Specifying the behavior desired of hybrid systems requires analytical modeling of physical phenomena. Similarly, testing them requires simulation of continuous systems. While numerous tools support later stages of developing simulation codes, there is still a large gap between analytical modeling and running simulation code, which significantly impedes the ability of scientists and engineers to develop novel hybrid systems. The support for partial derivative, in particular, is limited in present mainstream modeling and simulation languages as well as reachability analysis tools for hybrid systems. Either they do not provide such language construct, requiring the modeler to manually transform the model or its correctness is unclear.

In this thesis, we demonstrate that compile-time transformations can improve hybrid system formalisms by supporting partial derivatives and equational constraints. These improvements allow the user to express, among other things, the Euler-Lagrangian equation, and to capture practically relevant constraints that arise naturally in mechanical systems. Achieving this level of expressivity requires using a binding time-analysis (BTA), program differentiation, symbolic Gaussian elimination, and abstract interpretation using interval analysis. We give an operational semantics for the specialization process along with a declarative and algorithmic specifications of the binding-time analysis. A type safety theorem is given to show the correctness of the semantics for specialization. The declarative specification of binding-time analysis is used to prove soundness with respect to the specialization process. We also provide an open-source implementation demonstrating our approach.