On the existence of a negative semidefinite, antistabilizing solution to the discrete-time algebraic Riccati equation

In the problem of infimizing a not necessarily positive semidefinite quadratic cost subject to a linear dynamical constraint, it is usually expected that the existence of a lower bound to the cost is equivalent to the existence of a negative semidefinite, antistabilizing solution to the algebraic Riccati equation. By a counterexample, it is shown that this equivalence breaks down in the discrete-time case. This phenomenon, as well as the whole question of the existence of the appropriate solution to the algebraic Riccati equation, are investigated in detail.