  In this paper, a quadratic optimal control problem is considered for
second-order parabolic PDEs with homogeneous Dirichlet boundary conditions, in
which the "point" control function (depending only on time) constitutes a
source term. These problems involve choosing a control function (with or
without "peak-value" constraints) to approximately steer the solution of the
PDE in question to a desired function at the end of a prescribed (finite)
time-interval. To compute approximations to the desired optimal control
functions, semi-discrete, Galerkin approximations to the equation involved are
introduced and the corresponding (approximating) control problems are tackled.
It is shown that the sequences of solutions to both the constrained and
unconstrained approximating (finite-dimensional) control problems converge,
respectively, to the optimal solutions of the control problems involving the
original initial/boundary value problem. The solution to the unconstrained
approximating problem can be quite explicitly characterized, with the main
numerical step for its computation requiring only the solution of a Lyapunov
equation. Whereas approximate solutions to the constrained control problems can
be obtained on the basis of Lagrangian duality and piecewise linear
multipliers. These points are worked out in detail and illustrated by numerical
examples involving the heat equation (HEq).
