  This work is concerned with the design and effects of the synchronization
gains on the synchronization problem for a class of networked distributed
parameter systems. The networked systems, assumed to be described by the same
evolution equation in a Hilbert space, differ in their initial conditions. The
proposed synchronization controllers aim at achieving both the control
objective and the synchronization objective. To enhance the synchronization, as
measured by the norm of the pairwise state difference of the networked systems,
an adaptation of the gains is proposed. An alternative design arrives at
constant gains that are optimized with respect to an appropriate measure of
synchronization. A subsequent formulation casts the control and synchronization
design problem into an optimal control problem for the aggregate systems. An
extensive numerical study examines the various aspects of the optimization and
adaptation of the gains on the control and synchronization of networked 1D
parabolic differential equations.
