✐ ✐ Chapter 12 Solving optimal control problems described by PDEs by Czeslaw Bajer 1 , Andrzej My´ sli ´ nski 2 , Antoni ˙ Zochowski 2 , Bartlomiej Dyniewicz 1 and Dominik Pisarski 1 12.1 Introduction The computational methods used in practice do not allow an accurate representa- tion of the phenomena occurring in the processes described. Despite the effort and engagement of the knowledge from many disciplines we can not describe all the phenomena which occur. For this reason, we try to separate them from each other and focus our attention on one single phenomenon. Often we are forced to go back and accept the solution of simpler problems, which significantly deviates from our initial expectations. We make simplifications every step of the calculation. We must chose between statics and dynamics, linear or nonlinear description, and finally take a solution method which satisfy our requirements. Solution methods ap- plied to problems described by partial differential equations are treated in numerous papers. The efficiency and accuracy of solutions are main two features that are taken into account in numerical modelling. Numerical methods applied to such problems can be divided into two main groups: • methods based on the discretization of the differential equation (for example cen- tral difference method), • methods based on the discretization of the spatial and time domain of the problem (finite element method, space-time finite element method). Moreover, one method can be applied to space while the other to time. While many of simplifications are intuitive and the degree of approximation is assessed in a rather arbitrary manner, the degree of approximate validity of these mathematical methods can usually be estimated well. Hence, there are a large variety of computational tools. Mathematical methods that lead to numerical schemes can be divided into three groups: Strong form – description of the equations of motion is represented by a system of differential equations in space and time, supplemented by boundary and initial 1 Institute of Fundamental Research in Technology, Polish Academy of Sciences, 02-106 Warsaw 2 Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw 347