16,926 research outputs found
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
On Structured Realizability and Stabilizability of Linear Systems
We study the notion of structured realizability for linear systems defined
over graphs. A stabilizable and detectable realization is structured if the
state-space matrices inherit the sparsity pattern of the adjacency matrix of
the associated graph. In this paper, we demonstrate that not every structured
transfer matrix has a structured realization and we reveal the practical
meaning of this fact. We also uncover a close connection between the structured
realizability of a plant and whether the plant can be stabilized by a
structured controller. In particular, we show that a structured stabilizing
controller can only exist when the plant admits a structured realization.
Finally, we give a parameterization of all structured stabilizing controllers
and show that they always have structured realizations
System Level Synthesis
This article surveys the System Level Synthesis framework, which presents a
novel perspective on constrained robust and optimal controller synthesis for
linear systems. We show how SLS shifts the controller synthesis task from the
design of a controller to the design of the entire closed loop system, and
highlight the benefits of this approach in terms of scalability and
transparency. We emphasize two particular applications of SLS, namely
large-scale distributed optimal control and robust control. In the case of
distributed control, we show how SLS allows for localized controllers to be
computed, extending robust and optimal control methods to large-scale systems
under practical and realistic assumptions. In the case of robust control, we
show how SLS allows for novel design methodologies that, for the first time,
quantify the degradation in performance of a robust controller due to model
uncertainty -- such transparency is key in allowing robust control methods to
interact, in a principled way, with modern techniques from machine learning and
statistical inference. Throughout, we emphasize practical and efficient
computational solutions, and demonstrate our methods on easy to understand case
studies.Comment: To appear in Annual Reviews in Contro
Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems
The central objective of this paper is to develop reduced basis methods for
parameter dependent transport dominated problems that are rigorously proven to
exhibit rate-optimal performance when compared with the Kolmogorov -widths
of the solution sets. The central ingredient is the construction of
computationally feasible "tight" surrogates which in turn are based on deriving
a suitable well-conditioned variational formulation for the parameter dependent
problem. The theoretical results are illustrated by numerical experiments for
convection-diffusion and pure transport equations. In particular, the latter
example sheds some light on the smoothness of the dependence of the solutions
on the parameters
Optimal food price stabilization in a small open developing country
In poor countries, most governments implement policies aiming to stabilize the prices of staple foods, which often include storage and trade measures insulating their domestic market from the world market. It is of crucial importance to understand the precise motivations and efficiency of those interventions, because they can have consequences worldwide. This paper addresses those issues by analyzing the case of a small, open developing country confronted by shocks to both the crop yield and foreign price. In this model, government interventions may be justified by the lack of an insurance market for food prices. Considering this market imperfection, the authors design optimal public interventions through trade and storage policies. They show that an optimal trade policy largely consists of subsidizing imports and taxing exports, which benefits consumers at the expense of producers. Import subsidies alleviate the non-negativity of food storage. In other words, when stocks are exhausted, subsidizing imports prevents domestic price spikes. One striking result: an optimal storage policy on its own is detrimental to consumers, since its stabilizing benefits leak into the world market and it raises the average domestic price. By contrast, an optimal combination of storage and trade policies results in a powerful stabilizing effect for domestic food prices.Markets and Market Access,Economic Theory&Research,Emerging Markets,Access to Markets,Trade Policy
An Exponential Quantum Projection Filter for Open Quantum Systems
An approximate exponential quantum projection filtering scheme is developed
for a class of open quantum systems described by Hudson- Parthasarathy quantum
stochastic differential equations, aiming to reduce the computational burden
associated with online calculation of the quantum filter. By using a
differential geometric approach, the quantum trajectory is constrained in a
finite-dimensional differentiable manifold consisting of an unnormalized
exponential family of quantum density operators, and an exponential quantum
projection filter is then formulated as a number of stochastic differential
equations satisfied by the finite-dimensional coordinate system of this
manifold. A convenient design of the differentiable manifold is also presented
through reduction of the local approximation errors, which yields a
simplification of the quantum projection filter equations. It is shown that the
computational cost can be significantly reduced by using the quantum projection
filter instead of the quantum filter. It is also shown that when the quantum
projection filtering approach is applied to a class of open quantum systems
that asymptotically converge to a pure state, the input-to-state stability of
the corresponding exponential quantum projection filter can be established.
Simulation results from an atomic ensemble system example are provided to
illustrate the performance of the projection filtering scheme. It is expected
that the proposed approach can be used in developing more efficient quantum
control methods
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