TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE (2)

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1 ESAIM: COCV DOI:.5/cocv/4 ESAIM: Control, Optimisation and Calculus of Variations TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE Ross M. Adams, Rory Biggs and Claudiu C. Remsing Abstract. Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE, is detached feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space se. These reduced Hamilton Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis includes a study of the stability nature of all equilibrium states, as well as qualitative descriptions of all integral curves. Finally, the reduced Hamilton equations are explicitly integrated by Jacobi elliptic functions. Parametrisations for all integral curves are exhibited. Mathematics Subject Classification. 49J5, 93D5, E6, 53D7. Received September,. Revised June 4,. Published online July 4, 3.. Introduction A general left-invariant control affine system on the Euclidean group SE has the form ġ = ga + u B + + u l B l, where A, B,...,B l se, l 3. The elements B,...,B l are assumed to be linearly independent. Specific left-invariant optimal control problems on the Euclidean group SE, associated with the above mentioned control systems, have been studied by several authors see, e.g., [,,7,,3,5 8]. In this paper, we consider only two-input control systems, i.e., systems of the form ġ = g A + u B + u B. Any such homogeneous full-rank control system is detached feedback equivalent to the control system Σ : ġ = gu E + u E 3. Then again, any such inhomogeneous control system is detached feedback equivalent to exactly one of the control systems Σ : ġ = ge + u E + u E 3 and Σ,α : ġ = gαe 3 + u E + u E, α>. Here E,E and E 3 denote elements of the standard basis for se. In each typical case, we consider an optimal control problem with quadratic cost of the form ġ = g A + u B + u B, g SE, u=u,u R J = T g = g, gt =g T c u t+c u t dt min. Keywords and phrases. Left-invariant control system, detached feedback equivalence, Lie Poisson structure, energy-casimir method, Jacobi elliptic function. Department of Mathematics Pure and Applied, Rhodes University, Grahamstown, South Africa. dros@webmail.co.za; rorybiggs@gmail.com; c.c.remsing@ru.ac.za Article published by EDP Sciences c EDP Sciences, SMAI 3

2 948 R.M. ADAMS ET AL. Each problem is lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space se. Then the minus Lie Poisson structure on se is used to derive the equations for extrema cf. [3,, 4]. The stability nature of all equilibrium states for the reduced system is then investigated by the energy-casimir method. Also, a qualitative description of all integral curves of the reduced system is given. Finally, these equations are explicitly integrated by Jacobi elliptic functions. A brief description of this process is given now. First, we partition the set of initial conditions in terms of simple inequalities. Specifically, we distinguish between various ways that the level sets, defined by the constants of motion, intersect. When required, this set is further partitioned in order to facilitate integration. This enables one to distinguish between solution curves with different explicit expressions. In each case, the extremal equations are reduced to a separable differential equation and then transformed into standard form see, e.g., [4] or[5]. Thereafter, an integral formula is applied. Consequently, by use of the constants of motion and allowing for possible changes in sign, an explicit expression for the solution is obtained. The paper is organized as follows. In Section we review some basic facts regarding left-invariant control systems, optimal control, the energy-casimir method and Jacobi elliptic functions. In Section 3 we classify all two-input left-invariant control affine systems on SE and then introduce a general optimal control problem with quadratic cost to be considered for each equivalence class. In Section 4 a qualitative analysis of the reduced Hamiltonian systems is given and in Section 5 the reduced systems are explicitly integrated. A tabulation of integral curves is included as an appendix. We conclude the paper with a summary and a few remarks.. Preliminaries.. Invariant control systems and optimal control Invariant control systems on Lie groups were first considered in 97 by Brockett [8] and by Jurdjevic and Sussmann [3]. A left-invariant control system Σ is a smooth control system evolving on a real, finitedimensional Lie group G, whose dynamics Ξ : G U T G are invariant under left translations. The tangent bundle T G is identified with G g, where g is the Lie algebra of G. For the sake of convenience, we shall assume that G is a matrix Lie group. Also, for the purposes of this paper, we may assume that U = R l. Such a control system is described as follows cf. [3,, 4] ġ = Ξ g, u, g G, u R l. where Ξ g, u =gξ,u T g G. Admissible controls are bounded and measurable maps u : [,T] R l, whereas the parametrisation { map Ξ, : R l g is an embedding. The trace Γ = imξ, is a submanifold of g so that Γ = Ξu = Ξ,u:u R l} cf. [5,6]. A left-invariant control affine system is one whose parametrisation map is affine. For such a system, the trace Γ is an affine subspace of g. We say that the system has full rank if the Lie algebra generated by its trace, Lie Γ, coincides with g. Atrajectory for an admissible control u :[,T] R l is an absolutely continuous curve g :[,T] G such that ġt =gt Ξ,ut for almost every t [,T]. We shall denote a left-invariant control system Σ by G,Ξ see, e.g., [5, 6]. We say that a system Σ =G,Ξisconnected if its state space G is connected. Let Σ =G,ΞandΣ =G,Ξ be two connected full-rank systems with traces Γ g and Γ g, respectively. We say that Σ and Σ are locally detached feedback equivalent if there exist open neighbourhoods N and N of the unit elements and, respectively, and a diffeomorphism Φ = φ ϕ : N R l N R l such that φ = and T g φ Ξ g, u =Ξ φg,ϕu for g N and u R l. Two detached feedback equivalent systems have the same trajectories up to a diffeomorphism in the state space, which are parametrised differently by admissible controls. We recall the following result. Proposition. [6]. Σ =G,Ξ and Σ =G,Ξ are locally detached feedback equivalent if and only if there exists a Lie algebra isomorphism ψ : g g such that ψ Γ = Γ.

3 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 949 Now, consider an optimal control problem given by the specification of i a left-invariant control system Σ =G,Ξ, ii a cost function L : R l R, and iii boundary data, consisting of an initial state g G,atarget state g G and a terminal time T>. Explicitly, we want to minimize the functional J = T Lut dt over the trajectory-control pairs of Σ subject to the boundary conditions g = g, gt =g.. The Pontryagin Maximum Principle is a necessary condition for optimality which is most naturally expressed in the language of the geometry of the cotangent bundle T G of G cf. [3, ]. The cotangent bundle T G can be trivialized from the left such that T G = G g,where g is the dual space of the Lie algebra g. The dual space g has a natural Poisson structure, called the minus Lie Poisson structure, given by {F, G} p = p [df p, dgp] for p g and F, G C g. Note that df p is a linear function on g andsoisanelementof g. The Poisson space g, {, } is denoted by g. Each left-invariant Hamiltonian on the cotangent bundle T G is identified with its reduction on the dual space g. To an optimal control problem with fixed terminal time T Lut dt min.3 subject to. and., we associate, for each real number λ and each control parameter u R l, a Hamiltonian function on T G = G g : Hu λ ξ =λlu+ξ gξ,u = λlu+p Ξ,u, ξ =g, p T G. The Maximum Principle can be stated, in terms of the above Hamiltonians, as follows. Maximum Principle. Suppose the trajectory-control pair ḡ, ū defined over the interval [, T] is a solution for the optimal control problem..3. Then, there exists a curve ξ : [,T] T G with ξt Tḡt G,t [,T], and a real number λ, such that the following conditions hold for almost every t [,T]: λ, ξt,.4 ξt = H ūt λ ξt.5 Hūt λ ξt = max u Hλ u ξt = const..6 An optimal trajectory ḡ : [,T] G is the projection of an integral curve ξ of the time-varying Hamiltonian vector field H ūt λ defined for all t [,T]. A trajectory-control pair ξ,u defined on [,T] is said to be an extremal pair if ξ satisfies the conditions.4,.5 and.6. The projection ξ ofan extremal pair is called an extremal. An extremal curve is called normal if λ = and abnormal if λ =.In this paper, we shall be concerned only with normal extremals. Suppose the maximum condition.6 eliminates the parameter u from the family of Hamiltonians H u, and as a result of this elimination, we obtain a smooth function H without parameters on T G in fact, on g. Then the whole left-invariant optimal control problem reduces to the study of integral curves of a fixed Hamiltonian vector field H... The energy-casimir method The energy-casimir method [9] gives sufficient conditions for Lyapunov stability of equilibrium states for certain types of Hamilton Poisson dynamical systems cf. [6, ]. The method is restricted to certain types of systems, since its implementation relies on an abundant supply of Casimir functions.

4 95 R.M. ADAMS ET AL. The standard energy-casimir method states that if z e is an equilibrium point of a Hamiltonian vector field H associated with an energy function H and if there exists a Casimir function C such that z e is a critical point of H + C and d H + Cz e is positive or negative definite, then z e is Lyapunov stable. Ortega and Ratiu have obtained a generalisation of the standard energy-casimir method cf. [8, 9]. This extended version states that if C = λ C + + λ k C k,where λ,...,λ k R and C,...,C k are conserved quantities i.e., they Poisson commute with the energy function H, then definiteness of d λ H +Cz e,λ R is only required on the intersection subspace W =kerdhz e ker dc z e ker dc k z e..3. Jacobi elliptic functions Given the modulus k [, ], the basic Jacobi elliptic functions sn,k, cn,k and dn,k can be defined as snx, k =sinamx, k cnx, k =cosamx, k dnx, k = k sin amx, k where am,k=f,k is the amplitude and F ϕ, k = ϕ dt For the degenerate cases k = k sin t and k =, we recover the circular functions and the hyperbolic functions, respectively. The complementary modulus k and the number K are then defined as k = k and K = F π,k. The functions sn,kand cn,k are4k periodic, whereas dn,k isk periodic. Nine other elliptic functions are defined by taking reciprocals and quotients; in particular, we get nd,k= sn,k cn,k dn,k,sd,k= dn,k and cd,k= dn,k Simple elliptic integrals can be expressed in terms of appropriate inverse elliptic functions. The following formulas hold true see [4] or[5]: x x a x dt a t b t = a sn b x, b a, x b<a.7 dt a + t b t = a a +b sd +b b ab x, a, x b.8 +b dt a t t b = a dn a x, a b a, b x a Control systems on SE We consider two-input left-invariant control affine systems on SE. Such a system is fully specified by its parametrisation map Ξ,u=A + u B + u B. A system is said to be homogeneous if A B,B, i.e., the trace Γ is a linear subspace of se. In this paper, the notation, is used for the linear span of two vectors. Otherwise, the system is said to be inhomogeneous. A classification of all full-rank two-input systems, under detached feedback equivalence, is provided. We then introduce a general optimal control problem with diagonal cost to be considered for each equivalence class. 3.. The Euclidean group SE The Euclidean group SE = {[ ] } : v R v R,R SO

5 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 95 is a real three-dimensional connected matrix Liegroup. TheassociatedLiealgebraisgivenby se = x x 3 : x,x,x 3 R x x 3. Let E =, E =, E 3 = be the standard basis of se. The bracket operation is given by [E,E 3 ]=E,[E 3,E ]=E and [E,E ]=. With respect to this basis, the group of Lie algebra automorphisms of se is given by Aut se = x y v ςy ςx w : x, y, v, w R, x + y,ς= ± ς. We use the non-degenerate bilinear form x x 3, y y 3 = x y + x y + x 3 y 3 x x 3 y y 3 to identify se with se cf. [].Theneachextremalcurve p in se is identified with a curve P in se via the formula P t,x = ptx for all X se. Thus P t = P t P 3 t P t P 3 t where P i t = P t,e i = pte i =p i t, i =,, 3. Now consider a Hamiltonian H on se. The equations of motion take the following form or, explicitly, We note that C : se R, Cp =p + p 3.. Classification of systems ṗ i = p[e i, dhp], i =,, 3 ṗ = H p 3 p ṗ = H p 3 p ṗ 3 = H p p H p p is a Casimir function. It turns out that there is only one homogeneous two-input system on SE, up to equivalence. Furthermore, in the inhomogeneous case there are only two types. The characterisation of detached feedback equivalence in Proposition. is used to prove both these results.

6 95 R.M. ADAMS ET AL. Theorem 3.. Any full-rank homogeneous two-input system Σ is locally detached feedback equivalent to the system Σ with parametrisation Ξ,u=u E + u E 3. 3 Proof. Let the trace of Σ be given by Γ = i= b ie i, 3 i= c ie i First, as either b 3 or c 3,we may assume b 3.Then b Γ = b 3 E + b b 3 E + E 3, bc3 E +c bc3 b 3 E. Now let x = bc3 b 3 and y = c bc3 b 3 Then y x b b 3 ψ = x y b b 3 is a Lie algebra automorphism mapping Γ = E,E 3 to Γ. Theorem 3.. Any inhomogeneous two-input system Σ is locally detached feedback equivalent to exactly one of the following systems: Σ or Σ,α α> with respective parametrisations Ξ,u=E + u E + u E 3, Ξ,α,u=αE 3 + u E + u E. Proof. Let the trace of Σ be given by Γ = b a i E i + b i E i, i= i= 3 c i E i First, consider the case b 3 or c 3. We may assume b 3 andso i= Γ = a E + a E + b E + b E + E 3,c E + c E for some constants a i,b i,c i R, i =,. Now either c or c andso [ ][ ] [ ] c c v a = v c c a has a unique solution. Note that v = leads to a contradiction. Hence ψ = v c v c b v c v c b is a Lie algebra automorphism mapping Γ = E + E,E 3 to Γ. Next, consider the case b 3 = and c 3 =.Then Γ = a E + a E + a 3 E 3 + b E + b E, E + c E Since a 3 andeither b or b,wegetthat ψ = b a sgna 3 b α a b sgna 3 b α sgna 3 is a Lie algebra automorphism. If we set α = a 3,then ψ maps Γ,α = αe 3 + E,E to Γ. Finally, a simple argument shows that Σ is not equivalent to any system Σ,α,andthat Σ,α is not equivalent to Σ,β, for any α β, α, β >.

7 3.3. Left-invariant control problems TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 953 Henceforth, we consider only the systems Σ, Σ and Σ,α. In each of these typical cases, we shall investigate the optimal control problem corresponding to an arbitrary diagonal cost Lu = u + c u,where,c >. Specifically, we shall consider the left-invariant control problems: ġ = g u E + u E 3 g = g, gt =g T T J = c u t+c u t LiCP dt min and ġ = g E + u E + u E 3 g = g, gt =g T T J = c u t+c u t dt min ġ = g αe 3 + u E + u E g = g, gt =g T T J = c u t+c u t dt min. LiCP LiCP3 Remark 3.3. Each member of a significant subclass of left-invariant control problems on SE is equivalent to one of the above three problems, up to cost-equivalence [7]. If two cost-extended systems are cost-equivalent, then they have the same extremal trajectories, up to a Lie group isomorphism between their state spaces. The corresponding controls are mapped by an affine isomorphism. More specifically, any full-rank cost-extended system Ξ,u=u B + u B, Lu =u Qu is cost-equivalent to Ξ,u=u E + u E 3, L u =u + u. 3. Here Q R, Q is positive definite; a proof can be found in [7]. LiCP corresponds to 3.. On the other hand, any cost-extended system Ξ,u=A + u B + u B, Lu =u μ Q u μ where A/ B,B is cost-equivalent to one of the cost-extended systems Ξ,u=E + u E + u E 3, L,β u =u μ + β u μ 3. Ξ,α,u=αE 3 + u E + u E, L,β u =u + β u 3.3 where α, β > andβ. Here μ R, Q R,andQ is positive definite. LiCP corresponds to 3. with μ = μ =,whereas LiCP3 corresponds to 3.3. The following three results easily follow. Proposition 3.4. For the LiCP, the normal extremal control is given by u = p, u = c p 3,where H p = p + c p 3 and ṗ = c p p 3 ṗ = c p p ṗ 3 = p p.

8 954 R.M. ADAMS ET AL. Proposition 3.5. For the LiCP, the normal extremal control is given by u = p, u = c p 3,where H p =p + p + c p 3 and ṗ = c p p 3 ṗ = c p p ṗ 3 = p p. Proposition 3.6. For the LiCP3, the normal extremal control is given by u = p, u = c p,where H 3 p =αp 3 + p + c p and ṗ = αp ṗ = αp 3.6 ṗ 3 = c p p. 4. Qualitative analysis In this section a qualitative analysis of the reduced Hamilton Poisson systems is performed. The stability nature of every equilibrium state is determined. The vector fields H, H,and H 3 are shown to be complete. Subsequently, each maximal integral curve is described as a constant, periodic or bounded curve. 4.. Equilibrium states The equilibrium states for 3.4 are e μ =μ,,, eν =,ν, and e μ 3 =,,μ where μ, ν R, ν. Theorem 4.. The equilibrium states have the following behaviour: i Each equilibrium state e μ is stable. ii Each equilibrium state e ν is unstable. iii Each equilibrium state e μ 3 is stable. Proof. The linearization of the system is given by c p 3 c p c p 3 c p p p i Assume μ. The state e = e 3 is dealt with in iii. Let H χ = H + χc be an energy-casimir function, i.e., H χ p,p,p 3 = p + c p 3 + χp + p where χ C R. The derivative dh χ = [ p χp + p p +p χp + p ] c p 3 vanishes at e μ if χμ =. Then the Hessian at e μ d H χ μ,, = diag χμ +4μ χμ, + χμ, c is positive definite if χμ > and χμ =. The function χx = x μ x satisfies these requirements. Hence, by the standard energy-casimir method, e μ is stable.

9 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 955 ii The linearization of the system at e ν has eigenvalues λ =,λ,3 = ± ν cc Thus e ν is unstable. [ ] iii Let H λ = λ H+λ C.Then dh λ,,μ= λμ c and d H λ,,μ = diagλ, λ +λ, λ c. Suppose μ = andlet λ = λ =.Then dh λ,, = and d H λ,, is positive definite. On the other hand, suppose μ andlet λ =, λ =.Then dh λ,,μ= and d H λ,,μ=diag,,. Also, ker dhe μ 3 ker dceμ 3 =span{,,,,, } and so d H λ,,μ W W = diag, is positive definite. Hence, by the extended energy-casimir method, e μ 3 is stable. The equilibrium states for 3.5 are where μ, ν R, ν. e μ =μ,,, eμ =,μ, and e ν 3 =,,ν Theorem 4.. The equilibrium states have the following behaviour: i Each equilibrium state e μ is unstable if μ [,] and stable if μ,,. ii Each equilibrium state e μ is unstable. iii Each equilibrium state e ν 3 is stable. Proof. The linearization of the system is given by c p 3 c p c p 3 c p p p i Assume μ,. The linearization of the system at e μ has eigenvalues λ =,λ,3 = ± c μμ c Thus e μ is unstable. Now, assume μ = or μ =. Then the linearization of the system at e μ has eigenvalues λ,,3 =. Thus, as the geometric multiplicity is strictly less than the algebraic multiplicity, e μ is unstable. Assume μ,,. Let H χ = H + χc be an energy-casimir function, i.e., where χ C R. The derivative H χ p,p,p 3 =p + p + c p 3 + χp + p dh χ = [ +p χp + p p +p χp + p c p 3 ] vanishes at e μ if χμ = μ. Then the Hessian at eμ d H χ μ,, = diag 4μ χμ + χμ, + χμ, c = diag4μ χμ μ, μ, c ii is positive definite if χμ > 4μ The function χx = 3 8μ +x 3+8μ3 3 4μ x satisfies these requirements. Hence, by the standard energy-casimir method, e μ is stable. Assume μ.thecase μ = has already been dealt with. The linearization of the system at e μ has eigenvalues λ =,λ,3 = ± μ cc Thus e μ is unstable.

10 956 R.M. ADAMS ET AL. iii Let H λ = λ H + λ C,where λ =,λ =.Now dh = [ p c p 3 ] and dc = [ p p ]. Hence dh λ,,ν= and d H λ,,ν=diag,,. Also, ker dhe ν 3 ker dceν 3 =span{ ν,,c,,, } and so d H λ,,ν W W =diag ν, is positive definite. Hence, by the extended energy-casimir method, e ν 3 is stable. The equilibrium states for 3.6 are Theorem 4.3. Each equilibrium state e μ e μ =,,μ, μ R. is stable. Proof. Let H λ = λ H + λ C,where λ =,λ =.Now dh = [ p c p α ] and dc = [ p p ]. Hence dh λ,,μ= and d H λ,,μ = diag,,. Also, ker dhe μ ker dce μ =span{,,,,, } and so d H λ,,μ W W =diag, is positive definite. Hence, by the extended energy-casimir method, e μ is stable. 4.. Integral curves We give qualitative descriptions of the integral curves of H, H,and H 3.Let E, E,and E 3 set of equilibrium points for H, H,and H 3, respectively. denote the Proposition 4.4. The level sets C i = C c H i h i \E i, i =,, 3 are bounded embedded -submanifolds of se for c >, h >, h > c,and h 3 R. Some typical cases for these sets are graphed in Figs. 3. Proof. Let F : se \E R, p Cp,H p. Note that, as E is closed, se \E is open and thus an embedded 3-submanifold of se.wehave [ ] p p DF p = p c p 3 which has full rank unless p = and p 3 =. However,,p, E.Thus C = F c,h is an embedded -submanifold of se.for p =p,p,p 3 C,wehave p + p = c and p + c p 3 = h. Hence p c, p c,and p 3 h c.thus C is bounded. A similar argument shows that C and C 3 are bounded embedded -submanifolds of se. The conditions on c, h,and h are required such that the sets C, C,and C 3 are nonempty. As H i and C are constants of the motion, any non-constant integral curve p of Hi evolves on C i where c = Cp and h i = H i p. Moreover, as each C i is bounded, any integral curve lies in a compact subset of se. Hence see, e.g., []

11 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 957 Corollary 4.5. The vector fields H, H and H 3 are complete. We now describe the integral curves of H, H,and H 3.Let p be a maximal integral curve of a complete vector field on se.apoint p se is an ω-limit point of p if there exists a sequence t n such that t n and pt n p. Similarly, if there exists a sequence t n such that t n and pt n q,then q is an α-limit point. The set of all α-limit points of p is denoted lim α p; the set of all ω-limit points of p is denoted lim ω p. Theorem 4.6. Let p :R se be a non-constant maximal integral curve of Hi.Let Ci be the corresponding connected component of C i containing p.. Suppose p is an integral curve of H. a If C is closed or equivalently cl C E =, then p is periodic. b If C is not closed or equivalently cl C E, then p is bounded and the limit sets lim ω p and lim α p are singletons in E. In either case, p has image C.. Suppose p is an integral curve of H. a If C is closed or equivalently cl C E =, then p is periodic. b If C is not closed or equivalently cl C E, then p is bounded and the limit sets lim ω p and lim α p are singletons in E. In either case, p has image C. 3. If p is an integral curve of H3,then p is periodic and has image C3. Proof. First we show that Ci is closed if and only if cl Ci E i = for i =,, 3. If Ci is closed, then cl Ci = C i C c H i h i \E i and so cl Ci E i =. Conversely, suppose cl Ci E i =. Then clci se \E i and cl Ci C c H i h i, i.e., clci C i. However, as Ci is connected, cl Ci is connected. Thus cl Ci = C i. Wehavethat p evolvesonc.as C is a connected -manifold, it is diffeomorphic to either the circle S or the real line R. a Suppose C is closed. Then C is diffeomorphic to S. Let X be the push-forward to S of the restriction of H to C. X is complete and nonzero everywhere. As S is compact, X attains a positive minimum. Hence there is a finite interval [t,t ] in which any integral curve of X covers S. Hence any maximal integral curve of X is periodic. Consequently, p is periodic as it is diffeomorphic to a periodic integral curve and has image C. b Suppose C is not closed. As C is bounded, the limit sets lim α p and lim ω p are non-empty, connected, compact subsets of se cf. []. Also p is bounded. Now C is diffeomorphic to R. Let X be the push-forward to R of the restriction of H to C. X is complete and nonzero everywhere. Hence, we may assume Xq > for q R. Let q :R R be a maximal integral curve of X. For every compact interval [qt,q ] or [q,qt ] in R, X attains a positive minimum. Hence for every such interval there exists a T such that qt + T >q or qt T <q, respectively. From this we draw two conclusions. First, any maximal integral curve of X covers R. Second, if t n is a sequence in R such that t n ± and q :R R is a maximal integral curve, then qt n does not converge in R. Consequently, the image of p is C.Also,thereareno α- or ω-limit points of p in C. However, any α- or ω-limit point must be in cl C C c H h. We claim that lim α p C = and lim ω p C =. Suppose there exists p lim α p C or p lim ω p C.Then p cl C, p C,and p/ C. Hence C {p} is connected and C {p} C.Thus C {p} is contained in the connected component C of C containing p, a contradiction. So we have lim α p C c H h lim α p C c H h \E = and similarly for lim ω p. Hence lim α p, lim ω p C c H h E.

12 958 R.M. ADAMS ET AL. A simple calculation shows that C c H h E is finite and hence completely disconnected. Therefore, as lim α p and lim ω p are non-empty and connected, lim α p and lim ω p are singletons. For, a similar argument yields the result. 3 We show that C3 is closed by showing that cl C3 E 3 =. The result then follows in the same way as for a. Suppose there exists p cl C3 E 3.Then p C c H3 h 3 E 3. Hence, as p E 3, p =,,μ for some μ R. Hence, as p C c, we get c =,a contradiction. 5. Explicit integration The reduced Hamilton equations can be integrated by Jacobi elliptic functions. In fact, 3.6 can easily be integrated by trigonometric functions. In each of these cases, we obtain explicit expressions for the integral curves of H. Before producing these results, we outline the basic approach employed in obtaining them. In each case, Mathematica is utilised to facilitate calculations. A similar approach was used in [] for single-input systems on SE. Note 5.. In this section we consider only non-constant integral curves. First, we fix a Hamiltonian vector field H specifically 3.4 or3.5. We then partition the set of all initial conditions; this enables one to produce a single explicit expression for each subcase. The first separation is made by considering when the level surfaces, defined by the constants of motion C and H, are tangent to one another. This level of separation is sufficient for solving 3.4, but further partitioning made retrospectively is needed in solving 3.5. Next, we suppose that p : ε, ε se is an integral curve of H satisfying some appropriate conditions. We let h = H p and c = C p >. Then, as p solves 3.4 or3.5, we get that d dt p = ± c c p h p 5. d dt p = ± c c p h c p + p 5. respectively. In most cases, the respective separable differential equation is transformed into standard form see [4] or[5]. A formula for an elliptic integral is then applied to obtain an expression for p t or p t, respectively. Observe howeverthat 5. is already in standard form. Often, a good deal of further simplification is then performed. Next, by use of the constants of motion C and H, expressions for p t, p t and p 3 t are determined up to a choice of sign and organised so as to be smooth again involving further simplification. Accordingly, we get a prospective smooth integral curve p whose domain is extended to R. Insome special cases, the prospective integral curve p may be produced by a limiting process from other results already obtained, or by directly solving the differential equation, as is the case with 3.6. Then, by explicitly differentiating p, we verify for which choices of sign p is an integral curve of H. This is then further verified by solving the respective differential equation numerically for some suitable initial condition. Finally, we show that any other integral curve p : ε, ε se of H, evolving on C c H h, is identical to p up to a translation in the independent variable and an allowable choice of sign. Various properties of the Jacobi elliptic functions are involved in making the above mentioned calculations cf. [4, 5]. In particular, we use the periodicity properties e.g., snx + K, k =cdx, k, relations of squares cnx, k+dnx, k +dnx, k. e.g., k sn x, k =dn x, k and half-angle formulas e.g., cn x, k = We now produce the results for each typical case. A summary of these results may be found in the appendix. Only for Theorem 5. will a proof detailing the method used to obtain the result be provided. For the remaining results we omit details pertaining to finding a maximal integral curve they follow the same approach or are easy to obtain by straightforward integration and only verify that every other integral curve is identical up to a translation in the independent variable.

13 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 959 E 3 E 3 E 3 E E E E E E E 3 E 3 E 3 E E E E E E a c > h b c = h c c < h Figure. Typical cases of LiCP. 5.. Homogeneous systems There are three typical cases for the reduced extremal equations 3.4, corresponding to a c > h,b c = h and c c < h.infigure, we graph the level sets of H and C and their intersection for some suitable values of h,c, and c. The stable equilibrium points illustrated in blue and unstable equilibrium points illustrated in red, as presented in Theorem 4., are also plotted in each case. We begin our presentation of the integral curves of 3.4 with case a. In the following theorem, formula.7 is used to obtain a prospective integral curve. Theorem 5. case a. Suppose p : ε, ε se is an integral curve of H such that Hp = h >, Cp = c > and c > h. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where Here k = h c and Ω = c c p t =σ c dn Ωt,k p t = h sn Ωt,k p 3 t = σ c h cn Ωt,k Proof. Verification that p is an integral curve satisfying H p = h, Cp = c whenever c > h > is straightforward. We have p +p = c and p + c p 3 =h.thus p h and so

14 96 R.M. ADAMS ET AL. p = c p c h >. Let σ =sgnp we have σ. Furthermore, h c p 3 h c, p 3 = σ c h and p 3 K Ω =σ c h. Hence there exists t R such that p 3 t =p 3. Now p =h c c p 3 =h c c p 3 t = p t. Thus p = ± p t. However, p t = p t and p 3 t = p 3 t. Therefore there exists t R either t = t or t = t such that p 3 = p 3 t andp = p t. Also p = c p = c p t = p t. Hence, as sgn p = σ =sgn p t, we get p = p t. Thus p = pt. Consequently the integral curves t pt and t pt + t solve the same Cauchy problem, and therefore are identical. Next, by limiting c h in Theorem 5. and allowing for possible changes in sign, we get a prospective integral curve for case b. Proposition 5.3 case b. Suppose p : ε, ε se is an integral curve of H such that Hp = h >, Cp = c > and c = h. Then there exists t R and σ,σ {, } such that pt = pt + t for t ε, ε, where p t =σ σ c sech Ωt p t =σ c tanh Ωt Here Ω = c c p 3 t = σ cc sech Ωt Proof. Let σ = sgn p 3 and let σ = σ sgn p. We have p +p = c.thus c p c and so there exists t R such that p = p t. A simple computation then yields p = pt. It is also simple to verify that σ and σ providedthat p is not constant. Consequently the integral curves t pt and t pt + t solve the same Cauchy problem, and therefore are identical. Lastly, for case c, we obtain a prospective integral curve by use of formula.7. Theorem 5.4 case c. Suppose p : ε, ε se is an integral curve of H such that Hp = h >, Cp = c > and c < h. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where p t =σ c cn Ωt,k p t = c sn Ωt,k p 3 t = σ c h dn Ωt,k Here k = c h h and Ω = c Proof. Let σ = sgn p 3. We have p + p = c and p + c p 3 =h. Hence p c and so p 3 c h c >. Thus σ.also, c p c, p K Ω = c and p 3K Ω = c. Therefore there exists t [ K Ω, 3K Ω ] such that p = p t. We have p = c p = c p t = p t.

15 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 96 Thus p = ± p t. Now p t+ K Ω = p t and p t+ K Ω = p t. Hence there exists t R t = t or t = t + K Ω such that p = p t and p = p t. Next p 3 =h c c p =h c c p t = p 3 t. Hence, as sgn p 3 = σ =sgn p 3 t, p 3 = p 3 t. Thus p = pt. Consequently the integral curves t pt andt pt + t solve the same Cauchy problem, and therefore are identical. Remark 5.5. These three results are similar to those found by Sachkov [6]. However, our approach and formulation are different. The control problem LiCP, or rather, the associated sub-riemannian problem, is further studied in [7, 7, 8]. 5.. Inhomogeneous systems In order to separate qualitatively different cases of 3.5, we determine at which points and for what values of h,c, and c the cylinder C c and the paraboloid H h are tangent to one another. If they are tangent at a point p =p,p,p 3, then the gradients of the functions defining these level surfaces at p must be parallel, i.e., Cp = [ p p ] = r [ p c p 3 ] = r Hp for some r R. There are three distinct possibilities: p = p = p = p 3 = ± p h c = ± c c p 3 = p p = p 3 =. The first case corresponds to a constant solution and is therefore ignored. By back substitution into C and H, the third case yields h = c ; this motivates us to distinguish between the three signs for h c.observe however that the cases h = c and c = correspond to constant solutions, whereas the situation h < c is impossible. The second case only occurs when c c > and is hence distinguished from the case c c. Back substitution yields h = c + c, motivating another separation of three cases. However, not all combinations of these cases are possible. Lemma 5.6. If c c > and h c + c,then h > c. Proof. It suffices to show that c + c > c.now c + c > c c + c > 4c c c c c + c4 > c c > thus yielding the result. Some of the above mentioned cases are retrospectively further subdivided to facilitate integration. An index of the final subdivision of cases is provided in Table. In Figure, we graph the level sets of H and C and their intersection for each major case i.e., bychoosing some suitable values for h,c, and c. The stable equilibrium points illustrated in blue and unstable equilibrium points illustrated in red, as presented in Theorem 4., are also plotted in each case. We start our presentation of the integral curves of 3.5 by considering case ai. Formula.7 is utilised in the following theorem.

16 96 R.M. ADAMS ET AL. E3 E3 E E E E3 E E3 E E E3 E3 E E E E3 E E E3 E d Case a E E E c Case c E E b Case b a Case a E E3 E E E3 E3 E3 E E e Case b Figure. Typical cases of LiCP. E f Case ci E

17 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 963 Table. Index of typical cases for reduced extremals of LiCP. c c c c > Conditions h > h > h c c h = h c h < h c Index ai aii aiii h = h < bi c h = bii c <h < c h > c + c h = c + c c h > h < c +c h c = c c <h < c c a b ci cii ciii Theorem 5.7 case ai. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case ai are satisfied. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where p t = h δ h + δ sn Ωt,k c h + δ h δ sn Ωt,k Here δ = h c, Ω = p t = σ c δ p 3 t = σδ c k cn Ωt,k h + δ h δ sn Ωt,k dn Ωt,k h + δ h δ sn Ωt,k h+δ h +δ c, k = h δ h δ h +δ h +δ and k = δ h +δ h +δ Proof. Again, verification that p is a maximal integral curve satisfying H p = h and Cp = c whenever c c, h > c and h > h c is straightforward. Also, note that h + δ> h δ and so pt is defined for all t R. We have p + p = c, p + p + c p 3 = h and h > c.thus c p c.also p 3 = h p c p. Now min c p p c c p = c. Hence p 3 h c >. Hence p 3.Let σ =sgnp 3. Now p K Ω = c and p K Ω = c. Thus there exists t [ K Ω, K Ω ] such that p t =p. Next p = c p = c p t = p t. Hence p = ± p t. Now p t+ K Ω = p t and p t+ K Ω = p t. Thus there exists t R t = t or t = t + K Ω such that p = p t and p = p t. We have p 3 =c h c p c p =c h c p t c p t = p 3 t. Thus, as sgn p 3 = σ =sgn p 3 t, we get p 3 = p 3 t. Hence p = pt. Consequently the integral curves t pt and t pt + t solve the same Cauchy problem, and therefore are identical.

18 964 R.M. ADAMS ET AL. Next, by limiting δ h from the right in the above result, we get a prospective integral curve for case aii. The proof is omitted as it is essentially the same as for Theorem 5.7 K is replaced by π. Proposition 5.8 case aii. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case aii are satisfied. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where p t = h δ h + δ sinωt c h + δ h δ sinωt p t = σ c δ p 3 t = cosωt h + δ h δ sinωt σδ c h + δ h δ sinωt Here δ = h c c and Ω = h c It turns out that integral curves satisfying the conditions of case aiii or a take the same explicit expression. We shall provide a detailed proof for the result. Lemma 5.9. For cases aiii and a we have h + h c > and h + h c >. Theorem 5. cases aiii and a. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case aiii or a are satisfied. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where p t = h δ h + δ cn Ωt,k c h + δ h δ cn Ωt,k p t =σ c δ p 3 t =σδ c sn Ωt,k h + δ h δ cn Ωt,k dn Ωt,k h + δ h δ cn Ωt,k Here δ = h c δ, Ω = c and k = h δh +δ δ Proof. We start by explaining how the expression for p was found. A number of convenient assumptions are made implicitly and translations in the independent variable are discarded. We shall verify that p isa maximal integral curve defined for any h,c, and c satisfying the conditions of case aiii or a only at the end of the construction. Suppose p is an integral curve of H satisfying the conditions of aiii or a, where h = H p and c = C p. We transform 5. into standard form see, e.g., [4] or[5]. First, we can rewrite 5. as where d p dt = A p r + B p r A p r + B p r A = h + δ c δ > B = h δ c δ < A = h + δ δ B = h + δ δ r = h δ r = h + δ. < <

19 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 965 Then we have dt = d p A p r + B p r A p r + B p r We make a change of variable s = p r p r, which then yields t = r r A A p t r p t r ds B A + s B A s Applying formula.8, we get r r A A t = B A B A sd B A B B A B A A p t r p t r, B A B A B A Solving for p t we find r p t = B B B A B A sd B B B A B A sd r r B A B A t, r r B A B A t, Substituting the values for A,A,B,B,r,r and simplifying then yields p t = h h δ h + δk sd Ωt,k δ h + δ h δk sd Ωt,k B A B A B A r B A B A B A Now cn x+3k, k =k sd x, k. Thus, by making a suitable translation in t, we get the following prospective expression for p p t = h h δ h + δ cn Ωt,k δ h + δ h δ cn Ωt,k Next, as C pt = c,weget p t = c p t = δ h δ sn Ωt,k h + δ h δ cn Ωt,k Hence p t =σ c δ sn Ωt,k h + δ h δ cn Ωt,k for some σ {, }. By a similar computation, using the constant of motion H, weobtain p 3 t =σ 3 δ c dn Ωt,k h + δ h δ cn Ωt,k for some σ 3 {, }. We now show that p is an integral curve for certain values of σ we require that d dt p t = c p t p 3 t. Now and σ 3.From3.5, d dt p t c p t p 3 t = δ δ dn Ωt,ksnΩt,k h δ c σ σ 3 h + δ h δ cn Ωt,k

20 966 R.M. ADAMS ET AL. Therefore, if σ = σ 3 = σ {, }, then d dt p t = c p t p 3 t. For this choice of σ and σ 3,further d calculation shows that dt p t = c p t p 3 t and d dt p 3t = p t p t. This motivates p as a prospective integral curve. Elementary calculations show that p is defined over R when either set of conditions case aiii or a is satisfied. In particular, note that the denominator in each expression is strictly positive, the constants δ, k and Ω are real, and <k<. Thus p is a maximal integral curve of H for any h,c, and c satisfying the conditions of case aiii or a. Finally, we claim that any integral curve p as described in the statement of the theorem must be of the form pt = pt + t for some σ {, } and t R. ByTheorem4.6, p covers each connected component of H h C c for each σ {, }, respectively as illustrated in Figures a andd. Thus, as p H h C c by assumption, there must exists t R and σ {, } satisfying the conditions of the theorem. We now give a proof for this claim. We have p + p = c and p + p + c p 3 = h.thus c p c.also c p 3 = h p c p. Suppose the conditions of case aiii hold. Then min c p p c c p = c. Hence c p 3 h c > andsop 3. On the other hand, suppose the conditions of case a hold. Then min c p p c c p = c + c. Hence c p 3 h c + c > andsop 3. Let σ =sgnp 3. Now p = c and p K Ω = c.thusthereexists t [, K Ω ] such that p t =p. Next we have p = c p = c p t = p t. Therefore p = ± p t. Now p t = p t and p t = p t. Hence there exists t R t = t or t = t such that p = p t and p = p t. Next p 3 =c h c p c p =c h c p t c p t = p 3 t. Thus, as sgn p 3 = σ =sgn p 3 t, we get p 3 = p 3 t. Hence p = pt. Consequently the integral curves t pt and t pt + t solve the same Cauchy problem, and therefore are identical. Limiting processes were unsuccessful in producing a prospective integral curve for case b. Therefore, we resorted to integrating case bi independently by use of formula.9. Proposition 5. case bi. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case bi are satisfied. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where p t = h coshωt 3 + h cosh Ωt h sinhωt p t =σh c h cosh Ωt h Here Ω = h h c p 3 t =σ h c h coshωt cosh Ωt h

21 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 967 Proof. We have p + p = c, p + p + c p 3 = h and h = c.if p 3 =, then a simple calculation shows that p = and so p is constant. Hence p 3 if p is not constant. Let σ =sgnp 3. Now c p < c, p = c and lim t p t = c.thusthereexists t R such that p t =p. Next p = c p = c p t = p t. Hence p = ± p t. Now p t = p t and p t = p t. Thus there exists t R t = t t = t such that p = p t and p = p t. We have or p 3 =c h c p c p =c h c p t c p t = p 3 t. Thus, as sgn p 3 = σ =sgn p 3 t, we get p 3 = p 3 t. Hence p = pt. Consequently the integral curves t pt and t pt + t solve the same Cauchy problem, and therefore are identical. A simple calculation either limiting h from the left in the above case, or integrating 5. directly then yields a prospective integral curve for case bii. The proof is omitted as it is essentially the same as for Proposition 5.. Proposition 5. case bii. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case bii are satisfied. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where t c p t = c t + c t p t =σ c c t + c p 3 t =σc c c t + c For case c, the roots of the two quadratics in 5. need to be deinterlaced before this differential equation can be transformed into standard form. Consequently, the expression for this integral curve is quite involved. It turns out that integral curves satisfying the conditions of case ciii, take the same explicit expression as those of case c. The following result utilises formula.7. Theorem 5.3 cases c and ciii. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case c or ciii are satisfied. Then there exists t R such that pt = pt + t for t ε, ε, where c ρ c ρ +ρ + ρ ρ ρ cd Ωt,k p t =η ρ + ρ ρ ρ cd Ωt,k p t = η k sd Ωt,k + ndωt,k +kcd Ωt,k k ρ + ρ ρ ρ cd Ωt,k p 3 t = η 3 cn Ωt,k + ndωt,k k cd Ωt,k k ρ + ρ ρ ρ cd Ωt,k

22 968 R.M. ADAMS ET AL. Here and δ = c c ρ c + δ ρ h + c k = ρ c + δ + ρ ρ = +δ c + δ δ +c k ρ = ρ c + δ + ρ ρ = 4 c δ ρ c + δ c Ω = c + δ + ρ c c c + ρ η = δ + c c η = c + δρ + c + δ ρ η 3 = c δ ρ ρ ρ 4δ + c ρ 4 δ + c c Proof. Verification that p is an integral curve is quite involved. However, it is sufficient to show that p t = c p t p 3 t, p t + p t = c and p t+ p t + c p 3 t = h. We have p + p = c and p + p + c p 3 = h. Hence c p c.also, p + c p h. Thus p δ p + δ, and so p δ or p + δ. It turns out that c δ c + δ. Therefore c p δ. We have that p t = p t p t + 4K Ω = p t p t + 4K Ω = p t p t = p t p 3 t = p 3 t p t + 4K Ω p 3 t + 4K Ω = p t p t + 4K Ω = p t = p3 t p 3 t + 4K Ω = p3 t. Now p = c and p K Ω = δ. Hence there exists t [, K Ω ] such that p t =p. Then by using the constants of motion C and H weget p = ± p t andp 3 = ± p 3 t. Therefore there exists t R t = t, t = t, t = t + K Ω,or t = t + K Ω such that p = pt. Consequently the integral curves t pt and t pt + t solve the same Cauchy problem, and therefore are identical. We now proceed to case b. In this case, a prospective integral curve is found by limiting h c + c from the left in case a Thm. 5. and allowing for possible changes in sign. Proposition 5.4 case b. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case b are satisfied. Then there exists t R and σ,σ {, } such that pt = pt + t for t ε, ε, where p t = + c c σ c coshωt p t = c σ σ c c sinhωt σ c coshωt Here Ω = c c σ c c c p 3 t = σ c coshωt

23 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 969 Proof. We have p t + p t = c and p t + p t + c p 3 t = h. Hence we have p t + p t c c = c c p 3 t.let sp,p = p + p c c. Consider the restriction of s to the circle S = {p,p :p + p = c }; s S has exactly two roots p,p =, ± c c. We claim that p t and p 3 t foranyt ε, ε. Suppose p 3 t =forsomet.then p t = and so pt is a equilibrium point. Hence we have p is a constant trajectory, a contradiction. On the other hand suppose p t = for some t.then p t = c c and so p 3t =.Hence p isaconstant trajectory, a contradiction. Now sgn p t = σ and sgn p 3 t = σ σ.let σ = sgn p and σ = σ sgn p 3. We claim that there exists t R such that p = p t and sgnp = sgn p t. We need to consider two cases for p. First suppose p >.Then σ =. We have c c <p < c c, lim p t =σ c c t lim t p t = σ c c and p t > fort R. Thusthereexists t R satisfying the claim. Now suppose p <.Then σ =. Let α = Ω cosh c >. We have p α =σ c lim p t =σ c c t p α = σ c lim t p t = σ c c and p t < for t, α α, p t > for t α, α. Note that < α <α<. Wehave c p c.if p then there exists t [ α, α] satisfying the claim. If <p <,then c <p < c c or c c <p < c.ineither case there exists t, α α, satisfying the claim. By using the constant of motion C, we find p = p t. However, we have that sgn p = sgn p t and so p = p t. Next, by using the constant of motion H, we find p 3 = p 3 t. However sgn p 3 = σ σ =sgn p 3 t and so p 3 = p 3 t. Consequently the integral curves t pt and t pt + t solve the same Cauchy problem, and therefore are identical. Moving on to case ci, we again find a prospective integral curve by use of formula.9. Theorem 5.5 case ci. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case ci are satisfied. Then there exists t R and σ {, } such that pt = pt + t for t ε, ε, where h +δ h p t =h δ k δ h δ σ h + δ dn Ωt,k Here δ = h c, Ω = p t = k δ h δ p 3 t =σδk c h δh +δ c, k = k h δ σ h + δ dn Ωt,k cn Ωt,k k h δ σ h + δ dn Ωt,k sn Ωt,k k h δ σ h + δ dn Ωt,k δ h δh +δ and k = h+δh δ h δh +δ

24 97 R.M. ADAMS ET AL. Proof. We have p + p + c p 3 = h and p +p = c. Hence p + p h. Let sp,p =p + p h. Consider the restriction of s to the circle S = {p,p :p + p = c }; s S has exactly four roots p,p =α, ±α and p,p =β, ±β, where α = + c + c h α = h c + c h β = c + c h β = h + c + c h. Consequently, if p >,thenα p c. On the other hand, if p <,then c p β. It is easy to show, under the conditions of ci, that p.let σ = sgn p. Suppose p >, i.e., σ =. A somewhat involved computation yields p = α and p K Ω = c. Hence there exists t [, K Ω ] such that p t =p. On the other hand, suppose p <, i.e., σ =. Again, a somewhat involved computation yields p = β and p K Ω = c. Hence there again exists t [, K Ω ] such that p t =p. Finally, we claim that there exists t R such that pt =p. By using the constants of motion C and H, weget p = ± p t and p 3 = ± p 3 t. Now p t = p t p t + K Ω = p t p t + K Ω = p t p t = p t p t + K Ω = p t p t + K Ω = p t p 3 t = p 3 t p 3 t + K Ω = p3 t p 3 t + K Ω = p3 t. Therefore there exists t R t = t, t = t, t = t + K Ω,or t = t + K Ω such that p = pt. Consequently the integral curves t pt andt pt + t solve the same Cauchy problem, and therefore are identical. We are left to consider case cii. By limiting c h changes in sign, we obtain a prospective integral curve. in case ciii Thm. 5.3 and allowing for possible Proposition 5.6 case cii. Suppose p : ε, ε se is an integral curve of H such that Hp = h, Cp = c > and the conditions of case cii are satisfied. Then there exists t R such that pt = pt + t for t ε, ε, where p t = c + c 3+cosΩt c + cosωt p t = c c c sinωt c + cosωt p 3 t = c c c cosωt c + cosωt c c c Here Ω = Proof. We have p + p + c p 3 Let sp,p =p + p h. Consider the restriction of s to the circle S = {p,p :p + p = c }; s S has exactly three roots p,p = c, p,p = = h and p +p = c. Hence p + p h. c, ± c.

25 TWO-INPUT CONTROL SYSTEMS ON THE EUCLIDEAN GROUP SE 97 Consequently p = c or c p c. However, if p = c,then p = and p 3 =, i.e., p is a constant trajectory. Thus this case can be discarded. Now p = c and p π Ω = π c.thusthereexists t [, Ω ] such that p t =p. By using the constants of motion C and H, weget p = ± p t and p 3 = ± p 3 t. Now p t = p t p t = p t p 3 t = p 3 t p t + π Ω = p t p t + π Ω = p t p t + π Ω = p t p t + π Ω = p t p 3 t + π Ω = p3 t p 3 t + π Ω = p3 t. Therefore there exists t R t = t, t = t, t = t + π Ω,or t = t + π Ω such that p = pt. Consequently the integral curves t pt andt pt + t solve the same Cauchy problem, and therefore are identical. Finally, let us consider the invariant control problem LiCP3. There is only one typical case which is easy to integrate. Again, we graph the level sets of H and C and their intersection. E 3 E 3 E E E E Figure 3. Typical reduced extremal of LiCP3. Proposition 5.7. The reduced Hamilton equations 3.6 have the solutions p t = c sin αt+ t p t = c cos αt+ t p 3 t = h α c α where c = Cp, h = Hp and t R. sin αt+ t + c cos αt+ t

26 97 R.M. ADAMS ET AL. 6. Conclusion This paper shows that there are only three types of two-input left-invariant control affine systems on SE up to equivalence, and then studies a general optimal control problem with quadratic cost for each type. In each case the problem is reduced via the Pontryagin Maximum Principle to the study of a single Hamiltonian H on the Poisson space se. The stability nature of all equilibrium states of the reduced system is investigated by use of the energy-casimir method. H is shown to be complete. Also, every maximal integral curve is shown to be a constant, periodic or bounded curve. For each problem, explicit expressions for all integral curves of H are found, up to a choice of sign and a translation in the independent variable. This is achieved in most instances by reducing the given system of differential equations to a single separable differential equation and then transforming it into standard form. Thereafter, an application of an appropriate integral formula yields a solution involving Jacobi elliptic functions. Using the constants of motion C and H and allowing for possible changes in sign an explicit expression for an integral curve of H is then obtained. Finally, we verify that any other integral curve, satisfying the same partitioning conditions, is identical to the one produced up to a translation. We now have explicit expressions for all normal extremal controls. A natural next step would be to solve the equations on the base SE, i.e., to obtain expressions for the normal extremal trajectories. For the control problem LiCP, this was essentially accomplished in [6]. The inhomogeneous case has yet to be considered. Single-input systems evolving on SE have been considered in []. Specifically, it was shown that there are only two typical cases up to equivalence. Likewise, stability and explicit integration were addressed. The three-input case is a topic for future research. Appendix A. Tabulation of integral curves Table A.. Integral curves of H for reduced extremals of LiCP and LiCP3. Case a c > h p t =σ c dn Ωt,k p t = h sn Ωt,k p 3t = σ c h cn Ωt,k k = h c Ω = c c σ {, } Case b c =h p t =σ σ c sech Ωt p t =σ c tanh Ωt p 3t = σ c c sech Ωt, Ω = c c σ {, } σ {, } Case c c < h p t =σ c cn Ωt,k p t = c sn Ωt,k p 3t = σ c h dn Ωt,k k = c Ω = h h c LiCP3 p t = c sin αt+ t p t = c cos αt+ t p 3t = h α c α sin αt+ t + c cos αt+ t