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# Optimal Control of Networks: Energy Scaling and Open Challenges

## December 10 @ 2:00 pm - 3:30 pm

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Recent years have witnessed increased interest from the scientific community regarding the control of complex dynamical networks. Some common types of networks examined throughout the literature are power grids, communication networks, gene regulatory networks, neuronal systems, food webs, and social systems. Optimal control studies strategies to control a system that minimize a cost function, for example the energy that is required by the control action.

We show that by controlling the states of a subset of the nodes of a network, rather than the state of every node, the required energy to control a portion of the network can be reduced substantially. The energy requirements exponentially decay with the number of target nodes, suggesting that large networks can be controlled by a relatively small number of inputs, as long as the target set is appropriately sized.

An important observation is that the minimum energy solution of the control problem for a linear system produces a control trajectory that is nonlocal. However, when the network dynamics is linearized, the linearization is only valid in a local region of the state space and hence the question arises whether optimal control can be used. We provide a solution to this problem by determining the region of state space where the trajectory does remain local and so minimum energy control can still be applied to linearized approximations of nonlinear systems. We apply our results to develop an algorithm that determines a piecewise open-loop control signal for nonlinear systems. Applications include controlling power grid dynamics and the regulatory dynamics of the intracellular circadian clock.

This work is in collaboration with Isaac Klickstein and Afroza Shirin (UNM).

[1] I. Klickstein, A. Shirin, F. Sorrentino, “Energy Scaling of Targeted Optimal Control of Complex Networks”, Nature Communications, 8, 15145 (2017)

[2] I. Klickstein, A. Shirin, F. Sorrentino, “Locally Optimal Control of Complex Networks”, Phys. Rev. Lett., 119, 268301 (2017)

Speaker(s): Francesco Sorrentino,

Location:

Room: ECS 130

Bldg: Engineering/Computer Science (ECS)

University of Victoria

3800 Finnerty Rd

Victoria, British Columbia

V8P 5C2