Actuator Placement for Heterogeneous Complex Dynamical Networks with Long-Term Memory

We consider the Actuator Placement (AP) problem for heterogeneous complex dynamical networks. Initially, we propose a fractional order dynamical system for capturing longterm memory observed in complex network dynamics. Then, we formalize an energy and cost-efficient AP problem, wherein heterogeneous placement costs are assumed. A Gramian-based metric originating from the minimum control energy state transfer problem acts as the objective function and the total placement cost is upper bounded by a knapsack constraint. Leveraging recent advances in non-submodular optimization under knapsack constrains, we address the AP problem via a greedy algorithm with approximation guarantees that depend on quantities that measure how far the Gramianbased metric is from being submodular. From extensive experimental results for Erdos-Rényi, and Barabási-Albert compleẍ networks, we observe that the proposed algorithm achieves on average 95% of the global optimal objective value.