17 research outputs found
An indirect numerical method for a time-optimal state-constrained control problem in a steady two-dimensional fluid flow
This article concerns the problem of computing solutions to state-constrained
optimal control problems whose trajectory is affected by a flow field. This
general mathematical framework is particularly pertinent to the requirements
underlying the control of Autonomous Underwater Vehicles in realistic scenarii.
The key contribution consists in devising a computational indirect method which
becomes effective in the numerical computation of extremals to optimal control
problems with state constraints by using the maximum principle in Gamkrelidze's
form in which the measure Lagrange multiplier is ensured to be continuous. The
specific problem of time-optimal control of an Autonomous Underwater Vehicle in
a bounded space set, subject to the effect of a flow field and with bounded
actuation, is used to illustrate the proposed approach. The corresponding
numerical results are presented and discussed
Regular path-constrained time-optimal control problems in three-dimensional flow fields
This article concerns a class of time-optimal state constrained control
problems with dynamics defined by an ordinary differential equation involving a
three-dimensional steady flow vector field. The problem is solved via an
indirect method based on the maximum principle in Gamkrelidze's form. The
proposed computational method essentially uses a certain regularity condition
imposed on the data of the problem. The property of regularity guarantees the
continuity of the measure multiplier associated with the state constraint, and
ensures the appropriate behavior of the corresponding numerical procedure
which, in general, consists in computing the entire field of extremals for the
problem in question. Several examples of vector fields are considered to
illustrate the computational approach.Comment: 23 page
Instabilities of Higher-Order Parametric Solitons. Filamentation versus Coalescence
We investigate stability and dynamics of higher-order solitary waves in
quadratic media, which have a central peak and one or more surrounding rings.
We show existence of two qualitatively different behaviours. For positive phase
mismatch the rings break up into filaments which move radially to initial ring.
For sufficient negative mismatches rings are found to coalesce with central
peak, forming a single oscillating filament.Comment: 5 pages, 7 figure
Modulational instability of solitary waves in non-degenerate three-wave mixing: The role of phase symmetries
We show how the analytical approach of Zakharov and Rubenchik [Sov. Phys.
JETP {\bf 38}, 494 (1974)] to modulational instability (MI) of solitary waves
in the nonlinear Schr\"oedinger equation (NLS) can be generalised for models
with two phase symmetries. MI of three-wave parametric spatial solitons due to
group velocity dispersion (GVD) is investigated as a typical example of such
models. We reveal a new branch of neck instability, which dominates the usual
snake type MI found for normal GVD. The resultant nonlinear evolution is
thereby qualitatively different from cases with only a single phase symmetry.Comment: 4 pages with figure
Pulsed quadrature-phase squeezing of solitary waves in chi((2)) parametric waveguides
It is shown that coherent quantum simultons (simultaneous solitary waves at two different frequencies) can undergo quadrature-phase squeezing as they propagate through a dispersive chi((2)) waveguide. This requires a treatment of the coupled quantized fields including a quantized depleted pump field. A technique involving nonlinear stochastic parabolic partial differential equations using a nondiagonal coherent state representation in combination with an exact Wigner representation on a reduced phase space is outlined. We explicitly demonstrate that group-velocity matched chi((2)) waveguides which exhibit collinear propagation can produce quadrature-phase squeezed simultons. Quasi-phase-matched KTP waveguides, even with their large group-velocity mismatch between fundamental and second harmonic at 425 nm, can produce 3 dB squeezed bright pulses at 850 nm in the large phase-mismatch regime. This can be improved to more than 6 dB by using group-velocity matched waveguides