212 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
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
Model reduction methods for bilinear control systems are compared by means of
practical examples of Liouville-von Neumann and Fokker--Planck type. Methods
based on balancing generalized system Gramians and on minimizing an H2-type
cost functional are considered. The focus is on the numerical implementation
and a thorough comparison of the methods. Structure and stability preservation
are investigated, and the competitiveness of the approaches is shown for
practically relevant, large-scale examples
H2 optimal model reduction on general domains
Optimal model reduction for large-scale linear dynamical systems is studied.
In contrast to most existing works, the systems under consideration are not
required to be stable, neither in discrete nor in continuous time. As a
consequence, the underlying rational transfer functions are allowed to have
poles in general domains in the complex plane. In particular, this covers the
case of specific conservative partial differential equations such as the linear
Schr\"odinger and the undamped linear wave equation with spectra on the
imaginary axis.
By an appropriate modification of the classical continuous time Hardy space
, a new like optimal model reduction problem is
introduced and first order optimality conditions are derived. As in the
classical case, these conditions exhibit a rational Hermite
interpolation structure for which an iterative model reduction algorithm is
proposed. Numerical examples demonstrate the effectiveness of the new method
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