In this chapter the above design methodology is applied to systems governed by partial differential equations. The representation of the solution for initial and boundary value problems by Fourier series is the essential point, and leads to analysis of infinite-dimensional systems of differential equations. The chapter is concerned with the control problem for systems governed by parabolic equations, while the results can be extended for other types of partial differential equations. At the beginning, the system with infinite-dimensional control is considered; then the particularities of finite-dimensional control for distributed parameter systems are discussed.

13.1 One-Dimensional Heat System with Distributed Control

One-Dimensional Heat Equation

Let us consider a heating process described by a one-dimensional parabolic equation given by

(13.1)

where t is time, t > 0, z is the spatial variable, 0 < z < 1, x( z, 0) = x ( z) is the initial condition, [ ?x( z, t)/ ?z] _{z = 0} = 0 and [ ?x( z, t)/ ?z] _{z = 1} = 0 are the boundary conditions, c( t) is an unknown varying parameter, c( t) < c ₀ < ?, w( z, t) is a distributed external disturbance unavailable for measurement, u( z, t) is the distributed control, ? ² is a constant (we shall take