The author constructs optimal paths as one parameter groups defined in terms of matrix exponentiation, proves matrix groups are manifolds, explores the properties of the

Lie bracket, and uses maximal tori to prove several theorems about familiar compact matrix groups.

A unicycle model or a two-wheel car model, shown in Figure 1, is basically a three-dimensional nonholonomic system having two inputs and three states with depth-one Lie bracket.

To verify property (P2), it is sufficient to calculate the following Lie bracket of [G.

Recall that a Lie bracket on a vector space V is a bilinear binary product [*, *]: V x V [right arrow] V such that for all x, y, z [member of] V,

The brackets are said to be compatible if any linear combination of them is a Lie bracket.

where the

Lie bracket on Gr(G) is given by the group commutator.

To compute the

Lie Bracket amounts to compute the

Lie Bracket over [[universal].

Given two polynomials P, Q [member of] K<A>, their Lie product is the

Lie bracket [P, Q] = PQ - QP.

Our convention is that the

Lie bracket IX, Y] of vector fields on a manifold M is given by [X, Y]f = Y(Xf) -X(Yf) for f [element of] [C.

The tangent bundle (TM,t,M) with the usual

Lie bracket and [rho] equal to the identity map form a Lie algebroid.

theta]], ([for exists])X, Y [member of] [GAMMA]([theta]), where the first bracket is the

Lie bracket on X(M).

This covers finite-dimensional linear control systems, linear partial differential equations, controllability of nonlinear systems in finite dimensions, linearized control systems and fixed-point methods, iterated

Lie brackets, return methods, quasi-state deformation, power series expansion, Schrodinger equations, linear control systems in finite dimensions and applications to nonlinear control systems, stabilization of nonlinear control systems in finite dimensions, feedback design tools, and applications to some partial differential equations.