We analyze the lumpability of linear systems on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction operator onto a smaller state space in which a self-contained dynamical description exists. We obtain conditions for lumpability of dynamics defined by unbounded operators using the theory of strongly continuous semigroups. We also derive results from the dual space point of view using sun dual theory. Furthermore, we connect the theory of lumping to several results from operator factorization. We indicate several applications to particular systems, including delay differential equations.
The stability of functional differential equations under
delayed feedback is investigated near a Hopf bifurcation. Necessary
and sufficient conditions are derived for the stability of the
equilibrium solution using averaging theory. The results are used to
compare delayed versus undelayed feedback, as well as discrete
versus distributed delays. Conditions are obtained for which
delayed feedback with partial state information can yield stability
where undelayed feedback is ineffective. Furthermore, it is shown
that if the feedback is stabilizing (respectively, destabilizing),
then a discrete delay is locally the most stabilizing (resp.,
destabilizing) one among delay distributions having the same mean.
The result also holds globally if one considers delays that are symmetrically
distributed about their mean.
The Winfree model describes finite networks of phase oscillators. Oscillators interact by broadcasting pulses that modulate the frequencies of connected oscillators. We study a generalization of the model and its fluid-dynamical limit for networks, where oscillators are distributed on some abstract $\sigma$-finite Borel measure space over a separable metric space. We give existence and uniqueness statements for solutions to the continuity equation for the oscillator phase densities. We further show that synchrony in networks of identical oscillators is locally asymptotically stable for finite, strictly positive measures and under suitable conditions on the oscillator response function and the coupling kernel of the network. The conditions on the latter are a generalization of the strong connectivity of finite graphs to abstract coupling kernels.
We analyze stability of consensus algorithms in networks of
multi-agents with time-varying topologies and delays. The topology
and delays are modeled as induced by an adapted process and are rather
general, including i.i.d. topology processes, asynchronous consensus
algorithms, and Markovian jumping switching. In case the self-links are instantaneous, we prove that the network
reaches consensus for all bounded delays if the graph corresponding to the
conditional expectation of the coupling matrix sum across a finite
time interval has a spanning tree almost surely.
Moreover, when self-links are also delayed
and when the delays satisfy certain integer patterns, we
observe and prove that the algorithm may not reach consensus but
instead synchronize at a periodic trajectory, whose period depends
on the delay pattern. We also give a brief discussion on the
dynamics in the absence of self-links.