1,636 research outputs found
Galerkin approximations for the optimal control of nonlinear delay differential equations
Optimal control problems of nonlinear delay differential equations (DDEs) are
considered for which we propose a general Galerkin approximation scheme built
from Koornwinder polynomials. Error estimates for the resulting
Galerkin-Koornwinder approximations to the optimal control and the value
function, are derived for a broad class of cost functionals and nonlinear DDEs.
The approach is illustrated on a delayed logistic equation set not far away
from its Hopf bifurcation point in the parameter space. In this case, we show
that low-dimensional controls for a standard quadratic cost functional can be
efficiently computed from Galerkin-Koornwinder approximations to reduce at a
nearly optimal cost the oscillation amplitude displayed by the DDE's solution.
Optimal controls computed from the Pontryagin's maximum principle (PMP) and the
Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE
systems, are shown to provide numerical solutions in good agreement. It is
finally argued that the value function computed from the corresponding reduced
HJB equation provides a good approximation of that obtained from the full HJB
equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042
Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces
The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems
Kinetics and Mechanism of Metal Nanoparticle Growth via Optical Extinction Spectroscopy and Computational Modeling: The Curious Case of Colloidal Gold
An overarching computational framework unifying several optical theories to
describe the temporal evolution of gold nanoparticles (GNPs) during a seeded
growth process is presented. To achieve this, we used the inexpensive and
widely available optical extinction spectroscopy, to obtain quantitative
kinetic data. In situ spectra collected over a wide set of experimental
conditions were regressed using the physical model, calculating light
extinction by ensembles of GNPs during the growth process. This model provides
temporal information on the size, shape, and concentration of the particles and
any electromagnetic interactions between them. Consequently, we were able to
describe the mechanism of GNP growth and divide the process into distinct
genesis periods. We provide explanations for several longstanding mysteries,
for example, the phenomena responsible for the purple-greyish hue during the
early stages of GNP growth, the complex interactions between nucleation,
growth, and aggregation events, and a clear distinction between agglomeration
and electromagnetic interactions. The presented theoretical formalism has been
developed in a generic fashion so that it can readily be adapted to other
nanoparticulate formation scenarios such as the genesis of various metal
nanoparticles.Comment: Main text and supplementary information (accompanying MATLAB codes
available on the journal webpage
Tunneling decay in a magnetic field
We provide a semiclassical theory of tunneling decay in a magnetic field and
a three-dimensional potential of a general form. Because of broken
time-reversal symmetry, the standard WKB technique has to be modified. The
decay rate is found from the analysis of the set of the particle Hamiltonian
trajectories in complex phase space and time. In a magnetic field, the
tunneling particle comes out from the barrier with a finite velocity and behind
the boundary of the classically allowed region. The exit location is obtained
by matching the decaying and outgoing WKB waves at a caustic in complex
configuration space. Different branches of the WKB wave function match on the
switching surface in real space, where the slope of the wave function sharply
changes. The theory is not limited to tunneling from potential wells which are
parabolic near the minimum. For parabolic wells, we provide a bounce-type
formulation in a magnetic field. The theory is applied to specific models which
are relevant to tunneling from correlated two-dimensional electron systems in a
magnetic field parallel to the electron layer.Comment: 16 pages, 11 figure
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