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T. Levajković:
"Polynomial Chaos Expansion Approach for Stochastic Partial Differential Equations with Applications";
Leopold‐Franzens‐Universität Innsbruck, 2020.

In this thesis we study some types of stochastic partial differential equations (SPDEs) in the framework of white noise analysis and thier particular applications in optimal control. The thesis is divided in two parts: theore- tical results and applications. In the first part we developed the theore- tical framework for studying different classes of SPDEs with singular data. Particularly, we developed generalized Malliavin calculus on spaces of gene- ralized stochastic functions based on the chaos expansions. We solved different classes of stochastic evolution equations using the chaos expansion method and generalized some of these results to the related optimal control problem.
The second part of the thesis is devoted to applications. We study infinite dimensional stochastic linear quadratic optimal control problems related to evolution equations discussed in the previous chapter. We proved an optimal feedback synthesis along with well-posedness of the Riccati equation in a general setting. We provided a novel numerical framework for solving this type of control problems using the method of chaos expansions. We also presented an approximation framework for computing the solution of the stochastic linear quadratic control problem on Hilbert spaces. For the finite horizon case, we proved convergence results of the finite-dimensional problem to the infinite-dimensional one. In addition, we developed a stochastic treatment of unbounded control action problems arising in a general class of dynamical systems which exhibit singular estimates, but are not necessarily analytic. Moreover, in the same setting we present a regularization scheme for operator differential algebraic equations with noise disturbances. Finally, we combined a polynomial chaos expansion method with splitting methods for solving particular classes of SPDEs.

https://publik.tuwien.ac.at/files/publik_291480.pdf