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Optimal Control of Partial Integro-Differential Equations and Analysis of the Gaussian Kernel

  • An important field of applied mathematics is the simulation of complex financial, mechanical, chemical, physical or medical processes with mathematical models. In addition to the pure modeling of the processes, the simultaneous optimization of an objective function by changing the model parameters is often the actual goal. Models in fields such as finance, biology or medicine benefit from this optimization step. While many processes can be modeled using an ordinary differential equation (ODE), partial differential equations (PDEs) are needed to optimize heat conduction and flow characteristics, spreading of tumor cells in tissue as well as option prices. A partial integro-differential equation (PIDE) is a parital differential equation involving an integral operator, e.g., the convolution of the unknown function with a given kernel function. PIDEs occur for example in models that simulate adhesive forces between cells or option prices with jumps. In each of the two parts of this thesis, a certain PIDE is the main object of interest. In the first part, we study a semilinear PIDE-constrained optimal control problem with the aim to derive necessary optimality conditions. In the second, we analyze a linear PIDE that includes the convolution of the unknown function with the Gaussian kernel.

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Author:Lukas Aaron Zimmer
Referee:Ekkehard Sachs, Leonhard Frerick
Advisor:Ekkehard Sachs
Document Type:Doctoral Thesis
Date of completion:2018/09/10
Publishing institution:Universität Trier
Granting institution:Universität Trier, Fachbereich 4
Date of final exam:2018/08/27
Release Date:2018/11/27
GND Keyword:Integrodifferentialgleichung; Mathematik
Institutes:Fachbereich 4 / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoCC BY-NC-ND: Creative-Commons-Lizenz: Namensnennung, Nicht kommerziell, Keine Bearbeitungen 4.0 International

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