Henry Hermes

Nonlinear Controllability via Lie Theory

Complete system associated with given control, discussing trajectories uniform approximation and nonlinear controllability conditions based on linear partial differential equation

Nilpotent approximations of control systems and distributions

A constructive method is given to approximate the vector fields in a nonlinear control system, which is linear in the controls, by a system of similar form and on the same state space, with the describing vector fields of the approximating system generating a Lie algebra which has a certain, relevant, sub-algebra nilpotent. Given a k-dimensional distribution on

Control systems which generate decomposable Lie algebras

R^n

Lie Algebras of Vector Fields and Local Approximation of Attainable Sets

, say locally defined near zero, the method leads to an approximating k-distribution which has nilpotent basis, agrees with the original distribution at zero, and each derived distribution of the approximating distribution and original distribution also agree at zero.

Nilpotent bases for distributions and control systems

Let X, Y be real analytic vector fields on an n -manifold M , (ad X, Y ) = [ X, Y ]denote the Lie product, inductively(ad k + 1 X, Y ) = [ X , (ad k X, Y )]and ℒ 1 ={(ad v X, Y):v =0,…}. The attainable set at time t , denoted Download full-size image ( t , for the control system (1) x = X(x) + u(t) Y(x), x (0) = p , ¦ u(t) ¦ ⩽ 1, is contained in the integral manifold through p of the Lie algebra, L (ℒ 1 ), generated byℒ 1 , L (ℒ 1 ) is said to be decomposable at p if (essentially), when dim L ℒ 1 ( p = n , the solution of (1) can be written as(exp tX ) ∘ (exp F 1 (t, u) V 1 ) ∘ … ∘ (exp F n ( t, u) V n )( p ) with V 1 , …, V n ∈Lℒ 1 . This is true, for example, at any p , if L (ℒ 1 ) is nilpotent. The F i ( t, u ) are shown to satisfy a system on ℝ n of the form (2) x i = u(t) G i ( t, x 1 ,…, x i − 1 ), x i (0) = 0, i = 1,…, n , where, if L (ℒ 1 ) is nilpotent, each G i is a polynomial in x 1 , …, x i − 1 . In this case, reachability properties of (2) for controls being countably additive measures (impluses) reduce to the study of algebraic equations; homogeneity allows these to be considered on projective space, ℙ n , giving local results by Bezouts theorem. Similar results hold for controls in the unit ball of ℒ ℞ . Kreners results on approximating systems of the form (1) by systems which generate nilpotent Lie algebras can then be used to remove the condition that L (ℒ 1 ) be nilpotent. The method applies to show that, ifℒ m denotes the set of all products of k elements ofℒ 1 with k ⩽ m and dim spanℒ m +1 ( p )= dim spanℒ m for all odd m , then(exp tX)(p )∈ int Download full-size image ( t )for all t > 0.

Homogeneous feedback controls for homogeneous systems

Consider an analytic, n-dimensional control system described by

On Local Controllability

{{dx} / {dt}} = X(x) + u(t)Y(x)

Asymptotically stabilizing feedback controls and the nonlinear regulator problem

,

Asymptotically stabilizing feedback controls

x(0) = p

Local Controllability and Sufficient Conditions in Singular Problems. II

and let