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Technische Universität Darmstadt

1. Ludovici, Francesco. Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative.

Degree: PhD, 04 Department of MathematicsNumerical Analysis and Scientific ComputingOptimizationNonlinear Optimization, 2017, Technische Universität Darmstadt

URL: http://tuprints.ulb.tu-darmstadt.de/6781/

The aim of this thesis is the numerical analysis of optimal control problems governed by parabolic PDEs and subject
to constraints on the state variable and its first derivative. The
control is acting distributed in time only while the state constraints are considered point-wise in time and global in space;
this setting generates an optimization problem of semi-infinite type.
The consideration of a space-time discretization of the problem requires the analysis of the convergence of the discretized
solution toward the continuous one, as temporal and space mesh size tend to zero. This is based, at any level of discretization,
on a priori error
estimates for the solution of the parabolic differential equation which are obtained within this thesis.
One of the main challenge for state-constrained problem consists in the presence of a Lagrange multiplier appearing as a Borel
measure in the system of first-order optimality conditions. In particular, such measure enters the optimality system as data in
the adjoint equation affecting the regularity of the adjoint variable itself. Therefore, in the derivation of the convergence
rates the use of adjoint information has to be avoided.
When considering non-convex problems, the presence of local solutions and the need for second-order optimality
conditions require a different strategy compared to the convex case, making the analysis more involved. In particular, the
convergence of the discretized solution toward the continuous one is based on a so-called quadratic growth-condition, which arises
from the second-order optimality conditions.
The a priori error estimates for the PDEs are verified numerically.
*Advisors/Committee Members: Wollner, Winnifried (advisor), Vexler, Boris (advisor).*

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APA (6^{th} Edition):

Ludovici, F. (2017). Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative. (Doctoral Dissertation). Technische Universität Darmstadt. Retrieved from http://tuprints.ulb.tu-darmstadt.de/6781/

Chicago Manual of Style (16^{th} Edition):

Ludovici, Francesco. “Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative.” 2017. Doctoral Dissertation, Technische Universität Darmstadt. Accessed September 22, 2017. http://tuprints.ulb.tu-darmstadt.de/6781/.

MLA Handbook (7^{th} Edition):

Ludovici, Francesco. “Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative.” 2017. Web. 22 Sep 2017.

Vancouver:

Ludovici F. Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative. [Internet] [Doctoral dissertation]. Technische Universität Darmstadt; 2017. [cited 2017 Sep 22]. Available from: http://tuprints.ulb.tu-darmstadt.de/6781/.

Council of Science Editors:

Ludovici F. Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative. [Doctoral Dissertation]. Technische Universität Darmstadt; 2017. Available from: http://tuprints.ulb.tu-darmstadt.de/6781/