3,066 research outputs found
Controllability Metrics, Limitations and Algorithms for Complex Networks
This paper studies the problem of controlling complex networks, that is, the
joint problem of selecting a set of control nodes and of designing a control
input to steer a network to a target state. For this problem (i) we propose a
metric to quantify the difficulty of the control problem as a function of the
required control energy, (ii) we derive bounds based on the system dynamics
(network topology and weights) to characterize the tradeoff between the control
energy and the number of control nodes, and (iii) we propose an open-loop
control strategy with performance guarantees. In our strategy we select control
nodes by relying on network partitioning, and we design the control input by
leveraging optimal and distributed control techniques. Our findings show
several control limitations and properties. For instance, for Schur stable and
symmetric networks: (i) if the number of control nodes is constant, then the
control energy increases exponentially with the number of network nodes, (ii)
if the number of control nodes is a fixed fraction of the network nodes, then
certain networks can be controlled with constant energy independently of the
network dimension, and (iii) clustered networks may be easier to control
because, for sufficiently many control nodes, the control energy depends only
on the controllability properties of the clusters and on their coupling
strength. We validate our results with examples from power networks, social
networks, and epidemics spreading
On Submodularity and Controllability in Complex Dynamical Networks
Controllability and observability have long been recognized as fundamental
structural properties of dynamical systems, but have recently seen renewed
interest in the context of large, complex networks of dynamical systems. A
basic problem is sensor and actuator placement: choose a subset from a finite
set of possible placements to optimize some real-valued controllability and
observability metrics of the network. Surprisingly little is known about the
structure of such combinatorial optimization problems. In this paper, we show
that several important classes of metrics based on the controllability and
observability Gramians have a strong structural property that allows for either
efficient global optimization or an approximation guarantee by using a simple
greedy heuristic for their maximization. In particular, the mapping from
possible placements to several scalar functions of the associated Gramian is
either a modular or submodular set function. The results are illustrated on
randomly generated systems and on a problem of power electronic actuator
placement in a model of the European power grid.Comment: Original arXiv version of IEEE Transactions on Control of Network
Systems paper (Volume 3, Issue 1), with a addendum (located in the ancillary
documents) that explains an error in a proof of the original paper and
provides a counterexample to the corresponding resul
Unique Continuation for Stochastic Heat Equations
We establish a unique continuation property for stochastic heat equations
evolving in a bounded domain . Our result shows that the value of the
solution can be determined uniquely by means of its value on an arbitrary open
subdomain of at any given positive time constant. Further, when is
convex and bounded, we also give a quantitative version of the unique
continuation property. As applications, we get an observability estimate for
stochastic heat equations, an approximate result and a null controllability
result for a backward stochastic heat equation
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