Assignments:
Systems of ordinary integro-differential equations — that is to say, systems of equations involving unknown functions of one independent variable, their derivatives and their indefinite integrals — naturally arise in mathematical physics, engineering sciences, biology, epidemiology and other fields to model phenomena in which memory effects play a significant role. This class of functional systems is typically analysed using operator theory, operational calculus and functional analysis.
The study of linear systems of ordinary integro-differential equations has recently been initiated in algebra [1] and computer algebra [2] using the ring II of ordinary integro-differential operators with polynomial coefficients (i.e. the non-commutative polynomial ring in the derivative, the integral and the evaluation at the lower bound of the integral, subject to standard calculus identities), non-commutative ring theory, and Gröbner basis methods.
Furthermore, the ring II is known to be coherent but not Noetherian [1]. An algorithmic proof of this property was developed in [3] and implemented in the computer algebra system Maple. This result provides an effective elimination theory for linear ordinary integro-differential systems with polynomial coefficients — that is to say, it makes it possible to effectively eliminate chosen unknown functions from a given linear system of ordinary integro-differential equations [3].
The primary objective of the postdoctoral research is to build on the aforementioned results by studying and extending them to larger rings of coefficients, such as the ring of exponential polynomials, and applying them to parameter estimation problems in different practical system classes [6]. The results will be implemented.
Building on the ideas of differential algebra developed by Ritt and Kolchin, integral algebra and integral elimination were recently introduced to study nonlinear integral equations [4, 5] and have since been applied to parameter estimation problems. The second objective of the postdoctoral research is to build on these results by considering derivations to develop an integral-differential algebra. The aim is to implement the outcomes and develop applications for parameter estimation problems and to the identifiability of linear and nonlinear differential systems. The focus will be on well-defined classes of control systems that are useful in practice [6].
The last goal is to help develop an effective approach to the backstepping method, which was originally conceived in the field of control theory for partial differential equations [7]. This involves automating the different steps using chains of integral transformations, implementing them, and applying the results to different classes of hyperbolic partial differential systems [7].
References:
[1] V. V. Bavula. The algebra of integro-differential operators on an affine line and its modules. J. Pure and Applied Algebra, vol. 217, 495–529, 2013.
[2] G. Regensburger, M. Rosenkranz, J. Middeke. A skew polynomial approach to integro-differential operators. ISSAC'09: Proceedings of the International Symposium on Symbolic and Algebraic Computations, 287-294, 2009.
[3] T. Cluzeau, C. Pinto, A. Quadrat. The ring of polynomial ordinary integro-differential operators is an effective Cramer ring, to appear in the Journal of Symbolic Computation.
[4] F. Lemaire, L. Roussel. Contribution to integral elimination. Computer Algebra in Scientific Computing (CASC) 26th International Workshop, Sep 2024, Rennes, France. pp. 215-235.
[5] F. Lemaire, L. Roussel. Recent Advances on Integral Elimination. Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation (ISSAC ’25), pp. 249–257.
[6] G. Rance, Commande H paramétrique et application aux viseurs gyrostabilisés, PhD thesis, University of Paris Sud, 2018.
[7] J.-M. Coron, Control and nonlinearity, American Mathematical Soc., 2009.
Steering:
The person recruited will be in charge of Alban Quadrat (Inria Paris, Ouragan, IMJ-PRG, Sorbonne University).
