Section: Overall Objectives

Algebraic and geometric studies of functional systems

Systems of functional equations or simply functional systems are systems whose unknowns are functions, such as systems of ordinary (OD) or partial differential (PD) equations, of differential time-delay equations, of difference equations, of integro-differential equations, etc. [34], [35]. Functional systems play a fundamental role in the mathematical modeling of physical phenomena studied in natural science such as physics, or in engineering sciences such as mathematical systems theory control theory, signal processing, etc. [34], [35]. Numerical aspects of functional systems, especially OD and PD systems, have largely been studied in applied mathematics due to the importance of numerical simulation issues.

Complementary approaches, based on algebraic and differential or algebraic geometric methods, are usually upstream or help the numerical simulation of systems of functional systems. These methods tackle questions and problems such as algebraic preconditioning, elimination and simplification, completion to formal integrability or involution, computation of integrability conditions or compatibility conditions, index reduction, reduction of variables, choice of adapted coordinate systems based on symmetries, computation of first integrals of motion, conservation laws, and Lax pairs, study of Liouville integrability or of the asymptotic behavior of solutions at a singularity, etc. For more details, see [36], [41], [51], [67], [75], [76], [81], [85], [101], [104], [109] and the references therein.

Let us state a few interests of an algebraic approach for the study of functional systems:

• Algebraic methods are clearly suitable for an algorithmic study, and thus for the development of efficient algorithms implementable in computer algebra systems.

• It can be used to finely study the behavior of the solutions of a system with respect to unfixed model parameters $-$ which is usually a difficult numerical issue. Moreover, the boundaries of the zones in the parameter space over which the behavior of the solution changes can be algorithmically characterized by means of algebraic methods, which yields a safe use of numeric methods in each regular zones (symbolic-numeric methods).

• The existence of closed-form solutions can highly simplify certain problems studied in applications by avoiding the use of time-consuming optimization problems, and thus fits well with nowadays real-time applications.

The GAIA team aims to develop algebraic and geometric methods for the study of functional systems.