1,374 research outputs found
Krylov subspaces associated with higher-order linear dynamical systems
A standard approach to model reduction of large-scale higher-order linear
dynamical systems is to rewrite the system as an equivalent first-order system
and then employ Krylov-subspace techniques for model reduction of first-order
systems. This paper presents some results about the structure of the
block-Krylov subspaces induced by the matrices of such equivalent first-order
formulations of higher-order systems. Two general classes of matrices, which
exhibit the key structures of the matrices of first-order formulations of
higher-order systems, are introduced. It is proved that for both classes, the
block-Krylov subspaces induced by the matrices in these classes can be viewed
as multiple copies of certain subspaces of the state space of the original
higher-order system
Pade-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems
A standard approach to reduced-order modeling of higher-order linear
dynamical systems is to rewrite the system as an equivalent first-order system
and then employ Krylov-subspace techniques for reduced-order modeling of
first-order systems. While this approach results in reduced-order models that
are characterized as Pade-type or even true Pade approximants of the system's
transfer function, in general, these models do not preserve the form of the
original higher-order system. In this paper, we present a new approach to
reduced-order modeling of higher-order systems based on projections onto
suitably partitioned Krylov basis matrices that are obtained by applying
Krylov-subspace techniques to an equivalent first-order system. We show that
the resulting reduced-order models preserve the form of the original
higher-order system. While the resulting reduced-order models are no longer
optimal in the Pade sense, we show that they still satisfy a Pade-type
approximation property. We also introduce the notion of Hermitian higher-order
linear dynamical systems, and we establish an enhanced Pade-type approximation
property in the Hermitian case
Long-range forces in controlled systems
This thesis investigates new phenomena due to long-range forces and their effects
on different multi-DOFs systems. In particular the systems considered are metamaterials,
i.e. materials with long-range connections. The long-range connections
characterizing metamaterials are part of the more general framework of non-local
elasticity.
In the theory of non-local elasticity, the connections between non-adjacent particles
can assume different configurations, namely one-to-all, all-to-all, all-to-all-limited,
random-sparse and all-to-all-twin. In this study three aspects of the long-range
interactions are investigated, and two models of non-local elasticity are considered:
all-to-all and random-sparse.
The first topic considers an all-to-all connections topology and formalizes the mathematical
models to study wave propagation in long-range 1D metamaterials. Closed
forms of the dispersion equation are disclosed, and a propagation map synthesizes
the properties of these materials which unveil wave-stopping, negative group velocity,
instability and non-local effects. This investigation defines how long-range
interactions in elastic metamaterials can produce a variety of new effects in wave
propagation.
The second one considers an all-to-all connections topology and aims to define an
optimal design of the long-range actions in terms of spatial and intensity distribution
to obtain a passive control of the propagation behavior which may produces
exotic effects. A phenomenon of frequency filtering in a confined region of a 1D
metamaterial is obtained and the optimization process guarantees this is the best
obtainable result for a specific set of control parameters.
The third one considers a random-sparse connections topology and provides a new
definition of long-range force, based on the concept of small-world network. The
small-world model, born in the field of social networks, is suitably applied to a
regular lattice by the introduction of additional, randomly selected, elastic connections
between different points. These connections modify the waves propagation
within the structure and the system exhibits a much higher propagation speed and
synchronization. This result is one of the remarkable characteristics of the defined
long-range connections topology that can be applied to metamaterials as well as
other multi-DOFs systems. Qualitative experimental results are presented, and a
preliminary set-up is illustrated.
To summarize, this thesis highlights non-local elastic structures which display unusual
propagation behaviors; moreover, it proposes a control approach that produces
a frequency filtering material and shows the fast propagation of energy within a
random-sparse connected material
A Generalization of the Hopf-Cole Transformation
A generalization of the Hopf-Cole transformation and its relation to the
Burgers equation of integer order and the diffusion equation with quadratic
nonlinearity are discussed. The explicit form of a particular analytical
solution is presented. The existence of the travelling wave solution and the
interaction of nonlocal perturbation are considered. The nonlocal
generalizations of the one-dimensional diffusion equation with quadratic
nonlinearity and of the Burgers equation are analyzed
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