## LSU Historical Dissertations and Theses

#### Title

Lie Group and Control Theory.

1988

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Jimmie D. Lawson

#### Abstract

Let G be a Lie group, L(G) its Lie algebra. If S is a closed subsemigroup of G, we define its tangent set L(S) by L(S) = $\{$X $\in$ L(G): exp(tX) $\in$ S for t $\geq$ 0$\}$. It is a standard result that L(S) is a closed convex cone--indeed a special such called a Lie wedge. In specific examples it is generally not so difficult to determine L(S) given S. However, in the reverse direction, it is often quite difficult to compute precisely the semigroup generated by exp(L(S)) (the so-called infinitesimally generated subsemigroup) and determine whether it is all of S. In this view, both directions are proved for the case of S = $\{$nonsingular totally positive matrices$\}$ and the tangent wedges of various kinds of given semigroups are obtained. We show that every upper (or lower) triangular stochastic matrix can be factorized as a finite product of exponentials of upper (or lower) triangular extreme intensity matrices in a certain order and that the set of upper (or lower) triangular stochastic matrices forms a semigroup. This factorization is a characterization of embeddable matrices in a restricted sense. Also, this factorization contributes as a nice example to the Lie semigroup theory. In this way, appropriate conditions are obtained under which $\{$exp(tX)exp(sY): t,s $\geq$ 0$\}$ is a Lie subsemigroup. Extending this idea to the Lie algebra generated by $\{$X,Y,Z$\}$, we show that $\{$exp(p((t + s)X + sY + Z))exp(qX)exp(r(X + Y)): p,q,r $\geq$ 0, s,t $\leq$ 0$\}$ is a Lie subsemigroup under some appropriate conditions. Also, we prove that the embeddable matrices obtained by using a finite number of intensity matrices as control variables have approximate Bang-Bang representations. This result is extended to the well-known Chattering Principle.

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