L1minimization with magnitude constraints in the frequency domain

The L1optimal control problem with additional constraints on the magnitude of the closed loop frequency response at fixed frequency points is considered. This problem is known to be equivalent to a convex optimization subject to infinite dimensional LMI. It is shown that the LMI problem can be approximated arbitrarily well by an infinite dimensional Linear Program. The main result of the paper states that, for multiblock problems, the computation of lower bounds to the optimal cost, based on approximating the dual linear programming problem by sequences of finite support, may fail to converge to the optimal cost of the infinite dimensional problem as the dimension of the approximation increases.