BIBO Stability of Linear Control Systems on Lie Group Examples

Abstract

We develop a collection of nontrivial examples that illustrate and test recent stability results for linear control systems ($LCS$) on Lie groups. We treat the main structural classes: Abelian (${\mathbb{R}}^n$), nilpotent (Heisenberg), solvable non-nilpotent (rigid motions of the plane $SE\left(2\right)$), compact semisimple ($SO\left(3\right)$), noncompact semisimple ($SL(2,\mathbb{R})$ via Iwasawa decomposition) and mixed/Levi-type groups. The examples are designed to (i) show the sharpness of geometric boundedness criteria, (ii) exhibit typical failure modes (exponential escape, polynomial central drift, noncompact neutrals), and (iii) demonstrate how the canonical quotient and suitable outputs recover $BIBO$ stability. The executive framework ($ICS$ existence/uniqueness, canonical quotient $G/\Gamma$, $BIBO$ characterization, robustness and ISS-type bounds) is briefly recalled; the main part of the paper consists of detailed worked examples implementing the practical checklist for applying these theorems.

1. Introduction

This paper presents a worked collection of nontrivial examples that clarify the geometric stability phenomena for linear control systems on Lie groups, based on the structural results developed in [1]. There, the authors prove general theorems on internal boundedness, invariant control sets ($ICS$), canonical quotients $G/\Gamma$, and $BIBO$ characterization. Here we use that theory as a toolkit, sketch the needed proofs, and concentrate on concrete computations and geometric interpretation.

In Euclidean spaces, stability of linear systems is often decided directly from the spectrum of the system matrix [2,3,4,5,6,7,8]. On Lie groups, stability is inherently global: it depends not only on the eigenvalues of the drift derivation, but also on how the corresponding expanding, neutral, and contracting directions fit into the group structure. The flow of automorphisms generated by the drift splits the Lie algebra into expanding, neutral, and contracting parts, and the associated subgroups $G^{+},G^0,$ and $G^{-}$ describe exponential escape, neutral behavior, and exponential contraction. $ICS$ then provide compact “cores” that collect all bounded controlled trajectories, while a canonical closed normal subgroup $\Gamma$ absorbs the unstable and noncompact neutral directions so that the quotient $G/\Gamma$ becomes the natural stage for $BIBO$ analysis.

The main aim of this article is to show how these abstract objects appear and can be used in concrete Lie group models. For each example we identify the derivation, compute the splitting ${\mathfrak{g}}^{+},{\mathfrak{g}}^0,$ and ${\mathfrak{g}}^{-}$ and the subgroups $G^{+},G^0,$ and $G^{-}$, verify the Lie algebra rank condition, construct $\Gamma$ when there are escaping directions, describe the ICS and minimal compact trap, and analyze which homomorphic outputs are $BIBO$ by looking at how they interact with $G^{+}$ and the neutral subgroup [9,10,11]. We also show how standard variation-of-constants estimates lead to ISS-type bounds on the canonical quotient $G/\Gamma$ and how these bounds persist under small perturbations that do not change the sign pattern of the relevant eigenvalues.

The systems considered here are not mere “toy” examples. They include Hurwitz systems on ${\mathbb{R}}^n$, the Heisenberg group as a prototype nilpotent system with central directions, $SE\left(2\right)$ as a model for planar mechanics and robotics, $SO\left(3\right)$ and $SL(2,\mathbb{R})$ as basic semisimple groups, and a Levi-type solvable⋊compact group arising from visual-cortex modeling. In each case we push the structural results of [1] down to the level of explicit formulas and geometric pictures: where escapes occur, how $ICS$ look, what the quotient $G/\Gamma$ is in coordinates, and which outputs “see” only the non-escaping dynamics. The $SE\left(2\right)$ example, in particular, realizes a planar mobile robot with two controls (forward motion and heading) plus a contracting drift on the translational part; the canonical subgroup is the translation subgroup $\Gamma ={\mathbb{R}}^2$, the quotient $SE\left(2\right)/\Gamma \simeq SO\left(2\right)$ captures all homomorphic $BIBO$ outputs (the heading), and one obtains a unique bounded $ICS$ together with explicit ISS–type bounds for the position.

Beyond serving as an “example and application” companion to [1,12,13,14], the paper also extends the theory by proving new ISS-type estimates on $G^{-}$ and on the canonical quotient $G/\Gamma$ (Lemma 1 and Proposition 1), and by turning the abstract decomposition $(G^{\pm },G^0,\Gamma )$ into concrete, quantitatively usable tools for stability and $BIBO$ analysis on specific Lie group models. The structural theorems are taken from [1], but the detailed calculations, the explicit construction of $\Gamma$ and $G/\Gamma$ in each class, and the example-level robustness and ISS estimates are original and intended to make the abstract theory readily usable in concrete control problems. Following a reviewer’s suggestion we also sketch the proofs of the cited theorems.

The organization of the paper is as follows. Section 2 briefly recalls the framework (definitions, $ICS$, canonical quotient, and main theorems stated with the assumptions needed in the examples). Section 3, Section 4 and Section 5 each treat one example class: group definition, chosen derivation, verification of hypotheses, explicit computation of ${\mathfrak{g}}^{\pm }$ and ${\mathfrak{g}}^0$, and $G^{\pm }$ and $G^0$, determination of $\Gamma$ when relevant, description of the $ICS$ and minimal trap, and analysis of $BIBO$ outputs with quantitative remarks on ISS bounds and robustness. We work in finite dimensions with connected Lie groups, use left-invariant Riemannian metrics to measure boundedness, and take “outputs” to be group homomorphisms unless explicitly stated otherwise.

2. Framework and Key Definitions

This section fixes notation and recalls the geometric and control objects used throughout the paper. We collect standard facts and introduce precise meanings for terms used in the statements of the main results.

• System class and notation. Let G be a connected finite-dimensional Lie group with identity e and Lie algebra $\mathfrak{g}=T_{e}G$. We consider linear control systems ($LCS$) on G [15], of the form

  $${\mathrm{\Sigma }}_G:\dot{g}\left(t\right)=\mathcal{X}\left(g\left(t\right)\right)+{\displaystyle \sum _{j=1}^m}{u}_j\left(t\right)Y_j\left(g\left(t\right)\right),$$

  where

  (a)

  $\mathcal{X}$ is a linear vector field on G: its flow ${\phi }_t$ is a one-parameter subgroup of $\mathrm{Aut}\left(G\right)$ and the infinitesimal generator at the identity is a derivation $\mathcal{D}\in \mathrm{Der}\left(\mathfrak{g}\right)$;

  (b)

  $Y_1,\dots ,Y_m$ are left-invariant vector fields on G (identified with elements of $\mathfrak{g}$);

  (c)

  Admissible controls $u(·)=(u_1(·),\dots ,u_m(·))$ are measurable essentially bounded functions taking values in a fixed compact control set $U\subset {\mathbb{R}}^m$ with $0\in \mathrm{int}U$.

Global existence and uniqueness hold for such systems (complete vector fields); we denote solutions by $\varphi (t,g,u)$ for initial condition $g\in G$ and control $u\in \mathcal{U}$.

2.

Variation in constants and reduction to the identity [15]. A key identity is

$$\varphi (t,g,u)={\phi }_t\left(g\right)\varphi (t,e,u),t\ge 0,$$

which decomposes any trajectory into the drift action on g and a trajectory from the identity. Thus the dynamics are determined by the family $\left\{\varphi \right(t,e,u):t\ge 0,u\in \mathcal{U}\}$. The positive reachable set

$${\mathcal{O}}^{+}\left(g\right):=\{\varphi (t,g,u):t\ge 0,u\in \mathcal{U}\},$$

and in particular ${\mathcal{O}}^{+}\left(e\right)$, plays a central role; its (closure) boundedness underlies our global analysis.

3.

Dynamical splitting induced by the drift [16]. The derivation $\mathcal{D}\in \mathrm{Der}\left(\mathfrak{g}\right)$ associated with $\mathcal{X}$ has a real Jordan decomposition, yielding a $\mathcal{D}$-invariant splitting

$$\mathfrak{g}={\mathfrak{g}}^{+}\oplus {\mathfrak{g}}^0\oplus {\mathfrak{g}}^{-},$$

where ${\mathfrak{g}}^{\pm }$ collect generalized eigenspaces with positive/negative real part and ${\mathfrak{g}}^0$ those with zero real part. Integrating gives connected subgroups $G^{+},G^0,G^{-}\subset G$ (closed in our finite-dimensional setting). Under the drift ${\phi }_t$, $G^{+}$ expands, $G^{-}$ contracts, and $G^0$ is neutral; their interaction governs escape versus trapping.

4.

Lie algebra rank condition ($LARC$) and accessibility [17,18]. Let

$$\Delta =\mathrm{span}\{Y_1,\dots ,Y_m\}\subset \mathfrak{g}$$

be the span of control directions, and let ${\mathcal{L}}_0$ be the smallest $\mathcal{D}$-invariant Lie subalgebra of $\mathfrak{g}$ containing $\Delta$. We say ${\mathrm{\Sigma }}_G$ satisfies $LARC$ if ${\mathcal{L}}_0=\mathfrak{g}$. By Sussmann’s Orbit Theorem [19], we may assume $LARC$ without affecting stability. Then the system is locally accessible from every point: reachable sets, and in particular ${\mathcal{O}}^{+}\left(e\right)$, have nonempty interior, which is crucial for the existence of control sets with nonempty interior and for the concatenation and approximate controllability arguments used later.

5.

Control sets and invariant control sets ($ICS$) [20]. Control sets are maximal regions where approximate controllability holds and which are forward invariant in the closure sense. A set $\mathcal{C}\subset G$ is a control set if (i) each $x\in \mathcal{C}$ admits controls keeping trajectories arbitrarily close to $\mathcal{C}$ forward in time, and (ii) $\mathcal{C}\subset \overline{{\mathcal{O}}^{+}\left(x\right)}$ for all $x\in \mathcal{C}$. An invariant control set ($ICS$) additionally satisfies

$$x\in \mathcal{C},u\in \mathcal{U}⟹\varphi (t,x,u)\in \mathcal{C}\mathrm{for\; all} t\ge 0.$$

Bounded $ICS$ with nonempty interior provide compact trapping regions whose closures typically contain all bounded trajectories and play the role of global attractors for bounded control dynamics. Under $LARC$, such $ICS$ (when they exist) have nonempty interior and enjoy the robustness and concatenation properties used throughout the paper.

Definition 1.

Let ${\mathrm{\Sigma }}_G$ be a linear control system on G and let F:$G\to H$ be a Lie group homomorphism (the output map) with kernel $N=\ker F$. We say that ${\mathrm{\Sigma }}_G$ is $BIBO$ stable relative to F (or that F yields a $BIBO$ output) if for every bounded admissible control $u\in \mathcal{U}$ the output trajectory

$$y_u\left(t\right):=F\left(\varphi (t,e,u)\right)\in H,t\ge 0,$$

is a bounded with respect to any left-invariant Riemannian metric on H [1].

Definition 2.

When N is closed and invariant under the drift so that the system projects to $G/N\simeq F\left(G\right)$, ${\mathrm{\Sigma }}_G$ is $BIBO$ relative to F iff for every bounded control the projected trajectory $\{{\varphi }_{G/N}(t,\overline{e},u):t\ge 0\}$ is bounded in $G/N$ [1].

A set or trajectory is bounded if it is bounded in this metric. For outputs we focus on homomorphic outputs: let $F:G\to H$ be a Lie group homomorphism into a connected Lie group H. Boundedness of the output is exactly boundedness of the projected trajectory in the quotient $G/N\simeq F\left(G\right)$ where $N=\ker F$, provided N is closed and invariant under the drift so that the system projects. The main theorems relate this quotient-level boundedness to the relative position of $G^{+}$ and $G^0$ with respect to N and to the compactness of the projected neutral subgroup.

• Remarks and practical considerations.

• The decomposition $\mathfrak{g}={\mathfrak{g}}^{+}\oplus {\mathfrak{g}}^0\oplus {\mathfrak{g}}^{-}$ is intrinsic to the derivation $\mathcal{D}$ (hence to the linear drift) [21], and is compatible with conjugation by $G^0$; in applications one computes it via the real Jordan form of the linearization of $\mathcal{X}$ at the identity (or via the complex eigenstructure of $\mathcal{D}$).
• $LARC$ is a mild controllability hypothesis satisfied in most illustrative examples (full actuation or bracket-generation conditions). Failure of $LARC$ forces restriction to reachable submanifolds and requires a variant treatment.
• When working with homomorphic outputs F, two technical points are recurrent: (i) one needs $N=\ker F$ to be closed and invariant under the drift ${\phi }_t$ so that the dynamics project to $G/N$, and (ii) $LARC$ should descend to the quotient to apply the internal boundedness criteria there. These assumptions are explicit in the main theorems.
• The canonical subgroup $\Gamma$ used to classify all $BIBO$ outputs is built from the algebraic data above: it is the minimal closed normal $\phi$-invariant subgroup that contains $G^{+}$ and the noncompact parts of $G^0$; factoring by $\Gamma$ eliminates all obstructions to boundedness.

These notions provide the language used in the subsequent statements and the example catalog: theorems are formulated in terms of ${\mathcal{O}}^{+}\left(e\right)$, $G^{\pm },G^0$, $ICS$ and factorizations through quotients like $G/N$ or $G/\Gamma$.

3. Main Results Statements Sketch of the Proof and Discussion

We state the principal theorems from [1] precisely and expand on their meanings, immediate corollaries, typical hypotheses, and proof ideas (sketches only). This extended presentation is intended to clarify how to apply the results in examples and to expose the geometric mechanisms behind each statement.

In the sketch of the following theorem we use a fact that was showed in [1]: every element of the central subgroup $G^0$ whose positive orbit is bounded belongs to the recurrent set. We start with new results.

Lemma 1

(ISS estimate along $G^{-}$). Assume $G^{+}=\left\{e\right\}$, $G^0$ is compact, and the restriction of $\mathcal{D}$ to ${\mathfrak{g}}^{-}$ has spectrum contained in $\{\mathrm{Re}\lambda \le -\alpha \}$ for some $\alpha >0$. Let $u\in L^{\infty }([0,\infty ),U)$ and write the trajectory as $\varphi (t,e,u)=g^{-}\left(t\right)g^0\left(t\right)$ with $g^{-}\left(t\right)\in G^{-}$, $g^0\left(t\right)\in G^0$. Then there exist constants $C,\gamma >0$, depending only on α, the choice of left-invariant metric, and ${\sup }_{t\ge 0}\parallel u\left(t\right)\parallel$, such that

$$d\left(g^{-}\left(t\right),e\right)\le C{\mathrm{e}}^{-\alpha t}d\left(g^{-}\left(0\right),e\right)+\gamma {\parallel u\parallel }_{L^{\infty }\left([0,t]\right)}\mathit{for}\mathit{all}t\ge 0.$$

Proof of Lemma 1.

Fix a left-invariant Riemannian metric on G and restrict it to $G^{-}$. Since $G^{-}$ is a connected Lie subgroup of a finite-dimensional Lie group, there is a neighborhood U of e on which the exponential map

$${\exp }^{-}:{\mathfrak{g}}^{-}\to G^{-}$$

is a diffeomorphism, and the Riemannian distance is equivalent to the norm in ${\mathfrak{g}}^{-}$: there exist constants $c_1,c_2>0$ such that for all $z\in {\mathfrak{g}}^{-}$ with ${\exp }^{-}\left(z\right)\in U$,

$$c_1\parallel z\parallel \le d\left({\exp }^{-}\left(z\right),e\right)\le c_2\parallel z\parallel .$$

(1)

Write the trajectory as $\varphi (t,e,u)=g^{-}\left(t\right)g^0\left(t\right)$ with $g^{-}\left(t\right)\in G^{-}$, $g^0\left(t\right)\in G^0$. Since $G^0$ is compact and u is bounded, the factor $g^0\left(t\right)$ remains in a compact subset of $G^0$, and all coefficients obtained by projecting the control vector fields onto ${\mathfrak{g}}^{-}$ are uniformly bounded.

On U we define $z\left(t\right)\in {\mathfrak{g}}^{-}$ by $g^{-}\left(t\right)={\exp }^{-}\left(z\left(t\right)\right)$.

The linear control system and the $\mathcal{D}$-invariance of ${\mathfrak{g}}^{-}$ yield an ODE for $z\left(t\right)$ of the form

$$\dot{z}\left(t\right)={\mathcal{D}}^{-}z\left(t\right)+f\left(t,z\left(t\right)\right)+{\displaystyle \sum _{i=1}^m}{b}_i\left(t,z\left(t\right)\right)u_i\left(t\right),$$

(2)

where ${\mathcal{D}}^{-}{:=\mathcal{D}|}_{{\mathfrak{g}}^{-}}$, the functions f and $b_i$ are smooth, and $f(t,0)\equiv 0$. The spectral assumption on ${\mathcal{D}}^{-}$ implies the existence of a norm ${\parallel ·\parallel }_{-}$ on ${\mathfrak{g}}^{-}$ and constants $\alpha >0$, $K\ge 1$ such that the semigroup generated by ${\mathcal{D}}^{-}$ satisfies

$$\parallel e^{t{\mathcal{D}}^{-}}{\parallel }_{\mathcal{L}\left({\mathfrak{g}}^{-}\right)}\le K{\mathrm{e}}^{-\alpha t}\mathrm{for}\mathrm{all}t\ge 0.$$

Since $g^0\left(t\right)$ and $u\left(t\right)$ are bounded, there exists $M>0$ and a neighborhood of 0 in which

$${\parallel f(t,z)\parallel }_{\_}\le {M\parallel z\parallel }_{\_},{∥{\sum }_i{b}_i(t,z)u_i\left(t\right)∥}_{\_}\le M\parallel u\left(t\right)\parallel$$

for all $t\ge 0$ and all z such that ${\exp }^{-}\left(z\right)\in U$. For such $z\left(t\right)$, taking norms in (2) and using the variation-of-constants formula yields

$${\parallel z\left(t\right)\parallel }_{\_}\le K{\mathrm{e}}^{-\alpha t}{\parallel z\left(0\right)\parallel }_{\_}+K{\int }_0^t{\mathrm{e}}^{-\alpha (t-s)}\left({M\parallel z\left(s\right)\parallel }_{-}+M\parallel u\left(s\right)\parallel \right)ds.$$

By Grönwall’s inequality, there exist constants $C_1,{\gamma }_1>0$ (depending only on $K,M,\alpha$ and ${\parallel u\parallel }_{L^{\infty }}$) such that

$${\parallel z\left(t\right)\parallel }_{-}\le C_1{\mathrm{e}}^{-\alpha t}{\parallel z\left(0\right)\parallel }_{-}+{\gamma }_1{\parallel u\parallel }_{L^{\infty }\left([0,t]\right)}\mathrm{for}\mathrm{all}t\ge 0,$$

(3)

as long as $z\left(s\right)$ remains in the neighborhood where the estimates above hold.

To remove this local restriction, note that $G^{-}$ is diffeomorphic to ${\mathbb{R}}^{\dim {\mathfrak{g}}^{-}}$ and the drift is strictly contracting on $G^{-}$: in the absence of input, $g^{-}\left(t\right)$ converges exponentially to e. The input term is uniformly bounded by ${\parallel u\parallel }_{L^{\infty }}$, so standard ISS arguments for linear systems on ${\mathbb{R}}^n$ imply that $g^{-}\left(t\right)$ is globally bounded and eventually enters (and stays in) a compact neighborhood of e where the exponential coordinates and estimates above are valid. Thus (3) holds for all $t\ge 0$ (possibly with larger constants, which we absorb into $C_1,{\gamma }_1$).

Finally, combining (1) and (3) gives

$$d\left(g^{-}\left(t\right),e\right)\le c_2{\parallel z\left(t\right)\parallel }_{-}\le c_2{C}_1{\mathrm{e}}^{-\alpha t}{\parallel z\left(0\right)\parallel }_{-}+c_2{\gamma }_1{\parallel u\parallel }_{L^{\infty }\left([0,t]\right)}\le C{\mathrm{e}}^{-\alpha t}d\left(g^{-}\left(0\right),e\right)+\gamma {\parallel u\parallel }_{L^{\infty }\left([0,t]\right)},$$

for suitable constants $C,\gamma >0$ depending only on the system data and the chosen metric. This is the desired ISS estimate. □

Theorem 1

(Geometric characterization of internal boundedness, [1]). Let G be a connected Lie group and ${\mathrm{\Sigma }}_G$ a linear control system on G satisfying the $LARC$. The following conditions are equivalent:

1. 

$G=G^{-,0}$ and $G^0$ is a compact subgroup;

2. 

${\mathcal{O}}^{+}\left(e\right)$ is bounded;

3. 

For any $u\in \mathcal{U}$, the trajectory $\left\{\varphi \right(t,e,u),t\ge 0\}$ is bounded.

Under these conditions there exists a bounded $ICS$$\mathcal{C}$ with nonempty interior. The $ICS$ is unique and

$$K=\overline{{\mathcal{O}}^{+}\left(e\right)}=\overline{\mathcal{C}}$$

is the minimal compact positively invariant trapping set.

Sketch of the Proof.

We only outline the main ideas of each implication.

$(1\Rightarrow 2)$. From $G=G^{-,0}$ and the fact that $G^0$ is compact, one first chooses a left-invariant Riemannian metric $\varrho$ on $G^{-}$ which is invariant under conjugation by $G^0$. For the linear flow $\left({\phi }_t\right)$ restricted to $G^{-}$ one shows exponential contraction

$$\varrho ({\phi }_t\left(g\right),{\phi }_t\left(h\right))\le c{e}^{-\lambda t}\varrho (g,h),g,h\in G^{-},t\ge 0,$$

for some $c\ge 1$, $\lambda >0$ (this is the usual stable behavior on $G^{-}$).

Since the one-step reachable set ${\mathcal{O}}_1^{+}\left(e\right)$ is bounded, its closure is compact, and one can cover it by a compact $K\subset G^{-}$ times $G^0$: $\overline{{\mathcal{O}}_1^{+}\left(e\right)}\subset K{G}^0$. Writing an arbitrary time $t=m+r$ ($m\in \mathbb{N}$, $r\in [0,1)$) and using the co-cycle property, $\varphi (t,e,u)$ can be factored into products of one-step pieces ${\varphi }_{1,{\theta }_{m-j+r}u}\left(e\right)\in {\mathcal{O}}_1^{+}\left(e\right)$, each of which lies in $K{G}^0$. Projecting onto $G^{-}$ and using that $G^0$ normalizes $G^{-}$, one obtains a $G^{-}$-component ${\varphi }^{-}(t,e,u)\in G^{-}$ written as a product of conjugates of elements of K followed by exponentially contracting factors ${\phi }_j$. The contraction estimate and the $G^0$-invariance of $\varrho$ imply that all ${\varphi }^{-}(t,e,u)$ lie in a uniform ball $B_{G^{-}}(e,R)$, hence

$$\varphi (t,e,u)\in B_{G^{-}}(e,R)G^0$$

for all $t\ge 0$ and $u\in \mathcal{U}$, which yields boundedness of ${\mathcal{O}}^{+}\left(e\right)$.

$(2\Rightarrow 3)$ is immediate: if the positive orbit from e is bounded, then every trajectory starting at e for any control is contained in that bounded set.

$(3\Rightarrow 1)$ Assume now that every trajectory starting at e is bounded. Using $LARC$ and standard results on linear control systems, one knows that e belongs to the closure of the interior of ${\mathcal{O}}^{+}\left(e\right)$, so

$$V:=G^{+}{G}_1^{-,0}\cap \mathrm{int}{\mathcal{O}}^{+}\left(e\right)$$

is a nonempty open subset. For any $g\in V$ there exists a control u and time $\tau >0$ such that $g=\varphi (\tau ,e,u)$, and by concatenating u with the zero control one sees that the positive drift orbit $\{{\phi }_t\left(g\right)$:$t\ge 0\}$ is bounded.

As in the proof of the stability theorem for the drift, this implies that $g=g_1{g}_2$ with $g_1\in G^{-}$ and $g_2\in G^0$, and that the drift orbit of $g_2$ is bounded. As we explain before, $g_2$ must lie in the recurrent set $\mathcal{R}\left(\phi \right)$ of the drift, so that

$$V\subset G^{-}\mathcal{R}\left(\phi \right).$$

Since V is open and nonempty, $G^{-}\mathcal{R}\left(\phi \right)$ has nonempty interior, and by connectedness one concludes $G=G^{-}\mathcal{R}\left(\phi \right)$, i.e.,

$$G=G^{-,0}{\mathrm{and}\phi |}_{G^0}\mathrm{is}\mathrm{elliptic}$$

Finally, $G=G^{-,0}$ implies that $G^{-}$ is a normal, drift-invariant subgroup and $G/G^{-}\simeq G^0$. The induced linear system on $G/G^{-}$ has elliptic drift, and by a compactness result for elliptic flows one concludes that $G^0$ is compact. This yields condition (1). □

We present a concise single central theorem that characterizes $BIBO$ stability of linear control systems on connected Lie groups relative to homomorphic outputs. The statement uses hypotheses such a closed, $\phi$-invariant kernels, inheritance of $LARC$ to quotients, and a canonical normal subgroup $\Gamma$ removing expanding and noncompact neutral directions.

Homogeneous space. Before stating the theorem, we recall the notion of a quotient of a Lie group by a closed subgroup. Let G be a (finite-dimensional) Lie group and $N\subset G$ a closed subgroup. Two elements $g_1,g_2\in G$ are said to be equivalent modulo N if $g_1^{-1}{g}_2\in N$, and we write $g_1\sim g_2$. The set of equivalence classes

$$G/N:=\{gN:g\in G\},gN:=\{gn:n\in N\},$$

is called the quotient space (or homogeneous space) of G by N. Since N is closed, $G/N$ carries a natural smooth manifold structure making the quotient map

$$\pi :G⟶G/N,\pi \left(g\right)=gN,$$

a smooth submersion. If N is normal, then $G/N$ inherits a Lie group structure for which $\pi$ is a Lie group homomorphism; otherwise $G/N$ is a homogeneous G-space with a smooth left action

$$G\times G/N\to G/N,(g,hN)\mapsto \left(gh\right)N.$$

Given a linear control system ${\mathrm{\Sigma }}_G$ on G and a closed, $\phi$-invariant subgroup $N\subset G$, the quotient map $\pi$ allows us to project the dynamics: the drift flow and control vector fields descend to a well-defined linear control system ${\mathrm{\Sigma }}_{G/N}$ on $G/N$, whose trajectories satisfy

$$\pi \left(\varphi (t,g,u)\right)=\overline{\varphi }\left(t,\pi \left(g\right),u\right),t\ge 0.$$

Assume,

Hypothesis 1.

The Lie algebra rank condition holds: the smallest $\mathcal{D}$-invariant Lie subalgebra containing the control directions equals $\mathfrak{g}$.

Hypothesis 2.

For any Lie group homomorphism $F:G\to H$ (output), its kernel $N=\ker F$ is closed and φ-invariant: ${\phi }_t\left(N\right)\subset N$ for all $t\in \mathbb{R}$.

Hypothesis 3.

Under (H2) the projected control vector fields on $G/N$ satisfy $LARC$ on the quotient Lie algebra (equivalently, images of the control directions generate the quotient Lie algebra under the projected derivation).

Hypothesis 4.

There exists a maximal closed normal φ-invariant subgroup $\Gamma ◃G$ containing $G^{+}$ and the noncompact connected factors of $G^0$; equivalently, Γ is the smallest closed normal φ-invariant subgroup for which $G/\Gamma$ has no expanding directions and has compact neutral subgroup.

Write the real Jordan decomposition of $\mathcal{D}$ and define ${\mathfrak{g}}^{\pm }$ and ${\mathfrak{g}}^0$, and subgroups $G^{\pm }$ and $G^0$ as usual; for a quotient map $\pi :G\to G/N$ denote induced objects with bars.

Finally, assumption (H3) precisely requires that, under (H2), the projected control vector fields on $G/N$ satisfy the Lie algebra rank condition, so that accessibility properties are inherited by the quotient.

Theorem 2.

Let ${\mathrm{\Sigma }}_G$ a $LCS$ satisfy $H1$–$H4$. Let F:$G\to H$ be a Lie group homomorphism with closed φ-invariant $N=\ker F$. Then the following statements are equivalent:

(a) 

${\mathrm{\Sigma }}_G$ is $BIBO$ stable relative to F: for every bounded admissible control u the output trajectory $\left\{F\right(\varphi (t,e,u))$:$t\ge 0\}$ is bounded in H.

(b) 

The quotient system ${\mathrm{\Sigma }}_{G/N}$ in $G/N\simeq F\left(G\right)$ satisfies ${(G/N)}^{+}=\left\{\overline{e}\right\}$ and ${(G/N)}^0$ is compact.

(c) 

$G^{+}\subset N$ and $F\left(G^0\right)$ is a compact subgroup of $F\left(G\right)$.

Moreover, let Γ be the canonical subgroup from (H4). Then

(d) 

The canonical projection Π:$G\to G/\Gamma$ has the universal factorization property: any homomorphism F:$G\to H$ producing a $BIBO$-stable output factors uniquely through Π.

(e) 

On $G/\Gamma$ the internal $ICS$ theory applies: the quotient system admits a unique bounded invariant control set with nonempty interior, whose closure is the minimal compact positively invariant trapping set. Consequently, bounded outputs correspond precisely to bounded trajectories in $G/\Gamma$.

(f) 

Consider the Jordan decomposition. Under the additional spectral-continuity assumption (small perturbations of $\mathcal{D}$ and control directions do not change signs of real parts of eigenvalues in the quotient), the $BIBO$ condition above is robust: there exists an open neighborhood in the parameter space where equivalent statements (a)–(c) hold for perturbed systems.

Sketch of the Proof.

(a) ⇒ (b). Since $N=\ker F$ is closed and $\phi$-invariant, the system projects to a linear control system ${\mathrm{\Sigma }}_{G/N}$ on $G/N\simeq F\left(G\right)$ and

$$F\left(\varphi (t,e,u)\right)=\overline{\varphi }(t,\overline{e},u),$$

where $\overline{\varphi }$ is the flow of ${\mathrm{\Sigma }}_{G/N}$ and $\overline{e}$ is the identity class. $BIBO$ boundedness of outputs for all bounded u is exactly boundedness of the positive reachable set ${\mathcal{O}}_{G/N}^{+}\left(\overline{e}\right)$. By the internal stability theorem for linear systems on Lie groups (Theorem 1, applied to ${\mathrm{\Sigma }}_{G/N}$ under $H1$–$H3$), this boundedness holds if and only if the unstable subgroup of the quotient is trivial and the neutral subgroup is compact, i.e., ${(G/N)}^{+}=\left\{\overline{e}\right\}$ and ${(G/N)}^0$ is compact.

(b) ⇒ (a). The same identification shows that if ${(G/N)}^{+}=\left\{\overline{e}\right\}$ and ${(G/N)}^0$ is compact, then Theorem 1 guarantees a bounded invariant control set in $G/N$ and bounded reachable sets from $\overline{e}$. Thus every quotient trajectory $\overline{\varphi }(t,\overline{e},u)$ is bounded and therefore so is $F\left(\varphi \right(t,e,u\left)\right)$.

(b) ⇔ (c). Let $\pi :G\to G/N$ be the quotient map. Functoriality of the dynamical subgroups yields

$$\pi \left(G^{+}\right)={(G/N)}^{+},\pi \left(G^0\right)={(G/N)}^0.$$

Hence ${(G/N)}^{+}=\left\{\overline{e}\right\}$ iff $G^{+}\subset N$. Compactness of ${(G/N)}^0$ is equivalent to compactness of its image $\pi \left(G^0\right)\cong F\left(G^0\right)$ inside $F\left(G\right)$. Thus (b) and (c) are equivalent.

(d) Universal factorization. By construction in (H4), $\Gamma$ is the subgroup generated by all unstable directions and noncompact neutral directions. If $F:G\to H$ yields a $BIBO$-stable output, then by (a)–(c) its kernel must contain $G^{+}$ and all noncompact neutral directions; hence $\Gamma \subset \ker F$ and there exists a unique homomorphism $\tilde{F}:G/\Gamma \to H$ with $F=\tilde{F}\circ \mathrm{\Pi }$. Conversely, any homomorphism on $G/\Gamma$ produces a $BIBO$-stable output by (b), since $G/\Gamma$ satisfies ${\overline{G}}^{+}=\left\{\overline{e}\right\}$ and ${\left(\overline{G}\right)}^0$ compact.

(e) $ICS$ on $G/\Gamma$. Let $\overline{G}:=G/\Gamma$ and consider the induced system ${\mathrm{\Sigma }}_{\overline{G}}$. By (H4), $\overline{G}$ has no expanding directions and compact neutral subgroup, and by (H1)–(H3) $LARC$ descends to $\overline{G}$. Applying again Theorem 1 to ${\mathrm{\Sigma }}_{\overline{G}}$ yields the existence and uniqueness of a bounded invariant control set ${\mathcal{C}}_{\overline{G}}\subset \overline{G}$ with nonempty interior; its closure $K_{\overline{G}}=\overline{{\mathcal{C}}_{\overline{G}}}$ is the minimal compact positively invariant trapping set containing $\overline{e}$. Moreover, any output F factoring through $\mathrm{\Pi }$ is bounded iff the corresponding quotient trajectory in $\overline{G}$ is bounded, so bounded outputs are in bijection with bounded trajectories in $\overline{G}$.

(f) Robustness. The spectral Jordan decomposition of the derivation $\mathcal{D}$ is continuous under small perturbations. Under the assumption that no eigenvalue crosses the imaginary axis in the quotient and that compactness of the neutral subgroup persists, the conditions ${(G/N)}^{+}=\left\{\overline{e}\right\}$ and compact ${(G/N)}^0$ remain valid in a neighborhood in parameter space. Hence the equivalence (a)–(c) and $BIBO$ stability are robust under such perturbations. □

Remark 1.

Assume spectral continuity: small perturbations of $\mathcal{D}$ and control directions preserve sign pattern of relevant eigenvalues in the quotient. Then existence/uniqueness of $ICS$, canonical quotient classification and $BIBO$ properties persist under small perturbations. On $G/\Gamma$ one obtains ISS-type estimates combining exponential contraction on $G^{-}$ with bounded neutral dynamics.

1. 

Spectral continuity: In finite dimensions the real parts of eigenvalues vary continuously with $\mathcal{D}$; thus there exist open neighborhoods of parameters where the sign pattern (positive/zero/negative real parts) remains unchanged. When this is the case the algebraic decomposition ${\mathfrak{g}}^{\pm },{\mathfrak{g}}^0$ is stable and so is Γ.

2. 

Persistence: Under small perturbations that do not introduce new expanding directions in the quotient the $ICS$ in $G/\Gamma$ persists (upper semicontinuity of invariant sets plus contraction on $G^{-}$). Therefore $BIBO$ classification is robust to modeling errors and small control direction perturbations.

We finish with an ISS-type estimate on the canonical quotient.

Proposition 1.

Let ${\mathrm{\Sigma }}_G$ be a linear control system satisfying $H1$–$H4$, and let $\Gamma ◃G$ be the canonical subgroup from (H4). Consider the induced system ${\mathrm{\Sigma }}_{G/\Gamma }$ on the quotient $G/\Gamma$, and denote $\overline{e}:=\Gamma$ and by ${\mathcal{C}}_{\mathrm{can}}\subset G/\Gamma$ its (unique) bounded invariant control set with nonempty interior. Assume that on the Lie algebra of the contracting subgroup ${\mathfrak{g}}^{-}$ the restriction ${\mathcal{D}}^{-}{:=\mathcal{D}|}_{{\mathfrak{g}}^{-}}$ satisfies $\mathrm{spec}\left({\mathcal{D}}^{-}\right)\subset \{\mathrm{Re}\lambda \le -\lambda \}$ for some $\lambda >0$. Then there exist a left-invariant Riemannian metric on $G/\Gamma$ and constants $C,R>0$ such that, for every bounded admissible control $u\in L^{\infty }([0,\infty ),U)$ and all $t\ge 0$,

$$d\left({\varphi }_{G/\Gamma }(t,\overline{e},u),{\mathcal{C}}_{\mathrm{can}}\right)\le C{\mathrm{e}}^{-\lambda t}d\left(\overline{e},{\mathcal{C}}_{\mathrm{can}}\right)+R{\parallel u\parallel }_{L^{\infty }\left([0,t]\right)},$$

(4)

where d is the Riemannian distance on $G/\Gamma$.

Proof. 

By (H4) and the construction of $\Gamma$, all expanding directions and noncompact neutral directions are quotiented out in $G/\Gamma$. Thus the induced derivation $\tilde{\mathcal{D}}$ on $\tilde{\mathfrak{g}}:=Lie(G/\Gamma )$ has no eigenvalues with positive real part, and the neutral subalgebra ${\tilde{\mathfrak{g}}}^0$ integrates to a compact subgroup ${\tilde{G}}^0$. Moreover, the restriction of $\tilde{\mathcal{D}}$ to the contracting subspace ${\tilde{\mathfrak{g}}}^{-}\simeq {\mathfrak{g}}^{-}$ has spectrum contained in $\{\mathrm{Re}\lambda \le -\lambda \}$ by hypothesis.

On $G/\Gamma$ choose a left-invariant Riemannian metric that is adapted to the splitting $\tilde{\mathfrak{g}}={\tilde{\mathfrak{g}}}^{-}\oplus {\tilde{\mathfrak{g}}}^0$, in the sense that the decomposition is orthogonal at the identity and transported by left translations. Then the Riemannian distance d is equivalent (locally and, by compactness of ${\tilde{G}}^0$, also globally up to constants) to the product distance

$$d\left(({\tilde{g}}^{-},{\tilde{g}}^0),({\tilde{h}}^{-},{\tilde{h}}^0)\right)\simeq d^{-}\left({\tilde{g}}^{-},{\tilde{h}}^{-}\right)+d^0\left({\tilde{g}}^0,{\tilde{h}}^0\right),$$

where $d^{-}$ (resp. $d^0$) denotes the distance restricted to the contracting (resp. neutral) factor.

By the general control set theory under $LARC$ and compactness of the neutral factor (see, e.g., [20]), the invariant control set ${\mathcal{C}}_{\mathrm{can}}$ is saturated in the neutral directions and its “transverse” dynamics are governed by the contracting component. In particular, there exists a compact subset $K\subset {\tilde{G}}^0$ such that, up to changing coordinates along ${\tilde{G}}^0$,

$${\mathcal{C}}_{\mathrm{can}}=\{({\tilde{g}}^{-},{\tilde{g}}^0)\in {\tilde{G}}^{-}\times {\tilde{G}}^0:{\tilde{g}}^{-}\in {\mathcal{C}}^{-},{\tilde{g}}^0\in K\},$$

for some compact, positively invariant set ${\mathcal{C}}^{-}\subset {\tilde{G}}^{-}$ containing the identity. Consequently, the distance from a point $\tilde{g}=({\tilde{g}}^{-},{\tilde{g}}^0)$ to ${\mathcal{C}}_{\mathrm{can}}$ is equivalent to the distance from ${\tilde{g}}^{-}$ to ${\mathcal{C}}^{-}$:

$$d\left(\tilde{g},{\mathcal{C}}_{\mathrm{can}}\right)\simeq d^{-}\left({\tilde{g}}^{-},{\mathcal{C}}^{-}\right),$$

(5)

with equivalence constants depending only on the chosen metric and on K.

Now consider the trajectory of the quotient system $\tilde{\varphi }(t,\overline{e},u):={\varphi }_{G/\Gamma }(t,\overline{e},u)$ and decompose it as $\tilde{\varphi }(t,\overline{e},u)=({\tilde{g}}^{-}\left(t\right),{\tilde{g}}^0\left(t\right))$ with ${\tilde{g}}^{-}\left(t\right)\in {\tilde{G}}^{-}$, ${\tilde{g}}^0\left(t\right)\in {\tilde{G}}^0$. The dynamics of ${\tilde{g}}^{-}\left(t\right)$ are governed by the restriction ${\mathcal{D}}^{-}{:=\mathcal{D}|}_{{\mathfrak{g}}^{-}}$, perturbed by bounded terms coming from the control and from the (compact) neutral factor. By Lemma 1 (applied to the quotient system), there exist constants $C_1,{\gamma }_1>0$, depending only on $\lambda$, the metric on ${\tilde{G}}^{-}$, and the control range, such that

$$d^{-}\left({\tilde{g}}^{-}\left(t\right),e\right)\le C_1{\mathrm{e}}^{-\lambda t}{d}^{-}\left({\tilde{g}}^{-}\left(0\right),e\right)+{\gamma }_1{\parallel u\parallel }_{L^{\infty }\left([0,t]\right)}.$$

(6)

Since ${\mathcal{C}}^{-}$ is compact, there is a constant $M>0$ with $d^{-}({\tilde{g}}^{-}\left(0\right),{\mathcal{C}}^{-})\le M{d}^{-}({\tilde{g}}^{-}\left(0\right),e)$ and similarly $d^{-}({\tilde{g}}^{-}\left(t\right),{\mathcal{C}}^{-})\le M{d}^{-}({\tilde{g}}^{-}\left(t\right),e)$ for all t. Thus (6) implies

$$d^{-}\left({\tilde{g}}^{-}\left(t\right),{\mathcal{C}}^{-}\right)\le M{C}_1{\mathrm{e}}^{-\lambda t}{d}^{-}\left({\tilde{g}}^{-}\left(0\right),e\right)+M{\gamma }_1{\parallel u\parallel }_{L^{\infty }\left([0,t]\right)}.$$

Using again $d^{-}({\tilde{g}}^{-}\left(0\right),e)\le c{d}^{-}({\tilde{g}}^{-}\left(0\right),{\mathcal{C}}^{-})$ for some $c>0$, we obtain

$$d^{-}\left({\tilde{g}}^{-}\left(t\right),{\mathcal{C}}^{-}\right)\le C^{\prime }{\mathrm{e}}^{-\lambda t}{d}^{-}\left({\tilde{g}}^{-}\left(0\right),{\mathcal{C}}^{-}\right)+{\gamma }^{\prime }{\parallel u\parallel }_{L^{\infty }\left([0,t]\right)},$$

for suitable $C^{\prime },{\gamma }^{\prime }>0$.

Finally, combining this with the equivalence (5) gives

$$d\left(\tilde{\varphi }(t,\overline{e},u),{\mathcal{C}}_{\mathrm{can}}\right)\le C{\mathrm{e}}^{-\lambda t}d\left(\overline{e},{\mathcal{C}}_{\mathrm{can}}\right)+R{\parallel u\parallel }_{L^{\infty }\left([0,t]\right)},$$

for some constants $C,R>0$ depending only on the geometry of $G/\Gamma$, the spectral gap $\lambda$ on ${\mathfrak{g}}^{-}$, and the control range U. This is precisely the ISS-type estimate (4). □

Practical checklist for applications

To apply the theorems in concrete examples follow this checklist:

• Compute derivation $\mathcal{D}$ and its real Jordan decomposition;
• Determine ${\mathfrak{g}}^{\pm },{\mathfrak{g}}^0$ and integrate to $G^{\pm },G^0$;
• Verify $LARC$;
• Check $G^{+}$ and compactness of $G^0$: if $G^{+}=\left\{e\right\}$ and $G^0$ compact, bounded $ICS$ exist and internal boundedness follows; otherwise construct $\Gamma$ and consider quotient dynamics;
• For a candidate output F, verify $G^{+}\subset \ker F$ and compactness of $F\left(G^0\right)$ to decide $BIBO$; if not, consider altering F or quotienting out problematic directions.

For the stability analysis of an $LCS$ on a connected Lie group G, this extended presentation highlights both the logical structure of the main results and the geometric ideas required to execute them in examples. The examples collected in the sequel will follow this road map.

4. Examples of $\mathit{BIBO}$ Stability on Classes of Lie Groups

In this section, we develop several examples for different classes of groups.

4.1. Example on the Abelian Euclidean Group ${\mathbb{R}}^n$

Let $G=({\mathbb{R}}^n,+)$. Identify the Lie algebra with ${\mathbb{R}}^n$. Consider the linear control system

$$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),u\left(t\right)\in U\subset {\mathbb{R}}^m,$$

with $A\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times m}$ and U a compact convex set with $0\in \mathrm{int}U$ (e.g., the Euclidean unit ball). This is an $LCS$ on the abelian Lie group ${\mathbb{R}}^n$ (linear vector field given by $\mathcal{D}=A$ and left-invariant fields given by columns of B).

We present a concrete nontrivial stable choice and verify the hypotheses and conclusions: take $n=2$ and

$$A=\left(\begin{array}{cc}-1& 0\\ 0& -2\end{array}\right),B=I_2,U=\{u\in {\mathbb{R}}^2{:\parallel u\parallel }_{\infty }\le 1\}.$$

• Dynamical splitting. The derivation $\mathcal{D}=A$ has eigenvalues $-1,-2<0$, hence

  $${\mathfrak{g}}^{+}=\left\{0\right\},{\mathfrak{g}}^0=\left\{0\right\},{\mathfrak{g}}^{-}={\mathbb{R}}^2.$$
  Integrating gives $G^{+}=\left\{0\right\}$, $G^0=\left\{0\right\}$, $G^{-}={\mathbb{R}}^2$.
• $LARC$/accessibility. $\Delta =\mathrm{span}\left\{\mathrm{columns}\mathrm{of}B\right\}={\mathbb{R}}^2$, so ${\mathcal{L}}_0=\mathfrak{g}={\mathbb{R}}^2$ and $LARC$ holds.
• Boundedness of identity-based reachable set. For $x\left(0\right)=0$ and any measurable u with ${\parallel u\parallel }_{\infty }\le 1$, the solution is

  $$x\left(t\right)={\int }_0^t{e}^{A(t-s)}Bu\left(s\right)ds.$$
  Using $\parallel e^{At}\parallel \le C{e}^{-\lambda t}$ (here $C=1$, $\lambda =1$ because diagonal entries $-1,-2$), we bound for all $t\ge 0$

  $$\parallel x\left(t\right)\parallel \le {\int }_0^t\parallel e^{A(t-s)}\parallel \parallel u\left(s\right)\parallel ds\le {\parallel u\parallel }_{\infty }{\int }_0^{\infty }C{e}^{-\lambda s}ds={\displaystyle \frac{C}{\lambda }}{\parallel u\parallel }_{\infty }.$$
  With our numbers, $\parallel x\left(t\right)\parallel \le 1·{\parallel u\parallel }_{\infty }$. Hence the positive orbit ${\mathcal{O}}^{+}\left(0\right)$ is bounded (uniform bound independent of control length).
• Existence and uniqueness of $ICS$; minimal trap. By the geometric theorem, since $G^{+}=\left\{0\right\}$ and $G^0$ is compact (trivial), conditions hold and there exists a unique bounded $ICS$$\mathcal{C}\subset {\mathbb{R}}^2$ with nonempty interior. Here ${\mathcal{O}}^{+}\left(0\right)$ is itself bounded with nonempty interior (indeed interior arises from $LARC$), and $K=\overline{{\mathcal{O}}^{+}\left(0\right)}$ is the minimal compact positively invariant trapping set.
• $BIBO$ outputs and canonical quotient. Any linear output $y=Cx$ (with $C\in {\mathbb{R}}^{p\times 2}$) yields bounded outputs for every bounded control because the state remains bounded. The canonical subgroup $\Gamma$ is trivial (no obstructions), so $G/\Gamma =G$ and all homomorphic outputs factor trivially.
• Robustness/ISS estimates. Small perturbations $A+\Delta A$ with eigenvalues remaining in the left half-plane preserve the contraction rate. Quantitatively, if A is perturbed so the contraction rate becomes ${\lambda }^{\prime }>0$ and $\parallel e^{(A+\Delta A)t}\parallel \le C^{\prime }{e}^{-{\lambda }^{\prime }t}$, then for ${\parallel u\parallel }_{\infty }\le 1$ the same integral bound yields $\parallel x\left(t\right)\parallel \le C^{\prime }/{\lambda }^{\prime }$. This gives an $ISS$-type bound for the quotient (trivial here) and shows persistence of boundedness and $ICS$ under small perturbations.

Remark 2.

This example is nontrivial in that the control input affects all directions (full rank B) and the bounded control reachable set is a genuine bounded region in ${\mathbb{R}}^2$ (not the whole space), illustrating the geometric $ICS$ phenomenon in an abelian context. If A had zero eigenvalues (neutral directions) the neutral subgroup would be noncompact and internal boundedness would fail; this contrasts with the present Hurwitz case and highlights the necessity of $G^0$ compactness in the general theory.

On the other hand, is directly applicable to linear engineering problems; it confirms that classical spectral intuition (Hurwitz matrix) suffices to guarantee bounded $ICS$ and $BIBO$ for linear outputs. Practically, use the explicit ISS bounds to size actuators and set safety margins against disturbances and saturation.

4.2. Example on the Heisenberg Group $H_3$

Let $H_3$ be the real Heisenberg group with coordinates $(x,y,z)\in {\mathbb{R}}^3$ and group law

$$(x,y,z)·(x^{\prime },y^{\prime },z^{\prime })=\left(x+x^{\prime },y+y^{\prime },z+z^{\prime }+\frac{1}{2}(x{y}^{\prime }-y{x}^{\prime })\right).$$

Its Lie algebra ${\mathfrak{h}}_3$ has basis $\{X,Y,Z\}$ with $[X,Y]=Z$ and Z central. We use the left-invariant identification so that $X,Y,$ and Z correspond to the standard left fields.

Consider the linear control system on $H_3$

$$\dot{g}=\mathcal{X}\left(g\right)+u_1{Y}_X\left(g\right)+u_2{Y}_Y\left(g\right),$$

where $Y_X,Y_Y$ are the left-invariant fields associated with X and Y (controls act on the horizontal directions) and the linear drift $\mathcal{X}$ is the linear vector field corresponding to the derivation $\mathcal{D}\in \mathrm{Der}\left({\mathfrak{h}}_3\right)$ defined on the basis by

$$\mathcal{D}\left(X\right)=-X,\mathcal{D}\left(Y\right)=-2Y,\mathcal{D}\left(Z\right)=-3Z.$$

Any such triple satisfies the derivation identity because

$$\mathcal{D}\left(\right[X,Y\left]\right)=[\mathcal{D}X,Y]+[X,\mathcal{D}Y]=-1·Z+-2·Z=-3Z=\mathcal{D}\left(Z\right)$$

We analyze this $LCS$ with $U=\{u\in {\mathbb{R}}^2:\parallel u{\parallel }_{\infty }\le 1\}$.

• Dynamical splitting. The derivation $\mathcal{D}$ has eigenvalues $-1,-2,-3<0$, so ${\mathfrak{g}}^{+}=\left\{0\right\}$, ${\mathfrak{g}}^0=\left\{0\right\}$, ${\mathfrak{g}}^{-}={\mathfrak{h}}_3$. Hence $G^{+}=\left\{e\right\}$ and $G^0$ is trivial (compact). All directions contract under the drift.
• $LARC$ (accessibility). The control distribution $\Delta =\mathrm{span}\{X,Y\}$ together with bracket $[X,Y]=Z$ generates the full Lie algebra ${\mathfrak{h}}_3$. Moreover $\Delta$ is invariant under $\mathcal{D}$ up to span closure, so ${\mathcal{L}}_0={\mathfrak{h}}_3$ and $LARC$ holds.
• Boundedness of identity-based reachable set. Solutions from the identity obey the variation-of-constants formula. Since $\mathcal{D}$ has strictly negative real parts, the linearized flow $e^{t\mathcal{D}}$ is exponentially contracting: there exist $C,\lambda >0$ with $\parallel e^{t\mathcal{D}}\parallel \le C{e}^{-\lambda t}$. Consequently, for any bounded control u the identity-based trajectory satisfies an integral bound analogous to the Euclidean case (using BCH/coordinate charts or exponential coordinates): the state remains uniformly bounded for all $t\ge 0$, with a bound depending only on ${\parallel u\parallel }_{\infty }$, $C,\lambda$ and the control range. Thus ${\mathcal{O}}^{+}\left(e\right)$ is bounded.
• Existence and uniqueness of $ICS$; minimal trapping set. By the geometric theorem (internal boundedness), since ${\mathfrak{g}}^{+}=\left\{0\right\}$ and $G^0$ compact (trivial), there exists a unique bounded $ICS$$\mathcal{C}$ with nonempty interior. Its closure $K=\overline{{\mathcal{O}}^{+}\left(e\right)}$ is the minimal compact positively invariant set containing the identity; $\mathcal{C}$ organizes all bounded controlled trajectories.
• $BIBO$ outputs and canonical quotient. Because there are no unstable or noncompact neutral directions, the canonical subgroup $\Gamma$ is trivial and $G/\Gamma =H_3$. Any homomorphic output $F:H_3\to H$ thus yields bounded outputs for bounded controls (provided F is continuous), so $BIBO$ stability holds for all such outputs. In practice, interesting outputs project onto compact factors (when available) or annihilate contracting directions.
• Robustness and ISS estimates. Small perturbations of the derivation and control directions that keep the real parts of eigenvalues negative preserve the contraction property. Hence existence/uniqueness of the $ICS$ and boundedness are robust. On $H_3$ one may construct left-invariant metrics adapted to the grading so that estimates of the form

  $$d\left(\varphi (t,e,u),\mathcal{C}\right)\le C^{\prime }{e}^{-{\lambda }^{\prime }t}d(e,\mathcal{C})+R{\parallel u\parallel }_{\infty }$$

  hold, providing explicit $ISS$-type bounds.

Remark 3.

The Heisenberg group is step-two nilpotent and has a central direction; yet by choosing a derivation that contracts the center as well as the horizontal directions we obtain global contraction and internal boundedness. It illustrates that nilpotency alone does not forbid internal stability. What matters is the action of the drift (derivation) on the center and on the graded layers. On the other hand, if one had chosen $\mathcal{D}$ with zero eigenvalue on Z (so $\mathcal{D}\left(Z\right)=0$) then $G^0$ would contain a noncompact central direction and internal boundedness would fail despite contraction on X and Y.

The example demonstrates that nilpotency does not preclude internal stability if the drift contracts the central direction; relevant for nonholonomic systems with accumulated variables.

In robotics or phase-space models, emphasize how enforcing contraction on the central coordinate (via control or feedback) prevents polynomial drift.

4.3. Example on the Solvable Group $SE\left(2\right)$: Canonical Subgroup and a Mobile Robot Application

Definition of group and algebra. Let

$$SE\left(2\right)={\mathbb{R}}^2⋊SO\left(2\right)$$

with product $(R_{\theta },p)·(R_{{\theta }^{\prime }},p^{\prime })=(R_{\theta +{\theta }^{\prime }},p+R_{\theta }{p}^{\prime })$. Its Lie algebra $\mathfrak{se}\left(2\right)$ has basis $\{A,X,Y\}$ with

$$[A,X]=Y,[A,Y]=-X,[X,Y]=0,$$

where A generates infinitesimal rotations and $X,Y$ translations.

Left-invariant fields. In coordinates $(\theta ,x,y)$ the standard left-invariant fields are

$$A={\partial }_{\theta },X=\cos \theta {\partial }_x+\sin \theta {\partial }_y,Y=-\sin \theta {\partial }_x+\cos \theta {\partial }_y.$$

Linear drift and control system. Choose a derivation $\mathcal{D}$ given by

$$\mathcal{D}\left(A\right)=0,\mathcal{D}\left(X\right)=-\alpha X,\mathcal{D}\left(Y\right)=-\alpha Y,\alpha >0,$$

which satisfies the Leibniz rule. Let $\mathcal{X}$ be the associated linear vector field and consider the $LCS$ with controls only in the rotation and one translation direction:

$${\mathrm{\Sigma }}_{SE\left(2\right)}:\dot{g}=\mathcal{X}\left(g\right)+u_{1}X\left(g\right)+u_{2}A\left(g\right),$$

(7)

where $u=(u_1,u_2)$ is measurable and bounded with values in a compact set U (e.g., ${\parallel u\parallel }_{\infty }\le 1$).

• Accessibility/$LARC.$ Take $\Delta =\mathrm{span}\{X,A\}$. Using the Lie brackets,

  $$[A,X]=Y\in \mathrm{Lie}(\Delta ).$$
  Therefore $\mathrm{Lie}\{X,A\}$ is equal to the full algebra $\mathfrak{se}\left(2\right)$. Practically, the bracket produces the missing horizontal direction Y, so $LARC$ holds.
• Dynamical splitting. The matrix of $\mathcal{D}$ in the basis $\{A,X,Y\}$ is

  $$\mathcal{D}=\left(\begin{array}{ccc}0& 0& 0\\ 0& -\alpha & 0\\ 0& 0& -\alpha \end{array}\right).$$
  Thus

  $${\mathfrak{g}}^{+}=\left\{0\right\},{\mathfrak{g}}^{-}=\mathrm{span}\{X,Y\},{\mathfrak{g}}^0=\mathrm{span}\left\{A\right\}.$$
  The corresponding connected subgroups are

  $$G^{-}\cong {\mathbb{R}}^2\left(\mathrm{translations}\right),G^0\cong SO\left(2\right)(\mathrm{pure}\mathrm{rotations}),G^{+}=\left\{e\right\}.$$
  The algebraic boundedness condition $G=G^{-,0}$ with compact $G^0$ is satisfied.
• Explicit computation of the canonical subgroup. In our framework, the canonical subgroup $\Gamma \subset G$ is the smallest closed, connected, normal subgroup such that all homomorphic outputs $F:G\to H$ with BIBO behavior factor through the quotient $G/\Gamma$. In the present solvable case, the unstable (expanding) directions are absent ($G^{+}=\left\{e\right\}$) and the only source of potential unboundedness is the noncompact translational part $G^{-}\cong {\mathbb{R}}^2$. Since

  (a)

  The drift contracts $G^{-}$ at rate ${\mathrm{e}}^{-\alpha t}$;

  (b)

  The control acts along X and, via brackets, along Y;

  (c)

  The rotational subgroup $G^0\cong SO\left(2\right)$ is compact, the natural canonical subgroup that must be “factored out” in order to see purely bounded outputs is precisely the translational subgroup:

  $$\Gamma =G^{-}\cong {\mathbb{R}}^2=\left\{(\mathrm{Id},p)\in SE\left(2\right):p\in {\mathbb{R}}^2\right\}.$$

  Equivalently, at the Lie algebra level,

  $$\mathrm{Lie}(\Gamma )=\mathrm{span}\{X,Y\}.$$

  The canonical projection

  $${\pi }_{\Gamma }:SE\left(2\right)⟶SE\left(2\right)/\Gamma$$

  identifies all configurations that differ only by translation. Since $SE\left(2\right)/{\mathbb{R}}^2\simeq SO\left(2\right)$, we have a canonical isomorphism

  $$SE\left(2\right)/\Gamma \simeq SO\left(2\right),$$

  and the quotient dynamics coincide with the pure rotational component.

• Internal boundedness and ICS. By the main theorem (boundedness condition $G^{+}=\left\{e\right\}$ and compact $G^0$), the positive reachable set ${\mathcal{O}}^{+}\left(e\right)$ is bounded and there exists a unique bounded $ICS$$\mathcal{C}$ with nonempty interior. Moreover

  $$K=\overline{{\mathcal{O}}^{+}\left(e\right)}$$

  is the minimal compact positively invariant trapping set. The fact that Y is generated via brackets does not prevent interior for the control set because $LARC$ ensures reachable sets have nonempty interior.
• BIBO outputs and role of the canonical subgroup. Consider the rotation projection F:$SE\left(2\right)\to SO\left(2\right)$,

  $$F(R_{\theta },p)=R_{\theta },p\in {\mathbb{R}}^2.$$
  Its kernel is precisely the translation subgroup $N={\mathbb{R}}^2=\Gamma$ and its image $F\left(SE\right(2\left)\right)=SO\left(2\right)$ is compact. Therefore, for any solution $g\left(t\right)$ of (7) with bounded controls $u(·)\in L^{\infty }$, the output

  $$y\left(t\right)=F\left(g\left(t\right)\right)\in SO\left(2\right)$$

  remains in a compact set, and

  $$\underset{t\ge 0}{\sup }{d}_{SO\left(2\right)}(y\left(t\right),I)<\infty ,$$

  for any fixed Riemannian metric $d_{SO\left(2\right)}$ on $SO\left(2\right)$. In particular, F is BIBO with canonical subgroup $\Gamma ={\mathbb{R}}^2$.
  More generally, any homomorphism $F:SE\left(2\right)\to H$ that annihilates the translation subgroup $\Gamma$ (i.e., ${F|}_{\Gamma }\equiv e_H$) factors through $SE\left(2\right)/\Gamma \simeq SO\left(2\right)$ and is $BIBO.$ Conversely, any homomorphism that does not annihilate $\Gamma$ may exhibit unbounded outputs in the absence of the contracting drift.
• ISS-type estimate for the translational part. Let $g\left(t\right)=(R_{\theta \left(t\right)},p\left(t\right))\in SE\left(2\right)$ be a solution of (7) with bounded controls ${\parallel u\parallel }_{\infty }\le 1$. In exponential coordinates on ${\mathbb{R}}^2$ we write $p\left(t\right)=\left(x\right(t),y(t\left)\right)$. The derivation $\mathcal{D}$ produces a linear part

  $$\dot{p}\left(t\right)=-\alpha p\left(t\right)+B\left(\theta \left(t\right)\right)u_1\left(t\right),$$

  where $B\left(\theta \right)$ is a $2\times 1$ matrix of sines and cosines coming from the left-invariant field X. Since $|u_1\left(t\right)|\le 1$ and $\parallel B\left(\theta \right)\parallel$ is uniformly bounded on $SO\left(2\right)$, there exists a constant $b>0$ such that $\parallel B\left(\theta \left(t\right)\right)u_1\left(t\right)\parallel \le b$ for all $t\ge 0$.
  Solving the scalar differential inequality

  $${\displaystyle \frac{d}{dt}}\parallel p\left(t\right)\parallel \le -\alpha \parallel p\left(t\right)\parallel +b,$$

  and applying Grönwall’s lemma yields the explicit ISS-type bound

  $$\parallel p\left(t\right)\parallel \le {\mathrm{e}}^{-\alpha (t-t_0)}\parallel p\left(t_0\right)\parallel +{\displaystyle \frac{b}{\alpha }}\left(1-{\mathrm{e}}^{-\alpha (t-t_0)}\right)\le {\mathrm{e}}^{-\alpha (t-t_0)}\parallel p\left(t_0\right)\parallel +{\displaystyle \frac{b}{\alpha }}.$$

  (8)
  Thus translations are exponentially attracted to a bounded ball whose radius is proportional to the magnitude of the input and inversely proportional to the contraction rate $\alpha$. Since the rotational component remains on the compact group $SO\left(2\right)$, estimate (8) implies internal boundedness and yields explicit $BIBO$ gains for outputs that factor through $SO\left(2\right)$.
• Concrete mobile robot application and numerical simulation. Interpreting $SE\left(2\right)$ as the configuration space of a planar mobile robot (unicycle), $R_{\theta }$ is the heading and $p=(x,y)$ its position. System (7) corresponds to

  $$\dot{\theta }=u_2,\dot{p}=-\alpha p+R_{\theta }{e}_1{u}_1,$$

  where $e_1={(1,0)}^{\top }$ in body coordinates. The term $-\alpha p$ models a contracting drift (e.g., a virtual stabilizing field or friction-like effect), while $u_1$ and $u_2$ are forward velocity and steering rate.
  A simple simulation scenario demonstrating the theory:

  (a)

  Fix $\alpha >0$ and choose bounded controls, e.g.,

  $$u_1\left(t\right)=\sin t,u_2\left(t\right)=\cos t,t\ge 0.$$

  (b)

  Integrate the system from an initial condition $g\left(0\right)=(R_{{\theta }_0},p_0)$ with large position offset $\parallel p_0\parallel \gg 1$.

  (c)

  Record the translational state $p\left(t\right)$ and the rotational output $y\left(t\right)=R_{\theta \left(t\right)}\in SO\left(2\right)$.

  Numerically one observes:

  (a)

  The position $p\left(t\right)$ is driven into and remains within a bounded ball, in agreement with the analytic bound (8).

  (b)

  The heading $R_{\theta \left(t\right)}$ always lies in $SO\left(2\right)$ and is uniformly bounded; the output $F\left(g\left(t\right)\right)=R_{\theta \left(t\right)}$ is $BIBO.$

  (c)

  Changing $\alpha$ rescales the radius $b/\alpha$ in (8), providing a concrete design guideline for selecting $\alpha$ in practice.

  This simulation, together with the explicit computation of the canonical subgroup $\Gamma ={\mathbb{R}}^2$, illustrates how bracket-generated directions compensate for missing actuators while translational contraction yields a bounded $ICS$ and $BIBO$/ISS properties for physically relevant outputs.

Remark 4.

The $SE\left(2\right)$ example provides a solvable, noncompact case where

1.

The canonical subgroup is explicitly computed as the translation subgroup $\Gamma ={\mathbb{R}}^2$;

2.

The quotient $SE\left(2\right)/\Gamma \simeq SO\left(2\right)$ captures all homomorphic $BIBO$ outputs (e.g., heading measurements in mobile robotics);

3.

Simple numerical simulations of a unicycle-type robot with contracting drift confirm the theoretical ISS and $BIBO$ estimates.

• This offers a concrete and practically motivated illustration of the general framework.

4.4. Example A Compact Semisimple Case: $SO\left(3\right)$

Group and Lie algebra. Let

$$G=SO\left(3\right)=\{R\in {\mathbb{R}}^{3\times 3}:R^{\top }R=I,\det R=1\},$$

a compact, connected semisimple Lie group of dimension 3. Its Lie algebra $\mathfrak{so}\left(3\right)$ is identified with skew-symmetric $3\times 3$ matrices. Using the hat/isomorphism $\widehat{·}:{\mathbb{R}}^3\to \mathfrak{so}\left(3\right)$, we identify $\mathfrak{so}\left(3\right)\simeq ({\mathbb{R}}^3,\times )$. Fix an orthonormal basis $\{E_1,E_2,E_3\}$ of $\mathfrak{so}\left(3\right)$ (corresponding to unit rotation axes).

Left-invariant vector fields. For $V\in \mathfrak{so}\left(3\right)$ the left-invariant field is $Y_V\left(R\right)=RV$. We denote $Y_i:=Y_{E_i}$, $i=1,2,3$.

Linear drift (inner derivation) and $LCS$. On a compact semisimple group every derivation is inner. Pick $W\in \mathfrak{so}\left(3\right)$ (nonzero) and set $\mathcal{D}=\mathrm{ad}\left(W\right)$. The corresponding linear vector field $\mathcal{X}$ has flow of inner automorphisms ${\phi }_t\left(R\right)=\exp \left(tW\right)R\exp (-tW)$, an isometric action for any bi-invariant metric.

Consider the controlled system

$${\mathrm{\Sigma }}_{SO\left(3\right)}:\dot{R}=\mathcal{X}\left(R\right)+u_1{Y}_{V_1}\left(R\right)+u_2{Y}_{V_2}\left(R\right),$$

where $V_1,V_2\in \mathfrak{so}\left(3\right)$ are chosen so that $\mathrm{Lie}\{V_1,V_2,W\}=\mathfrak{so}\left(3\right)$ (this ensures $LARC$ when the drift is present). Controls $u_1$ and $u_2$ are measurable and bounded in a compact set U.

• Dynamics and spectral structure. The adjoint action $\mathrm{ad}\left(W\right)$ on $\mathfrak{so}\left(3\right)$ has purely imaginary eigenvalues (compact case). Consequently ${\mathfrak{g}}^{+}=\left\{0\right\}$, ${\mathfrak{g}}^0=\mathfrak{so}\left(3\right)$, and ${\mathfrak{g}}^{-}=\left\{0\right\}$. There are no expanding directions; the neutral subgroup equals the whole compact group.
• Boundedness and $ICS$. Because $SO\left(3\right)$ is compact, every trajectory of any continuous control system on $SO\left(3\right)$ is a bounded subset of $SO\left(3\right)$ trivially. Under $LARC$ (ensured by the choice of $V_1,V_2,$ and W or by including appropriate controls) the reachable sets have nonempty interior and the unique $ICS$ with nonempty interior is the whole group $SO\left(3\right)$. Hence the minimal compact positively invariant trapping set is $K=SO\left(3\right)$ itself.
• Nontriviality: controllability and nontrivial drift. The example is nontrivial because the drift $\mathcal{X}$ is an inner rotation by W (nonzero) and the controls are limited to two directions; controllability arises from the combination of drift and controlled fields, not from trivial full actuation. In particular, trajectories can explore the full group via concatenation and Lie brackets even though the drift is nonvanishing.
• $BIBO$ outputs. Any homomorphism $F:SO\left(3\right)\to H$ has compact image (continuous image of a compact set). Therefore every homomorphic output is $BIBO$ stable for bounded controls. The canonical subgroup $\Gamma$ is trivial (no expanding or noncompact neutral parts to remove), so $G/\Gamma =SO\left(3\right)$ and the universal classification is degenerate here: all homomorphic outputs are $BIBO$.
• Robustness and ISS remarks. Compactness gives immediate robustness: small perturbations of the drift or control directions do not create escape because the state space remains compact. One can still derive useful local ISS-type estimates for deviations from a reference trajectory using standard compactness arguments and linearization, but global exponential contraction terms (as in noncompact contractive cases) do not appear because ${\mathfrak{g}}^{-}$ is trivial.
• Practical interpretation. This setup models rigid body rotation with controlled torques about two axes plus a background steady rotation (the drift). The compactness of the state space implies boundedness of attitudes irrespective of bounded control magnitudes. The control design problem is then concerned with tracking or stabilization within the compact manifold rather than preventing escape.

Remark 5.

The $SO\left(3\right)$ example is a prototypical compact semisimple case: boundedness is automatic, the $ICS$ is the whole group under $LARC,$ and $BIBO$ stability holds for every homomorphic output. The nontrivial aspect stems from the interplay of a nonzero inner drift and limited control directions that nonetheless generate full accessibility via Lie brackets combined with drift.

Compactness makes boundedness trivial; applied interest lies in control, tracking, and observer design on the compact manifold.

1.

For attitude systems, emphasize exploiting compactness to simplify robustness proofs and observer design without quotienting.

2.

Suggest experiments with limited actuators (two axes) to demonstrate bracket-generation efficacy combined with nontrivial drift.

4.5. Example A Noncompact Semisimple Case $SL(2,\mathbb{R})$

Group and Iwasawa decomposition. Consider

$$G=SL(2,\mathbb{R})=\{g\in {\mathbb{R}}^{2\times 2}:\det g=1\},$$

a connected noncompact semisimple Lie group. The Iwasawa decomposition reads

$$SL(2,\mathbb{R})=KAN,$$

where

$$K=SO\left(2\right)=\left\{\left(\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right)\right\},A=\left\{\left(\begin{array}{cc}a& 0\\ 0& a^{-1}\end{array}\right):a>0\right\},N=\left\{\left(\begin{array}{cc}1& x\\ 0& 1\end{array}\right):x\in \mathbb{R}\right\}.$$

Every $g\in SL(2,\mathbb{R})$ factors uniquely as $g=kan$ with $k\in K,a\in A,n\in N$. The noncompact factors A (hyperbolic) and N (unipotent) are the sources of escape.

Lie algebra and basis. A standard basis of $\mathfrak{sl}(2,\mathbb{R})$ is

$$H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),E=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),$$

with brackets $[H,E]=2E,[H,F]=-2F,$ and $[E,F]=H$. Here H generates $\mathfrak{a}$, E generates $\mathfrak{n}$ and a compact generator sits in $\mathfrak{k}$ (e.g., $K_0=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$).

Linear drift and control directions. Take an inner derivation $\mathcal{D}=\mathrm{ad}\left(H\right)$ (i.e., drift $\mathcal{X}$ with flow ${\phi }_t\left(g\right)=\exp \left(tH\right)g\exp (-tH)$). Consider the $LCS$

$$\dot{g}=\mathcal{X}\left(g\right)+u_1{Y}_E\left(g\right)+u_2{Y}_F\left(g\right),$$

where $Y_E$ and $Y_F$ are left-invariant fields associated with E and F, and controls $u=(u_1,u_2)$ are bounded (compact range). This is a natural model: the drift $\mathrm{ad}\left(H\right)$ is hyperbolic (expanding in the E direction, contracting in the F direction), while controls act in the E and F directions.

Dynamical splitting and algebraic data. The derivation $\mathcal{D}=\mathrm{ad}\left(H\right)$ has eigenvalues $\{2,0,-2\}$ on the complexified algebra, so ${\mathfrak{g}}^{+}=\mathrm{span}\left\{E\right\},{\mathfrak{g}}^0=\mathrm{span}\left\{H\right\},$ and ${\mathfrak{g}}^{-}=\mathrm{span}\left\{F\right\}.$ Integrating gives $G^{+}=N$ (unstable unipotent), $G^0=A$ (neutral hyperbolic group), and $G^{-}$ the opposite unipotent subgroup. In particular $G^{+}\ne \left\{e\right\}$ and $G^0$ is noncompact (isomorphic to ${\mathbb{R}}_{>0}$), so the algebraic boundedness condition fails.

Accessibility ($LARC$). The control directions E and F with brackets generate $\mathfrak{sl}(2,\mathbb{R})$ (since $[E,F]=H$), so $LARC$ holds. Thus, reachable sets have interior in a neighborhood, and bracket motions can explore noncompact directions.

Internal boundedness and instability. Because $G^{+}$ is nontrivial (the N-factor) or $G^0$ noncompact (the A-factor), the geometric boundedness theorem indicates ${\mathcal{O}}^{+}\left(e\right)$ is unbounded: there exist bounded controls producing trajectories that escape to infinity along the $AN$ directions (e.g., flow of E amplified by the hyperbolic drift). Concretely, the drift ${\phi }_t\left(\exp \left(sE\right)\right)=\exp \left(e^{2t}sE\right)$ shows exponential growth in the N-coordinate for $t\to +\infty$, so trajectories can be driven to arbitrarily large N-component values.

$BIBO$ outputs and canonical quotient. Consider a homomorphic output $F:G\to H$. For $BIBO$ stability relative to F, the theorem requires $G^{+}\subset \ker F$ and $F\left(G^0\right)$ compact. In $SL(2,\mathbb{R})$ the unstable subgroup $G^{+}=N$ and neutral subgroup $G^0=A$ are noncompact and normally generate large normal closures. Since $SL(2,\mathbb{R})$ is (almost) simple, any normal subgroup containing a noncentral element is large (often the whole group), so the only homomorphisms that kill N and yield compact image are typically trivial (or factor through finite/central quotients). Therefore nontrivial homomorphic $BIBO$ outputs are essentially absent for generic hyperbolic drift: $BIBO$-stable homomorphisms are trivial or finite-image. The canonical subgroup $\Gamma$ generated by $G^{+}$ and noncompact parts of $G^0$ is (typically) all of $SL(2,\mathbb{R})$, so $G/\Gamma$ trivial and the universal quotient classification yields no nontrivial $BIBO$ outputs.

Robustness remarks. Small perturbations of the derivation that create or remove hyperbolicity alter boundedness qualitatively; hyperbolic splitting is not robustly removed unless eigenvalues cross the imaginary axis. Thus instability here is persistent under small perturbations that keep the sign of real parts.

Remark 6.

The $SL(2,\mathbb{R})$ example illustrates a prototypical semisimple noncompact situation: despite $LARC$ and strong controllability, the presence of genuine expanding/unbounded neutral factors prevents internal boundedness and severely restricts $BIBO$ homomorphic outputs. In physical terms, systems whose drift has hyperbolic scaling directions (dilations) cannot be kept uniformly bounded by merely bounded controls unless the output intentionally ignores (quotients out) those directions, which for simple groups is typically impossible nontrivially.

1.

Clearly illustrates fundamental limits: presence of (A,N) factors prevents boundedness and severely constrains nontrivial $BIBO$ homomorphic outputs.

2.

In applications (e.g., projective camera models or hyperbolic dynamics) advise designing outputs invariant to escaping modes or modifying the drift to introduce contraction.

3.

Propose numerical diagnostics to detect hyperbolic modes experimentally and to guide whether output redesign or plant modification is preferable.

4.6. Example Levi-Type Mixed Group

Group and Lie algebra. Construct a connected Lie group G as a semidirect product $G=R⋊K$ where

• $R\cong {\mathbb{R}}^3$ is the solvable radical with Lie algebra $\mathfrak{r}=\mathrm{span}X,Y,Z$ (abelian for simplicity);
• $K\cong SO\left(2\right)$ is the compact Levi factor with Lie algebra $\mathfrak{k}=\mathrm{span}J$;
• K acts on R by rotations in the ((Y,Z))-plane and trivially on X.

Thus the Lie algebra $\mathfrak{g}=\mathfrak{r}\oplus \mathfrak{k}$ has nonzero brackets

$$[J,Y]=Z,[J,Z]=-Y,$$

and all other brackets among $X,Y,$ and Z vanish (so $\mathfrak{r}$ is abelian).

Linear drift (derivation) and $LCS.$ Define a derivation $\mathcal{D}\in Der\left(\mathfrak{g}\right)$ by

$$\mathcal{D}\left(X\right)=0,\mathcal{D}\left(Y\right)=-\alpha Y,\mathcal{D}\left(Z\right)=-\alpha Z,\mathcal{D}\left(J\right)=0,$$

with parameter $\alpha >0$. The flow of the associated linear vector field $\mathcal{X}$ contracts the $(Y,Z)$ directions and leaves X and J neutral. Consider the $LCS$

$${\mathrm{\Sigma }}_G:\dot{g}=\mathcal{X}\left(g\right)+u_{1}Y\left(g\right)+u_{2}Z\left(g\right)+u_{3}J\left(g\right),$$

with bounded controls u (compact control range). We will also comment the case with only $u_1,u_2$ (no direct J-actuation).

• Dynamical splitting. Eigenvalues of $\mathcal{D}$: $-\alpha ,-\alpha ,0,0$ corresponding to $\{Y,Z\}$ (contracting) and $\{X,J\}$ (neutral). Therefore ${\mathfrak{g}}^{-}=\mathrm{span}\{Y,Z\}$, ${\mathfrak{g}}^0=\mathrm{span}\{X,J\}$, and ${\mathfrak{g}}^{+}=\left\{0\right\}$.
• Integrated subgroups: $G^{-}\simeq {\mathbb{R}}^2$ (contracting plane), $G^0\simeq \mathbb{R}\times SO\left(2\right)$ (product of the noncompact X-direction and compact rotations).
• Accessibility ($LARC$). With controls in $Y,Z,$ and J (or Y and Z together with the drift), the Lie algebra generated by control directions and $\mathcal{D}$ closes to $\mathfrak{g}$. In particular, $[J,Y]=Z$ and brackets with J generate the needed directions; hence $LARC$ holds for typical choices (we assume the control set ensures $LARC$).
• Internal boundedness: failure due to noncompact neutral direction. Although there are no expanding directions (${\mathfrak{g}}^{+}=0$), the neutral subgroup $G^0$ is noncompact because of the X-factor. By the main theorem internal boundedness fails: ${\mathcal{O}}^{+}\left(e\right)$ is unbounded in general because trajectories can drift along the noncompact neutral X direction (no contraction acts on X).
• Intuition: translations along X are not contracted; bounded controls in the other directions combined with neutral drift can produce unbounded excursions in the X-coordinate.
• Canonical quotient and $BIBO$ repair. The canonical subgroup $\Gamma$ generated by unstable and noncompact neutral parts must contain the noncompact X-direction. Here $\Gamma =\exp \left(\mathbb{R}X\right)$ (closed, normal, $\phi$-invariant). Consider the homomorphism (quotient) $\Pi :G\to G/\Gamma$. The quotient collapses the noncompact neutral X-direction, leaving $\overline{G}=G/\Gamma \cong ({\mathbb{R}}^2⋊SO\left(2\right))/\left(\mathrm{mod}X\right)\simeq \left({\mathbb{R}}^2\mathrm{contracted}\right)⋊SO\left(2\right)$ with contracted translational fiber and compact neutral factor $SO\left(2\right)$. By construction the induced system on $G/\Gamma$ satisfies the internal boundedness condition: ${\left(\overline{G}\right)}^{+}=\left\{e\right\}$ and ${\left(\overline{G}\right)}^0$ is compact. Hence the quotient admits a unique bounded $ICS$ and the closure of the reachable set there is compact.
• $BIBO$ outputs: explicit nontrivial example. Define an output homomorphism $F:G\to H$ by projecting out X: e.g., $F\left(\exp \right(sX\left)rk\right)=(r,k)$ where $r\in \exp \left(\mathrm{span}\right\{Y,Z\left\}\right)$, $k\in SO\left(2\right)$. Equivalently F factors through $\Pi :G\to G/\Gamma$. Since $G^{+}\subset \Gamma$ (trivial) and $F\left(G^0\right)$ is compact (image of $SO\left(2\right)$), the $BIBO$ theorem applies: ${\mathrm{\Sigma }}_G$ is $BIBO$ stable relative to F. Thus although the full state may drift unboundedly along X, the output that forgets the X-direction remains bounded for every bounded control. This shows a nontrivial rescue of $BIBO$ stability by appropriate output choice.
• Robustness and ISS remarks. The contracting directions Y and Z provide exponential decay; one can derive ISS-type bounds on the quotient $G/\Gamma$ combining contraction with the compactness of the $SO\left(2\right)$ factor. The $BIBO$ property and $ICS$ on the quotient persist under small perturbations of $\alpha$ and of control directions so long as no neutral direction becomes expanding. At the full G level, persistence of the unbounded neutral X-drift means internal boundedness cannot be recovered by small perturbations unless the drift is modified to contract X.

Remark 7.

1. Representative of mixed systems: a noncompact neutral in the radical breaks internal boundedness, but quotienting by Γ restores $BIBO$ for relevant outputs.

2.

Practical systems (e.g., $V1$ models, navigation with wheel slip) should prioritize output-observer designs that factor out the drifting coordinate; explicit construction of Γguides this selection.

3.

Recommend adding a concrete simulation (e.g., navigation with slip) showing how the output that ignores (X) remains bounded while full-state drift grows unbounded.

Application of the $\mathit{BIBO}$ theorem: quotient outputs

The noncompact $V1$ model on G is not internally stable, but it can be $BIBO$ stable relative to a suitable output map that quotients out the noncompact neutral direction.

Let F:$G\to H$ be the homomorphism that “forgets” the X-direction:

$$H:=(R^2⋊SO\left(2\right)),F\left(\exp \left(sX\right)rk\right):=rk,$$

so that $\ker F=\exp \left(RX\right)\simeq R$. This kernel is normal and invariant under ${\phi }_t$ because $\mathcal{D}X=0$.

Now

$$G^{+}=\left\{e\right\}\subset \ker F,F\left(G^0\right)=SO\left(2\right)\mathrm{is}\mathrm{compact}.$$

Hence the $BIBO$ theorem applies: for the linear control system on G and the output homomorphism F:$G\to H$ defined above, ${\mathrm{\Sigma }}_G$ is $BIBO$ stable relative to F.

Indeed, the kernel $\ker F=\exp \left(\mathbb{R}X\right)$ is $\phi$-invariant. Moreover $G^{+}=\left\{e\right\}\subset \ker F$, and $F\left(G^0\right)=SO\left(2\right)$ is compact. The conclusion follows directly from Theorem 2.

This example exhibits the main geometric message of the paper in a single picture:

• The contracting directions $(Y,Z)$ guarantee a stable component ($G^{-}$);
• The compact rotational symmetry $SO\left(2\right)$ is harmless (compact part of $G^0$);
• The only obstruction to internal stability is the noncompact neutral translation $\exp \left(\mathbb{R}X\right)$;
• Quotienting out that neutral drift yields genuine $BIBO$ stability.

Thus, in Levi-type groups, stability phenomena are governed exactly by: (i) the absence of expanding directions and (ii) compactness properties of the neutral factor globally for internal stability, and after projection for $BIBO.$

• Concluding remarks.

This Levi-type example illustrates the mixed behavior typical of many applied symmetry groups: a solvable radical providing contracting and noncompact neutral directions, and a compact Levi factor giving bounded neutral dynamics. The theory shows how quotienting out noncompact neutrals (or designing outputs that ignore them) yields meaningful $BIBO$ stability, while internal boundedness at the full group fails. This is a realistic model for systems where one degree of freedom exhibits uncontrolled drift (e.g., position offset) but measured outputs (ignore that drift) remain bounded and useful.

5. Conclusions and Future Works

In this paper we illustrated the general stability theory for linear control systems on connected Lie groups by means of a broad set of explicit examples. For each structural class (abelian, nilpotent, solvable, compact semisimple, noncompact semisimple, and mixed/Levi type) we computed the Jordan splitting of the drift derivation, identified the subgroups $G^{\pm }$ and $G^0$, and verified how the presence or absence of expanding directions and of noncompact neutral factors governs internal boundedness, existence of invariant control sets, and $BIBO$ stability. The canonical subgroup $\Gamma$ and the quotient $G/\Gamma$ emerged as the natural objects that isolate all intrinsically unbounded modes and provide a universal stage where $BIBO$ outputs are classified and ISS-type bounds hold. These examples show that the abstract results of [1] are effectively checkable in concrete models and that the geometric decomposition of the drift, together with suitable quotient or output choices, offers a systematic route to stability analysis and design on Lie groups.

The examples in this paper suggest several directions for further research.

This work connects directly with robust stability by expressing boundedness and $BIBO$ properties in terms of spectral and geometric data that are stable under small perturbations of the derivation and control directions. The decomposition $\mathfrak{g}={\mathfrak{g}}^{+}\oplus {\mathfrak{g}}^0\oplus {\mathfrak{g}}^{-}$ and the canonical subgroup $\Gamma$ persist as long as eigenvalues do not cross the imaginary axis, so the qualitative stability picture is robust. On the canonical quotient $G/\Gamma$ we obtain ISS-type estimates that give explicit quantitative robustness margins with respect to bounded disturbances. The examples show how the abstract Lie group framework in [22] yields practically usable, structurally robust stability criteria for linear control systems beyond the Euclidean setting.

Quantitative and numerical aspects. Refine ISS-type and robustness estimates in terms of the spectrum of $\mathcal{D}$ and the geometry of $G^{-}$, and design numerical tools that implement the checklist (computation of ${\mathfrak{g}}^{\pm }$ and ${\mathfrak{g}}^0$, detection of $\Gamma$, automatic $BIBO$ tests) for high-dimensional examples.

Applied models and nonlinear extensions. Apply the framework to concrete systems in mechanics, robotics, vision and neuroscience (e.g., $V1$ models, underactuated bodies, synchronization on networks), and explore extensions of the geometric mechanisms (splitting, canonical quotient, $ICS$) to suitable nonlinear invariant systems on Lie groups or homogeneous spaces.

6. Future Works

• Broader classes of Lie groups. Extend the present catalog to higher-dimensional solvable and nilpotent groups, nonconnected groups, and groups with nontrivial discrete center or lattice quotients, with a systematic study of how $\Gamma$ and $G/\Gamma$ behave under products, semidirect products and central extensions.
• Beyond homomorphic outputs. Develop $BIBO$ criteria for more general outputs (smooth, proper, polynomial), and investigate output feedback stabilization on Lie groups, including conditions under which a nonhomomorphic output can be reduced to a homomorphic one after suitable extension or quotienting.