An algebraic characterization of closed small attainability subspaces of delay systems

If small attainability subspaces of linear time delay systems are closed in a certain Sobolev space, the existence of Lagrange multipliers for optimal control to small solutions is guaranteed. This paper characterizes the required closedness property using an algebraic approach due to B. Jakubczyk. As a main result it turns out that closedness is—in an algebraic sense—generic in the variety of system matrices (A₀,A₁, B₀) with rank A₁ not greater than the dimension of the control space. This is in contrast to known results on closedness of attainability subspaces playing an analogous role for optimal control to fixed final states instead of small solutions.