## Section: New Results

### Mathematical Foundations of Physical Systems Modeling Languages

Participants : Albert Benveniste, Benoît Caillaud, Mathias Malandain.

Modern modeling languages for general physical systems, such as Modelica or Simscape, rely on Differential Algebraic Equations (DAE), i.e., constraints of the form $f(\dot{x},x,u)=0$. This facilitates modeling from first principles of the physics. This year we completed the development of the mathematical theory needed to sound, on solid mathematical bases, the design of compilers and tools for DAE based physical modeling languages.

Unlike Ordinary Differential Equations (ODE, of the form $\dot{x}=g(x,u)$), DAE exhibit subtle issues because of the notion of *differentiation index* and related *latent equations*—ODE are DAE of index zero for which no latent equation needs to be considered. Prior to generating execution code and calling solvers, the compilation of such languages requires a nontrivial *structural analysis* step that reduces the differentiation index to a level acceptable by DAE solvers.

Multimode DAE systems, having multiple modes with mode-dependent dynamics and state-dependent mode switching, are much harder to deal with. The main difficulty is the handling of the events of mode change. Unfortunately, the large literature devoted to the numerical analysis of DAEs does not cover the multimode case, typically saying nothing about mode changes. This lack of foundations causes numerous difficulties to the existing modeling tools. Some models are well handled, others are not, with no clear boundary between the two classes. Basically, no tool exists that performs a correct structural analysis taking multiple modes and mode changes into account.

In our work, we developed a comprehensive mathematical approach supporting compilation and code generation for this class of languages. Its core is the *structural analysis of multimode DAE systems,* taking both multiple modes and mode changes into account. As a byproduct of this structural analysis, we propose well sound criteria for accepting or rejecting models at compile time.

For our mathematical development, we rely on *nonstandard analysis,* which allows us to cast hybrid systems dynamics to discrete time dynamics with infinitesimal step size, thus providing a uniform framework for handling both continuous dynamics and mode change events.

A big comprehensive document has been written, which will be finalized and submitted next year.