Publications

The control of dynamical, networked systems continues to receive much attention across the engineering and scientific research fields. Of particular interest is the proper way to determine which nodes of the network should receive external control inputs in order to effectively and efficiently control portions of the network. Published methods to accomplish this task either find a minimal set of driver nodes to guarantee controllability or a larger set of driver nodes which optimizes some control metric. Here, we investigate the control of lattice systems which provides analytical insight into the relationship between network structure and controllability. First we derive a closed form expression for the individual elements of the controllability Gramian of infinite lattice systems. Second, we focus on nearest neighbor lattices for which the distance between nodes appears in the expression for the controllability Gramian. We show that common control energy metrics scale exponentially with respect to the maximum distance between a driver node and a target node.

We consider the FDA approved Glucose-Insulin-Glucagon nonlinear model which describes how the body responds to exogenously supplied insulin and glucagon in patients affected by Type I diabetes. Based on this model, we design infusion rates of either insulin (monotherapy) or insulin and glucagon (dual-therapy) that can optimally maintain the blood glucose level after consuming a meal within healthy limits and prevent the onset of both hypoglycemia and hyperglycemia. The problem is formulated as a nonlinear optimal control problem, which we solve using the numerical optimal control package $\mathcal{PSOPT}$. Interestingly, in the case of monotherapy, the optimal solution we find is nearly identical to the standard method of insulin based glucose regulation, which is to assume a variable amount of insulin half an hour before each meal. We also find that the optimal dual-therapy (using both insulin and glucagon) is better able to regulate glucose as compared to using insulin alone. We also propose a rule for both dosage and time of delivery for the dual-therapy. Our results, which are optimal, can serve as a reference to evaluate the relative performance of alternative control algorithms that use the blood glucose model.

It has recently been shown that the average energy required to control a subset of nodes in a complex network scales exponentially with the cardinality of the subset. While the mean scales exponentially, the variance of the control energy over different subsets of nodes is large and has as of yet not been explained. Here, we provide an explanation of the large variance as a result of both the length of the path that connects control inputs to the target nodes and the redundancy of paths of shortest length. Our first result provides an exact upper bound of the control energy as a function of path length between driver node and target node along an infinite path graph. We also show that the energy estimation is still very accurate even when finite size effects are taken into account. Our second result refines the upper bound that takes into account not only the length of the path, but also the redundancy of paths. We show that it improves the upper bound approximation by an order of magnitude or more. Finally, we lay out the foundations for a more accurate estimation of the control energy for the multi-target and multi-driver problem.

Symmetry in graphs which describe the underlying topology of networked dynamical systems plays an essential role in the synchronization patterns that can emerge. Many real networked systems have a very large number of symmetries. Often one wants to test new results on large sets of random graphs that are representative of the real networks of interest. Unfortunately, existing graph generating algorithms will seldom produce graphs with any symmetry, much less ones with desired symmetry patterns. Here, we present an algorithm that is able to generate graphs with any desired symmetry pattern. The algorithm can be coupled with other graph generating algorithms to tune the final graph's properties of interest such as the degree distribution.

The effects of molecularly targeted drug perturbations on cellular activities and fates are difficult to predict using intuition alone because of the complex behaviors of cellular regulatory networks. An approach to overcoming this problem is to develop mathematical models for predicting drug effects. Such an approach beckons for co-development of computational methods for extracting insights useful for guiding therapy selection and optimizing drug scheduling. Here, we present and evaluate a generalizable strategy for identifying drug dosing schedules that minimize the amount of drug needed to achieve sustained suppression or elevation of an important cellular activity/process, the recycling of cytoplasmic contents through (macro)autophagy. Therapeutic targeting of autophagy is currently being evaluated in diverse clinical trials but without the benefit of a control engineering perspective. Using a nonlinear ordinary differential equation (ODE) model that accounts for activating and inhibiting influences among protein and lipid kinases that regulate autophagy (MTORC1, ULK1, AMPK and VPS34) and methods guaranteed to find locally optimal control strategies, we find optimal drug dosing schedules (open-loop controllers) for each of six classes of drugs. Our approach is generalizable to designing monotherapy and multi therapy drug schedules that affect different cell signaling networks of interest.

In the field of complex networks and graph theory, new results are typically tested on graphs generated by a variety of algorithms such as the Erdos-Renyi model or the Barabasi-Albert model. Unforunately, most graph generating algorithms do not typically create graphs with symmetries, which have been shown to have an important role on the network dynamics. Here, we present an algorithm to generate graphs with prescribed symmetries. The algorithm can also be used to generate graphs with a prescribed equitable partition but possibly without any symmetry. We also use our graph generator to examine the recently raised question about the relation between the orbits of the automorphism group and a graph's minimal equitable partition.

The control of complex networks has generated considerable interest in a variety of fields from traffic management to neural systems. A commonly used metric to compare two particular control strategies that accomplish the same task is the control energy, the time-integral of the sum of squares of all control inputs. The minimum control energy problem determines the control input that lower bounds all other control inputs with respect to their control energies. Here, we focus on the infinite lattice graph with linear dynamics and analytically derive the expression for the minimum control energy in terms of the modified Bessel function. We then demonstrate that the control energy of the infinite lattice graph accurately predicts the control energy of finite lattice graphs.

We consider the problem of a dynamical network whose dynamics is subject to external perturbations ("attacks") locally applied to a subset of the network nodes. We assume that the network has an ability to defend itself against attacks with appropriate countermeasures, which we model as actuators located at (another) subset of the network nodes. We derive the optimal defense strategy as an optimal control problem. We see that the network topology as well as the distribution of attackers and defenders over the network affect the optimal control solution and the minimum control energy. We study the optimal control defense strategy for several network topologies, including chain networks, star networks, ring networkjs, and scale free networks.

It has recently been shown that the expected energy requirements of a control action applied to a complex network scales exponentially with the number of nodes that are targeted. While the exponential scaling law provides an adequate prediction of the mean required energy, it has also been shown that the spread of energy values for a particular number of targets is large. Here, we explore more closely the effect distance between driver nodes and target nodes and the magnitude of self-regulation has on the energy of the control action. We find that the energy scaling law can be written to include information about the distance between driver nodes and target nodes to more accurately predict control energy.

It has recently been shown that the minimum energy solution of the control problem for a linear system produces a control trajectory that is nonlocal. An issue then arises when the dynamics represents a linearization of the underlying nonlinear dynamics of the system where the lienarization is only valid in a local region of the state space. Here we provide a solution to the problem of optimally controlling a linearized system by deriving a time-varying set that represents all possible control trajectories parameterized by time and energy. As long as the control action terminus is defined within this set, the control trajectory is guaranteed to be local. If the desired terminus of the control action is far from the initial state, a series of local control actions can be performed in series, relinearizing the dynamics at each new position.

We have seen 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. Here, we see that the control energy can be reduced even more if the prescribed final states are not satisfied strictly. We introduce a new control strategy called balance control for which we set our objective function as a convex combination of two competitive terms: (i) the distance between the output final states at a given final time and given prescribed states and (ii) the total control energy expenditure over the given time period. We also see that the required energy for the optimal balanced control problem approximates the required energy for the optimal target control problem when the coefficient of the second term is very small. We validate our conclusions in model and real networks regardless of system size, energy restrictions, state restrictions, input node choices, and target node choices.

Recently it has been shown that the control energy required to control a dynamical complex network is prohibitively large when there are only a few control inputs. Most methods to reduce the control energy have focused on where, in the network, to place additional control inputs. Here, in contrast, we show that by controlling the states of a subset of the nodes of a network, rather than the state of every node, while holding the number of control signals constant, 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. We validate our conclusions in model and real networks to arrive at an energy scaling law to better design control objectives regardless of system size, energy restrictions, state restrictions, input node choices and target node choices.

Real-world systems are often composed of heterogeneous parts that interact in multiple ways with complicated patterns. The multilayer networks modeling framework describes such systems; studies implement this framework in epidemiology, social sciences, power transportation, economics and engineering. Our main contribution is twofold: (i) we define and compute the symmetries of multilayer networks and (ii) we study the emergence of cluster synchronization in these networks. We distinguish between symmetries that affect the types of allowed dynamics in the other layers (Dependent Layer Symmetries) and those that do not (Independent Layer Symmetries.) We then study stability of the cluster synchronous solution: we decouple the stability problem into a number of independent blocks, corresponding to the irreducible representations of the symmetry group, and then assess stability through a Master Stability Function. We see that the blocks associated with Dependent Layer Symmetries have a different structure than those arising in the study of single layer networks, which we show has a profound effect on the stability of the clusters involving these symmetries. Finally, we apply the developed theory to a fully analog experiment in which seven electronic oscillators of three kinds are coupled with two kinds of coupling.