An Initial Value Control Problem For The Burgers Equation

Abstract

Attached

In this thesis, we consider an initial value control problem governed by the Burgers equation. The cost functional is Min. J = Min.

where  is a given domain in the upper half plane and u is the solution to the Burgers equation with initial condition

By imposing certain conditions on g, we establish the differentiability of J with respect to uc and also prove a theorem providing upper and lower bounds for the controller uc,

We then compute the distributional derivative of the cost functional J with respect to uc, by representing the solution to the Burgers equation using Heaviside-Delta functions. We use this derivative to find the minimizing uc.

We choose u2 as the functional g satisfying all the assumptions and compute the cost functional and its derivative for some representative cases, by direct integration and do the same thing using the distributional derivative as well. By comparing the numerical results, we conclude the validity of our distributional approach.

Possible extensions for more general cases are also mentioned.