919 research outputs found
H_2-Optimal Decentralized Control over Posets: A State-Space Solution for State-Feedback
We develop a complete state-space solution to H_2-optimal decentralized
control of poset-causal systems with state-feedback. Our solution is based on
the exploitation of a key separability property of the problem, that enables an
efficient computation of the optimal controller by solving a small number of
uncoupled standard Riccati equations. Our approach gives important insight into
the structure of optimal controllers, such as controller degree bounds that
depend on the structure of the poset. A novel element in our state-space
characterization of the controller is a remarkable pair of transfer functions,
that belong to the incidence algebra of the poset, are inverses of each other,
and are intimately related to prediction of the state along the different paths
on the poset. The results are illustrated by a numerical example.Comment: 39 pages, 2 figures, submitted to IEEE Transactions on Automatic
Contro
Lectures on Linear Stability of Rotating Black Holes
These lecture notes are concerned with linear stability of the non-extreme
Kerr geometry under perturbations of general spin. After a brief review of the
Kerr black hole and its symmetries, we describe these symmetries by Killing
fields and work out the connection to conservation laws. The Penrose process
and superradiance effects are discussed. Decay results on the long-time
behavior of Dirac waves are outlined. It is explained schematically how the
Maxwell equations and the equations for linearized gravitational waves can be
decoupled to obtain the Teukolsky equation. It is shown how the Teukolsky
equation can be fully separated to a system of coupled ordinary differential
equations. Linear stability of the non-extreme Kerr black hole is stated as a
pointwise decay result for solutions of the Cauchy problem for the Teukolsky
equation. The stability proof is outlined, with an emphasis on the underlying
ideas and methods.Comment: 25 pages, LaTeX, 3 figures, lectures given at first DOMOSCHOOL in
July 2018, minor improvements (published version
Analytic and algebraic properties of Riccati equations:A survey
AbstractThis is a survey of recent results on the classical problems of the analytic properties of Riccati equations and algebraic properties of Riccati equations and applications to spatially distributed systems
Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics
A new method of deriving reduced models of Hamiltonian dynamical systems is
developed using techniques from optimization and statistical estimation. Given
a set of resolved variables that define a model reduction, the
quasi-equilibrium ensembles associated with the resolved variables are employed
as a family of trial probability densities on phase space. The residual that
results from submitting these trial densities to the Liouville equation is
quantified by an ensemble-averaged cost function related to the information
loss rate of the reduction. From an initial nonequilibrium state, the
statistical state of the system at any later time is estimated by minimizing
the time integral of the cost function over paths of trial densities.
Statistical closure of the underresolved dynamics is obtained at the level of
the value function, which equals the optimal cost of reduction with respect to
the resolved variables, and the evolution of the estimated statistical state is
deduced from the Hamilton-Jacobi equation satisfied by the value function. In
the near-equilibrium regime, or under a local quadratic approximation in the
far-from-equilibrium regime, this best-fit closure is governed by a
differential equation for the estimated state vector coupled to a Riccati
differential equation for the Hessian matrix of the value function. Since
memory effects are not explicitly included in the trial densities, a single
adjustable parameter is introduced into the cost function to capture a
time-scale ratio between resolved and unresolved motions. Apart from this
parameter, the closed equations for the resolved variables are completely
determined by the underlying deterministic dynamics
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