951 research outputs found
Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator
We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state. We give necessary and sufficient condition to have an exponential rate of convergence. Numerical simulations are presented to illustratethe theoretical results
Quantum Control Landscapes
Numerous lines of experimental, numerical and analytical evidence indicate
that it is surprisingly easy to locate optimal controls steering quantum
dynamical systems to desired objectives. This has enabled the control of
complex quantum systems despite the expense of solving the Schrodinger equation
in simulations and the complicating effects of environmental decoherence in the
laboratory. Recent work indicates that this simplicity originates in universal
properties of the solution sets to quantum control problems that are
fundamentally different from their classical counterparts. Here, we review
studies that aim to systematically characterize these properties, enabling the
classification of quantum control mechanisms and the design of globally
efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry,
Vol. 26, Iss. 4, pp. 671-735 (2007
On the exact boundary controllability of semilinear wave equations
We address the exact boundary controllability of the semilinear wave equation
posed over a bounded domain of
. Assuming that is continuous and satisfies the condition
for some small enough and some , we
apply the Schauder fixed point theorem to prove the uniform controllability for
initial data in . Then, assuming that is
in and satisfies the condition , we apply
the Banach fixed point theorem and exhibit a strongly convergent sequence to a
state-control pair for the semilinear equation
Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations
A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani et al. (Automatica 46(10), 1616-1625, 2010 ). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed)
Optimal trajectory tracking
This thesis investigates optimal trajectory tracking of nonlinear dynamical
systems with affine controls. The control task is to enforce the system state
to follow a prescribed desired trajectory as closely as possible. The concept
of so-called exactly realizable trajectories is proposed. For exactly
realizable desired trajectories exists a control signal which enforces the
state to exactly follow the desired trajectory. For a given affine control
system, these trajectories are characterized by the so-called constraint
equation. This approach does not only yield an explicit expression for the
control signal in terms of the desired trajectory, but also identifies a
particularly simple class of nonlinear control systems. Based on that insight,
the regularization parameter is used as the small parameter for a perturbation
expansion. This results in a reinterpretation of affine optimal control
problems with small regularization term as singularly perturbed differential
equations. The small parameter originates from the formulation of the control
problem and does not involve simplifying assumptions about the system dynamics.
Combining this approach with the linearizing assumption, approximate and partly
linear equations for the optimal trajectory tracking of arbitrary desired
trajectories are derived. For vanishing regularization parameter, the state
trajectory becomes discontinuous and the control signal diverges. On the other
hand, the analytical treatment becomes exact and the solutions are exclusively
governed by linear differential equations. Thus, the possibility of linear
structures underlying nonlinear optimal control is revealed. This fact enables
the derivation of exact analytical solutions to an entire class of nonlinear
trajectory tracking problems with affine controls. This class comprises
mechanical control systems in one spatial dimension and the FitzHugh-Nagumo
model.Comment: 240 pages, 36 figures, PhD thesi
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