Small gain theorem for distributed feedback control of Sturm-Liouville dynamics

This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H ∞ -norm is extended to be 2D-H ∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.