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Synchronization of Nonlinear Circuits in Dynamic Electrical Networks with General Topologies
Sufficient conditions are derived for global asymptotic synchronization in a
system of identical nonlinear electrical circuits coupled through linear
time-invariant (LTI) electrical networks. In particular, the conditions we
derive apply to settings where: i) the nonlinear circuits are composed of a
parallel combination of passive LTI circuit elements and a nonlinear
voltage-dependent current source with finite gain; and ii) a collection of
these circuits are coupled through either uniform or homogeneous LTI electrical
networks. Uniform electrical networks have identical per-unit-length
impedances. Homogeneous electrical networks are characterized by having the
same effective impedance between any two terminals with the others open
circuited. Synchronization in these networks is guaranteed by ensuring the
stability of an equivalent coordinate-transformed differential system that
emphasizes signal differences. The applicability of the synchronization
conditions to this broad class of networks follows from leveraging recent
results on structural and spectral properties of Kron reduction---a
model-reduction procedure that isolates the interactions of the nonlinear
circuits in the network. The validity of the analytical results is demonstrated
with simulations in networks of coupled Chua's circuits
(Non)local and (non)linear free boundary problems
We discuss some recent developments in the theory of free boundary problems,
as obtained in a series of papers in collaboration with L. Caffarelli, A.
Karakhanyan and O. Savin.
The main feature of these new free boundary problems is that they deeply take
into account nonlinear energy superpositions and possibly nonlocal functionals.
The nonlocal parameter interpolates between volume and perimeter functionals,
and so it can be seen as a fractional counterpart of classical free boundary
problems, in which the bulk energy presents nonlocal aspects.
The nonlinear term in the energy superposition takes into account the
possibility of modeling different regimes in terms of different energy levels
and provides a lack of scale invariance, which in turn may cause a structural
instability of minimizers that may vary from one scale to another
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