Abstract:
The aim of this thesis is to study a priori and a posteriori error analysis of nite element methods for elliptic and parabolic optimal control problems with measure data in a bounded convex domain in Rd(d = 2 or 3). The control problems governed by elliptic partial di eren- tial equation with measure data are used to model the potential of an electric eld with an electric charge distribution, whereas parabolic optimal control problems with measure data in space are used in the design and management of waste water treatment systems, mainly the disposal of sea outfalls discharging polluting e uent from a sewerage systems. On the other hand, parabolic optimal control problems with measure data in time appear in the optimality conditions of some optimal control problems with pointwise state constraints in time. The solution of the state equation of such type of problems exhibits low regularity due to the pres- ence of measure data which introduces some di culties for both theory and numerics of the nite element method. An e ort has been made in this thesis to investigate both a priori and a posteriori error analysis of nite element method for these control problems. The strategy optimize-then-discretize is employed for the approximations of these control problems.We rst analyze elliptic optimal control problem with measure data and prove the existence, uniqueness and regularity of the solution to the optimal control problem.
