Global stability of a class of difference equations on solvable Lie algebras

Abstract

Motivated by the ubiquitous sampled-data setup in applied control, we examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a solvable matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: (1) its Lie series expansion enjoys a type of strong convergence; (2) the origin is an equilibrium; (3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of the dynamics at the origin, in particular, global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability.

Appendix

1.1 Proof of Claim 1

Proof (Claim 1)

Fix the word length $\ell \ge 2$ and the number of letters in $\widetilde{X}$, $1 \le q \le \ell$. There are $n^q$ choices of letters in $\widetilde{X}$, $r^{\ell - q}$ choices of letters in $\widetilde{W}$, and $\left( {\begin{array}{c}\ell \\ q\end{array}}\right)$ ways to position the letters in $\widetilde{X}$. Thus, there are $\left( {\begin{array}{c}\ell \\ q\end{array}}\right) n^qr^{\ell - q}$ words of length $\ell$ with q letters in $\widetilde{X}$. First, recall from (14), that $u_i := \sum _{|\omega | \le i}c_\omega \otimes (P_i\bar{\omega }_{i - 1})$. Applying (17), we have

$$\begin{aligned} \Vert u_i[k]\Vert \le \left( \sum _{\begin{array}{c} 2 \le \ell \le i \\ 1 \le q \le \ell \end{array}}\max _{|\omega | = \ell }\{\Vert c_\omega \Vert \}{\left( {\begin{array}{c}\ell \\ q\end{array}}\right) }n^qr^{\ell - q}\mu ^{\ell - 1}\Vert \imath _{i - 1}\Vert ^\ell \alpha _{i - 1}^q\Vert \bar{X}_{i - 1}[0]\Vert ^q\beta ^{\ell - q}\right) \max _{\begin{array}{c} 2 \le \ell \le i \\ 1 \le q \le \ell \end{array}}\{\lambda _{i - 1}^qs^{\ell - q}\}^k, \end{aligned}$$

whose right side is bounded above by

$$\begin{aligned} \underbrace{\left( \sum _{\ell = 2}^i\max _{|\omega | = \ell }\{\Vert c_\omega \Vert \}\mu ^{\ell - 1}\Vert \imath _{i - 1}\Vert ^\ell \sum _{q = 1}^\ell \left( {\begin{array}{c}\ell \\ q\end{array}}\right) n^qr^{\ell - q}\alpha _{i - 1}^qM^{q - 1}\beta ^{\ell - q}\right) }_{=: \gamma _i} {\underbrace{\max _{\begin{array}{c} 2 \le \ell \le i \\ 1 \le q \le \ell \end{array}}\{\lambda _{i - 1}^qs^{\ell - q}\}}_{\lambda _i}}^k\Vert \bar{X}_i[0]\Vert . \end{aligned}$$

Since $0< \lambda _{i - 1} < 1$ and $s \ge 1$, the maximization is solved by $\ell = i$ and $q = 1$, thus, the maximization term is equal to $\lambda _i$. $\square$

1.2 Proof of Lemma 5

Proof (Lemma 5)

Using $\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1} + \imath _{i - 1} \circ P_{i - 1} = \mathrm {Id}_\mathfrak {g}$ and bilinearity of the Lie bracket,

$$\begin{aligned} P_i\omega = P_i[(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_1,[Y_2,[\ldots ,Y_{|\omega |}]\cdots ] + P_i[\imath _{i - 1} \circ P_{i - 1}Y_1,[Y_2,[\ldots ,Y_{|\omega |}]\cdots ], \qquad Y_j \in \widetilde{X} \cup \widetilde{W}. \end{aligned}$$

(31)

We next decompose the second letter of the first term in (31) with respect to $\imath _0 \circ P_0$ and invoke Lemma 2:

$$\begin{aligned} P_i[(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_1,[Y_2,[\ldots ,Y_{|\omega |}]\cdots ]= & {} P_i[(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_1,[\imath _0 \circ P_0 Y_2,[\ldots ,Y_{|\omega |}]\cdots ]\nonumber \\&+ P_i\underbrace{[\underbrace{(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_1}_{\in \mathfrak {h}^{(i)}},[\underbrace{(\mathrm {Id}_\mathfrak {g}- \imath _0 \circ P_0)Y_2}_{\in \mathfrak {h}^{(1)}},[\ldots ,Y_{|\omega |}]\cdots ]}_{\in \mathfrak {h}^{(i + 1)}},\nonumber \\ \end{aligned}$$

(32)

where membership in $\mathfrak {h}^{(i + 1)}$ follows from Lemma 1; the second term is zero, since $P_i\mathfrak {h}^{(i + 1)} = 0$. Decomposing the rest of the letters in (32) with respect to $\imath _0 \circ P_0$ yields

$$\begin{aligned} P_i[(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_1,[Y_2,[\ldots ,Y_{|\omega |}]\cdots ] = P_i[(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_1,[\imath _0 \circ P_0 Y_2,[\ldots ,\imath _0 \circ P_0 Y_{|\omega |}]\cdots ].\nonumber \\ \end{aligned}$$

(33)

Now decompose the second letter of the second term in (31) with respect to $\imath _{i - 1} \circ P_{i - 1}$:

$$\begin{aligned} P_i[\imath _{i - 1} \circ P_{i - 1}Y_1,[Y_2,[\ldots ,Y_{|\omega |}]\cdots ]= & {} P_i[\imath _{i - 1} \circ P_{i - 1}Y_1,[\overbrace{(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_2}^{\in \mathfrak {h}^{(i)}},[Y_3,[\ldots ,Y_{|\omega |}]\cdots ] \nonumber \\&\quad + P_i[\imath _{i - 1} \circ P_{i - 1}Y_1,[\imath _{i - 1} \circ P_{i - 1} Y_2,[Y_3,[\ldots ,Y_{|\omega |}]\cdots ]. \end{aligned}$$

(34)

We continue in a fashion similar to that following (31), the only noteworthy difference is the decomposition of $\imath _{i - 1} \circ P_{i - 1}Y_1$ with respect to $\imath _0 \circ P_0$. $\square$

Claim 3

For all $i \ge 1$, the following diagram commutes.

Proof (Claim 3)

From the definitions of $P_0$, $P_{i - 1}$, and $\imath _{i - 1}$, we have $\mathfrak {g}= {{\,\mathrm{Im}\,}}\imath _{i - 1} \oplus \mathfrak {h}^{(i)}$ and ${{\,\mathrm{Ker}\,}}P_0 = \mathfrak {h}\supseteq \mathfrak {h}^{(i)} = {{\,\mathrm{Ker}\,}}P_{i - 1}$. Then $P_0\mathfrak {g}= P_0 {{\,\mathrm{Im}\,}}\imath _{i - 1} \oplus P_0 \mathfrak {h}^{(i)} = P_0 {{\,\mathrm{Im}\,}}\imath _{i - 1}$. $\square$

It follows immediately from Claim 3 that $\imath _0 \circ P_0 \circ \imath _{i - 1} \circ P_{i - 1} = \imath _0 \circ P_0$. Thus, the decomposition process specified above yields

$$\begin{aligned} P_i\omega= & {} P_i[\imath _{i - 1} \circ P_{i - 1} Y_1,[\imath _{i - 1} \circ P_{i - 1} Y_2,[Y_3,[\ldots ,Y_{|\omega |}]\cdots ] \nonumber \\&\quad + P_i[(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_1,[\imath _0 \circ P_0 Y_2,[\ldots ,\imath _0 \circ P_0 Y_{|\omega |}]\cdots ] \nonumber \\&\quad + P_i[\imath _0 \circ P_0Y_1,[(\mathrm {Id}_\mathfrak {g}- \imath _{i - 1} \circ P_{i - 1})Y_2,[\imath _0 \circ P_0 Y_3,[\ldots ,\imath _0 \circ P_0 Y_{|\omega |}]\cdots ].\nonumber \\ \end{aligned}$$

(35)

Applying this process to the rest of the letters in the first word of (35) completes the proof. $\square$

1.3 Proof of Claim 2

Proof (Claim 2)

Suppose f satisfies (5). In particular, suppose there exists $\varrho _1 \le 1$ such that

$$\begin{aligned} \Vert X_1\Vert ,\ldots ,\Vert X_n\Vert ,\Vert W_1\Vert ,\ldots ,\Vert W_r\Vert < \varrho _1. \end{aligned}$$

On this domain, we have $\Vert \omega \Vert \le \mu ^{|\omega | - 1}\varrho _1^{|\omega |}$ and

$$\begin{aligned} \sum _\omega \mu ^{|\omega | - 1}\Vert c_\omega \Vert \varrho _1^{|\omega |} < \infty . \end{aligned}$$

We can rewrite this summation by grouping all words of the same length:

$$\begin{aligned} \sum _{\ell = 2}^\infty \mu ^{\ell - 1}\left( \sum _{|\omega | = \ell }\Vert c_\omega \Vert \right) \varrho _1^\ell , \end{aligned}$$

which can be viewed as a series over the single index $\ell$. Since this series converges, by the root test [33, Theorem 3.33], we have

$$\begin{aligned} \begin{aligned} \limsup _{\ell \rightarrow \infty }\root \ell \of {\mu ^{\ell - 1}\varrho _1^\ell \sum _{|\omega | = \ell }\Vert c_\omega \Vert }&= \varrho _1\limsup _{\ell \rightarrow \infty }\mu ^{1 - \frac{1}{\ell }}\limsup _{\ell \rightarrow \infty }\root \ell \of {\sum _{|\omega | = \ell }\Vert c_\omega \Vert } \\&= \varrho _1\mu \limsup _{\ell \rightarrow \infty }\root \ell \of {\sum _{|\omega | = \ell }\Vert c_\omega \Vert } \\&\le 1. \end{aligned} \end{aligned}$$

Let $0< \varrho _2 < \frac{\varrho _1}{\Vert \imath _0\Vert }$. Applying the root test to the series

$$\begin{aligned} \sum _\omega (\mu \Vert \imath _0\Vert )^{|\omega | - 1}|\omega |\Vert c_\omega \Vert \varrho _2^{|\omega |}, \end{aligned}$$

(36)

we have

$$\begin{aligned} \begin{aligned} \limsup _{\ell \rightarrow \infty }\root \ell \of {\ell (\mu \Vert \imath _0\Vert )^{\ell - 1}\varrho _2^\ell \sum _{|\omega | = \ell }\Vert c_\omega \Vert }&= \varrho _2\mu \Vert \imath _0\Vert \limsup _{\ell \rightarrow \infty }\root \ell \of {\ell }\limsup _{\ell \rightarrow \infty }\root \ell \of {\sum _{|\omega | = \ell }\Vert c_\omega \Vert } \\&= \varrho _2\mu \Vert \imath _0\Vert \limsup _{\ell \rightarrow \infty }\root \ell \of {\sum _{|\omega | = \ell }\Vert c_\omega \Vert } \\&< \varrho _1\mu \limsup _{\ell \rightarrow \infty }\root \ell \of {\sum _{|\omega | = \ell }\Vert c_\omega \Vert } \\&< 1, \end{aligned} \end{aligned}$$

which implies that (36) converges. Let $\varrho \le \varrho _2^2$, then for all $|\omega | \ge 2$, $\varrho ^{|\omega | - 1} < \varrho _2^{|\omega |}$. Then, by the comparison test [33, Theorem 3.25], if $\Vert \bar{X}_0\Vert ,\Vert \bar{W}_0\Vert \le \varrho$, then (29) converges. $\square$