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Optimal Control of Partial Integro-Differential Equations and Analysis of the Gaussian Kernel
(2018)
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.
Given a compact set K in R^d, the theory of extension operators examines the question, under which conditions on K, the linear and continuous restriction operators r_n:E^n(R^d)→E^n(K),f↦(∂^α f|_K)_{|α|≤n}, n in N_0 and r:E(R^d)→E(K),f↦(∂^α f|_K)_{α in N_0^d}, have a linear and continuous right inverse. This inverse is called extension operator and this problem is known as Whitney's extension problem, named after Hassler Whitney. In this context, E^n(K) respectively E(K) denote spaces of Whitney jets of order n respectively of infinite order. With E^n(R^d) and E(R^d), we denote the spaces of n-times respectively infinitely often continuously partially differentiable functions on R^d. Whitney already solved the question for finite order completely. He showed that it is always possible to construct a linear and continuous right inverse E_n for r_n. This work is concerned with the question of how the existence of a linear and continuous right inverse of r, fulfilling certain continuity estimates, can be characterized by properties of K. On E(K), we introduce a full real scale of generalized Whitney seminorms (|·|_{s,K})_{s≥0}, where |·|_{s,K} coincides with the classical Whitney seminorms for s in N_0. We equip also E(R^d) with a family (|·|_{s,L})_{s≥0} of those seminorms, where L shall be a a compact set with K in L-°. This family of seminorms on E(R^d) suffices to characterize the continuity properties of an extension operator E, since we can without loss of generality assume that E(E(K)) in D^s(L).
In Chapter 2, we introduce basic concepts and summarize the classical results of Whitney and Stein.
In Chapter 3, we modify the classical construction of Whitney's operators E_n and show that |E_n(·)|_{s,L}≤C|·|_{s,K} for s in[n,n+1).
In Chapter 4, we generalize a result of Frerick, Jordá and Wengenroth and show that LMI(1) for K implies the existence of an extension operator E without loss of derivatives, i.e. we have it fulfils |E(·)|_{s,L}≤C|·|_{s,K} for all s≥0. We show that a large class of self similar sets, which includes the Cantor set and the Sierpinski triangle, admits an extensions operator without loss of derivatives.
In Chapter 5 we generalize a result of Frerick, Jordá and Wengenroth and show that WLMI(r) for r≥1 implies the existence of a tame linear extension operator E having a homogeneous loss of derivatives, such that |E(·)|_{s,L}≤C|·|_{(r+ε)s,K} for all s≥0 and all ε>0.
In the last chapter we characterize the existence of an extension operator having an arbitrary loss of derivatives by the existence of measures on K.