09:50
Hybrid optimal control problems for partial differential equations
Dr. Sebastien Court | Karl-Franzens-Universität Graz | Austria
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Authors:
Dr. Sebastien Court | Karl-Franzens-Universität Graz | Austria
Prof. Dr. Karl Kunisch | Karl-Franzens-Universität Graz & RICAM | Austria
Dr. Laurent Pfeiffer | Karl-Franzens-Universität Graz | Austria
We provide optimality conditions and numerical results for hybrid optimal control problems governed by partial differential equations. These problems involve parameters to be determined, such as switching times to be optimized, at which the dynamics, the integral cost, and the bound constraints on the control can vary. The developed method to solve these problems is based on first and second-order optimality conditions. It is applied to different type of partial differential equations, such as semilinear parabolic equations, coupled systems or hyperbolic conservation laws. Numerical illustrations are presented, whose qualitative aspects underlie the original motivations. The latter ranges from the maximization of a population density at a time chosen to be optimal, to the creation, in a basin, of a wave whose the location is let free.
10:10
Boundary feedback stabilization of the isothermal Euler-equations
Prof. Dr. Martin Gugat | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) | Germany
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Author:
Prof. Dr. Martin Gugat | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) | Germany
The isothermal Euler-equations are a model of gas flow through pipelines.
This is a hyperbolic system of pdes, so in general the regularity of the solution can
decrease in finite time. We show that with sufficiently small initial data and a suitable boundary feedback law,
this does not happen, so the solution keeps the regularity of the initial data. In fact, the feedback law
stabilizes the system state exponentially fast to a stationary state.
The regularity of the solution is determined by the velocity of the gas, that is the solution of the quasilinear wave equation
\[
v_{tt} = (a^2 - v^2) \,v_{xx} - 2 \, v \, v_{tx} - 2 \, v_t \, v_x - 2 \, v \, (v_x)^2
-\theta \, |v| \left( v_t + \frac{3}{2} \, v\, v_x\right)
.
\]
For the corresponding closed--loop system on a finite space interval with Neumann-velocity feedback
at one end and Dirichlet boundary conditions at the other end, we consider transient
solutions locally around a given stationary state.
We introduce a strict $H^2$-Lyapunov function and show that if
\begin{enumerate}
\item the $C^2$--norm of the stationary state is sufficiently small,
\item the boundary feedback constant is chosen
sufficiently large and
\item the initial state is sufficiently small in $H^2$
\end{enumerate}
the Lyapunov function and also the $H^2$-norm of the difference between the
transient system state and the stationary state decay exponentially with time.
In this way the system state keeps its $H^2$ regularity
for all times.
10:30
All-at-once a posteriori error estimates for inverse problems for PDEs
M. Sc. Mario Luiz Previatti de Souza | Alpen-Adria Universität Klagenfurt | Austria
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Authors:
M. Sc. Mario Luiz Previatti de Souza | Alpen-Adria Universität Klagenfurt | Austria
Prof. Barbara Kaltenbacher | Alpen-Adria Universität Klagenfurt | Austria
This talk deals with a combined analysis of regularization and discretization of inverse problems in Banach spaces, specifically in the context of partial differential equations (PDEs). The relevant quantities - parameters and states - have to be discretized, e.g., by the finite element method, and the error due to this discretization has to be appropriately estimated and controlled by error estimators and mesh refinement. Thus one of the main challenges is here to take into account the interplay between mesh size, regularization parameter and data noise level. The focus on the PDEs setting is relevant to the adaptive discretization of the regularized problems. Hence, I will present a posteriori functional error estimates adapted to non smooth inverse problems in Banach spaces for Tikhonov minimization problems with different types of regularization functionals in a
all-at-once formulation, where we deal with it adding penalization terms. Such problems play a crucial role in numerous applications, ranging from medical imaging via nondestructive testing (e.g. electrical impedancy tomography) to geophysical prospecting (e.g. inverse water ground filtration), with the Banach space setting assigned by the inherent regularity of the sought coefficients as well as structural features such as sparsity. Since those problems are usually nonlinear, a natural first step to the treatment of nonlinear problems is to con-
sider successive local linearizations using previous estimates. In particular, we focus on Iteratively Regularized Gauss Newton type Methods.