AFFINE CONTROL SYSTEMS ON NON- COMPACT LIE GROUP

In this paper we deal with affine control systems on a non-compact Lie group cx+e group. First we study topological properties of the state space Ef(1) and the automorphism orbit of Ef(1). Affine control system, non-compact Lie group state space Ef(1). Affine control systems on the generalized Heisenberg Lie groups are studied. Affine algebra, automorphism. Introduction: The purpose of this paper affine control systems on some specific lie group is called cx+e group by relating to associated bilinear parts. Related to the affine control system on lie groups, in Ef(1). The authors Ayala and San Martin have the subalgebra of the Lie algebra Ef(G) generated by the vector fields of a linear control system the drift vector field X is an infinitesimal automorphism i.e.,(XK)K∈M is a one-parameter subgroup of Aut(G); have lifted the system itself to a right-invariant control system on Lie group Ef(1) for compact connected and non-compact semi-simple Lie group. The affine control systems on a non-compact Lie group cx+e group have been investigated and given characterization. 1. Affine Control Systems On Lie Groups If G is a connected Lie group with Lie algebra L(G), the affine group Ef(G) of G is the semi-direct product of Aut(G) with G itself i.e.,Ef(G) = Aut(G)× G. The group operation of Ef(G) . The identity element of Aut G and e denotes the neutral element of G, then the group identity of Ef G is 1, a and (Φ, Φ(𝑕)) In the invers of Φ, 𝑕 ∈ Ef G . Hence, h → 1, 𝑕 and Φ → (Φ, a) embed G into Ef G and Aut G into Af G respectively. Therefore, G and Aut G are subgroups of Ef G . The natural transitive action Ef G × G → G (Φ,𝑕1).𝑕2 → 𝑕1Φ(𝑕2) www.ijcrt.org © 2018 IJCRT | Volume 6, Issue 1 March 2018 | ISSN: 2320-2882 IJCRT1801632 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 720 Where (Φ,𝑕1) ∈ Ef(G) and 𝑕2 ∈ G. “Affine in the control” is used to describe class system. dx dt = n(x) +h(x)v is considered affine control. Theorem:1 Let = Ef 1 , D be an affine control system. Then, the state space Ef 1 is a locally compact Hausdorff space. Proof: Ef 1 is a Hausdorff space is a lie group. The compactness for a given x ∈ Ef 1 and neighborhood Z of x, the existence of some neighborhood Z of x such that. The topology on Ef 1 half plane is homomorphic to the standard topology of M. Therefore, ∀ x ∈ Ef(1), the neighborhood Z of x is homeomorphic to an open ball.For each neighborhood Z of x, there is neighborhood W of x such x ∈ W. Since W is also homeomorphic to an open ball the closure of U is a closed ball. Theorem:2 The automorphism orbit of the state space Ef(1) is dense. Proof: The set J = exp (cf(1) – [cf(1), cf(1)]) Aut(Ef(1))-orbit of Ef(1). The exponential mappingfrom the tangent plane to the surface of diffeomorphism. Then two elements 𝑕1,𝑕2 ∈ J the line segment 𝑕1𝑕2 which is parallel to [Ef(1), Ef(1)], Φ : J → J Defined by 𝑕1 → k1𝑕1+ k2=J,k1,k2 ∈ M Also it is possible to connect those segments with the perpendicular segments .Aut(Ef(1)) orbits open the center[Ef(1), Ef(1)] for any element x ∈ [Ef(1), Ef(1)] and every neighborhood Q (x, γ) of x have some element of Ef(1) different then x. Ef(1) – [Ef(1), Ef(1)] = Ef(1). Theorem:3 The affine control system Σc on the state space Ef(1) is not have any equilibrium point and the associated bilinear system www.ijcrt.org © 2018 IJCRT | Volume 6, Issue 1 March 2018 | ISSN: 2320-2882 IJCRT1801632 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 721 Σc = (Ef(1), De) is control on the Aut (Ef(1)) orbit. Proof: For the control not having equilibrium point is necessary. Now consider the associated bilinear system Σe= (Ef(1), De is control on the Aut(Ef(1)) orbit. Φδ :∂L(G) × L(G)→ ∂L(G) × L(G) Φδ = Id × 1 δ ∀D + X ∈ cf(1) = ∂L(G) × L(G),we have


Introduction:
The purpose of this paper affine control systems on some specific lie group is called cx+e group by relating to associated bilinear parts.
Related to the affine control system on lie groups, in Ef (1). The authors Ayala and San Martin have the subalgebra of the Lie algebra Ef(G) generated by the vector fields of a linear control system the drift vector field X is an infinitesimal automorphism i.e.,( ) ∈ is a one-parameter subgroup of Aut(G); have lifted the system itself to a right-invariant control system on Lie group Ef(1) for compact connected and non-compact semi-simple Lie group.
The affine control systems on a non-compact Lie group cx+e group have been investigated and given characterization.

Affine Control Systems On Lie Groups
If G is a connected Lie group with Lie algebra L(G), the affine group Ef(G) of G is the semi-direct product of Aut(G) with G itself i.e.,Ef(G) = Aut(G)× G. The group operation of Ef(G) .
The identity element of Aut and e denotes the neutral element of G, then the group identity of Ef 1, and ( −1 , −1 ( −1 )) In the invers of , ∈ Ef . Hence, h → 1, and → ( , ) embed G into Ef and Aut into Af respectively. Therefore, G and Aut are subgroups of Ef . The natural transitive action Where ( , 1 ) ∈ Ef(G) and 2 ∈ G.
"Affine in the control" is used to describe class system. = n(x) +h(x)v is considered affine control.

Theorem:1
Let = 1 , be an affine control system. Then, the state space Ef 1 is a locally compact Hausdorff space.

Proof:
Ef 1 is a Hausdorff space is a lie group. The compactness for a given x ∈ Ef 1 and neighborhood Z of x, the existence of some neighborhood Z of x such that. The topology on Ef 1 half plane is homomorphic to the standard topology of 2 .
Therefore, ∀ x ∈ Ef(1), the neighborhood Z of x is homeomorphic to an open ball.For each neighborhood Z of x, there is neighborhood W of x such x ∈ W. Since W is also homeomorphic to an open ball the closure of U is a closed ball.

Theorem:2
The automorphism orbit of the state space Ef(1) is dense.

Proof:
The set (1) (1)]. Since the state space is connected, the affine system is control on Ef(1).

Proof:
The mapping is 1-1 and onto its image.

Lemma:2
Let H be a generalized Heisenberg Lie group. Then there exist a dense Aut(H)-orbit.

Proof:
The Is an Aut(H)-orbit of H. The exponential map is a global diffeomorphism for simply connected nilpotent Lie groups. Two elements X,Y ∈ the line segment mod XY parallel to [H,H], can be connected via a line segment by taking once X as a initial point so that the function that connection : → defined by X → 1 X + 2 = Y, where 1 , 2 ∈ IM, is an automorphism. Actually it is possible to connect these segments with the perpendicular segments to each oyher via the same way.

Theorem:4
Let G be a non-compact connected Lie group and L(G) be its Lie algebra. Then, compact subsets of G are not -invariant, if the control system on G is an invariant system.

Proof:
For ∀x ∈ G, ∀X ∈ L(G) and ∀k ∈ IM, the differentiable curve (;x) : (c,e) ⊂ IM → G is defined (k,x) = (x). Assume that F ⊂ G is a compact and -invariant subset. Each vector field X ∈ L(G) is complete. Consider any open covering E = { \ i ∈ / + }. Therefore, ∀ x(k, ) is an open covering of K, since (x), ∀ ∈ . K is compact, therefore it can be covered by a finite subfamily of = { x(k, )| i ∈ / + }. Then, inverse images of the elements of covers IM, which is a contradiction.