Impulse-free interval-stabilization of switched differential algebraic equations

In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory. © 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
In this paper we consider switched differential algebraic equations (switched DAEs) of the following form: where σ : R → N is the switching signal and E p , A p ∈ R n×n , B p ∈ R n×m , for p, n, m ∈ N. In general, trajectories of switched DAEs exhibit jumps (or even impulses), which may exclude classical solutions from existence. Therefore, we adopt the piecewise-smooth distributional solution framework introduced in [1]. We study impulse-free stabilizability of (1) where impulsefree stabilizability means the ability to find for each initial value a control signal such the state converges towards zero and remains impulse free (see the forthcoming Definition 9).
Differential algebraic equations (DAEs) arise naturally when modeling physical systems with certain algebraic constraints on the state variables. Examples of applications of DAEs in electrical circuits (with distributional solutions) can be found in e.g. [2]. The algebraic constraints are often eliminated such that the system is described by ordinary differential equations (ODEs). In the case that a system undergoes abrupt structural changes a switched DAE is obtained. Examples of applications are electronic circuits ✩ This work was supported by the NWO Vidi-grant 639.032.733. * Corresponding author.
containing switches or mechanical systems with component failure. Since each mode generally has different algebraic constraints, there does not exist a switched ODE description of the system with a common state variable. This problem can be overcome by studying switched DAEs directly.
Several structural properties of switched DAEs have been studied recently. Among those are null-controllability [3], stability [4], stabilizability [5] and observability [6]. All of these studies allow for Dirac impulses in the state trajectory, whereas for some applications Dirac impulses are undesirable. Examples of such applications are electrical circuits containing switches where Dirac impulses in the voltage can damage components or cause electric sparks to occur at the switch [7]. Furthermore, in the case components need to be replaced, e.g. for maintenance reasons, or if new components are attached to an operational electrical circuit, some state components might, for safety reasons, be required to be stabilized in a impulse-free fashion. Impulse-controllability of switched DAEs, i.e., the ability to avoid Dirac impulses in the state trajectory by means on an input, has been studied in [8]. However, switched systems that are both impulse-controllable and stabilizable are not necessarily stabilizable with an impulse-free trajectory, as is shown in the following example.
Consider the electrical circuit given in Fig. 1. For maintenance reasons the capacitor and the component consisting of the operational amplifier combined with an inductor are disconnected at t = t 1 . In order to keep the network to which this circuit is connected running, the voltage source V 0 needs to remain constant. However, there is another controllable voltage source u available.  An example of an electrical circuit that is stabilizable and impulse-controllable, but not stabilizable without Dirac impulses.
Since the system is operational at t = t 0 it is assumed that the state at t 0 is consistent. Defining the state as x = [ V L I L V C ,I C ,V 0 ], we obtain that for t ∈ [t 0 , t 1 ) the system is described by Eq. (2), whereas for t ∈ [t 1 , ∞) it is described by Eq. (3).
The current through the resistors is given by I R = u/R and hence RI L = u. After opening the switch, the voltage over the resistor is zero and thus I R = I L = 0. Hence for a non-zero input u at t − 1 , we obtain that I L jumps to zero at t + 1 and consequently a Dirac impulse occurs in V L = Lİ L . However, if the input is brought to zero smoothly, no Dirac impulses occur and hence the system is impulse-controllable. However, since the amount of charge stored on the capacitor is given by q = C (V 0 − u), we have for a nonzero u at t 1 , that the capacitor is charged and is unable to discharge, since the current I C = 0. The capacitor can be discharged before t 1 , but that requires a nonzero u at t − 1 , which produces a Dirac impulse yet stabilizes the state of the components. Hence we have an example of a system which is impulse controllable and stabilizable, but not stabilizable with an impulse free trajectory.
Motivated by this example, this paper considers stabilization of switched DAEs where Dirac impulses are to be avoided, so called impulse-free stabilization.
The outline of the paper is as follows: notations and results for non-switched DAEs are presented in Section 2. In Section 3 the definition of impulse-free stabilizability is given, together with the introduction of the concept of interval-stabilizability. Results on impulse controllability and (impulse-free) stabilizability are given in Section 4. Finally, the paper ends with a conclusion in Section 5.
Notation The sets of natural, real and complex numbers are denoted by N, R and C, respectively, C + = {λ ∈ C | ℜ(λ) ⩾ 0} denotes the closed right-half complex plane. For a vector x ∈ R n , |x| is its Euclidean norm; e i ∈ R n is the vector of all zeros except for a one in position i. Let X , Y ⊆ R n be vector spaces and consider the linear map A : X → Y. Then, im A = {Ax | x ∈ X } and ker A = {x ∈ X | Ax = 0}, respectively. The inverse image of a subspace V ⊆ Y is given as

Properties and definitions for regular matrix pairs
In the following, we consider regular matrix pairs (E, A), i.e.
for which the polynomial det(sE − A) is not the zero polynomial.
Recall the following result on the quasi-Weierstrass form [9].
× R n×n is regular if, and only if, there exist invertible matrices S, T ∈ R n×n such that , where J ∈ R n 1 ×n 1 , 0 ⩽ n 1 ⩽ n, is some matrix and N ∈ R n 2 ×n 2 , n 2 := n − n 1 , is a nilpotent matrix.
The matrices S and T can be calculated by using the so-called Wong sequences [9,10]: The Wong sequences are nested and get stationary after finitely many iterations. The limiting subspaces are defined as follows: For any full rank matrices V , W with im V = V * and im W = W * , the matrices T := [V , W ] and S := [EV , AW ] −1 are invertible and (4) holds.
Based on the Wong sequences we define the following projectors and selectors.

Definition 2.
Consider the regular matrix pair (E, A) with corresponding quasi-Weierstrass form (4). The consistency projector of (E, A) is given by the differential selector and impulse selector are given by respectively.
In all three cases the block structure corresponds to the block structure of the quasi-Weierstrass form. Furthermore we define Note that all the above defined matrices do not depend on the specifically chosen transformation matrices S and T ; they are uniquely determined by the original regular matrix pair (E, A). An important feature for DAEs is the so called consistency space, defined as follows: and the augmented consistency space is defined as In order to express (augmented) consistency spaces in terms of the Wong limits we introduce the following notation for matrices

Distributional solutions
The  It can be shown (see e.g. [12]) that the space D pwC ∞ can be equipped with a multiplication, in particular, the multiplication of a piecewise-constant function with a piecewise-smooth distribution is well defined and the switched DAE (1) can be interpreted as an equation within the space of piecewise-smooth distributions. Hence the following solution behavior (depending on σ ) is well defined: and restrictions of x and u to intervals, are well defined as well.
Given the notation x I for the restriction of x to the interval I ⊆ R, it is shown in [1] that the initial trajectory problem (ITP) (7) has a unique solution for any initial trajectory if, and only if, the matrix pair (E, A) is regular. As a direct consequence, the switched DAE (1) with regular matrix pairs is also uniquely solvable (with piecewise-smooth distributional solutions) for any switching signal with locally finitely many switches.

Properties of DAEs
For the rest of this section we are considering the DAE Recall the following definitions and characterization of (impulse) controllability [11].

Proposition 5.
The reachable space of the regular DAE (8) defined as It is easily seen that the reachable space for (8) coincides with the (null-)controllable space, i.e.  (8) is given by } .
In particular, the DAE (8) The impulse controllable space can be characterized as follows [13].

Definition 8. The DAE (8) is stabilizable if for all initial conditions
x 0 ∈ R n there exists a solution (x, u) of the ITP (7) such that According to [7] if the input u(·) is sufficiently smooth, trajectories of (8) are continuous on the open interval (t 0 , ∞) and given by In particular, all trajectories can be written as the sum of an autonomous part x aut (t, t 0 ; x 0 ) = e A diff t Π (E,A) x 0 and a controllable part x u (t, t 0 ) as follows: This decomposition remains valid for switched DAEs when evaluated at the initial condition at time t − 0 ; the impulsive part of x at the initial time t 0 is then given by

Stabilizability concepts
The concepts introduced in the previous section are now utilized to investigate impulse free stabilizability of switched DAEs. In order to use the piecewise-smooth distributional solution framework and to avoid technical difficulties in general, we only consider switching signals from the following class i.e. we exclude an accumulation of switching times (see [1]). By further excluding infinitely many switches in the past and by appropriately relabeling the matrices we can assume that and that for the first switching instant t 1 it holds that t 1 > t 0 := 0. After some results relating interval-wise properties to global properties in the remainder of this section, we will restrict our attention to the bounded interval (t 0 , t f ) for some t f > 0.
As a consequence there are only finitely many switches in this interval, say n ∈ N, and for notation convenience we let t n+1 = t f .
Roughly speaking, in classical literature on non-switched systems, a linear system is called stabilizable if every trajectory can be steered towards zero as time tends to infinity. This definition can readily be applied to switched DAEs. Hence we will define impulse free stabilizability for switched DAEs in a similar fashion as follows, based on the definition of stabilizability in [5].
Definition 9 (Impulse-free Stabilizability). The switched DAE (1) with switching signal (10) is stabilizable if the corresponding solution behavior B σ is stabilizable in the behavioral sense on In the case of switched DAEs, it is reasonable to assume that there are an infinite amount of switching instances as time tends to infinity. This poses a problem when it comes to verifying conditions for stabilizability in a finite amount of steps. To overcome this problem, we investigate stabilizability on a bounded interval. Therefore we introduce the following definition of (impulse-free) interval stabilizability.
One should note that a solution on some interval is not necessarily a part of a solution on a larger interval. Consequently, stabilizability does not always imply interval stabilizability. The stabilizable, since the only global solution is the zero solution.
However, on the interval [t 1 , s) there are nonzero solutions which do not converge towards zero.
Furthermore according to Definition 10 it is required that the norm of the state is smaller at the end of an interval. This means that (impulse-free) interval stability could depend on the length of the interval considered instead of the asymptotic behavior of the system. An unstable oscillating system is thus possibly (impulse-free) interval stable and an asymptotically stable oscillating system is not necessarily (impulse-free) interval stable, depending on the choice of interval. However, under the following uniformity assumption on the switched DAE we can conclude global stabilizability.
Assumption 11 (Uniform Interval-stabilizability). Consider the switched system (1) with switching signal σ . Let τ 0 := t 0 and assume that there exists an unbounded, strictly increasing is decreasing and converging to zero as t → ∞.
that the system is (impulse-free) (τ i−1 , τ i ) -stabilizable with KL function β i for which additionally it holds that for some uniform α ∈ (0, 1) and M ⩾ 1.
The proof of Proposition 12 is along the same lines as the proof of Proposition 8 in [14].

Impulse-free stabilization and controllability
Assumption 11 can be verified for a general class of systems such as systems with periodic switching and systems with a finite amount of modes. Therefore we turn our attention to finding necessary and sufficient conditions for interval stabilizability.
As follows from Definition 10, for any initial condition x 0 , there needs to exist a solution on [t 0 , t f ) that is impulse-free and satisfies the stability property. Hence we will first discuss necessary and sufficient conditions for a switched DAE to have impulse free solutions for any initial condition x 0 on a bounded interval, i.e. impulse controllability for switched DAEs. Once these conditions are discussed, we will investigate under which conditions these impulse-free solutions are satisfying the stability property.
In the remainder of this section we will use to denote the corresponding matrices and subspaces associated to the ith mode.

Impulse controllability
As mentioned above, we will first investigate the concept of impulse controllability of a switched DAE, of which the definition is formalized as follows.
Definition 13. The switched DAE (1) with some fixed switching

Remark 14.
As an alternative for Definition 13, impulse controllability could also be defined in terms of arbitrary initial values x 0 ∈ R n . This would result in the immediate necessary condition that the first mode of a switched DAE needs to be impulse controllable. However, given a higher index DAE, Dirac impulses cannot be avoided for initial conditions in (C imp 0 ) ⊥ . Therefore it is reasonable to consider initial conditions in C imp 0 Considering the linearity of solutions and the fact that initial conditions in ker E 0 result in trajectories that jump to zero in an impulse free manner, the initial conditions of interest are those contained in V (E 0 ,A 0 ,B 0 ) .

Remark 15.
If the interval (t 0 , t f ) does not contain a switch, then the corresponding switched DAE is always impulse controllable on that interval due the definition of the augmented consistency space in terms of smooth (in particular, impulse free) solutions. This seems counter intuitive, because the active mode on that interval is not necessarily impulse controllable; however, recall that impulse controllability for a single mode (see Definition 6) is formulated in terms of the ITP (7), which can be interpreted as a switched system with one switch at t 1 = 0. In fact, letting t 0 = −ε, t f = ε, (E 0 , A 0 , B 0 ) = (I, 0, 0) and (E 1 , A 1 , B 1 ) = (E, A, B) . Therefore we consider the largest set of points from which the impulse controllable space of the next mode can be reached impulse freely from the preceding mode. To that extent we define the following sequence of subspaces regarding the switched DAE (1) with switching signal (10): . Note furthermore, that the definition is backwards in time; the sequences start with the last mode n and end with the initial mode 0. With these sets, we can prove the following lemma.

Lemma 16.
Consider the (interval restricted) switched DAE The proof is similar to the proof of Lemma 19 in [8] and therefore omitted.
Corollary 17. Consider the switched system (1) with switching signal (10). The system is impulse controllable if and only if The proof is similar to the proof of Theorem 21 in [8] and therefore omitted.
Example 18. Consider the example given in the introduction on the interval (0, t f ) with a switch at t = t 1 . The matrices (E 0 , A 0 , B 0 ) correspond the system matrices given in (2) and (E 1 , A 1 , B 1 ) are the system matrices given in (3) , can be calculated that and hence we can conclude that the system is impulsecontrollable.

Impulse-free stabilizability
As shown in the introduction, a switched DAE which is impulse controllable and stabilizable is not necessarily impulsefree stabilizable. However, impulse-controllability is an obvious necessary condition for impulse-free stabilizability. In order to stabilize a state on a bounded interval in an impulse-free way, there needs to exist an impulse-free solution in the first place. To that extent, we will make the following standing assumptions throughout the rest of this section:

The switched DAE (1) is impulse-controllable. 2. The initial condition is consistent, i.e. x(t
Under these assumptions, we will derive necessary and sufficient conditions for impulse-free stabilizability. The approach taken is as follows. First we consider the space of points that can be reached in an impulse free way from an initial value x 0 . It will then be shown that this space is an affine subspace. We then consider an element of this affine subspace with minimal norm; if this norm is smaller than the norm of the corresponding initial value, we can conclude interval stabilizability. Towards this goal, we consider the following sequence of (affine) subspaces (defined forward in time) For x 0 = 0 we drop the dependency on x 0 , i.e.

Remark 19. Note that the above
, the latter is defined as the space of all points that can be reached in an impulse-free way, i.e., it is the union of K f i (x 0 ) over all The intuition behind the sequence is as follows: K f 0 (x 0 ) are all values for x u (t − 1 , x 0 ) which can be reached in an impulse free (in fact, smooth) way during the initial mode 0. Now, inductively, we calculate the set K f i (x 0 ) of points which can be reached just before the switching time t i+1 by first considering the points K f i−1 (x 0 ) which can be reached in an impulse free way just before t i , then pick those which can be continued in mode i impulse-freely by intersecting them with C imp i , propagate this set forward according to the evolution operator and finally add the reachable space of mode i. This intuition is verified by the following lemma.
Lemma 20. Consider the switched system (1) on some bounded interval (t 0 , t f ) with the switching signal given by (10). Then for all i = 0, 1, . . . , n and x 0 ∈ V (E 0 ,A 0 ,B 0 ) if (x, u) is an impulse free solution on (t 0 , t f ). To that extent, consider an impulse-free solutions (x, u) of (1) on (t 0 , t 1 ), which by definition satisfies the solution formula (9), i.e., for some η 0 ∈ R 0 and x 0 ∈ V (E 0 ,A 0 ,B 0 ) . This shows that x u (t − 1 , x 0 ) ∈ K f 0 (x 0 ). We proceed inductively by assuming that the statement holds for i > 0 and prove the statement for i + 1.
Let (x, u) be an impulse-free solution on (t 0 , t i+1 ). Then we is impulse-free on (t 0 , t i+1 ), it follows that ξ i can be reached impulse-freely from x 0 and hence ξ i−1 ∈ K f i−1 (x 0 ). This proves that x u (t − i+1 , x 0 ) ∈ K f i (x 0 ). In the following we will prove that for all elements of K f i (x 0 ) there exists an impulse-free solution (x, u) with initial condition x u (t + 0 , x 0 ) = x 0 . We will again prove this inductively. Therefore, consider ξ 0 ∈ K f 0 (x 0 ). Then for some η 0 ∈ R 0 we have we have that there exists aũ such that xũ(t, x 0 ) is impulse-free on [t 0 , t 1 ). Then it follows from the solution formula (9) that If we define u =û+ũ it then follows from linearity of solutions that x u (t − 1 , x 0 ) = ξ 0 and is impulse-free on (t 0 , t 1 ). Assuming that the statement holds for i > 0 we continue by proving the statement for i + 1.
It follows from the induction assumption that there exists an Lemma 20 gives rise to another characterization of impulse controllability, which follows as a corollary.
Corollary 23. Consider the switched system (1) on some interval (t 0 , t f ) with the switching signal given by (10) and the sequence of affine subspaces K f i (x 0 ) given by (13).
Proof. If the system is impulse controllable, then for every initial condition x 0 there exists an impulse free solution (x, u) on (t 0 , t f ). Therefore and hence K f n+1 (x 0 ) ̸ = ∅. Conversely, if K n (x 0 ) ̸ = ∅, then let ξ ∈ K f n+1 (x 0 ). By definition there exists an impulse free solution This holds for every x 0 ∈ V (E 0 ,A 0 ,B 0 ) and hence (1) is impulse controllable. ■ Note that in contrast to Corollary 17 the computations in Corollary 23 run forward in time. Hence this result is useful in the case that not all modes are determined yet and the next mode is to be chosen. If Corollary 17 would be used, all computations would need to be redone, whereas with a forward computation only parts need to be redone.
In the following we will show that K f i (x 0 ) is an affine shift of K f f and hence K f i (x 0 ) is an affine subspace. In proving this statement, we will use some general results which can be found in the Appendix.

Lemma 24.
Consider the switched system (1) with switching signal (10) and assume it is impulse-controllable. The impulse-freereachable space from x 0 at t i is an affine shift from the impulse-free reachable space, i.e., there exists a matrix M i , such that Proof. First we simplify the notation introducing the following Then we prove the statement inductively. The statement holds trivially for n = 0, for K f 0 = Y 0 x 0 + R 0 and hence we assume that the statement holds for n. Since we assumed that the system is impulse controllable,  (14) exists only in case that the system is impulse-controllable, otherwise M i would also need to map to the empty set. In the case M i does exist, this matrix can be chosen independently of x 0 . It is however not necessarily unique, because M i+1 is dependent on N i obtained from Proposition 42 in a nonunique way. It follows from Lemma 43 from the Appendix that N i can be any matrix for which Thus, from the proof of Lemma 24 together with Lemma 43 from the Appendix the following constructive result can be obtained.
Corollary 25. Consider the switched system (1) with switching signal (10) and assume it is impulse-controllable.
Π 0 . Then for any choice of N i satisfying (15), a matrix M i+1 satisfying (14) can be calculated sequentially as follows: Remark 26. In order to compute an N i that satisfies (15) we can invoke Lemma 44 from Appendix. This means that given projectors onto R i and C imp i+1 , an N i that satisfies the conditions (15) can be constructed by solving for Q i and defining N i := Π C imp i+1 Q i . Since the existence of a solution of (16) is guaranteed by the assumption of impulsecontrollability, such a matrix equation can be solved using a linear programming solver.
Since K f i (x 0 ) contains all the states that can be reached from x 0 in an impulse free way, it follows that the norm of the state with minimal norm is given by the distance dist(K f i (x 0 ), 0). The computation of this distance is straightforward, because K f i (x 0 ) is an affine subspace. It follows from elementary linear algebra that the distance between an affine subspace and the origin, is equal to the norm of any element projected to the orthogonal complement of the vector space associated to the affine subspace. In the case of K f i (x 0 ) we would need to project onto (K f i ) ⊥ with a projector Π (K f i ) ⊥ . An important property of these projectors is that their restriction to the corresponding augmented consistency space is well defined.
Lemma 27. Consider the DAE (1) with switching signal (10). For as was to be shown. ■ Consequently, the following result follows.
Lemma 28. Consider the DAE (1) with switching signal (10) and assume it is impulse-controllable. For any M i satisfying (14) we have that It follows that we can consider Π ( K f i ) ⊥ M i as a linear map from the initial condition x 0 to the state with minimal norm in K f i (x 0 ). This allows us to formulate the following characterization of impulse-free stabilizability, which is independent of the initial condition x 0 and independent of any coordinate system. Theorem 29. Consider the switched DAE (1) with switching signal (10) and assume it is impulse controllable. Then the system is impulse-free interval-stabilizable on (t 0 , t f ) if and only if Proof. It follows from Lemma 28 that Π ( From this we can conclude that there exists a class KL function β(|x 0 |, t f − t 0 ) such that the system is impulse-free interval stabilizable in the sense of Definition 8. Conversely, if the system is impulse-free interval stabilizable, then there exists a trajectory for each initial condition This means that for the operator Π (K f n ) ⊥ M n that maps |x 0 | to the element with minimal norm that can be reached in an impulse-free way it must hold that which proves the result. ■ For many applications it is not sufficient to reduce the norm of the state, but it is necessary to control the state to zero without any Dirac impulses occurring. If a state can be steered to zero in an impulse free way, we call this state impulse-free null-controllable. A formal definition of this concept is as follows.
Definition 30. Consider the system (1) with switching signal (10). An initial condition x 0 is called impulse-free null-controllable if there exists an input u such that x u (t − f , x 0 ) = 0 and the trajectory is impulse-free. We call the system impulse-free null- Using the method from the previous section, the following characterization can readily be stated.

An initial value x 0 is impulse-free null-controllable, if and only if for
Proof. If an initial condition is impulse-free null-controllable, there exists an input u such that x u (t − f , x 0 ) = 0 and the trajectory is impulse free. This means that 0 ∈ K f n+1 (x 0 ). As a consequence As a direct consequence we can state the following result.
Corollary 32. Consider the switched system (1) with switching signal (10) and assume it is impulse controllable. Then the system is impulse-free null-controllable on (t 0 , t f ) if, and only if, for some i ∈ {0, 1, . . . , n} Proof. If the system is impulse-null controllable, we have that K f i (x 0 ) ⊆ K f i for all x 0 . Then it follows that which implies that K

Impulsive stabilizability and impulse-controllability
In the case that Dirac impulses are allowed in the trajectory similar results as in the above can be formulated. The crucial condition for impulse-free trajectories is that the state is in the impulse controllable space of the next mode at each switching instance. If this condition is dropped, a similar lemma as Lemma 20 can be formulated after considering the following sequence of sets For x 0 = 0 we drop the dependency on x 0 , i.e.
Lemma 33. Consider the switched system (1) on some bounded interval (t 0 , t f ) with the switching signal given by (10). Then for all i = 0, 1, . . . , ñ Proof. The proof is along similar lines as the proof of Lemma 20 when C Proof. Denote Y i = e A diff i (t i+1 −t i ) Π i for shorthand notation. Then for i = 0 we haveM 0 = Y 0 satisfies (18). Hence assume the statement holds for i. Then if we defineM i+1 = Y iMi for i + 1 we have that Proof. The proof follows the proof of Theorem 29 analogously. ■ As was already shown in the introduction, not every stabilizable system that is also impulse-controllable, is automatically impulse-free stabilizable. This can be explained by viewing K f i (x 0 ) andK f i (x 0 ) as affine subspaces. Note that since every state that can be reached impulse-free from x 0 is by definition also an element ofK f i (x 0 ). This leads to the following result.
Lemma 37. Consider the switched system (1) with switching signal (10) and assume the system is impulse-controllable. Then Proof. This follows immediately from Lemmas 20 and 33. ■ As a consequence, we can state the following corollary.
Corollary 38. Consider the system (1) with switching signal (10) and assume it is impulse-controllable. Then for any M i satisfying (14) we havẽ . This means that for any Given that a system is impulse-controllable and stabilizable, we have that there exists an M i satisfying (14) and we know that ∥Π (K f n ) ⊥ M n ∥ 2 < 1. However, the system is impulse-free stabilizable if and only if ∥Π (K f n ) ⊥ M n ∥ 2 < 1. This is however not implied by the statement that ∥Π (K f n ) ⊥ M n ∥ 2 < 1. Indeed, since Example 39. Again consider the example given in the introduction on the interval (0, t f ) with a switch at t = t 1 . The matrices (E 0 , A 0 , B 0 ) correspond the system matrices given in (2) and (E 1 , A 1 , B 1 ) are the system matrices given in (3). Then it follows from the algorithm (17) that the reachable space of the switched systemK ] .
From which it follows that ∥Π (K f 1 ) ⊥M1 ∥ = 1 √ 2 < 1 and hence the system is stabilizable. However, the impulse-free reachable space K f 1 can be calculated from (13) and is given by From which it follows that ∥Π (K f 1 ) ⊥M1 ∥ = √ 2 √ 2 = 1 and hence the system is not impulse-free stabilizable.
Remark 40. In the case V 0 becomes a control input after the switch the system would be null-controllable, but not impulsefree null-controllable. Furthermore, since the state of the initial condition can be reduced via an impulse-free trajectory, the system would also become impulse-free (interval) stabilizable. However, since there is no way of discharging the capacitor, it follows that there exists no input such that lim t→∞ x(t) = 0.
Remark 41. All the results on stabilizability in this paper can be applied to switched ordinary differential equations (ODEs) without difficulty. In the case of a switched ODE we have E i = I, Note that all solutions are trivially impulse-free, hence, impulse-free stabilizability is equivalent to stabilizability.

Conclusion
In this paper stabilization of switched differential algebraic equations was considered, where Dirac impulses in both the input and state-trajectory were to be avoided. Necessary and sufficient conditions for the existence of impulse-free solutions were given, followed by characterizations of (impulse-free) interval stabilizability. The results rely on the fact that the points that can be reached from an initial condition form an affine subspace. It followed that the system is (impulse-free) interval stabilizable if and only if the operator that maps the initial condition to the element of minimal norm (that can be reached in an impulse-free manner) has a norm strictly smaller than one.
A natural future direction of research would be the investigation of controllers achieving interval stabilizability for switched systems. The theory established in this paper could be used as starting point in the search (for feedback) controllers. Furthermore, a natural extension would be to consider stabilizability properties of switched systems with unknown switching signals.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Given the subspaces V, S and the matrix M, a matrix N satisfying the conditions of Lemma 43 can constructively be computed.