Infinite-dimensional convex optimization in optimal and robust control theory

Many engineering problems can be shown to be equivalent to solving semidefinite programs (SPs), i.e., convex optimization problems involving linear matrix inequalities (LMIs). Powerful computation tools are available for such problems in the finite-dimensional case. However, the problems arising in optimal and robust control theory are often infinite dimensional, and so adequate computation tools are not available. The key to tackling such problems with finite computation tools is to have a primal-dual formulation of the problem without duality gap. In this paper we study infinite-dimensional SPs and present a lifting technique to recast SPs as parameterized linear programs (LPs). This enables the wealth of theoretical tools available for infinite-dimensional LPs to be extended to infinite-dimensional SPs. In particular, we develop some new sufficient conditions for the lack of a duality gap for infinite-dimensional SPs and give an exact characterization of the primal and dual problems for these cases. Both primal and dual problems are formed as infinite-dimensional SP problems, with finite truncations to each giving upper and lower bounds, respectively, on the exact solution to the infinite-dimensional problem. Thus, these results can form the basis of practical computation schemes for infinite-dimensional problems, which require only finite-dimensional computation tools. To illustrate the power of these tools we apply the results to two previously unsolved optimization problems, namely minimizing the l₁ norm of a closed-loop system subject to bounds on the frequency response magnitude at a finite number of points and/or bounds on the H₂ norm