On the complexity of the path-following method for a tracking problem governed by parabolic equations

Abstract

The complexity of optimization methods applied to an infinite-dimensional problem in a great manner depends on the quality of finite-dimensional approximations. In this work we consider a tracking problem for a linear parabolic equation. The boundary control is assumed to have the form of a linear combination of shape-like functions. We do not consider any discretization of the differential equation. It is suppose that the solution admits a spectral representation via Fourier-like rapidly converging series involving eigenvalues and eigenfunc- tions of the elliptic operator, and, as a consequence, it can be rapidly calculated with machine accuracy. We show that, in this setting, the tracking problem admits an effective approxima- tion by finite-dimensional optimization problems. The proof of the approximation theorem uses the maximum principle for parabolic equations. Based on our approximation theorem we obtain a complexity bound for the path-following method applied to the tracking problem governed by a linear parabolic equation. The result is illustrated by a series of examples showing the efficiency of the obtained complexity bound.