Symbolic Transformations of Dynamical Models
Gleb Pogudin · HAL (Le Centre pour la Communication Scientifique Directe) · 2024
This thesis is focused on symbolic algorithms for dynamical models defined by differential or difference equations. Such algorithms aim at complementing traditional numerical tools, they are exact and often operate on the level of symbolic expressions. In the context of differential and difference equations, perhaps the most well-known symbolic algorithms are the ones for finding closed-form solutions which are available in many scientific software packages. However, a large portion (if not majority) of the equations appearing in the modeling literature do not admit such solutions. This fact does not render the symbolic methods useless. On the contrary, there is a number of ways to transform a model on the symbolic level to facilitate its further analysis. In this thesis, we discuss the following problems of this type:- eliminating a subset of the variables (for example, the latent ones) from a model;- assessing structural identifiability of parameters, that is, checking if the parameter values can be inferred uniquely from input-output data, and transforming a model into a one with better identifiability properties; - performing exact model reduction, that is mapping a model into a one of lower dimension without introducing approximation errors; - quadratizing a model, that is embedding the model into a one with at most quadratic nonlinearities. The results we present for these problems range from theorems and theoretical algorithms to practical software implementations and case studies.