
These facts should not surprise us in view of Proposition A.5 below.
We conclude this technical section with a curious remark. As shown by the following example:
even for matrix functions as in Proposition A.3, one may have
Nevertheless, one can prove the following fact.
Proposition A.5
Letbe asquare-matrix-valued function. Assume thatis orthogonal for all t. Thenis constant if and only if so is.
Proof
Let us first prove that if is constant then is constant as well. As above, for the sake of simplicity, we drop the explicit indication of the dependence of A on t.
Clearly, we have and, since Proposition A.3 yields , we also have . Now, using these facts we get
Observe also that
because . Thus, , which implies that is a constant matrix.
Conversely, if is constant, and a similar proof shows that is constant too. □
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Acknowledgements
The authors thank both anonymous referees for their careful reading of the manuscript and many precious suggestions.
The authors have been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Bisconti, L., Calamai, A. & Spadini, M. Periodic solutions of semi-explicit differential-algebraic equations with time-dependent constraints. Bound Value Probl 2014, 179 (2014). https://doi.org/10.1186/s13661-014-0179-2
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DOI: https://doi.org/10.1186/s13661-014-0179-2