350 research outputs found
On Three Generalizations of Contraction
We introduce three forms of generalized contraction (GC). Roughly speaking,
these are motivated by allowing contraction to take place after small
transients in time and/or amplitude. Indeed, contraction is usually used to
prove asymptotic properties, like convergence to an attractor or entrainment to
a periodic excitation, and allowing initial transients does not affect this
asymptotic behavior.
We provide sufficient conditions for GC, and demonstrate their usefulness
using examples of systems that are not contractive, with respect to any norm,
yet are GC
Non-Euclidean Contraction Theory for Robust Nonlinear Stability
We study necessary and sufficient conditions for contraction and incremental
stability of dynamical systems with respect to non-Euclidean norms. First, we
introduce weak pairings as a framework to study contractivity with respect to
arbitrary norms, and characterize their properties. We introduce and study the
sign and max pairings for the and norms, respectively.
Using weak pairings, we establish five equivalent characterizations for
contraction, including the one-sided Lipschitz condition for the vector field
as well as matrix measure and Demidovich conditions for the corresponding
Jacobian. Third, we extend our contraction framework in two directions: we
prove equivalences for contraction of continuous vector fields and we formalize
the weaker notion of equilibrium contraction, which ensures exponential
convergence to an equilibrium. Finally, as an application, we provide (i)
incremental input-to-state stability and finite input-state gain properties for
contracting systems, and (ii) a general theorem about the Lipschitz
interconnection of contracting systems, whereby the Hurwitzness of a gain
matrix implies the contractivity of the interconnected system
Biological control via "ecological" damping: An approach that attenuates non-target effects
In this work we develop and analyze a mathematical model of biological
control to prevent or attenuate the explosive increase of an invasive species
population in a three-species food chain. We allow for finite time blow-up in
the model as a mathematical construct to mimic the explosive increase in
population, enabling the species to reach "disastrous" levels, in a finite
time. We next propose various controls to drive down the invasive population
growth and, in certain cases, eliminate blow-up. The controls avoid chemical
treatments and/or natural enemy introduction, thus eliminating various
non-target effects associated with such classical methods. We refer to these
new controls as "ecological damping", as their inclusion dampens the invasive
species population growth. Further, we improve prior results on the regularity
and Turing instability of the three-species model that were derived in earlier
work. Lastly, we confirm the existence of spatio-temporal chaos
Lipschitz regularized gradient flows and latent generative particles
Lipschitz regularized f-divergences are constructed by imposing a bound on
the Lipschitz constant of the discriminator in the variational representation.
They interpolate between the Wasserstein metric and f-divergences and provide a
flexible family of loss functions for non-absolutely continuous (e.g.
empirical) distributions, possibly with heavy tails. We construct Lipschitz
regularized gradient flows on the space of probability measures based on these
divergences. Examples of such gradient flows are Lipschitz regularized
Fokker-Planck and porous medium partial differential equations (PDEs) for the
Kullback-Leibler and alpha-divergences, respectively. The regularization
corresponds to imposing a Courant-Friedrichs-Lewy numerical stability condition
on the PDEs. For empirical measures, the Lipschitz regularization on gradient
flows induces a numerically stable transporter/discriminator particle
algorithm, where the generative particles are transported along the gradient of
the discriminator. The gradient structure leads to a regularized Fisher
information (particle kinetic energy) used to track the convergence of the
algorithm. The Lipschitz regularized discriminator can be implemented via
neural network spectral normalization and the particle algorithm generates
approximate samples from possibly high-dimensional distributions known only
from data. Notably, our particle algorithm can generate synthetic data even in
small sample size regimes. A new data processing inequality for the regularized
divergence allows us to combine our particle algorithm with representation
learning, e.g. autoencoder architectures. The resulting algorithm yields
markedly improved generative properties in terms of efficiency and quality of
the synthetic samples. From a statistical mechanics perspective the encoding
can be interpreted dynamically as learning a better mobility for the generative
particles
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