Mathematics 697CV - ST-Calc of Var & Opt Cntrl Thy

We will introduce the classical concepts and techniques of the calculus of variations, that is, the extremization of functionals. The standard problem in the calculus of variations concerns functionals that are integrals of an integrand depending on the unknown function and its first derivatives. We will first consider one-dimensional problems, for which the generic form may be interpreted as Hamilton's principle of least action in classical mechanics. We will consider necessary conditions (Euler-Lagrange equations) and some sufficient conditions (Jacobi theory) for extrema. Then we will study Hamilton-Jacobi theory, which provides a comprehensive characterization of all extremals. Optimal control theory may be thought of as a natural generalization of the calculus of variations, which it contains as a special case. We will introduce the basic concepts of optimal control and several motivating problems from science. The general theory will be expressed in terms of both the Pontryagin maximum principle and the Bellman equation. Some interesting particular problems will be solved. If time permits, some multidimensional problems will be discussed and connected to elliptic partial differential equations. Pre-requisites: Advanced calculus, undergraduate real analysis, differential equations.

Open to Graduate students only.