Wednesday, February 28, 1996, 1:30 p.m.

Hector J. Sussmann

Department of Mathematics, Rutgers University

**Geometry and Nonlinear Control, 300 Years After Johann
Bernoulli's Brachistochrone Problem**

In 1696, Johann Bernoulli challenged the mathematical community by proposing the "brachistochrone problem". This event is often taken to mark the birth of the Calculus of Variations, but Bernoulli's question was in fact the first optimal control problem ever studied. On the 300th anniversary of the birth of optimal control, we illustrate the power of the methods of modern optimal control theory, especially when coupled with differential-geometric ideas and tools, by means of a number of examples, including the brachistochrone problem, as well as more recent problems such as that of the equivalence of a control system to linear one, that of the structure of optimal trajectories, and the Markov-Dubins-Reeds-Shepp problem on shortest paths subject to a curvature constraint.

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