Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels

Fractional optimal control problems via a wide class of fractional operators with a general analytic kernel are introduced. Necessary optimality conditions of Pontryagin type for the considered problem are obtained after proving a Gronwall type inequality as well as results on continuity and differentiability of perturbed trajectories. Moreover, a Mangasarian type sufficient global optimality condition for the general analytic kernel fractional optimal control problem is proved. An illustrative example is discussed.

Assumptions on the Schwartz Function

This chapter introduces two classes of degenerate Schwartz functions which significantly simplify the computations and arguments of both the analytic kernel and the geometric kernel functions. It first restates the kernel identity in terms of un-normalized kernel functions before stating the assumptions on the Schwartz function and claiming that these assumptions can be “added” to the kernel identity without losing the generality. It then considers some simple properties of the assumptions and proceeds by discussing the two classes of degenerate Schwartz functions. In the first case, a non-archimedean local field and a non-degenerate quadratic space are described. In the second case, since all the data are unramified, the lemma can be verified by explicit computations.

Mordell-Weil Groups and Generating Series

This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before introducing a theorem, which is an identity between the analytic kernel and the geometric kernel. It then defines the generating series and uses it to describe the geometric kernel. It also presents a theorem, which is an identity formulated in terms of projectors, and reviews some basic notations and results on Shimura curves. Other topics covered include the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves, normalization of the geometric kernel, and the analytic kernel function. The chapter concludes with an analysis of the kernel identity implied in the first theorem.

Derivative of the Analytic Kernel

This chapter computes the derivative of the analytic kernel. It first decomposes the kernel function into a sum of infinitely many local terms indexed by places v of Fnonsplit in E. Each local term is a period integral of some kernel function. The chapter then considers the v-part for non-archimedean v. An explicit formula is given in the unramified case, and an approximation is presented in the ramified case assuming the Schwartz function is degenerate. An explicit result of the v-part for archimedean v is also introduced. The chapter proceeds by reviewing a general formula of holomorphic projection, and estimates the growth of the kernel function in order to apply the formula. It also computes the holomorphic projection of the analytic kernel function and concludes with a discussion of the holomorphic kernel function.