Intrinsic Difficulties in Using the Doubly-Infinite Time Axis for Input-Output Control Theory

We point out that the natural definitions of stability and causality in input-output control theory lead to certain inconsistencies when inputs and outputs are allowed to have support on the doubly-infinite time-axis. In particular, linear time-invariant systems with right-half plane poles cannot be considered to be both causal and stabilizable. In contrast, there is no such conflict when the semi-infinite time axis is used.