 
Summary: The Center for Control, Dynamical Systems, and Computation
University of California at Santa Barbara
Winter 2007 Seminar Series
Presents
The Role of Broken Extremals in Nonholonomic
Optimization
Roger Brockett
Harvard University
Friday, February 23th, 2007 3:00pm4:00pm
1001 LSB Rathmann Auditorium
Abstract:
We will discuss a problem which has a deceptively simple description. We want to transfer a nons
ingular matrix $X(0)$ to a second nonsingular matrix $X(1)$ under the assumption that the matrix
evolves according to $\dot{X}=UX$ with $U(t)=U^T(t)$. This system is controllable on the space of
nonsingular matrices with positive determinant and the first order necessary conditions associated
with minimizing $$\eta = \int _0^T U(t) \; dt $$ subject to the condition that $U$ should steer the
system from $X(0)=X_0$ to $X(1)=X_1$ imply that $U$ should take the form $$U(t) = e^{\Omega
t}He^{\Omega t }$$ with $\Omega = \Omega ^T $ and $H=H^T$, both constant. What makes this
problem interesting, however, is the abundance of conjugate points, the non uniqueness of the solu
tions of the first order necessary conditions and the possible necessity for broken extremals. These
