Applications of generalized smooth functions

Wider research context/Theoretical framework: The main aim of the proposed project is to develop several applications in the framework of generalized smooth functions (GSF; Colombeau's theory). The nonlinear theory of GSF has recently emerged as a minimal extension of Colombeau's theory that allows for more general domains for generalized functions, resulting in the closure with respect to composition and a better behavior on unbounded sets. Moreover, new general existence results was recently proved, such as the Banach fixed point theorem and a corresponding Picard-Lindelöf theorem for partial differential equations.

Research objectives: In this research proposal, we will consider the following theoretical developments with related applications. In the first part of the project, we want to extend Newton's method and Pontryagin's principle using Banach fixed point theorem on complete spaces of GSF defined on functionally compact sets and their generalized valued norms. The second work package plans the development of singular Hamiltonian mechanics and of the corresponding symplectic geometry using closure properties of sheaves of GSF. We finally plan to develop several ways to implement and visually represent GSF: from the classical embedding of Schwartz's distributions to numerical solutions of singular nonlinear differential equations having generalized smooth solutions.

Approach/Methods: The proposal thus aims at showing the flexibility of generalized smooth function theory in these classical applications, and at widely extending well known results in different fields of mathematical analysis. The basic methods rely on the applycations of results already proved for GSF, such as the Banach fixed point theorem or closure properties of the corresponding topos. It also fits well into the research interests of the international community of Colombeau generalized functions, where the interest for numerical implementation, applications of Colombeau's theory and its diffusion was clearly perceived in several conferences.

Innovation: Innovative aspects of the proposal are: a broad development of control theory with applications of the Pontryagin's principle to singular problems; a general solution of the extension of classical mechanics to singular models which is well grounded on generalized differential geometry; a flexible and easy-to-use Matlab toolbox to visualize and enhance intuition of students and researchers in the use and properties of GSF. Potential users can be hence foreseen both in theoretical and applied mathematics, in physics and engineering applications.

Primary researchers involved: The applicant and Prof. M. Kunzinger are the senior researchers working on this project and, for each one of the three work plans, we already explicitly indicate one PhD candidate having optimal mathematical competences for the development of applications to physics and numerical calculus. The employment of another PhD candidate is planned.