Invariant Control Systems on Lie Groups

Abstract

This is a survey of our research (conducted over the last few years) on invariant control systems, the associated optimal control problems, and the associated Hamilton–Poisson systems. The focus is on equivalence and classification. State space and detached feedback equivalence of control systems are characterized in simple algebraic terms; several classes of systems (in three dimensions, on the Heisenberg groups, and on the six-dimensional orthogonal group) are classified. Equivalence of cost-extended systems is shown to imply equivalence of the associated Hamilton–Poisson systems. Cost-extended systems of a certain kind are reinterpreted as invariant sub-Riemannian structures. A classification of quadratic Hamilton–Poisson systems in three dimensions is presented. As an illustrative example, the stability and integration of a typical system is investigated.

Appendix: Three-Dimensional Lie Algebras and Groups

There are eleven types of three-dimensional real Lie algebras; in fact, nine algebras and two parametrized infinite families of algebras (see, e.g., [71, 78, 82]). In terms of an (appropriate) ordered basis $\,\left (E_{1},E_{2},E_{3}\right )$, the commutation operation is given by

$$ \displaystyle\begin{array}{rcl} [E_{2},E_{3}]& =& n_{1}E_{1} - aE_{2} {}\\{} [E_{3},E_{1}]& =& aE_{1} + n_{2}E_{2} {}\\{} [E_{1},E_{2}]& =& n_{3}E_{3}. {}\\ \end{array} $$

The structure parameters a, n ₁, n ₂, n ₃ for each type are given in Table 7.1.

A classification of the three-dimensional (real connected) Lie groups can be found in [84]. Let G be a three-dimensional (real connected) Lie group with Lie algebra $\,\mathfrak{g}$.

• 1. If $\,\mathfrak{g}\,$ is Abelian, i.e., $\,\mathfrak{g}\cong 3\mathfrak{g}_{1}$, then G is isomorphic to $\,\mathbb{R}^{3}$, $\,\mathbb{R}^{2} \times \mathbb{T}$, $\,\mathbb{R} \times \mathbb{T}$, or $\,\mathbb{T}^{3}$.

• 2. If $\,\mathfrak{g}\cong \mathfrak{g}_{2.1} \oplus \mathfrak{g}_{1}$, then G is isomorphic to $\,\mathsf{Aff}\,(\mathbb{R})_{0} \times \mathbb{R}\,$ or $\,\mathsf{Aff}\,(\mathbb{R})_{0} \times \mathbb{T}$.

• 3. If $\,\mathfrak{g}\cong \mathfrak{g}_{3.1}$, then G is isomorphic to the Heisenberg group H ₃ or the Lie group $\,\mathsf{H}_{3}^{{\ast}} = \mathsf{H}_{3}/\mathop{\mathrm{Z}}\nolimits (\mathsf{H}_{3}(\mathbb{Z}))$, where $\,\mathop{\mathrm{Z}}\nolimits (\mathsf{H}_{3}(\mathbb{Z}))\,$ is the group of integer points in the centre $\,\mathop{\mathrm{Z}}\nolimits (\mathsf{H}_{3})\cong \mathbb{R}\,$ of H ₃.

• 4. If $\,\mathfrak{g}\cong \mathfrak{g}_{3.2}$, $\,\mathfrak{g}_{3.3}\,$, $\,\mathfrak{g}_{3.4}^{0}$, $\,\mathfrak{g}_{3.4}^{a}$, or $\,\mathfrak{g}_{3.5}^{a}$, then G is isomorphic to the simply connected Lie group G _3. 2, G _3. 3, G _3. 4 ⁰ = SE (1, 1), G _3. 4 ^a, or G _3. 5 ^a, respectively. (The centres of these groups are trivial.)

• 5. If $\,\mathfrak{g}\cong \mathfrak{g}_{3.5}^{0}$, then G is isomorphic to the Euclidean group SE (2), the n-fold covering SE _n (2) of SE ₁(2) = SE (2), or the universal covering group $\,\widetilde{\mathsf{SE}}\,(2)$.

• 6. If $\,\mathfrak{g}\cong \mathfrak{g}_{3.6}$, then G is isomorphic to the pseudo-orthogonal group SO (2, 1)₀, the n-fold covering A _n  of SO (2, 1)₀, or the universal covering group $\,\widetilde{\mathsf{A}}$. Here $\,\mathsf{A}_{2}\cong \mathsf{SL}\,(2, \mathbb{R})$.

• 7. If $\,\mathfrak{g}\cong \mathfrak{g}_{3.7}$, then G is isomorphic to either the unitary group SU (2) or the orthogonal group SO (3).

Among these Lie groups, only H ₃ ^∗, A _n , $n\geqslant 3$, and $\,\widetilde{\mathsf{A}}\,$ are not matrix Lie groups.

7.1.1 Matrix Representations for Solvable Groups

We have the following parametrizations of the solvable three-dimensional matrix Lie groups and their Lie algebras. (We omit the Abelian groups.)

An appropriate ordered basis for the Lie algebra in each case is given by setting (x, y, z) = (1, 0, 0) for E ₁, (x, y, z) = (0, 1, 0) for E ₂, and (x, y, z) = (0, 0, 1) for  E ₃.

7.1.2 Matrix Representations for Semisimple Groups

7.1.2.1 Matrix Lie Groups with Algebra $\,\mathfrak{g}_{3.6}$

There are only two connected matrix Lie groups with Lie algebra $\,\mathfrak{g}_{3.6}$, namely the pseudo-orthogonal group SO (2, 1)₀ and the special linear group $\,\mathsf{SL}\,(2, \mathbb{R})$; $\,\mathsf{SL}\,(2, \mathbb{R})\,$ is a double cover of SO (2, 1)₀.

The pseudo-orthogonal group

$$\displaystyle{\mathsf{SO}\,(2,1) =\{ g \in \mathbb{R}^{3\times 3}\,:\, g^{\top }Jg = J,\,\det g = 1\}}$$

has two connected components. Here $\,J = \mathop{\mathrm{diag}}\nolimits (1,1,-1)$. The identity component of SO (2, 1) is $\,\mathsf{SO}\,(2,1)_{0} = \left \{g \in \mathsf{SO}\,(2,1)\,:\, g_{33}> 0\right \}\,$ where $\,g = \left [\begin{array}{*{10}c} g_{ij} \end{array} \right ]\,$(for g ∈ SO (2, 1)). Its Lie algebra is given by

$$\displaystyle\begin{array}{rcl} \mathfrak{s}\mathfrak{o}\,(2,1)& =& \{A \in \mathbb{R}^{3\times 3}\,:\, A^{\top }J + JA = 0\} {}\\ & =& \left \{\left [\begin{array}{*{10}c} 0 & z &y \\ -z& 0 &x\\ y &x & 0 \end{array} \right ]\,:\, x,y,z \in \mathbb{R}\right \}.{}\\ \end{array}$$

On the other hand, the special linear group is given by

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

Its Lie algebra is given by

$$\displaystyle{\mathfrak{s}\mathfrak{l}\,(2, \mathbb{R}) = \left \{\left [\begin{array}{*{10}c} \frac{x} {2} & \frac{y-z} {2} \\ \frac{y+z} {2} & -\frac{x} {2} \end{array} \right ]\,:\, x,y,z \in \mathbb{R}\right \}.}$$

7.1.2.2 Matrix Lie Groups with Algebra $\,\mathfrak{g}_{3.7}$

There are exactly two connected Lie groups with Lie algebra $\,\mathfrak{g}_{3.7}$; both are matrix Lie groups. The special unitary group and its Lie algebra are given by

$$\displaystyle\begin{array}{rcl} \mathsf{SU}\,(2)& =& \left \{g \in \mathbb{C}^{2\times 2}\,:\, gg^{\dag } = \mathbf{1},\,\det g = 1\right \} {}\\ \mathfrak{s}\mathfrak{u}\,(2)& =& \left \{\left [\begin{array}{*{10}c} \frac{i} {2}x & \frac{1} {2}(iz + y) \\ \frac{1} {2}(iz - y)& -\frac{i} {2}x \end{array} \right ]\,:\, x,y,z \in \mathbb{R}\right \}{}\\ \end{array}$$

SU (2) is a double cover of the orthogonal group SO (3). The orthogonal group SO (3) and its Lie algebra are given by

$$\displaystyle\begin{array}{rcl} \mathsf{SO}\,(3)& =& \left \{g \in \mathbb{R}^{3\times 3}\,:\, gg^{\top } = \mathbf{1},\,\det g = 1\right \} {}\\ \mathfrak{s}\mathfrak{o}\,(3)& =& \left \{\left [\begin{array}{*{10}c} 0 &-z& y \\ z & 0 &-x\\ -y & x & 0 \end{array} \right ]\,:\, x,y,z \in \mathbb{R}\right \}.{}\\ \end{array}$$

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Note 6

Again, an appropriate ordered basis for the Lie algebra in each case is given by setting (x, y, z) = (1, 0, 0) for E ₁, (x, y, z) = (0, 1, 0) for E ₂, and (x, y, z) = (0, 0, 1) for E ₃.

7.1.3 Automorphism Groups

A standard computation yields the automorphism group for each three-dimensional Lie algebra (see, e.g., [57]). With respect to the given ordered basis (E ₁, E ₂, E ₃) , the automorphism group of each solvable Lie algebra has parametrization:

$$\displaystyle\begin{array}{rcl} \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.1})\;&:& \;\left [\begin{array}{*{10}c} yw - vz&x& u \\ 0 & y & v \\ 0 & z &w\\ \end{array} \right ]\qquad \qquad \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{2.1} \oplus \mathfrak{g}_{1})\;:\; \left [\begin{array}{*{10}c} x&y &u\\ y &x & v \\ 0 & 0 & 1\\ \end{array} \right ] {}\\ \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.2})\;&:& \;\left [\begin{array}{*{10}c} u&x&y \\ 0 &u&z\\ 0 & 0 & 1\\ \end{array} \right ]\qquad \qquad \qquad \qquad \quad \,\,\,\mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.3})\;:\; \left [\begin{array}{*{10}c} x&y& z\\ u & v &w \\ 0 &0& 1\\ \end{array} \right ] {}\\ \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.4}^{0})\;&:& \;\left [\begin{array}{*{10}c} x&y &u\\ y &x & v \\ 0 & 0 & 1\\ \end{array} \right ],\left [\begin{array}{*{10}c} x & y & u \\ -y&-x& v \\ 0 & 0 &-1\\ \end{array} \right ]\qquad \quad \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.4}^{a})\;:\; \left [\begin{array}{*{10}c} x&y &u\\ y &x & v \\ 0 & 0 & 1\\ \end{array} \right ] {}\\ \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.5}^{0}))\;&:& \;\left [\begin{array}{*{10}c} x &y &u\\ -y &x & v \\ 0 & 0 & 1\\ \end{array} \right ],\left [\begin{array}{*{10}c} x& y & u \\ y &-x& v \\ 0 & 0 &-1\\ \end{array} \right ]\qquad \quad \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.5}^{a})\;:\; \left [\begin{array}{*{10}c} x &y &u\\ -y &x & v \\ 0 & 0 & 1\\ \end{array} \right ]{}\\ \end{array}$$

For the semisimple Lie algebras, we have

$$\displaystyle{\mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.6}) = \mathsf{SO}\,(2,1)\qquad \text{and}\qquad \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g}_{3.7}) = \mathsf{SO}\,(3).}$$

In several cases, for a connected Lie group G, we wish to find the subgroup of linearized group automorphisms $\,d\mathop{\mathsf{Aut}}\nolimits (\mathsf{G}) =\{ T_{\mathbf{1}}\phi \,:\,\phi \in \mathop{\mathsf{Aut}}\nolimits (\mathsf{G})\}\leqslant \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g})$. If G is simply connected, then $\,d\mathop{\mathsf{Aut}}\nolimits (\mathsf{G}) = \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g})$. If G is not simply connected, then the subgroup $\,d\mathop{\mathsf{Aut}}\nolimits (\mathsf{G})\,$ may be determined by use of the following result.

Proposition 5 (cf. [58])

Let  G be a connected Lie group with Lie algebra $\,\mathfrak{g}\,$ and let $\,q:\widetilde{ \mathsf{G}} \rightarrow \mathsf{G}$ be a universal covering. If $\,\psi \in \mathop{\mathsf{Aut}}\nolimits (\mathfrak{g})$, then $\,\psi \in d\,\mathop{\mathsf{Aut}}\nolimits (\mathsf{G})\,$ if and only if ϕ(kerq) = kerq, where $\,\phi \in \mathop{\mathsf{Aut}}\nolimits (\widetilde{\mathsf{G}})\,$ is the unique automorphism such that T ₁ ϕ = (T ₁ q)⁻¹ ⋅ ψ ⋅ T ₁ q.

7.1.4 Casimir Functions for Lie–Poisson Spaces

An exhaustive list of Casimir functions (not necessarily globally defined), for low-dimensional Lie algebras, was obtained by Patera et al. [86]. For each three-dimensional Lie–Poisson space $\,\mathfrak{g}_{-}^{{\ast}}\,$ (associated to the three-dimensional Lie algebra $\,\mathfrak{g}$) we exhibit its Casimir function:

$$\displaystyle\begin{array}{rcl} \mathfrak{g}_{2.1} \oplus \mathfrak{g}_{1}\;&:& \;C(\,p) = p_{3}\qquad \qquad \quad \,\,\,\mathfrak{g}_{3.1}\;:\; C(\,p) = p_{1} {}\\ \mathfrak{g}_{3.2}\;&:& \;C(\,p) = p_{1}e^{\frac{p_{2}} {p_{1}} }\qquad \qquad \mathfrak{g}_{3.3}\;:\; C(\,p) = \frac{p_{2}} {p_{1}} {}\\ \mathfrak{g}_{3.4}^{0}\;&:& \;C(\,p) = p_{ 1}^{2} - p_{ 2}^{2}\qquad \,\,\,\,\mathfrak{g}_{ 3.4}^{\alpha }{}_{ 1\neq \alpha>0}\;:\; C(\,p) = \frac{\tfrac{1} {2}p_{1} + \tfrac{1} {2}p_{2}} {(\pm \tfrac{1} {2}p_{1} \mp \tfrac{1} {2}p_{2})^{\frac{\alpha -1} {\alpha +1} }} {}\\ \mathfrak{g}_{3.5}^{0}\;&:& \;C(\,p) = p_{ 1}^{2} + p_{ 2}^{2}\qquad \qquad \mathfrak{g}_{ 3.5}^{\alpha }{}_{ \alpha>0}\;:\; C(\,p) = (\,p_{1}^{2} + p_{ 2}^{2})\left (\frac{p_{1} - ip_{2}} {p_{1} + ip_{2}}\right )^{i\alpha } {}\\ \mathfrak{g}_{3.6}\;&:& \;C(\,p) = p_{1}^{2} + p_{ 2}^{2} - p_{ 3}^{2}\qquad \mathfrak{g}_{ 3.7}\;:\; C(\,p) = p_{1}^{2} + p_{ 2}^{2} + p_{ 3}^{2}. {}\\ \end{array}$$

On the trivial Lie–Poisson space $\,(3\mathfrak{g}_{1})_{-}^{{\ast}}$, every function is a Casimir function. Here $\,p \in \mathfrak{g}^{{\ast}}\,$ is written in coordinates as p = p ₁ E ₁ ^∗ + p ₂ E ₂ ^∗ + p ₃ E ₃ ^∗ where (E ₁ ^∗, E ₂ ^∗, E ₃ ^∗) is the dual of the ordered basis (E ₁, E ₂, E ₃). Note that only $\,3\mathfrak{g}_{1}$, $\,\mathfrak{g}_{2.1} \oplus \mathfrak{g}_{1}$, $\,\mathfrak{g}_{3.1}$, $\,\mathfrak{g}_{3.4}^{0}$, $\,\mathfrak{g}_{3.5}^{0}$, $\,\mathfrak{g}_{3.6}$, and $\,\mathfrak{g}_{3.7}\,$ admit globally defined Casimir functions.