621 research outputs found
Asymptotic Stability of POD based Model Predictive Control for a semilinear parabolic PDE
In this article a stabilizing feedback control is computed for a semilinear
parabolic partial differential equation utilizing a nonlinear model predictive
(NMPC) method. In each level of the NMPC algorithm the finite time horizon open
loop problem is solved by a reduced-order strategy based on proper orthogonal
decomposition (POD). A stability analysis is derived for the combined POD-NMPC
algorithm so that the lengths of the finite time horizons are chosen in order
to ensure the asymptotic stability of the computed feedback controls. The
proposed method is successfully tested by numerical examples
A HJB-POD approach for the control of nonlinear PDEs on a tree structure
The Dynamic Programming approach allows to compute a feedback control for
nonlinear problems, but suffers from the curse of dimensionality. The
computation of the control relies on the resolution of a nonlinear PDE, the
Hamilton-Jacobi-Bellman equation, with the same dimension of the original
problem. Recently, a new numerical method to compute the value function on a
tree structure has been introduced. The method allows to work without a
structured grid and avoids any interpolation.
Here, we aim to test the algorithm for nonlinear two dimensional PDEs. We
apply model order reduction to decrease the computational complexity since the
tree structure algorithm requires to solve many PDEs. Furthermore, we prove an
error estimate which guarantees the convergence of the proposed method.
Finally, we show efficiency of the method through numerical tests
Nonlinear model order reduction via Dynamic Mode Decomposition
We propose a new technique for obtaining reduced order models for nonlinear
dynamical systems. Specifically, we advocate the use of the recently developed
Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the
nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix
that correlates spatial features while simultaneously associating the activity
with periodic temporal behavior. With this decomposition, one can obtain a
fully reduced dimensional surrogate model and avoid the evaluation of the
nonlinear term in the online stage. This allows for an impressive speed up of
the computational cost, and, at the same time, accurate approximations of the
problem. We present a suite of numerical tests to illustrate our approach and
to show the effectiveness of the method in comparison to existing approaches
An Efficient Policy Iteration Algorithm for Dynamic Programming Equations
We present an accelerated algorithm for the solution of static
Hamilton-Jacobi-Bellman equations related to optimal control problems. Our
scheme is based on a classic policy iteration procedure, which is known to have
superlinear convergence in many relevant cases provided the initial guess is
sufficiently close to the solution. In many cases, this limitation degenerates
into a behavior similar to a value iteration method, with an increased
computation time. The new scheme circumvents this problem by combining the
advantages of both algorithms with an efficient coupling. The method starts
with a value iteration phase and then switches to a policy iteration procedure
when a certain error threshold is reached. A delicate point is to determine
this threshold in order to avoid cumbersome computation with the value
iteration and, at the same time, to be reasonably sure that the policy
iteration method will finally converge to the optimal solution. We analyze the
methods and efficient coupling in a number of examples in dimension two, three
and four illustrating its properties
A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations
In this paper we study the approximation of a distributed optimal control
problem for linear para\-bolic PDEs with model order reduction based on Proper
Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the
basis functions are obtained upon information contained in time snapshots of
the parabolic PDE related to given input data. In the present work we show that
for POD-MOR in optimal control of parabolic equations it is important to have
knowledge about the controlled system at the right time instances. For the
determination of the time instances (snapshot locations) we propose an
a-posteriori error control concept which is based on a reformulation of the
optimality system of the underlying optimal control problem as a second order
in time and fourth order in space elliptic system which is approximated by a
space-time finite element method. Finally, we present numerical tests to
illustrate our approach and to show the effectiveness of the method in
comparison to existing approaches
Error estimates for a tree structure algorithm solving finite horizon control problems
In the Dynamic Programming approach to optimal control problems a crucial
role is played by the value function that is characterized as the unique
viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well
known that this approach suffers of the "curse of dimensionality" and this
limitation has reduced its practical in real world applications. Here we
analyze a dynamic programming algorithm based on a tree structure. The tree is
built by the time discrete dynamics avoiding in this way the use of a fixed
space grid which is the bottleneck for high-dimensional problems, this also
drops the projection on the grid in the approximation of the value function. We
present some error estimates for a first order approximation based on the
tree-structure algorithm. Moreover, we analyze a pruning technique for the tree
to reduce the complexity and minimize the computational effort. Finally, we
present some numerical tests
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