Topological entropy of switched nonlinear and interconnected systems

A general upper bound for topological entropy of switched nonlinear systems is constructed, using an asymptotic average of upper limits of the matrix measures of Jacobian matrices of strongly persistent individual modes, weighted by their active rates. A general lower bound is constructed as well, using a similar weighted average of lower limits of the traces of these Jacobian matrices. In a case of interconnected structure, the general upper bound is readily applied to derive upper bounds for entropy that depend only on “network-level” information. In a case of block-diagonal structure, less conservative upper and lower bounds for entropy are constructed. In each case, upper bounds for entropy that require less information about the switching signal are also derived. The upper bounds for entropy and their relations are illustrated by numerical examples of a switched Lotka–Volterra ecosystem model.

1. In this subsection, an inequality between two vectors or matrices of the same size, or between a vector or matrix and a scalar, is to be interpreted entrywise (e.g., \( A \ge 0 \) means that A is a nonnegative matrix).

2. Note that \( \tilde{A}^\textrm{N}\) has to be a Metzler matrix; thus, its eigenvalue with the largest real part is real [10, Th. 10.2, p. 167].

3. We denote by \( 0< t_1< t_2 < \cdots \) the sequence of switches and let \( t_0:= 0 \), with \( \sigma (t) = 1 \) on \( [t_{2k}, t_{2k+1}) \) and \( \sigma (t) = 2 \) on \( [t_{2k+1}, t_{2k+2}) \).

4. Specifically, following the proof of [2, Th. 3] S contains the \( \omega \)-limit set of (50) if for all \( p \in {\mathcal {P}}\) and \( x \in \mathbb {R}_{\ge 0}^n \backslash (S \cup \{0\}) \), we have \( {\textbf{1}}_n^\top f_p(x) < 0 \), i.e., \( r_p^\top x + x^\top A_p x < 0 \). Note that \( r_p^\top x + x^\top A_p x = r_p^\top x + x^\top (A_p + A_p^\top )\, x/2 \le (r_p + \lambda _{\max }(A_p + A_p^\top )\, x/2)^\top x \), which is negative if \( 2 r_p^i + \lambda _{\max }(A_p + A_p^\top )\, x_i < 0 \) for all \( i \in \{1, \ldots , n\} \) as \( x \in \mathbb {R}_{\ge 0}^n \backslash \{0\} \).

5. In this example, we use the set S defined by (52) which contains the \( \omega \)-limit set for every switched system (50) satisfying (51). For a given family of coefficients, one can usually construct more precise over-approximation of the \( \omega \)-limit set and thus obtain less conservative upper bounds for entropy, such as those in [51, Example 3.6].

6. As shown in “Appendix A,” their main difference is that, in the proof of Lemma 5, the variational arguments are applied to the line segment connecting two solutions instead of the one connecting two initial states, which results in the different locations of convex hulls in, e.g., \( {{\overline{\eta }}}_\sigma (t) \) and \( {{\overline{\eta }}}_\sigma ^\textrm{alt}(t) \).

7. In our construction we ignore the case when (K) falls between integer multiples of \(( \theta )\), but if that happens then the construction can be slightly modified by taking an arbitrary \(( \bar{x} \in K )\) and defining the grid by \(( G(\theta ) := \{\bar{x} + (k_1 \theta _1, \ldots , k_n \theta _n) \in K: k_1, \ldots , k_n \in \mathbb {Z}\} )\) so that it is nonempty.

8. Also, the lower bound (66) does not require a condition similar to (63). For this reason, in [51, Lemma 2.3] the requirement that all \( \theta _i \) are nonincreasing in T is not actually needed for the lower bound [51, eq. (13)].

9. There was a typo in [50, Lemma 2] where [50, eq. (17)] should be replaced with (63), as the inequality in (63) needs to hold for each \( i \in \{1, \ldots , n\} \) instead of the upper limits of the sum.

This work was supported in part by the National Science Foundation Grant CMMI-2106043 and in part by the National Science Foundation Grant EPCN-1608880.

Authors and Affiliations

1. Department of Electrical and Computer Engineering, Rutgers University-New Brunswick, Piscataway, NJ, 08854, USA

   Guosong Yang

2. Coordinated Science Laboratory, University of Illinois Urbana-Champaign, Urbana, IL, 61801, USA

   Daniel Liberzon

3. Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA

   João P. Hespanha

Corresponding author

Correspondence to Guosong Yang.

This paper is dedicated to Prof. Eduardo D. Sontag on the occasion of his 70th birthday. As detailed later in the paper, our work is heavily influenced by Prof. Sontag’s research contributions to the areas of contractive nonlinear and interconnected systems.

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Appendix A Proofs of Lemmas 4 and 5

Lemmas 4 and 5 are established based on similar properties of a linear time-varying (LTV) system

with a piecewise-continuous, matrix-valued function \( A: \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}^{n \times n} \). The solution to (A1) at time \( t \ge 0 \) with initial state v is given by \( \xi (t, v) = \Phi _A(t, 0)\, v \), where \( \Phi _A(t, 0) \) is the state-transition matrix and satisfies Coppel’s inequalities [14, Th. 2.8.27, p. 34]

and Liouville’s formula [9, Th. 4.1, p. 28]

Proof of Lemma 4

We prove Lemma 4 by writing the Jacobian matrix of a solution to the switched nonlinear system (2) with respect to initial state \( J_x \xi _\sigma (t, x) \) as a matrix solution to the LTV system (A1) with a suitable function A(t) , using variational arguments from nonlinear systems analysis (see, e.g., [24, Sect. 4.2.4]). For all \( v \in \mathbb {R}^n \), we have \( J_x \xi _\sigma (0, v) = I \) and

for all \( t \ge 0 \) that are not switches. Hence, for each fixed \( v \in \mathbb {R}^n \), the matrix \( J_x \xi _\sigma (t, v) \) is equal to the state-transition matrix \( \Phi _A(t, 0) \) of (A1) with \( A(t) = J_x f_{\sigma (t)}(\xi _\sigma (t, v)) \).

First, given arbitrary initial states \( x, {{\bar{x}}} \in K \), let

Then \( \nu (\rho ) \in {{\,\textrm{co}\,}}(K) \) and \( \nu '(\rho ) = {{\bar{x}}} - x \) for all \( \rho \in [0, 1] \). Hence,

for all \( t \ge 0 \), where the second inequality follows from the second inequality in (A2) with \( A(t) = J_x f_{\sigma (t)}(\xi _\sigma (t, \nu (\rho ))) \) and \( \Phi _A(t, 0) = J_x \xi _\sigma (t, \nu (\rho )) \), and the last equality follows from the transformation (58). Hence, (56) holds.

Second, for each fixed \( v \in \mathbb {R}^n \), Liouville’s formula (A3) with \( A(t) = J_x f_{\sigma (t)}(\xi _\sigma (t, v)) \) and \( \Phi _A(t, 0) = J_x \xi _\sigma (t, v) \) implies that

Then

for all \( t \ge 0 \), where the last equality follows from the transformation (58). Hence, (57) holds. \(\square \)

Proof of Lemma 5

We prove Lemma 5 by writing the difference between two solutions to the switched nonlinear system (2) as a solution to the LTV system (A1) with a suitable function A(t) , using variational arguments similar to those in the proof of Lemma 4. Given arbitrary initial states \( x, {{\bar{x}}} \in K \), let

Then \( \nu (t, \rho ) \in {{\,\textrm{co}\,}}(\xi _\sigma (t, K)) \) and \( \partial _\rho \nu (t, \rho ) = \xi _\sigma (t, {{\bar{x}}}) - \xi _\sigma (t, x) \) for all \( \rho \in [0, 1] \) and \( t \ge 0 \). Hence,

for all \( t \ge 0 \) that are not switches. Therefore, \( \xi _\sigma (t, {{\bar{x}}}) - \xi _\sigma (t, x) \) is the solution to (A1) with \( A(t) = \int _{0}^{1} J_x f_{\sigma (t)}(\nu (t, \rho )) \textrm{d}\rho \) at time t with initial state \( {{\bar{x}}} - x \). Consequently, (A2) implies that

for all \( t \ge 0 \). Moreover, as the matrix measure \( \mu \) is a convex function, Jensen’s inequality [3, Th. 11.24, p. 417] implies that

and

for all \( t \ge 0 \). Then (59) follows from the transformation (58). \(\square \)

Appendix B Proof of Lemma 6

1. 1.

   If (62) holds, then

   $$\begin{aligned} K \subset \bigcup _{x \in G(\theta )} R(x) \subset \bigcup _{x \in G(\theta )} B_{f_\sigma }(x, \varepsilon , T). \end{aligned}$$

   Hence, \( G(\theta ) \) is \( (T, \varepsilon ) \)-spanning following the definition (4), and thus,

   $$\begin{aligned} S(f_\sigma , \varepsilon , T, K) \le \#G(\theta ) \le \prod _{i=1}^{n} \bigg ( \bigg \lfloor \frac{2 r_2}{\theta _i} \bigg \rfloor + 1 \bigg ) \le \prod _{i=1}^{n} \bigg ( \frac{2 r_2}{\theta _i} + 1 \bigg ). \end{aligned}$$

   If (62) holds for all \( T \ge 0 \) and \( \varepsilon > 0 \), then the definition of entropy (5) implies that

   $$\begin{aligned} h(f_\sigma , K)&\le \limsup _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \sum _{i=1}^{n} \frac{\log (2 r_2/\theta _i + 1)}{T} \nonumber \\&\le \limsup _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \sum _{i=1}^{n} \frac{\log (1/\theta _i)}{T} \nonumber \\&\quad \, + \sum _{i=1}^{n} \limsup _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \frac{\log (\theta _i + 2 r_2)}{T}, \end{aligned}$$

   (B4)

   where the last inequality holds as the upper limit is a subadditive function. Moreover, for all \( i \in \{1, \ldots , n\} \), the summands in the last term in (B4) satisfy

   $$\begin{aligned} \begin{aligned}&\limsup _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \frac{\log (\theta _i + 2 r_2)}{T} \le \limsup _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \frac{\max \{\log (2\theta _i),\, \log (4 r_2)\}}{T} \\&\qquad = \max \left\{ \limsup _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \frac{\log (2 \theta _i)}{T},\, \lim _{T \rightarrow \infty } \frac{\log (4 r_2)}{T} \right\} \\&\qquad = \max \left\{ \limsup _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \frac{\log \theta _i}{T},\, 0 \right\} , \end{aligned} \end{aligned}$$

   where the inequality holds as the logarithm is an increasing function and \( \theta _i, r_2 > 0 \), and the last equality holds in part because \( r_2 \) is independent of T. Then (63) implies (64).^{Footnote 9}

2. 2.

   If (65) holds, then for all distinct points \( x, {{\bar{x}}} \in G(\theta ) \), we have \( {{\bar{x}}} \notin B_{f_\sigma }(x, \varepsilon , T) \) as \( {{\bar{x}}} \notin R(x) \). Hence, \( G(\theta ) \) is \( (T, \varepsilon ) \)-separated following the definition (6), and thus,

   $$\begin{aligned} N(f_\sigma , \varepsilon , T, K) \ge \#G(\theta ) \ge \prod _{i=1}^{n} \bigg \lfloor \frac{2 r_1}{\theta _i} \bigg \rfloor \ge \prod _{i=1}^{n} \max \left\{ \frac{2 r_1}{\theta _i} - 1,\, 1 \right\} . \end{aligned}$$

   If (65) holds for all \( T \ge 0 \) and \( \varepsilon > 0 \), then the alternative definition of entropy (7) implies that

   $$\begin{aligned} h(f_\sigma , K)&\ge \liminf _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \sum _{i=1}^{n} \frac{\log (\max \{2 r_1/\theta _i - 1,\, 1\})}{T} \nonumber \\&\ge \liminf _{\varepsilon \searrow 0} \limsup _{T \rightarrow \infty } \sum _{i=1}^{n} \frac{\log (1/\theta _i)}{T} \nonumber \\&\quad +\, \sum _{i=1}^{n} \liminf _{\varepsilon \searrow 0} \liminf _{T \rightarrow \infty } \frac{\log (\max \{2 r_1 - \theta _i,\, \theta _i\})}{T}, \end{aligned}$$

   (B5)

   where the last inequality holds as the lower limit is a superadditive function, and for two functions \( g, {{\bar{g}}}: \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}\), we have

   $$\begin{aligned} \limsup _{T \rightarrow \infty } (g(T) + {{\bar{g}}}(T)) \ge \limsup _{T \rightarrow \infty } g(T) + \liminf _{T \rightarrow \infty } {{\bar{g}}}(T). \end{aligned}$$

   Moreover, for all \( i \in \{1, \ldots , n\} \), the summands in the last term in (B5) satisfy

   $$\begin{aligned} \begin{aligned} \liminf _{\varepsilon \searrow 0} \liminf _{T \rightarrow \infty } \frac{\log (\max \{2 r_1 - \theta _i,\, \theta _i\})}{T}&\ge \liminf _{\varepsilon \searrow 0} \liminf _{T \rightarrow \infty } \frac{\log (\max \{r_1,\, \theta _i\})}{T} \\&\ge \liminf _{\varepsilon \searrow 0} \liminf _{T \rightarrow \infty } \frac{\log r_1}{T} = 0, \end{aligned} \end{aligned}$$

   where the first inequality holds as the logarithm is an increasing function and \( r_1, \theta _i > 0 \), and the equality holds as \( r_1 \) is independent of T. Hence, (66) holds. \(\square \)

Appendix C Proof of Lemma 7

We prove only the lower bound (69) here; the upper bound (68) and lower bound (70) can be established using similar arguments (the omitted proof can be found in [52]). For brevity, we define the following functions on \( \mathbb {R}_{\ge 0} \):

with \( {{\bar{a}}}^i(0):= \max \{a_{\sigma (0)}^i(0),\, 0\} \). Also, note that for two continuous functions \( g, {{\bar{g}}}: \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}\), we have

First, we eliminate modes that are not persistent. The definition of \( {\mathcal {P}}_\infty \) in (11) implies that \( t_\infty := \sup \{t \ge 0: \sigma (t) \in {\mathcal {P}}\backslash {\mathcal {P}}_\infty \} \) is finite. Then

for all \( i \in \{1, \ldots , m\} \) and \( T > t_\infty \), where the inequality follows from (C6). Hence,

Second, we eliminate modes that are persistent but not strongly persistent. For all \( i \in \{1, \ldots , m\} \) and \( p \in {\mathcal {P}}_\infty \), as \( {\check{a}}_p^i = \liminf _{t \rightarrow \infty :\, \sigma (t) = p} a_p^i(t) \) are finite, \( {{\check{\alpha }}}_p^i:= \inf _{t \ge 0} a_p^i(t) {{\,\mathrm{\mathbbm {1}}\,}}_p(\sigma (t)) \) are finite as well. Then

for all \( i \in \{1, \ldots , m\} \) and \( T > 0 \), where the first inequality follows from (C6). Moreover, for all \( p \notin {\mathcal {P}}_\infty ^+\), as \( \rho _p(t) \ge 0 \) for all \( t \ge 0 \) and \( {{\hat{\rho }}}_p = \limsup _{t \rightarrow \infty } \rho _p(t) = 0 \), we have \( {{\hat{\rho }}}_p = \lim _{t \rightarrow \infty } \rho _p(t) = 0 \). Then

Hence,

Finally, as \( {\mathcal {P}}_\infty ^+\) is a finite set, the lower limit in the definition of \( {\check{a}}_p^i \) implies that, for each \( \delta > 0 \), there exists a large enough \( t_\delta \ge 0 \) such that

For all \( i \in \{1, \ldots , m\} \) and \( t \ge 0 \), consider

If \( t > t_\delta \), then

Otherwise \( t \in [0, t_\delta ] \), and hence,

Combining the two cases, we obtain

Then

for all \( T > 0 \), where the inequality follows in part from (C6). Hence,

Then (69) holds as \( \delta > 0 \) is arbitrary. \(\square \)

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