Asymptotic holdability

Given an autonomous linear control system and a linear subspace , we say that an initial state is asymptotically holdable in if the trajectory can be held arbitrarily near on any specified compact initial interval, subject to an L1bound on input energy which depends on the initial state and the interval. It is shown that asymptotic holdability is equivalent to holdability obtaining under a certain impulse control process. A geometric characterization is given.