Abstract
The cross-Gramian matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W_{c0} (T)</tex> for linear single-input single-output (SISO) dynamical systems has been related to the controllability and observability Gramians via <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W^{2}_{c0}(T)= W_{c}(T)W_{0}(T)</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T= \infty</tex> . The result is now proved for any arbitrary time interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> , and is extended to symmetric multivariable systems which have symmetric transfer function matrices.
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