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GAIA - 2018

Section: Research Program

Effective algebra

To develop a computational study of problems coming from control theory, signal processing, and multidisciplinary domains, parts of algebraic theories must be studied within an effective approach: methods and theoretical results must be made algorithmic based on computer algebra techniques appropriated for efficient implementations in computer algebra systems.

Polydisc Nullstellensatz & effective version of a theorem of Deligne

The works on stability and stabilization problems of multidimensional systems, developed in the former ANR MSDOS (2014–2018), have shown the importance for developing an effective version of the module theory over the ring of rational functions without poles in the closed unit polydisc of n [90], [47]. The stabilizability (resp. the existence of a doubly coprime factorization) of a multidimensional system is related to a module-theoretical property (projectivity, resp. freeness) that has to be algorithmically verified prior to compute stabilizing controllers (resp. the standard Youla-Kucˇera parametrization of all stabilizing controllers). Based on the works [89], [90], in [47], we have recently proved that the stabilizability condition is related to the development of an algorithmic proof of the so-called Polydisc Nullstellensatz [49], a natural extension of Hilbert's Nullstellensatz for the above-mentioned ring (see e.g. [40]). In addition, the existence of a doubly coprime factorization is related to a theorem obtained by (the Fields medalist) Deligne with a non-constructive proof [90]. This theorem can be seen as an extension of the famous Quillen-Suslin theorem (Serre's conjecture) [77]. Based on our experience of the first implementation of the Quillen-Suslin theorem in the computer algebra system (Maple ) [62], we aim to develop this effective framework as well as a dedicated Maple package.

Effective version of Spencer's theory of formal integrability of PD systems

A differential geometric counterpart of differential algebra and differential elimination theory [101], [76] is the so-called Spencer's theory of formal integrability and involutive PD systems [85], [104]. For linear PD systems, this theory can be seen as an intrinsic approach to Janet or Gröbner bases for noncommutatibe polynomial rings of PD operators. No complete algorithmic study of Spencer's theory has been developed yet. We aim to develop it as well as to implement it. The understanding of the connections between the different differential elimination theories (Janet [102] or Gröbner bases [40], Thomas decomposition [102], differential algebra [101], [76], Spencer's theory [85], exterior differential systems [51], etc.) will also be investigated. On a longer term, applications of Spencer's theory to Lie pseudogroups and their applications in mathematical physics (e.g. variational formulations based on Lie (pseudo)groups) [86] will be investigated and implemented.

Rings of integro-differential operators & integro-differential algebra

The main contribution of this axis is the development of effective elimination theories for both linear and nonlinear systems of integro-differential equations.

Linear systems of integro-differential equations

The rings of integro-differential operators are more complex than the purely differential case [96], [97] due to the existence of zero-divisors or the fact of having a coherent ring instead of a Noetherian ring [39]. We want to develop an algorithmic study of these rings. Following the direction initiated in [95] for the computation of zero divisors, we first want to develop algorithms for the computation of left/right kernels and left/right/generalized inverses of matrices with entries in such rings, and to use them to develop a module-theoretic approach to linear systems of integro-differential equations. Following [95], standard questions addressed within the computer algebra community such as the computation of rational/exponential/hyperexponential/etc. solutions will also be addressed. Moreover, famous Stafford's results [105], algorithmically studied in [96], [97] for rings of PD operators, are known to still hold for rings of integro-differential operators [39]. Their algorithmic extensions will be investigated and our corresponding implementation will be extended accordingly. Finally, following [93], [95], an algorithmically study of rings of integro-differential-delay operators will be further developed as well as their applications to the equivalence problem of differential constant/varying/distributed delay systems (e.g. Artstein's reduction, Fiagbedzi-Pearson's transformation) and their applications to control theory.

Nonlinear systems of integro-differential equations

Integro-differential algebra is an extension of Ritt-Kolchin's differential algebra [101], [76] that also includes integral operators. This extension is now attracting more attention in mathematics, computer algebras, and control theory. This new type of algebras will be algorithmically studied for integro-differential nonlinear systems. To do that, concepts such as integro-differential ideals and varieties have to be introduced and studied for developing an integro-differential elimination theory which extends the current differential elimination theory [45], [46], [72]. A Maple prototype will first be developed and then a C library when experience will be gained.