Applications of generalized smooth functions

The main aim of the proposed project is to develop several applications in the framework of generalized smooth functions (GSF). These functions are useful to describe rapidly changing phenomena such as breaking of steel structures, switching of electrical circuits, motion through different or granular media, quantum mechanics, etc. In this setting, new general existence results for differential equations was recently proved. In this research proposal, we will consider the following theoretical developments with related applications: 1. Solution of equations using Newton`s method and Pontryagin`s principle for optimal control. 2. Development of mechanics with GSF. 3. Visual representation of GSF by programming a Matlabs toolbox. Innovative aspects of the proposal are hence a broad development of optimal control theory; an extension of classical mechanics to models with generalized functions; 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 hence be foreseen both in theoretical and applied mathematics, in physics and engineering applications. Primary researchers involved are PhD P. Giordano and Prof. M. Kunzinger. The employment of a PhD candidate is planned.

The main aim of the proposed project was to develop several applications in the framework of generalized smooth functions (GSF). These functions are useful to describe rapidly changing phenomena such as breaking of steel structures, car crashing, switching of electrical circuits, motion through different or granular media, quantum mechanics, etc. In this setting, new general existence results for differential equations was recently proved. In this research proposal, we considered the following theoretical developments with related applications: 1. Solution of equations using Newton's method and Pontryagin's principle for optimal control. 2. Extension of manifolds with infinitesimal and infinite points and development of mechanics with GSF. 3. Visual representation of GSF by programming Mathematica notebooks. Innovative aspects of the proposal are hence a broad development of optimal control theory; an extension of classical mechanics to models with generalized functions; flexible and easy-to-use Mathematica notebooks to visualize and enhance intuition of students and researchers in the use and properties of GSF. Potential users can hence be foreseen both in theoretical and applied mathematics, in physics and engineering applications. Primary researchers involved are PhD P. Giordano and Prof. M. Kunzinger. We employed a PhD candidate, that will close the thesis before the end of 2024, and a postdoc that now has a permanent position as researcher at Martin Luther University Halle-Wittenberg (DE). The workpackage 2 allows us to start a future research collaboration with Japanese universities.