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\large{\bf MATHEMATICS 642:577\ ---\ Fall 2001}

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\large{Mathematical Topics in Systems Theory}

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\large{\it Instructor: H. J. Sussmann}

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\large{\bf CALCULUS OF VARIATIONS AND OPTIMAL CONTROL}

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\medbreak\noindent The course will present a self-contained introduction to
the classical calculus of variations and optimal control for problems
involving curves.  We will first discuss the statements of various
necessary conditions for an optimum, starting with the Euler-Lagrange
equation and then moving on to the Legendre and Weierstrass necessary
conditions, and the Pontryagin Maximum Principle in both its classical
version---for continuously differentiable maps---and more recent versions
that assume a lot less smoothness. Examples of applications to various
optimal control problems (including classical ones such as Johann
Bernoulli's brachystochrone, as well as more modern ones such as the
problems of Zermelo, Fuller, and Markov-Dubins-Reeds-Shepp) will be
discussed, to show how the techniques are actually used to get concrete
information about the solutions. All this will be done before we give the
proofs of the various necessary conditions, because (a)~the goal of this
first part of the course will be to acquaint the students with the
statements and how they are used, and (b)~there is no need to spend a lot
of time doing proofs of partial results that will be reproved later in the
course in a more general setting.

Once we get to the most general statement currently\footnote{In this
rapidly evolving area, ``currently''
means ``at the time the course is taught.'' The most general statement
available at that time 
may be more general than the one available now, in May 2001.}
available, we will shift gears and present a detailed proof, based on 
the use of suitable theories of generalized differentials.  This part will
be preceded by an introduction to the basic concepts of nonsmooth analysis 
needed for the proof, and in particular to J. Warga's theory of derivate
containers of Lipschitz maps and its extension, the theory of
semidifferentials.

We will then resume the discussion of examples and of the use of the
necessary conditions by showing how the natural language for translating
the consequences of the necessary conditions into useful information is
that of differential geometry, especially when systematic use is made of
intrinsic, covariant objects such as Lie brackets of vector fields. This
will be done by discussing various ``intrinsic'' ways to restate the
Maximum Principle and using them to prove theroems about the qualitative
properties of solutions.

Finally, if time permits, we will discuss sufficient conditions, fields of
extremals, and regular synthesis.

\medbreak\noindent {\it The material to be presented in the course will
combine the contents of several papers that are available in the
instructor's Web page, http://www.math.rutgers.edu/$\,\tilde{\,}$sussmann
(click on ``Papers available on the Web''), especially Papers No. 63, 62,
61, 58, 57, 53, 52, 51, 50, 49, 42, and 40.}



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