Q. Does the above sketch correspond to a developed area of mathematical investigation?
More of a "developing area" than a "developed" one, maybe. Here are some papers and web articles on Differential Logic that may have a bearing on your question:
- Differential Logic : Introduction
- Differential Logic and Dynamic Systems
- Differential Propositional Calculus @ PlanetMath (read under "page image" mode)
- Differential Propositional Calculus @ PlanetPhysics (try this if the other fails to load)