ABSTRACT
In the previous chapter we presented control results for the infinite dimensional problem and controls were given in terms of operators and functions satisfying appropriate smoothness constraints. One cannot of course implement such operators in practice. Indeed, for these controls to be implemented, the problem must be discretized and a sequence of finite dimensional LQR problems considered. As in previous chapters of this book we consider discretization in the context of Galerkin approximations and approximate solutions are sought in finite dimensional subspaces VN ⊂ V ⊂ H. The bases for these subspaces can consist of modes, splines, or finite elements which satisfy convergence criteria to be discussed in this section. We point out that the inclusion of VN in V may be too restrictive for some approximation methods such as finite differences, and certain spectral and collocation approximations. In such cases, a relaxation of hypotheses in the manner discussed in [BKregulator] for the bounded control input analysis can be employed. We also remark that VN is not required to be in DH(A). This is important when choosing a basis for VN and permits the use of linear splines in second-order problems and cubic splines in fourth-order systems (see, e.g., Chapter 12).
