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.
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