E extracta mathematicae Vol. 32, Núm. 2, 213 – 238 (2017) Invariant Control Systems on Lie Groups: A Short Survey Rory Biggs, Claudiu C. Remsing Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, South Africa rory.biggs@up.ac.za Department of Mathematics, Rhodes University, 6140 Grahamstown, South Africa c.c.remsing@ru.ac.za Presented by Dikran Dikranjan Received March 6, 2016 Abstract: This is a short survey of our recent research on invariant control systems (and their associated optimal control problems). We are primarily concerned with equivalence and classification, especially in three dimensions. Key words: Invariant control affine systems, detached feedback equivalence, optimal control. AMS Subject Class. (2010): 93B27, 22E60, 49J15, 53C17. 1. Introduction Geometric control theory began in the late 1960s with the study of (nonlin- ear) control systems by using concepts and methods from differential geometry (cf. [9,50,73]). A smooth control system may be viewed as a family of vector fields (or dynamical systems) on a manifold, smoothly parametrized by a set of controls. An integral curve corresponding to some admissible control func- tion (from some time interval to the set of controls) is called a trajectory of the system. The first basic question one asks of a control system is whether or not any two points can be connected by a trajectory: this is known as the controllability problem. Once one has established that two points can be connected by a trajectory, one may wish to find a trajectory that minimizes some (practical) cost function: this is known as the optimality problem. The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 317721. Also, the first author would like to acknowledge the financial support of the Claude Leon Foun- dation towards this research. 213 214 r. biggs, c. c. remsing A significant subclass of control systems rich in symmetry are those evolv- ing on Lie groups and invariant under left translations; for such a system the left translation of any trajectory is a trajectory. This class of systems was first considered in 1972 by Brockett [35] and by Jurdjevic and Sussmann [53]; it forms a natural geometric framework for various (variational) prob- lems in mathematical physics, mechanics, elasticity, and dynamical systems (cf. [9,33,50,51]). In the last few decades substantial work on applied nonlin- ear control has drawn attention to invariant control affine systems evolving on matrix Lie groups of low dimension (see, e.g., [52,67,69,70] and the references therein). This paper serves as a short survey of our recent research on (the equiv- alence of) left-invariant control systems and the associated optimal control problems. Ideas and key results from several papers published over the last couple of years are reexamined and restructured; some elements are also rein- terpreted. The first aspect we address is the equivalence of control systems. Both state space and (detached) feedback equivalence are characterized in simple algebraic terms. The classification problem in three dimensions is re- visited. The second aspect we address is the equivalence of invariant optimal control problems, or rather, their associated cost-extended systems. One asso- ciates to each cost-extended system, via the Pontryagin Maximum Principle, a quadratic Hamilton–Poisson system on the associated Lie–Poisson space. Equivalence of cost-extended systems implies equivalence of the associated Hamilton–Poisson systems. Additionally, the subclass of drift-free systems with homogeneous cost are reinterpreted as invariant sub-Riemannian struc- tures. An extended version of this survey will appear in [32]. Throughout, we make use of the classification of three-dimensional Lie groups and Lie algebras; relevant details are given in the appendix. 2. Invariant control systems and their equivalence 2.1. Invariant control affine systems. A ℓ-input left-invariant control affine system Σ on a (real, finite-dimensional, connected) Lie group G consists of a family of left-invariant vector fields Ξu on G, affinely parame- trized by controls u ∈ Rℓ. Such a system is written as ġ = Ξu(g) = Ξ(g,u) = g(A + u1B1 + · · · + uℓBℓ), g ∈ G, u ∈ Rℓ. Here A,B1, . . . ,Bℓ are elements of the Lie algebra g with B1, . . . ,Bℓ linearly independent. The “product” gA denotes the left translation T1Lg · A of invariant control systems on lie groups 215 A ∈ g by the tangent map of Lg : G → G, h 7→ gh. (When G is a matrix Lie group, this product is simply a matrix multiplication.) Note that the dynamics Ξ : G×Rℓ → TG are invariant under left translations, i.e., Ξ (g,u) = g Ξ (1,u). Σ is completely determined by the specification of its state space G and its parametrization map Ξ (1, ·). When G is fixed, we specify Σ by simply writing Σ : A + u1B1 + · · · + uℓBℓ. The trace Γ of a system Σ is the affine subspace Γ = A + Γ0 = A + ⟨B1, . . . ,Bℓ⟩ of g. (Here Γ0 = ⟨B1, . . . ,Bℓ⟩ is the subspace of g spanned by B1, . . . ,Bℓ.) A system Σ is called homogeneous if A ∈ Γ0, and inhomoge- neous otherwise; Σ is said to be drift free if A = 0. Also, Σ is said to have full rank if its trace generates the whole Lie algebra, i.e., Lie(Γ) = g. The admissible controls are piecewise continuous maps u(·) : [0,T ] → Rℓ. A trajectory for an admissible control u(·) is an absolutely continuous curve g(·) : [0,T ] → G such that ġ(t) = g(t) Ξ (1,u(t)) for almost every t ∈ [0,T]. We say that a system Σ is controllable if for any g0,g1 ∈ G, there exists a trajectory g(·) : [0,T ] → G such that g(0) = g0 and g(T) = g1. If Σ is controllable, then it has full rank. For more details about invariant control systems see, e.g., [9,50,53,71]. 2.2. Equivalence of systems. The most natural equivalence relation for control systems is equivalence up to coordinate changes in the state space. This is called state space equivalence (see [47]). State space equivalence is well understood. It establishes a one-to-one correspondence between the tra- jectories of the equivalent systems. However, this equivalence relation is very strong. In the (general) analytic case, Krener characterized local state space equivalence in terms of the existence of a linear isomorphism preserving iter- ated Lie brackets of the system’s vector fields ([58], see also [9,72,73]). Another fundamental equivalence relation for control systems is that of feedback equivalence. Two feedback equivalent control systems have the same set of trajectories (up to a diffeomorphism in the state space) which are parametrized differently by admissible controls. Feedback equivalence has been extensively studied in the last few decades (see [68] and the references therein). There are a few basic methods used in the study of feedback equiv- alence. These methods are based either on (studying invariant properties of) associated distributions or on Cartan’s method of equivalence ([41]) or in- spired by the Hamiltonian formalism ([47]); also, another fruitful approach is closely related to Poincaré’s technique for linearization of dynamical systems. 216 r. biggs, c. c. remsing Feedback transformations play a crucial role in control theory, particularly in the important problem of feedback linearization ([48]). The study of feedback equivalence of general control systems can be reduced, by a simple trick, to the case of control affine systems ([47]). For a thorough study of the equivalence and classification of (general) control affine systems, see [39]. We consider state space equivalence and feedback equivalence in the con- text of left-invariant control affine systems ([29], see also [18]). Character- izations of state space equivalence and (detached) feedback equivalence are obtained in terms of Lie group isomorphisms. Furthermore, the classification of systems on the three-dimensional Lie groups is treated. 2.2.1. State space equivalence. Two systems Σ and Σ′ are called state space equivalent if there exists a diffeomorphism ϕ : G → G′ such that, for each control value u ∈ Rℓ, the vector fields Ξu and Ξ′u are ϕ-related, i.e., Tgϕ · Ξ (g,u) = Ξ′ (ϕ(g),u) for g ∈ G and u ∈ Rℓ. We have the following simple algebraic characterization of this equivalence. Theorem 2.1. ([29], see also [58]) Two full-rank systems Σ and Σ′ are state space equivalent if and only if there exists a Lie group isomorphism ϕ : G → G′ such that T1ϕ · Ξ (1,u) = Ξ′ (1,u) for all u ∈ Rℓ. Proof sketch. Suppose that Σ and Σ′ are state space equivalent. By composition with a left translation, we may assume ϕ(1) = 1. As the elements Ξu(1), u ∈ Rℓ generate g and the push-forward ϕ∗Ξu of the left-invariant vector fields Ξu are left invariant, it follows that ϕ is a Lie group isomorphism satisfying the requisite property (cf. [18]). Conversely, suppose that ϕ : G → G′ is a Lie group isomorphism as prescribed. Then Tgϕ · Ξ(g,u) = T1(ϕ ◦ Lg) · Ξ(1,u) = T1(Lϕ(g) ◦ ϕ) · Ξ(1,u) = Ξ′(ϕ(g),u). Remark. If ϕ is defined only between some neighbourhoods of identity of G and G′, then Σ and Σ′ are said to be locally state space equivalence. A characterization similar to that given in Theorem 2.1, in terms of Lie algebra automorphisms, holds ([29]). In the case of simply connected Lie groups, local and global equivalence are the same (as dAut(G) = Aut(g)). State space equivalence is quite a strong equivalence relation. Hence, there are so many equivalence classes that any general classification appears to be very difficult if not impossible. However, there is a chance for some reasonable classification in low dimensions. We give an example to illustrate this point. invariant control systems on lie groups 217 Example 2.1. ([1]) Any two-input inhomogeneous full-rank control affine system on the Euclidean group SE (2) is state space equivalent to exactly one of the following systems Σ1,αβγ : αE3 + u1(E1 + γ1E2) + u2(βE2) Σ2,αβγ : βE1 + γ1E2 + γ2E3 + u1(αE3) + u2E2 Σ3,αβγ : βE1 + γ1E2 + γ2E3 + u1(E2 + γ3E3) + u2(αE3). Here α > 0,β ̸= 0 and γ1,γ2,γ3 ∈ R, with different values of these parameters yielding distinct (non-equivalent) class representatives. Note. A full classification (under state space equivalence) of systems on SE (2) appears in [1], whereas a classification of systems on SE (1,1) appears in [11]. For a classification of systems on SO (2,1)0, see [30]. 2.2.2. Detached feedback equivalence. We specialize feedback equivalence in the context of invariant systems by requiring that the feed- back transformations are compatible with the Lie group structure (cf. [18]). Two systems Σ and Σ′ are called detached feedback equivalent if there ex- ist diffeomorphisms ϕ : G → G′ and φ : Rℓ → Rℓ such that, for each control value u ∈ Rℓ, the vector fields Ξu and Ξ′φ(u) are ϕ-related, i.e., Tgϕ · Ξ (g,u) = Ξ′ (ϕ(g),φ(u)) for g ∈ G and u ∈ Rℓ. We have the following simple algebraic characterization of this equivalence in terms of the traces Γ = im Ξ(1, ·) and Γ′ = im Ξ′(1, ·) of Σ and Σ′. Theorem 2.2. ([29]) Two full-rank systems Σ and Σ′ are detached feed- back equivalent if and only if there exists a Lie group isomorphism ϕ : G → G′ such that T1ϕ · Γ = Γ′. Proof sketch. Suppose Σ and Σ′ are detached feedback equivalent. By composing ϕ with an appropriate left translation, we may assume ϕ(1) = 1′. Hence T1ϕ · Ξ(1,u) = Ξ′(1′,φ(u)) and so T1ϕ · Γ = Γ′. Moreover, as the elements Ξu(1), u ∈ Rℓ generate g and the push-forward of the left-invariant vector fields Ξu are left invariant, it follows that ϕ is a group isomorphism (cf. [18]). On the other hand, suppose there exists a group isomorphism ϕ : G → G′ such that T1ϕ · Γ = Γ′. Then there exists a unique affine isomorphism φ : Rℓ → Rℓ ′ such that T1ϕ · Ξ(1,u) = Ξ′(1′,φ(u)). As with state space equivalence, by left-invariance and the fact that ϕ is a Lie group isomorphism, it then follows that Tgϕ · Ξ(g,u) = Ξ′(ϕ(g),φ(u)). 218 r. biggs, c. c. remsing Remark. If ϕ is defined only between some neighbourhoods of identity of G and G′, then Σ and Σ′ are said to be locally detached feedback equivalent. A characterization similar to that given in Theorem 2.2, in terms of Lie algebra automorphisms, holds. As for state space equivalence, in the case of simply connected Lie groups local and global equivalence are the same (as dAut(G) = Aut(g)). Detached feedback equivalence is notably weaker than state space equiva- lence. To illustrate this point, we give a classification, under detached feedback equivalence, of the same class of systems considered in Example 2.1. Example 2.2. ([23]) Any two-input inhomogeneous full-rank control affi- ne system on SE (2) is detached feedback equivalent to exactly one of the following systems Σ1 : E1 + u1E2 + u2E3 Σ2,α : αE3 + u1E1 + u2E2. Here α > 0 parametrizes a family of class representatives, each different value corresponding to a distinct non-equivalent representative. 2.2.3. Classification in three dimensions. We exhibit a classifica- tion, under detached feedback equivalence, of the full-rank systems evolving on unimodular three-dimensional Lie groups (i.e., the classical Abelian, Heisen- berg, Euclidean, semi-Euclidean, pseudo-orthogonal and orthogonal groups). We shall restrict our discussion to the simply connected groups. A represen- tative is identified for each equivalence class. Systems on the Euclidean group and the orthogonal group are discussed as typical examples. Details on the classification of three-dimensional Lie groups and their Lie algebras (along with standard ordered bases), as well as the corresponding automorphisms groups, can be found in Appendix A. Note. A classification, under detached feedback equivalence, of all (full- rank) control systems on three-dimensional Lie groups appears in [27] (see also [19,21–23] and [24,26]). On higher dimensional Lie groups, a classification of control systems on the orthogonal group SO (4) was obtained in [4] (see also [2]). Controllability of the respective systems is also addressed in these papers. invariant control systems on lie groups 219 We start with the solvable groups; the classification procedure is as fol- lows. Firstly, the group of automorphisms is determined (see Appendix A). Equivalence class representatives are then constructed by considering the ac- tion of an automorphism on the trace of a typical system. Lastly, one verifies that none of the representatives are equivalent. Theorem 2.3. ([22,23]) Suppose Σ is a full-rank system evolving on a simply connected unimodular solvable Lie group G. Then G is isomorphic to one of the groups listed below and Σ is detached feedback equivalent to exactly one of accompanying (full-rank) systems on that group. 1. On R3, we have the systems Σ(2,1) : E1 + u1E2 + u2E3 Σ (3,0) : u1E1 + u2E2 + u3E3. 2. On H3, we have the systems Σ(1,1) : E2 + uE3 Σ (2,0) : u1E2 + u2E3 Σ (2,1) 1 : E1 + u1E2 + u2E3 Σ (2,1) 2 : E3 + u1E1 + u2E2 Σ(3,0) : u1E1 + u2E2 + u3E3. 3. On SE (1,1), we have the systems Σ (1,1) 1 : E2 + uE3 Σ (1,1) 2,α : αE3 + uE2 Σ(2,0) : u1E2 + u2E3 Σ (2,1) 1 : E1 + u1E2 + u2E3 Σ (2,1) 2 : E1 + u1(E1 + E2) + u2E3 Σ (2,1) 3,α : αE3 + u1E1 + u2E2 Σ(3,0) : u1E1 + u2E2 + u3E3. 4. On S̃E (2), we have the systems Σ (1,1) 1 : E2 + uE3 Σ (1,1) 2,α : αE3 + uE2 Σ(2,0) : u1E2 + u2E3 Σ (2,1) 1 : E1 + u1E2 + u2E3 Σ (2,1) 2,α : αE3 + u1E1 + u2E2 Σ (3,0) : u1E1 + u2E2 + u3E3. Here α > 0 parametrizes families of distinct (non-equivalent) class represen- tatives. 220 r. biggs, c. c. remsing Proof. We treat, as typical case, only item (4). The group of linearized automorphisms of S̃E (2) is given by dAut(S̃E (2)) =     x y u−σy σx v 0 0 σ   : x,y,u,v ∈ R, x2 + y2 ̸= 0, σ = ±1   . Let Σ be a single-input inhomogeneous system with trace Γ = A + Γ0 ⊂ s̃e (2). Suppose E∗3(Γ 0) ̸= {0}. (Here E∗3 is the corresponding element of the dual basis.) Then Γ = a1E1 + a2E2 + ⟨b1E1 + b2E2 + E3⟩. Thus ψ =   a1 a1 b1−a1 a2 b2 0 0 1   is an automorphism such that ψ · Γ(1,1)1 = Γ. So Σ is equivalent to Σ (1,1) 1 . On the other hand, suppose E∗3(Γ 0) = {0}. Then Γ = a1E1 + a2E2 + a3E3 + ⟨b1E1 + b2E2⟩ with a3 ̸= 0 (as Lie(Γ) = s̃e(2)). Hence ψ =   b2 sgn(a3) b1 a1 a3 sgn(a3) −b1 sgn(a3) b2 a2a3 sgn(a3) 0 0 sgn(a3)   is an automorphism such that ψ · Γ(1,1)2,α = Γ, where α = a3 sgn(a3). Let Σ be a two-input homogeneous system with trace Γ = ⟨B1,B2⟩. Then Σ̂ : B1 +⟨B2⟩ is a (full-rank) single-input inhomogeneous system. Therefore, there exists an automorphism ψ such that ψ · (B1 + ⟨B2⟩) equals either E2+⟨E3⟩ or αE3+⟨E2⟩. Hence, in either case, we get ψ·⟨B1,B2⟩ = ⟨E2,E3⟩. Thus Σ is equivalent to Σ(2,0). The classification for the two-input inhomogeneous systems follows simi- larly. If Σ is a three-input system, then clearly it is equivalent to Σ(3,0). Most pairs of systems cannot be equivalent due to different homogeneities or different number of inputs. As the subspace ⟨E1,E2⟩ is invariant (under the action of automorphisms), Σ (1,1) 1 is not equivalent to any system Σ (1,1) 2,α . For A ∈ s̃e (2) and ψ ∈ dAut(SE) (2), we have that E∗3(ψ · αE3) = ±α. Thus Σ (1,1) 2,α and Σ (1,1) 2,α′ are equivalent only if α = α ′. For the two-input inhomogeneous systems, similar arguments hold. We now proceed to the semisimple Lie groups; the procedure for classifi- cation is similar to that of the solvable groups. However, here we employ an invariant control systems on lie groups 221 invariant bilinear product ω (the Lorentzian product and the dot product, respectively); the inhomogeneous systems are (partially) characterized by the level set {A ∈ g : ω(A,A) = α} that their trace is tangent to. Theorem 2.4. ([21]) Suppose Σ is a full-rank system evolving on a sim- ply connected semisimple Lie group G. Then G is isomorphic to one of the groups listed below and Σ is detached feedback equivalent to exactly one of accompanying (full-rank) systems on that group. 1. On à = S̃L (2,R), we have the systems Σ (1,1) 1 : E3 + u(E2 + E3) Σ (1,1) 2,α : αE2 + uE3 Σ (1,1) 3,α : αE1 + uE2 Σ (1,1) 4,α : αE3 + uE2 Σ (2,0) 1 : u1E1 + u2E2 Σ (2,0) 2 : u1E2 + u2E3 Σ (2,1) 1 : E3 + u1E1 + u2(E2 + E3) Σ (2,1) 2,α : αE1 + u1E2 + u2E3 Σ (2,1) 3,α : αE3 + u1E1 + u2E2 Σ (3,0) : u1E1 + u2E2 + u3E3. 2. On SU (2), we have the systems Σ(1,1)α : αE2 + uE3 Σ (2,0) : u1E2 + u2E3 Σ(2,1)α : αE1 + u1E2 + u2E3 Σ (3,0) : u1E1 + u2E2 + u3E3. Here α > 0 parametrizes families of distinct (non-equivalent) class represen- tatives. Proof. We consider only item (2), i.e., systems on the unitary group SU (2). (The proof for item (1), although more involved, is similar.) The group of linearized automorphisms of SU (2) is dAut(SU (2)) = SO (3) = {g ∈ R3×3 : gg⊤ = 1, detg = 1}. The dot product • on su (2) is given by A • B = a1b1 + a2b2 + a3b3. (Here A = ∑3 i=1 aiEi and B = ∑3 i=1 biEi.) The level sets Sα = {A ∈ su (2) : A • A = α} are spheres of radius √ α (and are preserved by automorphisms). The group of automorphisms acts transitively on each sphere Sα. The critical point C•(Γ) (at which an inhomogeneous affine subspace is tangent to a sphere Sα) is given by C•(Γ) = A − A • B B • B B C•(Γ) = A − [ B1 B2 ] [B1 • B1 B1 • B2 B1 • B2 B2 • B2 ]−1 [ A • B1 A • B2 ] . 222 r. biggs, c. c. remsing Critical points behave well under the action of automorphisms, i.e., ψ·C•(Γ) = C•(ψ · Γ) for any automorphism ψ. (The critical point of Γ is well defined as it is independent of parametrization.) Let Σ be a single-input inhomogeneous system with trace Γ. There exists an automorphism ψ such that ψ · Γ = αsinθE1 + αcosθE2 + ⟨E3⟩, where α = √ C•(Γ) • C•(Γ). Hence ψ′ =  cosθ − sinθ 0sinθ cosθ 0 0 0 1   is an automorphism such that ψ′ · ψ · Γ = Γ(1,1)α . Let Σ be a two-input homogeneous system with trace Γ = ⟨B1,B2⟩. Then Σ̂ : B1 +⟨B2⟩ is a (full-rank) single-input inhomogeneous system. Therefore, there exists an automorphism ψ such that ψ · (B1 + ⟨B2⟩) = αE2 + ⟨E3⟩. Hence ψ · ⟨B1,B2⟩ = ⟨E2,E3⟩. Thus Σ is equivalent to Σ(2,0). Let Σ be a two-input inhomogeneous system with trace Γ. We have C•(Γ) • C•(Γ) = α2 for some α > 0. As C•(Γ1,α) • C•(Γ1,α) = α2, there exists an automorphism ψ such that ψ · C•(Γ) = C•(Γ1,α). Hence ψ · Γ and Γ1,α are both equal to the tangent plane of Sα2 at ψ · C•(Γ), and are therefore identical. If Σ is a three-input system, then it is equivalent to Σ(3,0). Lastly, we note that none of the representatives obtained are equivalent. (Again, we first distinguish representatives in terms of homogeneity and num- ber of inputs.) As α2 = C•(Γ (1,1) α ) • C•(Γ (1,1) α ) (resp. α 2 = C•(Γ (2,1) α ) • C•(Γ (2,1) α )) is an invariant quantity, the systems Σ (1,1) α and Σ (1,1) α′ (resp. Σ (2,1) α and Σ (2,1) α′ ) are equivalent only if α = α ′. 3. Invariant optimal control We consider the class of left-invariant optimal control problems on Lie groups with fixed terminal time, affine dynamics, and affine quadratic cost. Formally, such problems are given by ġ = g (A + u1B1 + · · · + uℓBℓ) , g ∈ G, u ∈ Rℓ (1) g(0) = g0, g(T) = g1 (2) J = ∫ T 0 (u(t) − µ)⊤ Q(u(t) − µ) dt −→ min. (3) invariant control systems on lie groups 223 Here G is a (real, finite-dimensional) connected Lie group with Lie alge- bra g, A, B1, . . . , Bℓ ∈ g (with B1, . . . , Bℓ linearly independent), u = (u1, . . . ,uℓ) ∈ Rℓ, µ ∈ Rℓ, and Q is a positive definite ℓ × ℓ matrix. To each such problem, we associate a cost-extended system (Σ,χ). Here Σ is the control system (1) and the cost function χ : Rℓ → R has the form χ(u) = (u − µ)⊤ Q(u − µ). Each cost-extended system corresponds to a fam- ily of invariant optimal control problems; by specification of the boundary data (g0,g1,T), the associated problem is uniquely determined. Optimal control problems of this kind have received considerable attention in the last few decades. Various physical problems have been modelled in this manner, such as optimal path planning for airplanes, motion planning for wheeled mobile robots, spacecraft attitude control, and the control of underactuated underwater vehicles ([61,67,75]); also, the control of quantum systems and the dynamic formation of DNA ([37,43]). Many problems (as well as sub-Riemannian structures) on various low-dimensional matrix Lie groups have been considered by a number of authors (see, e.g., [15, 16, 34, 49, 52, 54, 63,65,66,69,70]). We introduce a form of equivalence for problems of the form (1)–(2)–(3), or rather, the associated cost-extended systems (cf. [20, 28]). Cost equiva- lence establishes a one-to-one correspondence between the associated optimal trajectories, as well as the associated extremal curves. Via the Pontryagin Maximum Principle, we associate to each cost-extended systems a quadratic Hamilton–Poisson systems on the associated Lie–Poisson space. We show that cost equivalence of cost-extended systems implies equivalence of the associ- ated Hamiltonian systems. In addition, we reinterpret drift-free cost-extended systems (with homogeneous cost) as invariant sub-Riemannian structures. 3.1. Pontryagin Maximum Principle. The Pontryagin Maximum Principle provides necessary conditions for optimality which are naturally ex- pressed in the language of the geometry of the cotangent bundle T∗G of G (see [9,40,50]). The cotangent bundle T∗G can be trivialized (from the left) such that T∗G = G × g∗; here g∗ is the dual of the Lie algebra g. To an op- timal control problem (1)–(2)–(3) we associate, for each real number λ and each control parameter u ∈ Rℓ a Hamiltonian function on T∗G = G × g∗: Hλu (ξ) = λχ(u) + ξ (Ξu(g)) = λχ(u) + p(Ξu(1)), ξ = (g,p) ∈ T∗G. (4) 224 r. biggs, c. c. remsing We denote by H⃗λu the corresponding Hamiltonian vector field (with respect to the symplectic structure on T∗G). In terms of the above Hamiltonians, the Maximum Principle can be stated as follows. Maximum Principle. Suppose the controlled trajectory (ḡ(·), ū(·)) de- fined over the interval [0,T] is a solution for the optimal control problem (1)–(2)–(3). Then, there exists a curve ξ(·) : [0,T] → T∗G with ξ(t) ∈ T∗ ḡ(t) G, t ∈ [0,T ], and a real number λ ≤ 0, such that the following conditions hold for almost every t ∈ [0,T ] : (λ,ξ(t)) ̸≡ (0,0) (5) ξ̇(t) = H⃗λū(t)(ξ(t)) (6) Hλū(t) (ξ(t)) = maxu Hλu (ξ(t)) = constant. (7) An optimal trajectory, g(·) : [0,T] → G is the projection of an integral curve ξ(·) of the (time-varying) Hamiltonian vector field H⃗λ ū(t) . A trajectory-control pair (ξ(·),u(·)) is said to be an extremal pair if ξ(·) satisfies the conditions (5), (6), and (7). The projection ξ(·) of an extremal pair is called an extremal. An extremal curve is called normal if λ < 0 and abnormal if λ = 0. For the class of optimal control problems under consideration, the max- imum condition (7) eliminates the parameter u from the family of Hamil- tonians (Hu); as a result, we obtain a smooth G-invariant function H on T∗G = G × g∗. This Hamilton–Poisson system on T∗G can be reduced to a Hamilton–Poisson system on the (minus) Lie–Poisson space g∗−, with Poisson bracket given by {F,G} = −p([dF(p),dG(p)]). Here F,G ∈ C∞(g∗) and dF(p),dG(p) are elements of the double dual g∗∗ which is canonically identified with the Lie algebra g. 3.2. Equivalence of cost-extended systems. Let (Σ,χ) and (Σ′,χ′) be two cost-extended systems. (Σ,χ) and (Σ′,χ′) are said to be cost equivalent if there exist a Lie group isomorphism ϕ : G → G′ and an affine isomorphism φ : Rℓ → Rℓ such that Tgϕ · Ξ(g,u) = Ξ′(ϕ(g),φ(u)) and χ′ ◦ φ = rχ for g ∈ G, u ∈ Rℓ and some r > 0. Equivalently, (Σ,χ) and (Σ′,χ′) are cost equivalent if and only if there exist a Lie group isomorphism ϕ : G → G′ and an affine isomorphism φ : Rℓ → Rℓ such that T1ϕ·Ξ(1,u) = Ξ′(1′,φ(u)) and χ′ ◦ φ = rχ for some r > 0. Accordingly: invariant control systems on lie groups 225 • If (Σ,χ) and (Σ′,χ′) are cost equivalent, then Σ and Σ′ are detached feedback equivalent. • If two full-rank systems Σ and Σ′ are state space equivalent, then (Σ,χ) and (Σ′,χ) are cost equivalent for any cost χ. • If two full-rank systems Σ and Σ′ are detached feedback equivalent with respect to a feedback transformation φ, then (Σ,χ◦φ) and (Σ′,χ) are cost equivalent for any cost χ. Remark. The cost-preserving condition χ′ ◦φ = rχ is partially motivated by the following considerations. Each cost χ on Rℓ induces a strict partial ordering u < v ⇐⇒ χ(u) < χ(v). It turns out that χ and χ′ induce the same strict partial ordering on Rℓ if and only if χ = rχ′ for some r > 0. The dynamics-preserving condition Tgϕ · Ξ(g,u) = Ξ′(ϕ(g),φ(u)) is just that of detached feedback equivalence (on full-rank systems). Let (g(·),u(·)) be a controlled trajectory, defined over an interval [0,T ], of a cost-extended system (Σ,χ). We say that (g(·),u(·)) is a virtually optimal controlled trajectory (shortly VOCT) if it is a solution for the associated optimal control problem with boundary data (g(0),g(T),T). Similarly, we say that (g(·),u(·)) is an extremal controlled trajectory (shortly ECT) if it satisfies the necessary conditions of the Pontryagin Maximum Principle (with λ ≤ 0). Clearly, any VOCT is an ECT. A map ϕ × φ defining a cost equivalence between two cost-extended systems establishes a one-to-one correspondence between their respective VOCTs (and ECTs). Proposition 3.1. ([20,28]) Suppose ϕ×φ defines a cost equivalence be- tween (Σ,χ) and (Σ′,χ′). Then 1. (g(·),u(·)) is a VOCT if and only if (ϕ ◦ g(·),φ ◦ u(·)) is a VOCT; 2. (g(·),u(·)) is an ECT if and only if (ϕ ◦ g(·),φ ◦ u(·)) is an ECT. One can classify the cost-extended systems corresponding to a given in- variant control system by use of the following result. (We denote by Aff (Rℓ) the group of affine isomorphisms of Rℓ.) Proposition 3.2. ([20,28]) Let (Σ,χ) and (Σ,χ′) are two cost-extended systems (with identical underlying control system Σ) and let TΣ = { φ ∈ Aff (Rℓ) : ∃ψ ∈ dAut(G), ψ · Γ = Γ, ψ · Ξ(1,u) = Ξ(1,φ(u)) } 226 r. biggs, c. c. remsing be the group of feedback transformations leaving Σ invariant. (Σ,χ) and (Σ,χ′) are cost equivalent if and only if there exists an element φ ∈ TΣ such that χ′ = rχ ◦ φ for some r > 0. Example 3.1. ([20], cf. Theorem 2.3, item 4) On SE (2) , any full-rank two-input drift-free cost-extended system (Σ,χ) with homogeneous cost (i.e., Ξ(1,0) = 0 and χ(0) = 0) is cost equivalent to (Σ(2,0),χ(2,0)) : { Σ : u1E2 + u2E3 χ(u) = u21 + u 2 2. Example 3.2. (cf. Theorem 2.3, item 2) Any controllable cost-extended system on H3 is C-equivalent to exactly one of the cost-extended systems ( Σ(2,0),χ (2,0) 1 ) : { Σ(2,0) : u1E2 + u2E3 χ (2,0) 1 (u) = u 2 1 + u 2 2( Σ(2,0),χ (2,0) 2 ) : { Σ(2,0) : u1E2 + u2E3 χ (2,0) 2 (u) = (u1 − 1) 2 + u22 ( Σ(2,1),χ(2,1)α ) : { Σ(2,1) : E1 + u1E2 + u2E3 χ(2,1)α (u) = (u1 − α) 2 + u22( Σ(3,0),χ (3,0) α ) : { Σ(3,0) : u1E1 + u2E2 + u3E3, χ (3,0) α (u) = (u1 − α1)2 + (u2 − α2)2 + u23. Here α,α1,α2 ≥ 0 parametrize families of (non-equivalent) class representa- tives. Note. Several examples of classification under cost-equivalence can be found in [12,14,17,28] 3.3. Pontryagin lift. To any cost-extended system (Σ,χ) on a Lie group G we associate, via the Pontryagin Maximum Principle, a Hamilton– Poisson system on the associated Lie–Poisson space g∗− (cf. [9, 50, 71]). We show that equivalence of cost-extended systems implies equivalence of the associated Hamilton–Poisson systems. invariant control systems on lie groups 227 Note. The Pontryagin lift may be realized as a contravariant functor be- tween the category of cost-extended control systems and the category of Hamilton–Poisson systems ([28], see also [40]). A quadratic Hamilton–Poisson system (g∗−,HA,Q) is specified by HA,Q : g ∗ → R, p 7→ p(A) + Q(p). Here A ∈ g and Q is a quadratic form on g∗. If A = 0, then the system is called homogeneous; otherwise, it is called inhomogeneous. (When g∗− is fixed, a system (g∗−,HA,Q) is identified with its Hamiltonian HA,Q.) To each function H ∈ C∞(g∗), we associate a Hamiltonian vector field H⃗ on g∗ specified by H⃗[F] = {F,H}. A function C ∈ C∞(g∗) is a Casimir function if {C,F} = 0 for all F ∈ C∞(g∗), or equivalently C⃗ = 0. A linear map ψ : g∗ → h∗ is a linear Poisson morphism if {F,G} ◦ ψ = {F ◦ ψ,G ◦ ψ} for all F,G ∈ C∞(h∗). Linear Poisson morphisms are exactly the dual maps of Lie algebra homomorphisms. Let (E1, . . . ,En) be an ordered basis for the Lie algebra g and let (E∗1, . . . ,E ∗ n) denote the corresponding dual basis for g ∗. We write elements B ∈ g as column vectors and elements p ∈ g∗ as row vectors. When- ever convenient, linear maps will be identified with their matrices. If we write elements u ∈ Rℓ as column vectors as well, then we can express Ξu(1) = A+u1B1+· · ·+uℓBℓ as Ξu(1) = A+Bu, where B = [ B1 · · · Bℓ ] is a n × ℓ matrix. The equations of motion for the integral curve p(·) of the Hamiltonian vector field H⃗ corresponding to H ∈ C∞(g∗) then take the form ṗi = −p([Ei,dH(p)]). Let (Σ,χ) be a cost-extended system with Ξu(1) = A + Bu, χ(u) = (u − µ)⊤Q(u − µ). By the Pontryagin Maximum Principle we have the following result. Proposition 3.3. (cf. [20,50,59]) Any normal ECT (g(·),u(·)) of (Σ,χ) is given by ġ(t) = Ξ(g(t),u(t)), u(t) = Q−1 B⊤ p(t) ⊤ + µ where p(·) : [0,T] → g∗ is an integral curve for the Hamilton–Poisson system on g∗− specified by H(p) = p (A + Bµ) + 1 2 p B Q−1 B⊤ p⊤. (8) 228 r. biggs, c. c. remsing We say that two quadratic Hamilton–Poisson systems (g∗−,G) and (h ∗ −,H) are linearly equivalent if there exists a linear isomorphism ψ : g∗ → h∗ such that the Hamiltonian vector fields G⃗ and H⃗ are ψ-related, i.e., Tpψ ·G⃗(p) = H⃗(ψ(p)) for p ∈ g∗. Proposition 3.4. The following pairs of Hamilton–Poisson systems (on g∗−, specified by their Hamiltonians) are linearly equivalent: 1. HA,Q ◦ ψ and HA,Q, where ψ : g∗− → g∗− is a linear Lie–Poisson auto- morphism; 2. HA,Q and HA,rQ, where r > 0; 3. HA,Q and HA,Q + C, where C is a Casimir function. Theorem 3.1. ([28]) If two cost-extended systems are cost equivalent, then their associated Hamilton–Poisson systems, given by (8), are linearly equivalent. Proof. Let (Σ,χ) and (Σ′,χ′) be cost-extended systems with Ξu(1) = A + Bu and Ξ′u(1) = A ′ + B′ u′, respectively. The associated Hamilton– Poisson systems (on g∗− and (g ′)∗−, respectively) are given by H(Σ,χ)(p) = p(A + Bµ) + 1 2 pBQ−1 B⊤ p⊤ H(Σ′,χ′)(p) = p(A ′ + B′ µ′) + 1 2 pB′ Q′−1 B′⊤ p⊤. Suppose ϕ × φ defines a cost equivalence between (Σ,χ) and (Σ′,χ′), where φ(u) = Ru + φ0 and R ∈ Rℓ×ℓ. We have χ′ ◦ φ = rχ for some r > 0. A simple calculation yields T1ϕ·A = A′+B′ φ0, Rµ+φ0 = µ′, T1ϕ·B = B′R, RQ−1 R⊤ = r (Q′)−1. Thus (H(Σ,χ)◦(T1ϕ)∗)(p) = p(A′+B′ µ′)+ r2 pB ′ (Q′)−1 B′⊤ p⊤. Here (T1ϕ) ∗ : (g′)∗ → g∗ is the dual of the linear map T1ϕ. Hence, the vector fields associ- ated with H(Σ′,χ′) and H(Σ,χ)◦(T1ϕ)∗, respectively, are related by the dilation δ1/r : (g ′)∗ → (g′)∗, p 7→ 1 r p (Proposition 3.4). Moreover, the vector fields associated with H(Σ,χ) ◦ (T1ϕ)∗ and H(Σ,χ), respectively, are related by the linear Poisson isomorphism (T1ϕ) ∗ (Proposition 3.4). Consequently 1 r (T1ϕ) ∗ defines a linear equivalence between ((g′)∗−, H(Σ′,χ′)) and (g ∗ −, H(Σ,χ)). invariant control systems on lie groups 229 Remark. The converse of Theorem 3.1 is not true in general. In fact, one can construct cost-extended systems with different number of inputs but equivalent Hamiltonians (see, e.g., [28]). In Example 3.2 we gave a classification of the cost extended systems on H3. Each Hamiltonian system ((h3) ∗ −,H), where H is a positive definite quadratic form, can be realized as the Hamiltonian system (8) associated to some cost-extended system. Hence, by Theorem 3.1, we get the following result. Example 3.3. Any quadratic Hamilton–Poisson systems ((h3) ∗ −,H), where H is a positive definite quadratic form, is linearly equivalent to the system on (h3) ∗ − with Hamiltonian H ′(p) = 1 2 (p21 + p 2 2 + p 2 3). 3.4. Sub-Riemannian structures. Left-invariant sub-Riemannian (and, in particular, Riemannian) structures on Lie groups can naturally be as- sociated to drift-free cost-extended systems with homogeneous cost. We show that if two cost-extended systems are cost equivalent, then the associated sub-Riemannian structures are isometric up to rescaling. A left-invariant sub-Riemannian manifold is a triplet (G,D,g), where G is a (real, finite-dimensional) connected Lie group, D is a nonintegrable left- invariant distribution on G, and g is a left-invariant Riemannian metric on D. More precisely, D(1) is a linear subspace of the Lie algebra g of G and D(g) = gD(1); the metric g1 is a positive definite symmetric bilinear from on g and gg(gA,gB) = g1(A,B) for A,B ∈ g, g ∈ G. When D = TG (i.e., D(1) = g) then one has a left-invariant Riemannian structure. An absolutely continuous curve g(·) : [0,T ] → G is called a horizontal curve if ġ(t) ∈ D(g(t)) for almost all t ∈ [0,T ]. We shall assume that D satisfies the bracket generating condition, i.e., D(1) has full rank; this condition is necessary and sufficient for any two points in G to be connected by a horizontal curve. A standard argument shows that the length minimization problem ġ(t) ∈ D(g(t)), g(0) = g0, g(T) = g1,∫ T 0 √ g(ġ(t), ġ(t)) −→ min is equivalent to the energy minimization problem, or invariant optimal control 230 r. biggs, c. c. remsing problem: ġ = Ξu(g), u ∈ Rℓ g(0) = g0, g(T) = g1∫ T 0 χ(u(t))dt −→ min. (9) Here Ξu(1) = u1B1 +· · ·+uℓBℓ where B1, . . . ,Bℓ are some linearly indepen- dent elements of g such that ⟨B1, . . . ,Bℓ⟩ = D(1); χ(u(t)) = u(t)⊤Qu(t) = g1(Ξu(t)(1),Ξu(t)(1)) for some ℓ × ℓ positive definite (symmetric) matrix Q. More precisely, energy minimizers are exactly those length minimizers which have constant speed. In other words, the VOCTs of the cost-extended sys- tem (Σ,χ) associated with (9) are exactly the (constant speed) minimizing geodesics of the sub-Riemannian structure (G,D,g); the normal (resp. abnor- mal) ECTs of (Σ,χ) are the normal (resp. abnormal) geodesics of (G,D,g). Accordingly, to a (full-rank) cost-extended system (Σ,χ) on G of the form Σ : u1B1 + · · · + uℓBℓ, χ(u) = u⊤Qu we associate a sub-Riemannian structure (G,D,g) specified by D(1) = Γ = ⟨B1, . . . ,Bℓ⟩ , g1(u1B1+· · ·+uℓBℓ,u1B1+· · ·+uℓBℓ) = χ(u). Let (G,D,g) and (G′,D′,g′) be two sub-Riemannian structures associated to (Σ,χ) and (Σ′,χ′), respectively. Theorem 3.2. (Σ,χ) and (Σ′,χ′) are cost equivalent if and only if there exists a Lie group isomorphism ϕ : G → G′ such that ϕ∗D = D′ and g = rϕ∗g′ for some r > 0. Proof. Suppose ϕ × φ defines a cost equivalence between (Σ,χ) and (Σ′,χ′), i.e., ϕ∗Ξu = Ξ ′ φ(u) and χ′ ◦φ = rχ for some r > 0. As T1ϕ·Ξu(1) = Ξφ(u)(1), it follows that T1ϕ · D(1) = D′(1). Hence, as ϕ is a Lie group isomorphism, by left invariance we have ϕ∗D = D′. Furthermore rχ(u) = χ′(φ(u)) ⇐⇒ r g1(Ξu(1),Ξu(1)) = g′1(Ξ ′ φ(u)(1),Ξ ′ φ(u)(1)) (10) ⇐⇒ r g1(Ξu(1),Ξu(1)) = g′1(T1ϕ · Ξu(1),T1ϕ · Ξu(1)). Hence, as ϕ is a Lie group isomorphism, by left invariance we have r g = ϕ∗g′. Conversely, suppose ϕ∗D = D′ and g = rϕ∗g′. We have T1ϕ · D(1) = D′(1) and so T1ϕ·Γ = Γ′. Hence there exists a unique linear map φ : Rℓ → Rℓ invariant control systems on lie groups 231 such that T1ϕ · Ξu(1) = Ξ′φ(u)(1). Thus ϕ × φ defines a detached feedback equivalence between Σ and Σ′. By (10), it follows that χ′ ◦ φ = rχ. Thus (Σ,χ) and (Σ′,χ′) are cost equivalent. Remark. For (sub-Riemannian) Carnot groups and invariant Riemannian structures on nilpotent Lie groups, any isometry is the composition of a left translation and a Lie group isomorphism (see [36,45,55] and [60,76], respec- tively). Recently, this has been shown to generalize to any nilpotent metric Lie group ([56]). Hence, at least for these classes, if (G,D,g) and (G′,D′,g′) are isometric, then (Σ,χ) and (Σ′,χ′) are cost equivalent. Analogous to Example 3.1, we have the following classification of sub- Riemannian structures on the Euclidean group SE (2). Example 3.4. On SE (2), any left-invariant sub-Riemannian structure (D,g) isometric up to rescaling to the structure (D̄, ḡ) with orthonormal frame (E2,E3); here E2 and E3 are viewed as left-invariant vector fields. 4. Final remarks As already mentioned, a complete classification of the invariant control affine systems in three dimensions was obtained in [27] (see also [21–23]). There is no complete classification of the cost-extended systems in three di- mensions. However, there are classifications of the invariant sub-Riemannian structures ([8]) and invariant Riemannian structures ([44]). Classifications in four dimensions (and beyond) are also topics for future research. In order to find the extremal trajectories for a cost-extended system, one needs to integrate the associated Hamilton–Poisson system (see Proposi- tion 3.3). In the last decade or so several authors have considered quadratic Hamilton–Poisson systems on low-dimensional Lie–Poisson spaces (see, e.g., [3,6,10,16,74]). To our knowledge there is currently no general classification of the quadratic Hamilton–Poisson systems in three dimensions. A first attempt towards such a classification appears in [31] (see also [5,7,13,25,38]). A. 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., [57, 62, 64]). In terms of an (appropriate) ordered basis (E1,E2,E3), the commutation 232 r. biggs, c. c. remsing operation is given by [E2,E3] = n1E1 − aE2 [E3,E1] = aE1 + n2E2 [E1,E2] = n3E3. The structure parameters a,n1,n2,n3 for each type are given in Table 1. a n1 n2 n3 U n im o d u la r N il p o te n t C o m p l. S o lv . E x p o n en ti a l S o lv a b le S im p le Connected Groups 3g1 0 0 0 0 • • • • • R3, R2 × T, R × T2, T3 g2.1 ⊕ g1 1 1 −1 0 • • • Aff (R)0 × R, Aff (R)0 × T g3.1 0 1 0 0 • • • • • H3, H∗3 g3.2 1 1 0 0 • • • G3.2 g3.3 1 0 0 0 • • • G3.3 g03.4 0 1 −1 0 • • • • SE (1,1) ga3.4 a>0 a ̸=1 1 −1 0 • • • G a 3.4 g03.5 0 1 1 0 • • S̃E (2), SEn(2), SE (2) ga3.5 a>0 1 1 0 • • G a 3.5 g3.6 0 1 1 −1 • • Ã, An, SL (2,R), SO (2,1)0 g3.7 0 1 1 1 • • SU (2), SO (3) Table 1: Three-dimensional Lie algebras A classification of the three-dimensional (real, connected) Lie groups can be found in [42]. Let G be a three-dimensional (real, connected) Lie group with Lie algebra g. 1. If g is Abelian, i.e., g ∼= 3g1, then G is isomorphic to R3, R2 × T, R × T, or T3. 2. If g ∼= g2.1 ⊕ g1, then G is isomorphic to Aff (R)0 × R or Aff (R)0 × T. 3. If g ∼= g3.1, then G is isomorphic to the Heisenberg group H3 or the Lie group H∗3 = H3/Z(H3(Z)), where Z(H3(Z)) is the group of integer points in the centre Z(H3) ∼= R of H3. invariant control systems on lie groups 233 4. If g ∼= g3.2, g3.3 , g03.4, g a 3.4, or g a 3.5, then G is isomorphic to the simply connected Lie group G3.2, G3.3, G 0 3.4 = SE (1,1), G a 3.4, or G a 3.5, respec- tively. (The centres of these groups are trivial.) 5. If g ∼= g03.5, then G is isomorphic to the Euclidean group SE (2), the n-fold covering SEn(2) of SE1(2) = SE (2), or the universal covering group S̃E (2). 6. If g ∼= g3.6, then G is isomorphic to the pseudo-orthogonal group SO (2,1)0, the n-fold covering An of SO (2,1)0, or the universal covering group Ã. Here A2 ∼= SL (2,R). 7. If g ∼= g3.7, then G is isomorphic to either the unitary group SU (2) or the orthogonal group SO (3). Among these Lie groups, only H∗3, An, n ≥ 3, and à are not matrix Lie groups. Automorphism groups. A standard computation yields the automor- phism group for each three-dimensional Lie algebra (see, e.g., [46]). With respect to the given ordered basis (E1,E2,E3) , the automorphism group of each solvable Lie algebra has parametrization: Aut(g3.1) :  yw − vz x u0 y v 0 z w   Aut(g2.1 ⊕ g1) :  x y uy x v 0 0 1   Aut(g3.2) :  u x y0 u z 0 0 1   Aut(g3.3) :  x y zu v w 0 0 1   Aut(g03.4) :  x y uy x v 0 0 1   ,   x y u−y −x v 0 0 −1   Aut(ga3.4) :  x y uy x v 0 0 1   Aut(g03.5) :   x y u−y x v 0 0 1   ,  x y uy −x v 0 0 −1   Aut(ga3.5) :   x y u−y x v 0 0 1   234 r. biggs, c. c. remsing For the semisimple Lie algebras, we have Aut(g3.6) = SO (2,1) = { g ∈ R3×3 : g⊤ diag(1,1,−1)g = diag(1,1,−1), detg = 1 } Aut(g3.7) = SO (3) = { g ∈ R3×3 : gg⊤ = 1, detg = 1 } . References [1] R.M. Adams, R. Biggs, C.C. Remsing, Equivalence of control systems on the Euclidean group SE(2), Control Cybernet. 41 (3) (2012), 513 – 524. [2] R.M. Adams, R. Biggs, C.C. 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