2,591 research outputs found
Stability and Performance Verification of Optimization-based Controllers
This paper presents a method to verify closed-loop properties of
optimization-based controllers for deterministic and stochastic constrained
polynomial discrete-time dynamical systems. The closed-loop properties amenable
to the proposed technique include global and local stability, performance with
respect to a given cost function (both in a deterministic and stochastic
setting) and the gain. The method applies to a wide range of
practical control problems: For instance, a dynamical controller (e.g., a PID)
plus input saturation, model predictive control with state estimation, inexact
model and soft constraints, or a general optimization-based controller where
the underlying problem is solved with a fixed number of iterations of a
first-order method are all amenable to the proposed approach.
The approach is based on the observation that the control input generated by
an optimization-based controller satisfies the associated Karush-Kuhn-Tucker
(KKT) conditions which, provided all data is polynomial, are a system of
polynomial equalities and inequalities. The closed-loop properties can then be
analyzed using sum-of-squares (SOS) programming
Spectral Analysis for Matrix Hamiltonian Operators
In this work, we study the spectral properties of matrix Hamiltonians
generated by linearizing the nonlinear Schr\"odinger equation about soliton
solutions. By a numerically assisted proof, we show that there are no embedded
eigenvalues for the three dimensional cubic equation. Though we focus on a
proof of the 3d cubic problem, this work presents a new algorithm for verifying
certain spectral properties needed to study soliton stability. Source code for
verification of our comptuations, and for further experimentation, are
available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.Comment: 57 pages, 22 figures, typos fixe
Approximations of pseudo-differential flows
Given a classical symbol of order zero, and associated semiclassical
operators we prove that the flow of is well approximated, in time by a
pseudo-differential operator, the symbol of which is the flow of
the symbol A similar result holds for non-autonomous equations, associated
with time-dependent families of symbols This result was already used,
by the author and co-authors, to give a stability criterion for high-frequency
WKB approximations, and to prove a strong Lax-Mizohata theorem. We give here
two further applications: sharp semigroup bounds, implying nonlinear
instability under the assumption of spectral instability at the symbolic level,
and a new proof of sharp G\r{a}rding inequalities.Comment: Final version, to appear in Indiana Univ. Math.
Control Strategies for the Fokker-Planck Equation
Using a projection-based decoupling of the Fokker-Planck equation, control
strategies that allow to speed up the convergence to the stationary
distribution are investigated. By means of an operator theoretic framework for
a bilinear control system, two different feedback control laws are proposed.
Projected Riccati and Lyapunov equations are derived and properties of the
associated solutions are given. The well-posedness of the closed loop systems
is shown and local and global stabilization results, respectively, are
obtained. An essential tool in the construction of the controls is the choice
of appropriate control shape functions. Results for a two dimensional double
well potential illustrate the theoretical findings in a numerical setup
A stability criterion for high-frequency oscillations
We show that a simple Levi compatibility condition determines stability of
WKB solutions to semilinear hyperbolic initial-value problems issued from
highly-oscillating initial data with large amplitudes. The compatibility
condition involves the hyperbolic operator, the fundamental phase associated
with the initial oscillation, and the semilinear source term; it states roughly
that hyperbolicity is preserved around resonances.
If the compatibility condition is satisfied, the solutions are defined over
time intervals independent of the wavelength, and the associated WKB solutions
are stable under a large class of initial perturbations. If the compatibility
condition is not satisfied, resonances are exponentially amplified, and
arbitrarily small initial perturbations can destabilize the WKB solutions in
small time.
The amplification mechanism is based on the observation that in frequency
space, resonances correspond to points of weak hyperbolicity. At such points,
the behavior of the system depends on the lower order terms through the
compatibility condition.
The analysis relies, in the unstable case, on a short-time Duhamel
representation formula for solutions of zeroth-order pseudo-differential
equations.
Our examples include coupled Klein-Gordon systems, and systems describing
Raman and Brillouin instabilities.Comment: Final version, to appear in M\'em. Soc. Math. F
Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr\"odinger equation
We establish that the quadratic non-linear Schr\"odinger equation where u: \R \times \R \to \C, is locally well-posed in
when and ill-posed when . Previous work of Kenig,
Ponce and Vega had established local well-posedness for . The local
well-posedness is achieved by an iteration using a modification of the standard
spaces. The ill-posedness uses an abstract and general argument
relying on the high-to-low frequency cascade present in the non-linearity, and
a computation of the first non-linear iterate.Comment: 28 pages, no figures, to appear, J.Func. Anal. Some minor gaps filled
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