Stationary trajectories, singular Hamiltonian systems and ill-posed interconnection

We study Hamiltonian systems, namely, systems comprising of trajectories which are `stationary´ with respect to a quadratic performance index: they play a central role in many optimal control problems. A typical assumption in the literature is that of `regularity´: the resulting first-order dynamical system is a regular state space system and not a singular descriptor system. In this paper we show that the first order representation of a Hamiltonian is a singular descriptor system if and only if the interconnection of a related MIMO system G(s) with its dual (i.e. its adjoint) is ill-posed. We address the possibility of existence of inadmissible initial conditions, i.e. initial conditions that give rise to impulsive solutions. We characterize conditions on G(s) under which the corresponding singular Hamiltonian system has inadmissible initial conditions. Under suitable simplifying assumptions, which amount to studying an extreme case of ill-posedness, our main result states that there exist no inadmissible initial conditions if and only if the skew-symmetric part of the first moment about s = ∞ of the transfer matrix G(s) is nonsingular; a condition we show that is opposite to that for G(s) to be an all-pass filter. As a corollary, ill-posed interconnection of a square MIMO system with odd number of inputs (in particular, SISO systems) with its adjoint always contains in inadmissible initial conditions.