Stabilizability and lower spectral radius for linear switched systems with singular matrices
September 22, 2025
Abstract
We investigate the stabilizability of linear discrete-time switched systems with singular matrices, focusing on the spectral radius in this context. A new lower bound of the stabilizability radius is proposed, which is applicable to any matrix set. Based on this lower bound, more relationships between the stabilizability radius and joint spectral subradius are established. Detailed analysis of the stabilizability radius of a special kind of two-dimensional switched system, consisting of a singular matrix and a matrix with complex eigenvalues, is presented. The Hausdorff dimensions of the parameter sets such that the stabilizability radius of these systems equals a constant are also presented. Other properties of switched systems with singular matrices are also discussed along with examples.
1 Introduction↩︎
The switched system is widely studied in theory and applications. Let us mention [1] and [2] for comprehensive descriptions of formation and stability of such systems. Ref. [3] contains more recent works on continuous time switched linear systems. There are many aspects of research on switched systems. For example, Ref. [4] consider the control of switched linear systems, the Lyapunov analysis of switched systems considered in [5] and [6] and Ref. [7] is about applications of switched systems to epidemiological spread.
Despite the simple form of the system itself, the stability and stabilizability of such systems are hard to determine, even for linear ones. The discrete-time linear switched system of the given matrix set \(\mathcal{M} \subseteq \mathbb{R}^{n \times n}, \; n \geq 2\) is given by \[\label{eq5} x(t+1) = M_{\sigma(t)}x(t), \quad x(0)=x_0,\tag{1}\] where \(M_{\sigma(t)} \in \mathcal{M}\), \(\sigma(t) \in \{1, \dots, m\}\) and \(m\) is the cardinality of \(\mathcal{M}\). There are many general results on the stability and stabilizability of such system (for example, see [2]). However, we are not aware of any study focusing on the case where the matrix set \(\mathcal{M}\) contains singular matrices. In the literature, the term “singular switched system” refers to a special kind of switched system with many known results (see [8] and [9]), which is different from the one we consider in this paper.
To describe the stability and stabilizability of system (1 ), the joint spectral properties are widely used. Let \(\mathcal{M}^T\) denote the set that contains all \(M_{\sigma(t)} \cdots M_{\sigma(1)}\), the joint spectral radius of system (1 ) is defined as \[\hat{\rho}(\mathcal{M}) = \lim_{T \rightarrow \infty} \sup_{A \in \mathcal{M}^T} \|A\|^{1/T},\] for any norm. The joint spectral subradius is defined as \[\check{\rho}(\mathcal{M}) = \lim_{T \rightarrow \infty} \inf_{A \in \mathcal{M}^T} \|A\|^{1/T},\] for any norm. A new radius proposed in [10], called the stabilizability radius, is defined as follows. \[\tilde{\rho} = \sup_{x_0 \in \mathbb{R}^n} \inf \{ \lambda \geq 0 \, | \, \exists \sigma(\cdot), c > 0 \; s.t. \; \|x_\sigma(t)\| \leq c \lambda^t \|x_0\| \, \forall t \geq 0\},\] These three radii trivially satisfy \(\tilde{\rho}\leq\check{\rho}\leq\hat{\rho}\) and describe different properties of the switched system with different terminology in the literature. In this paper, we have
A switched system is called stable if \(\hat{\rho}(\mathcal{M}) < 1\) (see, for example, [11]).
A switched system is called uniformly stabilizable or open loop stabilizable in the literature if \(\check{\rho}(\mathcal{M}) < 1\) (see, for example, [11]).
A switched system is called pointwise stabilizable if \(\tilde{\rho}(\mathcal{M}) < 1\) (proposition 2.5 in [10]).
Stanford first showed that pointwise stabilizability is different from uniform stabilizability in [12].
The stabilizability radius \(\tilde{\rho}\), first introduced in [10], had many implicit studies beforehand, for example, Ref. [13] considered this problem in measurement scheduling and other cases, and Ref. [14] considered this problem with applications in HIV drug control. For non-singular matrix sets, \(\tilde{\rho}\) has been proved to have several lower bounds in [15].
In applications mentioned above, the matrix set could contain singular matrices. In this paper, we have a new lower bound of the stabilizability radius for any finite matrix set. In the meantime, we do a detailed analysis of \(\tilde{\rho}\) of a two dimensional switched system with singular matrices, and show how to compute the stability radius \(\tilde{\rho}\) using continued fractions and Diophantine approximation. We also show that for the case \(\tilde{\rho} = 0\), the set of such \(\mathcal{M}\) has zero Hausdorff dimension. Other properties of systems are mentioned as they appear.
Some notations are used in this paper with their well-known meanings. The area denotes the \(n-1\) dimensional Lebesgue measure of a set in \(\mathbb{R}^n\). For two functions \(f\) and \(g\), we write \(f(x) \sim g(x)\) as \(x \rightarrow 0\) if \(\lim_{x \rightarrow 0} f(x)/g(x) = 1\). The notation \(\| \cdot \|\) will now specifically denote the Euclidean norm in \(\mathbb{R}^n\).
In section \(2\), we show the \(\tilde{\rho}\) is lower bounded by \(\check{\rho}\) divided by the cardinality of \(\mathcal{M}\) and give one more situation when \(\tilde{\rho}=\check{\rho}\). Section \(3\) is dedicated to the stabilizability radius of the two dimensional switched system with a singular matrix and a matrix with complex eigenvalues. We also show the Hausdorff dimension of the parameter set such that \(\tilde{\rho} = c, 0 \leq c < 1\) is zero. Section \(4\) contains several examples of switched systems with singular matrices about relationship between smallest singular value and \(\tilde{\rho}\). One strategy of finding the optimal switching law is also mentioned.
2 Lower Bound↩︎
2.1 Lower bound formation↩︎
Let \(A_t\) denote the product \(M_{\sigma(t)} \cdots M_{\sigma(1)}\). Suppose \(s^2_1(A_t), \dots, s^2_n(A_t)\) are ordered eigenvalues of \(A^T_tA_t\) and \(v_1, \dots, v_n\) are corresponding orthonormal eigenvectors, where \(s_1\) is smallest singular value of \(A_t\). We have two nice properties of \(v_i\).
\(A_tv_i\) are orthogonal for \(i=1, \dots, n\).
\(\|A_t v_i\| =s_i(A_t)\).
Suppose \(s_1, \dots, s_\kappa\) are zero singular values, then \(v_1, \dots, v_\kappa\) form an orthonormal basis for \(\mathbf{Ker}(A_t)\). Let \((x_1, \dots, x_n)\) denote the coordinates of \(x \in \mathbb{R}^n\) in terms of \(v_1, \dots, v_n\).
Start from the unit sphere. If the \(t\) iteration of the unit sphere under any \(A_t\) is a subset of a ball centered at the origin, we know that \(\|x(t)\| \leq \lambda \|x_0\|\), where \(\lambda\) is the radius of the ball. Let \(B_r\) denote the ball with radius \(r\) centered at the origin and \(S^{n-1}\) denote the \(n\) dimensional unit sphere. We consider the intersection of the pre-image of \(A_t\) and \(S^{n-1}\), i.e. \(S_{r,A_t} = \{x \in S^{n-1} | \|A_t x\| \leq r\}\). As \(A_t v_j = 0\) for \(j=1, \dots,\kappa\), then for any \(x \in S_{r,A_t}\), we have \[\|A_t x\|^2 = \|\sum_{i=1}^{n} x_i A_tv_j \|^2 = \sum_{i=\kappa+1}^{n} s^2_i(A_t) x_i^2 \leq r^2.\] For any nonzero \(A_t\), consider the set \(S' = \{x \in S^{n-1} | |x_n| \leq r/s_n(A_t) \}\). Since \[\sum_{i=\kappa+1}^{n} s^2_i(A_t) x_i^2 \geq s^2_n(A_t) x_n^2,\] we have \(S_{r,A_t} \subseteq S'\), and \(S'\) is a spherical segment. The area of \(S'\) can be obtained using the area of spherical caps. A formula for area of caps is given by Lemma 1.
Lemma 1. ((1) in [16])The surface area of the cap of the unit sphere \(S^{n-1}\), i.e. \(T_n(h)=\{x \in S^{n-1}|x_1 \geq h\}\) with \(x_1\) be one of coordinates of \(x\), is \[|T_n(h)| = \frac{1}{2} |S^{n-1}| I\left(1-h^2;\frac{n-1}{2},\frac{1}{2}\right),\] where \(I(h;\cdot, \cdot)\) denotes the regularized incomplete beta function, defined as \[I(h;a,b) = \frac{\int_0^{h} t^{a-1}(1-t)^{b-1}\text{d}t}{\int_0^1 t^{a-1}(1-t)^{b-1}\text{d}t}, \quad 0 < h < 1.\]
A useful equality of \(I(h;\cdot, \cdot)\) is given by \[I\left(h;\frac{n-1}{2},\frac{1}{2}\right) = 1-I\left(1-h;\frac{1}{2},\frac{n-1}{2}\right).\] One can prove this by changing variables in the integral definition of \(I(h;\cdot, \cdot)\).
Lemma 2. The area of \(S_{r,A_t}\) is upper bounded by \[|S_{r,A_t}| \leq |S^{n-1}|I\left(\frac{r^2}{s^2_n\left(A_t\right)}; \frac{1}{2}, \frac{n-1}{2}\right).\] For any \(t \in \mathbb{N}^+\).
Proof. The area of \(S'\) is given by \[|S'| = |S^{n-1}| - 2 \left|T_n\left(\frac{r}{s_n\left(A_t\right)}\right)\right|.\] Hence \[|S_{r,A_t}| \leq |S^{n-1}|\left(1- I\left(1-\frac{r^2}{s^2_n\left(A_t\right)}; \frac{n-1}{2}, \frac{1}{2}\right)\right) = |S^{n-1}|I\left(\frac{r^2}{s^2_n\left(A_t\right)}; \frac{1}{2}, \frac{n-1}{2}\right).\] Thus we have a bound of \(|S_{r,A_t}|\). ◻
Theorem 1. The stabilizability radius satisfies \[\frac{\check{\rho}}{m} \leq \tilde{\rho} \leq \check{\rho}.\]
Proof. The \(\tilde{\rho} \leq \check{\rho}\) is from definition of \(\tilde{\rho}\). For a given \(T\), the area of the union of \(S_{r,A_T}\) under all \(A_T\) have an upper bound given by \[\left|\cup_\sigma S_{r,A_T}\right| \leq \sum_{\sigma} |S_{r,A_T}| \leq \sum_{\sigma} |S^{n-1}|I\left(\frac{r^2}{s^2_n(A_T)}; \frac{1}{2}, \frac{n-1}{2}\right)\] The sum above is over all possible value of \(\sigma(t)\) for \(t=1, \dots, T\). Since \(I(h,1/2,(n-1)/2)\) is the cumulative distribution function of the beta distribution, \(I(h,1/2,(n-1)/2)\) monotonically increases for \(0 \leq h \leq 1\). Let \(S(T)\) denotes the smallest \(s_n(A_T)\) for all \(\sigma\), then we have \[I\left(\frac{r^2}{s^2_n(A_T)};\frac{1}{2}, \frac{n-1}{2}\right) \leq I \left(\frac{r^2}{S(T)^2};\frac{1}{2}, \frac{n-1}{2}\right).\] Then \[\left|\bigcup_\sigma S_{r,A_T}\right| \leq m^T |S^{n-1}|I\left(\frac{r^{2}}{S(T)^2}; \frac{1}{2}, \frac{n-1}{2}\right).\] From the definition of \(\tilde{\rho}\), for any \(\lambda > \tilde{\rho}\), there exists a positive constant \(c\) such that the union of \(S_{c\lambda^T,A_T}\) covers the unit sphere. Hence \[\label{eq2} |S^{n-1}| = \left|\bigcup_\sigma S_{c\lambda^T,A_T}\right| \leq m^T |S^{n-1}|I\left(\frac{c^2\lambda^{2T}}{S(T)^2}; \frac{1}{2}, \frac{n-1}{2}\right).\tag{2}\] Rearranging (2 ), we have a bound of \(\lambda\) given by \[I\left(\frac{c^2\lambda^{2T}}{S(T)^2}; \frac{1}{2}, \frac{n-1}{2}\right) \geq \frac{1}{m^{T}}.\] Consider the limit \[\lim_{T \rightarrow \infty} S(T)^{1/T}.\] An alternative way to write this limit is \[\liminf_{T \rightarrow \infty} \{s_n(A_T)^{1/T} \;|\; \sigma(t) \in \{1,\dots,m\}, t \in\{1,\dots,T\}\}.\] Then this is the \(\check{\rho}\) of the matrix set \(\mathcal{M}\) (see for example Theorem 1.1 in [11]).
Now, we know the asymptotic behavior of \(I(h;1/2,(n-1)/2)\) when \(h\) tends to \(0\), allowing us to obtain a lower bound of \(\tilde{\rho}\) in terms of \(\check{\rho}\). Notice that \[I(h;1/2,(n-1)/2) = C\int_0^{h} t^{-\frac{1}{2}}(1-t)^{\frac{n-3}{2}}\text{d}t,\] for a constant \(C\). Let \(g(x)\) denote the integrand, then for any \(n \geq 2\), we have as \(x \rightarrow 0\), \[g(x) \sim x^{-\frac{1}{2}}.\] Namely, \[\int_{0}^{h} g(x) \text{d}x \sim 2\sqrt{h}.\] Then for large enough \(T\), we have \[2C\frac{c\lambda^{T}}{S(T)} \geq \frac{1}{m^T},\] which is \[(2Cc)^{1/T}\lambda \geq \frac{S(T)^{1/T}}{m}.\] Let \(T\) tend to infinity, then we have for any \(\lambda >\tilde{\rho}\), \[\lambda \geq \frac{\check{\rho}}{m}.\] Let \(\lambda\) tend to \(\tilde{\rho}\), then \[\tilde{\rho} \geq \frac{\check{\rho}}{m}.\] ◻
In general this is not an accurate bound, especially for large \(m\).
Example 1. (Based on an example first given in [12]) Switched system with \(\mathcal{M}=\{M_1,M_2\}\) such that \[M_1=\left(\begin{matrix} \frac{1}{2} &0 \\ 0 &2 \end{matrix}\right), \quad M_2=\left(\begin{matrix} \cos \frac{\pi}{6} &-\sin \frac{\pi}{6}\\ \sin \frac{\pi}{6} & \cos \frac{\pi}{6} \end{matrix}\right),\]
For Example 1, we have \(\check{\rho}=1\). From Theorem 1, we know that \(\tilde{\rho} \geq 1/2\). However, the lower bound given by Theorem \(2\) in [15] already gives us \(\tilde{\rho} \geq \sqrt{2}/2\). The advantage of the bound in Theorem 1 is that it works for matrix sets with singular matrices, especially for those system with known \(\check{\rho}\), while lower bounds in [15] doesn’t work for such matrix sets.
Example 2. Switched system with \(\mathcal{M}=\{M_1, M_2\}\) such that \[M_1=\left(\begin{matrix} \frac{1}{2} &0 &0 \\ 0 &2 &0 \\ 0 &0 &0 \end{matrix}\right), \quad M_2=\left(\begin{matrix} \cos \frac{\pi}{6} & -\sin \frac{\pi}{6} & 0\\ \sin \frac{\pi}{6} & \cos \frac{\pi}{6} & 0\\ 0 & 0 & 1 \end{matrix}\right).\]
In Example 2, \(M_1\) projects any three dimensional vector onto the \((x_1,x_2)\) plane. The \(\check{\rho}(\mathcal{M})\) of this system equals the \(\check{\rho}\) of Example 1. As \(M_1\) is singular, any previous lower bound in the literature doesn’t work for this system. From Theorem 1, we obtain the nontrivial bound \(\tilde{\rho} \geq 1/2\).
2.2 Joint spectral subradius and stabilizability radius↩︎
From Theorem 1, we know that the stabilizability radius \(\tilde{\rho}\) is equivalent to the joint spectral subradius \(\check{\rho}\). An immediate corollary of Theorem 1 is that when \(\tilde{\rho}=0\), we have \(\check{\rho}=0\). This describes the case when a matrix set pointwise converges to zero quickly enough, we know it converges to zero uniformly.
Notice that for a given \(\mathcal{M}\), the optimal switching law of \(\check{\rho}\) and \(\tilde{\rho}\) not necessarily contain every matrix in \(\mathcal{M}\). To avoid such phenomena, we have following definition.
Definition 1. In this paper, a matrix set \(\mathcal{M}\) is called irreducible if for any \(\mathcal{M}_1 \subset \mathcal{M}\), we have \[\tilde{\rho}(\mathcal{M}_1) > \tilde{\rho}(\mathcal{M}).\]
Not all linear discrete time switching systems are irreducible. However, it is sufficient to only consider the stabilizability radius of irreducible matrix sets. This is because an irreducible subset with the same stabilizability radius exists in any reducible matrix set (Corollary 3.4 in [17]).
Proposition 3. The \(\tilde{\rho} = \check{\rho}\) if the matrix set is irreducible and contains a singular matrix with one dimensional image.
Proof. For this irreducible matrix set, we know that the optimal sequence of \(\tilde{\rho}\) contains at least one singular matrix. We can always obtain the optimal sequence of matrices in two steps for such situations. The first step is finding the optimal sequence before applying the singular matrix, and then the second step is finding the optimal sequence starts from the image of the singular matrix. The first part of sequence has no influence on the stabilizability radius, because it doesn’t repeat afterwards. Then it is sufficient to find the optimal sequence only based on vectors in the image of the singular matrix, which have same direction. Consequently, this optimal sequence doesn’t depend on the initial vectors. ◻
Corollary 1. We have \(\tilde{\rho}=\check{\rho}\) for any two dimensional irreducible matrix set with at least one singular matrix.
However, for arbitrary matrix set with singular matrices, we don’t have such properties. Consider Example 2, we have \(\check{\rho} = 1\) but this system is pointwise stabilizable as it behaves the same as Example 1 (proved in [12]).
3 Two dimensions↩︎
3.1 A singular and a rotation matrix↩︎
Inspired by an example in [12], we focus on a two dimensional switched system in this section.
System 1. Consider a two dimensional switched system with a set of matrices \(\mathcal{M}=\{M_1, M_2\}\) with \(M_1\) a singular matrix and \(M_2\) a matrix with two complex eigenvalues, namely \[\label{eq4} M_1=\left(\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}\right), \quad M_2=\left(\begin{matrix} b_{11} & b_{12}\\ b_{21} & b_{22} \end{matrix}\right),\qquad{(1)}\] where \(a_{ij}, b_{ij} \in \mathbb{R}\), \(a_{11}a_{22}-a_{12}a_{21}=0\) and \((b_{11}-b_{22})^2 +4b_{12}b_{21} < 0\).
Let \(\lambda_1\) and \(\lambda_2\), \(v_1\) and \(v_2\) denote two eigenvalues and eigenvectors of \(M_1\), \(\lambda_3\) and \(\overline{\lambda}_3\) denote two eigenvalues of \(M_2\). If \(\lambda_1=\lambda_2=0\), \(M_1\) is nilpotent, then System 1 becomes trivial. In the following we use \(\lambda_2 \in \mathbb{R}\setminus\{0\}\) and \(\lambda_1 = 0\).
The matrix \(M_2\) is similar to its real Jordan form (see for example Theorem 3.4.1.5. in [18]), \[J=|\lambda_3| \left(\begin{matrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{matrix}\right)\] where \(\cos \alpha = (\lambda_3 + \overline{\lambda}_3)/2|\lambda_3|\). Thus, there exists an invertible matrix \(P\) such that \(M_2=PJP^{-1}\). Let \(M'_1=P^{-1}M_1P\), then the system 1 is stabilizable if and only if the following switched system is stabilizable.
System 2. Consider a two dimensional switched system with a set of matrices \(\mathcal{M}=\{M'_1, J\}\) such that \[M'_1=\left(\begin{matrix} a'_{11} & a'_{12} \\ a'_{21} & a'_{22} \end{matrix}\right), \quad J=|\lambda_3|\left(\begin{matrix} \cos \alpha & \sin \alpha\\ -\sin \alpha & \cos \alpha \end{matrix}\right),\] where \(a_{ij} \in \mathbb{R}\), \(a'_{11}a'_{22}-a'_{12}a'_{21}=0\).
Lemma 4. The stabilizability radius of System 1 is given by \[\tilde{\rho} =|\lambda_3| \inf_{l \in \mathbb{N}} \left| \frac{\lambda_2}{\lambda_3} \frac{\sin(l\alpha - \beta)\pi}{\sin \beta \pi} \right|^{1/(l+1)},\] where \(\beta \pi=\arccos|\left<v_1,v_2\right>|\).
Proof. To reduce the norm of \(x\), the general approach is to rotate (apply \(J\)) as close to \(v_1\) as possible, then apply \(M'_1\). Consider the switching law in the following general form \[A_t = J^{t_k} M'_1 J^{t_{k-1}} \cdots M'_1 J^{t_1},\] where \(t_i \in \mathbb{N}^{+}\) for \(i \in {2, \dots, k-1}\), \(t_k,\; t_1 \in \mathbb{N}\) and \[\sum_{i=1}^{k} t_i +k-1= t.\] For any vector \(x \in \mathbb{R}^2\), consider the polar coordinates of \(x\), i.e. \(x = (r,\; \theta \pi)\), we have \[J^t x= (|\lambda_3|^t r,\; (\theta -t\alpha) \pi).\] Without losing generality, we assume \(v_1\) is the \(x\) axis, Then we have \[M'_1 x= \left(\left|\lambda_2 r \frac{\sin \theta \pi}{\sin \beta \pi}\right|,\; \beta \pi +k\pi\right),\; k \in \mathbb{Z}.\] Notice that after the first time applying \(M'_1\), the trajectory starts from \(v_2\), Let \(t_i =l\) for \(i= 2 \dots k-1\) and \(\|M'_1J^{t_1}x\| = C \|x\|\), we have \[\label{eq7} \|x_{\sigma,x_0}(t) \| = \left|\lambda_2 |\lambda_3|^l \frac{\sin (\beta - l\alpha) \pi}{\sin \beta \pi}\right|^{k-2} C \|x_0\|.\tag{3}\] Since the factor of RHS in (3 ) is independent from \(x_0\), by taking the infimum over \(l\) we obtain the stabilizability radius. ◻
To find the infimum in Lemma 4, we need to know \(\|l \alpha - \beta \|_N\) for \(l \in \mathbb{N}\) with \(\| \cdot \|_N\) from number theory denoting the distance to the closest integer. Now we introduce several lemmas from [19]. The continued fraction of \(\alpha\) is given by \(\alpha = [a_0; a_1, a_2, \dots ]\). Let \[\frac{p_k}{q_k} = [a_0;a_1, \dots , a_k] \quad (k \geq 0),\] and \(D_k = q_k \alpha - p_k\), we have the following results.
Lemma 5. ([19] Lemma (3.1)) For any \(\alpha \in \mathbb{R} \setminus \mathbb{Q}\), we have \(l \in \mathbb{N}\) satisfy that \[q_K \leq l < q_{K+1}\] for some \(q_K\) of \(\alpha\). There exists a unique sequence \(\{c_{k+1}\}^{\infty}_{k=0}\) such that \[l= \sum_{k=0}^{\infty} c_{k+1} q_k.\] where \(c_{k+1}\) satisfy that
\(c_{k+1} = 0, \; \forall k > K\),
\(0 \leq c_1 < a_1\) and \(0 \leq c_{k+1} \leq a_{k+1}, \forall k \geq 1\),
\(c_k =0\) whenever \(c_{k+1} =a_{k+1}, \forall k \geq 1\).
Lemma 6. ([19] Lemma (3.2)) For any \(\alpha \in [0,1) \setminus \mathbb{Q}\) and \(\theta \in [-\alpha, 1-\alpha)\), there exists a unique integer sequence \(\{b_{k+1}\}^{\infty}_{k=0}\) such that \[\theta = \sum_{k=0}^{\infty} b_{k+1} D_k.\] where \(b_{k+1}\) satisfy that
\(0 \leq b_1 < a_1\) and \(0 \leq b_{k+1} \leq a_{k+1}, \forall k \geq 1\),
\(b_k =0\) whenever \(b_{k+1} =a_{k+1}, \forall k \geq 1\).
Details of \(c_{k+1}\) and \(b_{k+1}\) could be found in [19].
Lemma 7. ([19] Lemma (4.3)) Consider any \(\alpha \in [0,1) \setminus \mathbb{Q}\) and \(\theta \in [-\alpha, 1-\alpha)\), and suppose that \[\|l\alpha - \theta \|_N > 0, \; \forall l \in \mathbb{N}.\] There exists a smallest integer \(k_0\) such that \(c_{k_0+1} \neq b_{k_0+1}\) and we define \[\Sigma = \sum_{k=k_0}^{\infty} \delta_{k+1} D_k,\] where \(\delta_{k+1} = c_{k+1}-b_{k+1}\) with \(c_{k+1}\) and \(b_{k+1}\) from Lemma 5 and 6. Then we have \[\|l\alpha - \theta \|_N = \|\Sigma\|_N.\]
Then we have the following theorem.
Theorem 2. For System 1 with a given \(\beta\), we can obtain the stabilizability radius \(\tilde{\rho}\) for the following three cases.
For any \(\alpha\) such that \(\|l \alpha -\beta\|_N =0\) for some \(l\), we have \(\tilde{\rho}=0\).
For any \(\alpha \in \mathbb{Q}\) and not in case \(1\), let \(\alpha = p/q\), we have \[\tilde{\rho} = |\lambda_3|\min_{l} \left| \frac{\lambda_2}{\lambda_3} \frac{\sin(l\alpha - \beta)\pi}{\sin \beta \pi} \right|^{1/(l+1)},\; l \in \{0,\dots,q-1\}\]
For any \(\alpha \in (0,1) \setminus \mathbb{Q}\) and not in case \(1\), there exists a sequence \(\{l_n\}^{\infty}_{n=1}\) such that \[\tilde{\rho} = |\lambda_3|\inf_{l_n} \left| \frac{\lambda_2}{\lambda_3} \frac{\sin(l_n\alpha - \beta)\pi}{\sin \beta \pi} \right|^{1/(l_n+1)}.\] Each \(l_n\) is given by \[l_n= \sum_{k=0}^{n} b_{k+1} q_k,\] where \(b_{k+1}\) is obtained from Lemma 6 by letting \(\theta = \beta\) when \(\alpha \in (0,1-\beta)\), or \(\theta = \beta -1\) when \(\alpha \in [1-\beta,1)\).
Proof. Case \(1\) and \(2\) are easily obtained from Lemma 4. For case 3, choosing \[l_n = \sum_{k=0}^{n} b_{k+1} q_k,\] with \(b_{k+1}\) from Lemma 6 by letting \(\theta = \beta\) when \(\alpha \in (0,1-\beta)\), or \(\theta = \beta -1\) when \(\alpha \in [1-\beta,1)\). Then we have \(\delta_{k+1} = 0\) for all \(k \leq n\). Then \(l_n\alpha\) is the best approximation of \(\beta\) for any \(l \leq l_n\) by Lemma 7, the infimum of \(\tilde{\rho}\) reached by one of \(l_n\) or when \(n\) tends to infinity. ◻
Remark 1. The sequence \(\{l_n\}\) described in Theorem 2 is non-decreasing and unbounded, but not necessarily increasing.
It is worth mentioning that \(\tilde{\rho} = \check{\rho}\) in this case, as shown in the proof of Lemma 4. Then the lower bound from Theorem 1 gives us a trivial inequality \(\tilde{\rho} \geq \tilde{\rho}/2\).
3.2 Zero stability radius↩︎
There is one interesting question about System 1: ‘When would \(\tilde{\rho}=0\)?’ For a given \(\beta\), we have the following two cases of \(\tilde{\rho}\) for different \(\alpha\).
The infimum is reached by a finite \(l\).
The infimum is reached when \(n\) tends to infinity.
In the first case, \(\tilde{\rho}= 0\) if and only if \(\|l\alpha - \beta\|_N =0\) for some integer \(l\). Let \(L_\beta(0)\) denotes the set of \(\alpha\) such that the second case holds, it is clear that \(L_\beta(0)\) only contains irrational \(\alpha\), which all belong to the third case in Theorem 2. The Hausdorff dimension of \(L_\beta(0)\) is given by the following more general lemma.
Lemma 8. For a given \(\beta\) and a constant \(c \in [0,1)\), let \(L_\beta(c)\) be the set of \(\alpha\) such that \(\tilde{\rho}=c\) and the infimum in Lemma 4 is reached when \(n\) tends to infinity. Then we have that \(L_\beta(c)\) is a zero Hausdorff dimension set.
Proof. For each \(\alpha \in L_\beta(c)\), we have \[\tilde{\rho} =|\lambda_3| \liminf_{n \rightarrow \infty} \left| \frac{\lambda_2}{\lambda_3} \frac{\sin (l_n\alpha-\beta)\pi}{\sin \beta \pi} \right|^{1/(l_n+1)} = c,\] with \(\{l_n\}\) from the third case in Theorem 2. Let \(R_n\) denote \(\|l_n\alpha -\beta\|_N\), as \(\sin x \sim x\) when \(x \rightarrow 0\), there exists a infinite subsequence \(\{l_{n_i}\}\) such that \[\lim_{i \rightarrow \infty} R_{n_i}^{1/(l_{n_i}+1)} =c.\] Then there exists a \(c <c_1< 1\) such that for any \(\alpha\) in \(L_\beta(c)\), we have \(R_{n_i} = o(c_1^{l_{n_i}})\) when \(i\) tends to infinity.
For any \(0<s<1\) and \(\epsilon>0\), let \(\mathcal{L}(\alpha)\) denotes the smallest integer of each \(\alpha\) satisfy that
\(\mathcal{L}(\alpha) \in \{l_{n_i}\}\),
\(\|\mathcal{L}(\alpha)\alpha -\beta\|_N < c_1^{\mathcal{L}(\alpha)}\),
\(\left(t^{1-s}c_1^{st}\right)'|_{t=(\mathcal{L}(\alpha)-1)}<0\)
\(2c_1^{\mathcal{L}(\alpha)}/\mathcal{L}(\alpha) < \epsilon\),
\(2^s(-s\ln c_1)^{s-2} \Gamma(2-s,-s \ln c_1(\mathcal{L}(\alpha)-1)) < \epsilon\),
where \(\Gamma(a,b)=\int_{b}^{\infty}t^{a-1}e^{-t} \text{d}t\) is the incomplete gamma function. Such \(\mathcal{L}(\alpha)\) exists because \(\Gamma(a,b) \rightarrow 0\) as \(b \rightarrow \infty\). Let \(\mathcal{L}=\inf \{\mathcal{L}(\alpha) | \alpha \in L_\beta(c)\}\), then for any \(\alpha \in L_\beta(c)\), by (a) and (b), there exists an \(i\) such that \(l_{n_i}>\mathcal{L}\) and \[\label{e:c1} \|l_{n_i}\alpha -\beta\|_N < c_1^{l_{n_i}}.\tag{4}\] Then there exists an integer \(j\) such that \[\left\|\alpha-\frac{\beta + j}{l_{n_i}}\right\| < \frac{c_1^{l_{n_i}}}{l_{n_i}},\] Then we have \[\label{eq1} \alpha \subseteq \left(\frac{\beta + j}{l_{n_i}}-\frac{c_1^{l_{n_i}}}{l_{n_i}},\frac{\beta + j}{l_{n_i}}+\frac{c_1^{l_{n_i}}}{l_{n_i}}\right).\tag{5}\] The \(l_{n_i}\) and \(j\) in (5 ) depend on \(\alpha\). Let \(I(\alpha)\) denote the interval in (5 ), then we have a cover of \(L_\beta(c)\) given by \[\label{eq8} L_\beta(c) \subseteq \bigcup_{\alpha \in L_\beta(c)} I(\alpha) \subseteq \bigcup_{k=\mathcal{L}}^{\infty} \bigcup_{j=0}^{k-1} \left(\frac{\beta + j}{k}-\frac{c_1^k}{k},\frac{\beta + j}{k}+\frac{c_1^k}{k}\right).\tag{6}\] Let \(I(k,j)\) be the \(j\)th interval of \(k\) in 6 , by (d) we have \(|I(k,j)| = 2c_1^k/k < \epsilon\). By (c) and (e), we have \[\sum_{k=\mathcal{L}}^{\infty}k|I(k,j)|^s < 2^s\int_{\mathcal{L}-1}^{\infty}t^{1-s}c_1^{st} \text{d}t =2^s(-s\ln c_1)^{s-2} \Gamma(2-s,-s \ln c_1 (\mathcal{L}-1)) < \epsilon.\] Then for any \(0<s<1\), the Hausdorff measure of \(L_\beta(c)\) is zero, \(L_\beta(c)\) is a zero Hausdorff dimension set. ◻
Theorem 3. For a given \(\beta\), let \(S_\beta(c)\) be the set of \(\alpha\) such that \(\tilde{\rho}=c\), then \(S_\beta(c)\) is a zero Hausdorff dimension set.
Proof. We have \[S_\beta(c) = \{\alpha | \exists l, \, \|l\alpha-\beta\|_N =c\} \cup L_\beta(c).\] For any \(l \in \mathbb{N}\), there are only finite \(\alpha\) satisfies that \(\|l\alpha-\beta\|_N =0\). Hence \(S_\beta(c)\) is the union of a countable set and a set with zero Hausdorff dimension. ◻
For rational \(\beta\), the \(\alpha\) in \(L_\beta(0)\) are related to a familiar set of numbers.
Proposition 9. Let \(L\) be the set of all Liouville numbers in \((0,1)\). For rational \(\beta\) we have \(L_\beta(0) \subset L\).
Proof. For every positive integer \(j\), a Liouville number \(x\) satisfies that there exist integers \(p\) and \(q\) with \(q > 1\) such that \[0 < \|x - \frac{p}{q}\|_N < \frac{1}{q^j}.\] Let \(c_1=1/2\) in Equation (4 ) and \(\beta=p_0/q_0\). For any \(\alpha \in L_\beta(0)\) and any \(j\), there exists a sufficiently large \(i\) such that \(2^{l_{n_i}}l_{n_i} > (q_0 l_{n_i})^j\), i.e. \[\|\alpha-\frac{p}{q_0 l_{n_i}}\|_N = \frac{1}{l_{n_i}}\|l_{n_i} \alpha - \frac{p_0}{q_0} \|_N < \frac{1}{2^{l_{n_i}}l_{n_i}} < \frac{1}{(q_0 l_{n_i})^j},\] for some \(p \in \mathbb{Z}\). ◻
It is well known that the set of Liouville numbers has zero Hausdorff dimension (see Theorem \(2.4\) in [20] for example). This result coincides with Lemma 8.
Lemma 8 is also closely related to the finiteness property w.r.t. \(\tilde{\rho}\) of System 1. For general switched systems, case \(1\) in the beginning of section \(3.2\) can be characterized by the following definition.
Definition 2. A matrix set \(\mathcal{M}\) has finiteness property w.r.t. \(\tilde{\rho}\) if there exists a finite matrix sequence \(A_t = M_{\sigma(t)} \cdots M_{\sigma(1)}\) such that \(\tilde{\rho}^{t}\) is the spectral radius of \(A_t\).
This finiteness property w.r.t. \(\hat{\rho}\) has many known results ([21], for example). For system \(1\) with a given \(M_1\), \(\cup_{c \in [0,1)} S_\beta(c)\) is the set of \(\alpha\) such that this system is pointwise stabilizable. \(\cup_{c \in [0,1)} S_\beta(c) \setminus \cup_{c \in [0,1)} L_\beta(c)\) is the set of \(\alpha\) such that this system is pointwise stabilizable and has finiteness property w.r.t. \(\tilde{\rho}\). We know each \(S_\beta(c)\) and \(L_\beta(c)\) have zero Hausdorff dimension.
3.3 A special case↩︎
It is helpful to consider a special case of System 1 by letting \(a=2\) and \(b=c=d=0\), i.e the following example
Example 3. Consider the switched system with the matrix set \(\mathcal{M} = \{M_1, M_2\}\) such that \[\label{eq3} M_1 = \left(\begin{matrix} 2 &0 \\ 0 &0 \end{matrix}\right), \quad M_2 = \left(\begin{matrix} \cos \alpha \pi &\sin \alpha \pi\\ -\sin \alpha \pi &\quad \cos \alpha \pi \end{matrix}\right).\qquad{(2)}\]
Applying Lemma 10 with \(\beta = 1/2\), we have
Corollary 2. The stabilizability radius of Example 3 is given by \[\tilde{\rho} = \inf_{l \in \mathbb{N}} \left| 2 \cos l\alpha\pi \right|^{1/(l+1)}.\]
One vital property of Example 3 is that this system is pointwise stabilizable for most \(\alpha\). Figure 1 shows the \(\tilde{\rho}\) in terms of \(\alpha\) for Example 3. Each point in figure with \(\alpha\) as a sample from \(U(0,1)\) and \(\tilde{\rho}\) from Corollary 2. Notice that rational \(\alpha\) with an even denominator gives us \(\tilde{\rho}= 0\) from Corollary 2, so we randomly smaple \(\alpha\) in this way to avoid having too many zero values of \(\tilde{\rho}\).
Proposition 10. For any \(\alpha \in (0,1)\), we have \(\tilde{\rho} \leq 1\) and the equality only hold when \(\alpha = 1/3 \text{ or } 2/3\).
Proof. The Diophantine approximation of Example 3 is given by \(R(\alpha) = \|l\alpha - 1/2\|_N\). For irrational \(\alpha\), \(R(\alpha)\) could be arbitrarily small so there exists an \(l_0\) such that \(\|l_0\alpha - 1/2\|_N < 1/6\).
For rational \(\alpha\), it is known that the smallest nontrivial coprime integer of \(2\) is \(3\), so we have \(R(\alpha) \leq R(1/3) = R(2/3)\), i.e. there exists \(l\) such that \[\|l\alpha - \frac{1}{2}\|_N \leq \|\frac{l}{3} - \frac{1}{2}\|_N = \frac{1}{6}, \; \forall \text{ rational } \alpha \in (0,1).\] Combining the two cases, we have \(R(\alpha) \leq 1/6\) for any \(\alpha \in (0,1)\). Then we have \[\tilde{\rho} \leq \left|2 \sin \frac{\pi}{6} \right| = 1.\] The equality holds if and only if \(R(\alpha) = 1/6\), i.e. \(\alpha = 1/3\) or \(2/3\). ◻
Then we know for any \(\alpha \in (0,1) \setminus \bigcup_{c \in [0,1)} L_\beta(c)\), Example 3 has the finiteness property w.r.t. \(\tilde{\rho}\), and each \(L_\beta(c)\) has zero Hausdorff dimension. However the set \(\bigcup_{c \in [0,1)} L_\beta(c)\) is not necessarily a zero Hausdorff dimension set.
4 Some examples↩︎
From section \(3\), we know that when \(\alpha = 1/3\), Example 3 is unstabilizable. However, we can obtain a stabilizable system from Example 3 by increasing the smallest singular value.
Example 4. Switched system with \(\mathcal{M}=\{M_1, M_2\}\) similar to an example in [12] such that \[\label{eq6} M_1=\left(\begin{matrix} 2 &0 \\ 0 &1/2 \end{matrix}\right), \quad M_2 = \left(\begin{matrix} \cos \pi/3 &\sin \pi/3\\ -\sin \pi/3 &\quad \cos \pi/3 \end{matrix}\right),\qquad{(3)}\]
Lemma 11. Example 4 is pointwise stabilizable.
Proof. For any initial vector \(x_0 = (\cos \theta, \sin \theta)^T\) with \(\theta \in [0, \pi]\), we have one of following norm is less than \(1\).
\(\|A_2A_1x_0\|\).
\(\|A_2A_1A_1x_0\|\).
\(\|A_2A_1A_2x_0\|\).
\(\|A_2A_1A_2A_1x_0\|\).
\(\|A_2A_1A_1A_2x_0\|\).
Figure 2 shows the range of \(\theta\) such that each of the above norm is less than \(1\). One can obtain the precise range of \(\theta\) by solving the inequalities that the norm is less than \(1\), which is not shown here.
◻
Then to make a system stabilizable, we don’t necessarily need the smallest singular value as small as possible.
We know that the vector is projected to a lower dimensional subspace after applying a singular matrix. One general way of constructing a optimal sequence is to focus on the movement of the image of singular matrices. For example, after applying \(M_2\) once, Example 3 only describes movement of the one dimensional image in \(\mathbb{R}^2\). For some singular matrices, we can minimize the dimension of the image first.
Example 5. system with \(\mathcal{M}=\{M_1,M_2\}\) such that \[M_1=\left(\begin{matrix} 1 &0 &0\\ 0 &0 &1\\ 0 &0 &0 \end{matrix}\right), \quad M_2=\left(\begin{matrix} \cos\frac{\pi}{5} &0 &\sin\frac{\pi}{5}\\ 0&1 &0\\ -\sin\frac{\pi}{5}&0 &\cos\frac{\pi}{5}\\ \end{matrix}\right).\]
In Example 5, the image of \(M^2_1\) is a one dimensional subspace and the \(M^2_1\) have the same largest singular value as \(M_1\). To find the optimal sequence, we can first apply \(M_1\) twice to reduce the dimension of the image, then apply \(M_2\) to rotate the vector in the image subspace. Sometimes, matrix combinations can be used to reduce the dimension of the image.
Example 6. Switched system with \(\mathcal{M}=\{M_1,M_2,M_3\}\) such that \[M_1=\left(\begin{matrix} 1 &0 &0\\ 0 &1 &0\\ 0 &0 &0 \end{matrix}\right), \quad M_2=\left(\begin{matrix} 1 &0 &0\\ 0 &0&-1\\ 0 &1 &0 \end{matrix}\right), \quad M_3=\left(\begin{matrix} \cos\frac{\pi}{5} &0 &\sin\frac{\pi}{5}\\ 0&1 &0\\ -\sin\frac{\pi}{5}&0 &\cos\frac{\pi}{5}\\ \end{matrix}\right).\]
We know the image of the matrix combination \(M_1M_2M_1\) in Example 6 is a one dimensional subspace. To stabilize this system, we can apply \(M_1M_2M_1\) to any initial vectors first. However, maximizing the dimension of kernel by some matrix sequence is not always the best strategy to find the optimal sequence.
Example 7. Switched system with \(\mathcal{M}=\{M_1,M_2,M_3\}\), \[M_1=\left(\begin{matrix} 2 &0 &0\\ 0 &0 &\frac{1}{2}\\ 0 &0 &0 \end{matrix}\right), \quad M_2=\left(\begin{matrix} 1 &0 &0\\ 0 &0&-1\\ 0 &1 &0 \end{matrix}\right), \quad M_3=\left(\begin{matrix} \cos\frac{\pi}{5} &0 &\sin\frac{\pi}{5}\\ 0&1 &0\\ -\sin\frac{\pi}{5}&0 &\cos\frac{\pi}{5}\\ \end{matrix}\right).\]
Using computer programs, it’s easy to check that the optimal sequence does not start with maximizing the dimension of the kernel, i.e. \(M^2_1\). For example, when \(t=10\), the matrix combination \(M_3M_2M_1M_3M_2M_3M_1M^2_3M_2\) has smallest spectral radius.
5 Conclusion↩︎
In this paper, two different approaches to understand the stabilizability radius of linear discrete time switching systems are presented. First, we showed that in general, the stabilizability radius is lower bounded by the joint spectral subradius divided by the number of matrices. This result leads to more relationships between the stabilizability radius and joint spectral subradius. Then we focused on the two dimensional switched system with a singular matrix and a matrix with complex eigenvalues, in which we presented a method to compute the exact stabilizability radius. The parameter set is a zero Hausdorff dimension set when the stabilizability radius equals a constant, such as \(0\), are presented using this method.
Although most works of linear discrete time switching systems in the literature focus on general results, we presented some examples to show systems with singular matrices are also interesting. The singular matrices function as projections in systems, which could sometimes simplify the optimal sequence of matrices. However, it could also lead to more complex behaviors. For instance, in Example 3, the set of \(\alpha\) such that the stabilizability radius equals \(0\) has zero Hausdorff dimension and dense in \((0,1)\). In Example 7, we need to consider more matrix combinations to find the optimal sequence.
There are still many open questions about the stabilizability radius of linear discrete time switching systems. Such as finding a more precise lower bound of the stabilizability radius, or give more situations when the stabilizability radius equals the joint spectral subradius. In particular, for switched systems with singular matrices, could we find a general method to decide when would stabilizability radius equals zero? Can we say more about the finiteness property w.r.t. the stabilizability radius of System 1 or other systems?
Acknowledgments↩︎
The authors are grateful to Alan Haynes for pointing them to Ref. [19]. For the purpose of open access, the author(s) has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.