Schauder Decompositions, Approximations and Control Problems

Abstract

A Schauder decomposition for a Banach space X is a sequence $\{ P_n \} $ of finite rank continuous projections such that (a) $P_n P_m = P_m P_n = P_{\min \{ m,n\} } $ and (b) $\lim _n P_n x = x$ for each x in X. Schauder decompositions can be used to approximate the solution to optimal control problems defined on X. For example, let S and Tdenote continuous linear operators from X into itself; let u be a point in the range of S and let p be a continuous seminorm on X The problem: \[\begin{array}{*{20}c} {{\text{(I)}}\qquad } & \begin{gathered} {\text{find }} x ({\text{ and }}c) {\text{ in }} X {\text{ such that (a) }}S(x) = u,{\text{ (b) }}c = x - Tx, \hfill \\ {\text{(c) }} p(c) {\text{ is a minimum,}} \hfill \\ \end{gathered} \\ \end{array} \] can be discretized to the problem: \[\begin{array}{*{20}c} {{\text{(II)}}\qquad } & \begin{gathered} {\text{find }} x_n ({\text{and }} c_n ) {\text{ in the range of }}P_n {\text{ such that (a) }}Sx_n = P_n u, \hfill \\ {\text{(b) }}c_n = x_n - P_n Tx_n ,{\text{(c) }}p(c_n ){\text{ is a minimum.}} \hfill \\ \end{gathered} \\ \end{array} \] We discuss conditions under which the minima found in solving (II) converge to the minimum in (I) as $n \to \infty $. Then we illustrate our theory by computing approximate solutions to the problem: \[\begin{array}{*{20}c} {{\text{(III)}}\qquad } & \begin{gathered} {\text{find functions }}x\, ({\text{and }}c){\text{ such that (a) }}x(t){\text{ is given for }}t \hfill \\ {\text{in }}[0,\frac{1}{3}) \cup [\frac{2}{3},1],{\text{ (b) }}c(t) = x(t) - \int_0^t {x(s)} ds, \hfill \\ {\text{(c) }}\int_0^1 {| {c(t)} |} ^2 dt{\text{ is a minimum.}} \hfill \\ \end{gathered} \\ \end{array} \]