Time-optimal control of linear diffusion processes using Galerkin's method

Abstract

In the paper, attention has been drawn to the method of weighted residuals for modelling distributed- parameter systems governed by partial differential equations with mixed initial and boundary conditions. Galerkin's method, a particular form of the method of weighted residuals, has been considered in detail, and it is shown how lumped-parameter ‘state equations’ can be obtained to approximately describe the behaviour of many distributed-parameter systems with distributed and boundary inputs. The method is applied to determine approximations to the time-optimal control of linear systems involving the 1-dimensional heat-conduction equation and a single boundary input. It is shown that the eigenvalues associated with Galerkin's state equations are real if the eigenvalues associated with the distributed-parameter system are real. This is advantageous in the time-optimal control problems considered. The results obtained for approximations to time-optimal switching instants are compared with published work for those cases for which results are available in the literature. It is seen that there is very good agreement between the results.