  In this paper we consider distributed adaptive stabilization for uncertain
multivariable linear systems with a time-varying diagonal matrix gain. We show
that uncertain multivariable linear systems are stabilizable by diagonal matrix
high gains if the system matrix is an H-matrix with positive diagonal entries.
Based on matrix measure and stability theory for diagonally dominant systems,
we consider two classes of uncertain linear systems, and derive a threshold
condition to ensure their exponential stability by a monotonically increasing
diagonal gain matrix. When each individual gain function in the matrix gain is
updated by state-dependent functions using only local state information, the
boundedness and convergence of both system states and adaptive matrix gains are
guaranteed. We apply the adaptive distributed stabilization approach to
adaptive synchronization control for large-scale complex networks consisting of
nonlinear node dynamics and time-varying coupling weights. A unified framework
for adaptive synchronization is proposed that includes several general design
approaches for adaptive coupling weights to guarantee network synchronization.
