L_infty-instability of large-scale interconnected systems using orthogonal decomposition and exponential weighting

In this paper, we give an explicit characterization of the orthogonal complement of the "set of stabilizing inputs" for an unstable convolution operator. This characterization involves the right-coprimne factorization of the transfer function matrix of the convolution operator. Using this characterization, we then explicitly demonstrate specific inputs that would cause
L_2
-instability in a given large-scale interconnected system, using the technique of orthogonal decomposition of the input space. Finally, using the technique of exponential weighting, we give a general result that can be used to establish the
L_{\infty}
-instability of a given system by proving that an associated system is
L_2
-unstable. In this way, one can obtain
L_{\infty}
-instability criteria corresponding to virtually every known criterion for
L_2
-instability that uses the technique of orthogonal decomposition.