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Scientific foundations

Functional systems [bibcite key=Aczel] are systems whose unknowns are functions, such as  systems of  ordinary/partial differential equations, of differential time-delay equations, of difference equations, of integro-differential equations. These systems play a fundamental role in the mathematical modeling of physical phenomena studied in mathematical physics, engineering sciences, mathematical biology, etc. Numerical simulations are usually based on mathematical models defined by functional equations. They are also the cornerstone of many domains, for instance in mathematical physics, mathematical systems theory, control theory and signal processing.

Numerical aspects of functional systems, especially differential systems, have been widely studied in applied  mathematics due to numerical simulation issues. Complementary approaches, based on algebraic and differential geometric methods, are usually upstream or help the numerical simulation. These methods tackle a different range
of questions and problems such as algebraic preconditioning, elimination and simplification, completion to formal integrability or involution, computation of integrability conditions and compatibility conditions, index reduction, reduction of variables, choice of adapted coordinate systems based on symmetries, computation of first integrals of motion and conservation laws, study of the (asymptotic) behavior of solutions at a singularity, or of its dependency with respect to unfixed parameters. Although not yet very popular in applied mathematics, algebraic and differential geometric methods have been developed in the computer algebra community over the past years mostly driven by applications.

Differential elimination techniques [bibcite key=Boulier1,Boulier2] based on differential algebra [bibcite key=Ritt,Kolchin] for differential systems, Gr\”obner/Janet basis methods [bibcite key=Grobner,Robertz] for noncommutative polynomial rings of functional operators [bibcite key=CQR05] for linear functional systems are remarkable examples of algebraic and differential geometric techniques. They form the algorithmic “engines” at the basis of recent effective versions of algebraic theories developed in mathematics. A major source of motivation for the development of the effective study of algebraic theories in the computer algebra community is represented by control theory issues. In the same vein as what happened in language theory with the pioneering work of Schutzenberger or in algebraic coding theory, certain problems studied in control theory or signal processing can be better understood and finely studied by means of algebraic or geometric structures, and by means of algebraic or differential geometric techniques. To effectively study the original problems, computer algebra methods allow one to design efficient algorithms that can be implemented in symbolic-numeric softwares. Two main advantages of  an algebraic approach for the study of functional systems are:  first, when is possible, it can be used to finely study the behavior of the solutions of the system with respect to unfixed model parameters $-$ which is usually a difficult numerical issue. Secondly, the existence of closed-form solutions can highly simplify certain problems, for instance  by avoiding the use of time-consuming optimization problems.

The interplay between control theory and computer algebra has a long history. The first main paper on Gr\”obner
bases [bibcite key=Buchberger] written by their creators, Buchberger, was published in Bose’s book [bibcite key=Bose] on control theory of multidimensional systems since they play a fundamental role in multidimensional systems theory. They were the first main applications of Gröbner bases outside the field of algebraic geometry. The differential algebra approach to nonlinear control theory, developed by Fliess, Pommaret, Diop, etc.
[bibcite key=Fliess, Diop2,Diop3,Pom2], was the main motivation for the effective study of differential algebra (differential elimination theory, triangular sets, regular chains) [bibcite key=Boulier1,Boulier2] and its implementations in two packages (diffalg, DifferentialAlgebra)  of the commercial computer algebra system Maple. Within the effective differential algebra approach to nonlinear control systems, observability, identifiability, parameter estimation, etc. have received appealing characterizations and effective tests,  implementable, e.g. in Maple, were successfully developed.

Linear control theory [bibcite key=Kailath] and multidimensional systems theory [bibcite key=Bose] have recently been profoundly developed due to the so-called behavior and module approaches, developed by Oberst, Willems, Fliess, Pommaret, etc. [bibcite key=Oberst,Willems,FliessMounier,Pom2,AQ06]. Based on ideas of algebraic analysis [bibcite key=Bjork,Kashiwara], system properties of those systems $-$ such as controllability, parametrizability, differential flatness, internal stabilization, etc. $-$ are intrinsically characterized by means of properties of certain algebraic structures (modules). To effectively check the latter properties, we recently had to develop effective versions of two important algebraic theories, namely module theory [bibcite key=McConnellRobson] and homological algebra [bibcite key=Rotman], based on functional elimination techniques (i.e. Gr\”obner or Janet basis techniques for noncommutative polynomial rings of functional operators useful in practice such as differential, shift, delay or difference operators) [bibcite key=CQR05],[bibcite key=CQ08}] Dedicated packages, written in Maple and Mathematica, are now available to both control theory and constructive algebra communities [bibcite key=OreModules, OreMorphisms,OreAlgebraicAnalysis]. They can also be used for educational issues (learn mathematical theories by computing).
Finally, within the above effective approach, stability and stabilization problems of time-delay systems and multidimensional systems have recently been initiated in [bibcite key=BQR15], [bibcite key=BCMQ17].

The problems of estimating parameters of a system is a fundamental issue in control theory and signal processing since the system parameters are usually badly known. In [bibcite key=Fliess3], an algebraic approach to identification was proposed by Fliess and Sira-Ramirez based on the combination of techniques coming from algebraic analysis (rings of differential operators), differential elimination theory (annihilators) and operatorial calculus (e.g. Laplace transform, convolution). This approach, which was at the core of the project-teams Alien and Non-A [bibcite key=U16], yields closed-form solutions for the parameters in terms of integrals of the outputs and inputs of the system. The use of integrals helps to filter the noise on the output of the system. This approach has recently been extended to larger classes of signals (e.g. expansions of a signal on an orthogonal polynomial basis, signals defined by means of special functions) [bibcite key=UQ16]. An effective version of this approach has also been initiated based on module theory and differential elimination techniques [bibcite key=AQ17]. The parameter estimation problem plays a central role in the parametric robust control for gyrostabilized platforms recently developed with Safran Electronics & Defense [bibcite key=Rance], and in biological problems such as the synchronization of biological systems  (measurements and interpretation of bivalves mollusks behavior, online detection of coastal water pollution) [bibcite key=Ahmed]  and high performance tactile interactions [bibcite key=Uecc16].

Finally, the symbolic-numeric methods for functional systems start to be mature enough to tackle some real-life applications studied by industrial companies [bibcite key=AQAQ14,Rance], [bibcite key=RBAA17,RBAAR17,RBQQ18,DQZ17].

Scientific objectives

The scientific objectives of the GAIA team are fourfold.

1. Computer algebra:

Thanks to our leading expertise in differential elimination theory for nonlinear differential systems based on  effective differential algebra methods [bibcite key=Boulier1,Boulier2] and for linear functional systems based on Gröbner basis methods [bibcite key=CQR05], we aim to further reinforce it by attacking the remaining technical obstacles and new classes of functional systems as well, notably those coming from interesting applications in engineering  sciences.

The main computational bottleneck of effective differential algebra methods is the computation of greatest common divisors of multivariate polynomials. Progresses in this direction will yield significant computational improvements for the implementations of differential algebra methods, such as {\tt BLAD} \cite{BLAD}. Expertise in both differential polynomials and polynomials available from {\sc GAIA}’s members will be useful for this achievement of this task.

Based on regular chains of differential index 0 obtained by the effective differential algebra approach developed in [bibcite key=Boulier1,Boulier2], we want to develop numerical integrators for nonlinear systems of ordinary differential equations (ODEs) (which includes differential algebraic equations).

Spencer’s approach to formal integrability and involution of partial differential systems [bibcite key=Pom1] will be studied within an effective viewpoint, a dedicated implementation will be developed and its applications to mathematical physics, such as Lie pseudogroups approach to nonlinear elasticity, electromagnetism and relativity  [bibcite key=Pom3], will be studied. Connections between differential algebra, Spencer theory, Gröbner or Janet bases [bibcite key=Robertz] will be investigated as well.

The main contribution in this project’s component will be the development of effective elimination theories for linear and nonlinear systems of integro-differential equations. The first steps towards an elimination theory for linear (resp. nonlinear) systems of integro-differential equations have been done in [bibcite key=QR18] (resp. in [bibcite key=Boulier3]) based on the study of rings of integro-differential operators (resp. extensions of ideas coming
from differential algebra). These works show the feasibility of this objective and a considerable amount of remaining work. Thanks to applications in control theory, the case of linear systems of integro-differential (constant or varying) delay equations will also be studied following the approach developed in [bibcite key=AQ15].

In addition, an algorithmic study of certain problems concerning quasipolynomials (i.e. polynomials in a complex
variable $s$ and in $e^{-\tau_i \, s}$, where $\tau_i > 0$) $-$ the core of the study of linear differential time-delay systems (Laplace transform of the characteristic function) $-$ will be developed based on polynomial techniques and on the development of a numerical but certified version of the Newton-Puiseux algorithm [bibcite key=BPQ18], [bibcite key=PR15].

2. Effective algebra:

To develop effective applications in control theory, signal processing, mathematical physics and multidisciplinary domains, the proposed algebraic theories must be studied within an effective approach: methods and theorems must be made constructive based on computer algebra results and implemented in computer algebra systems such as  Maple or Mathematica. We shall focus on three algebraic theories.

First, we shall further develop our leading expertise in effective algebraic analysis  [bibcite key=CQR05],[bibcite key=CQ08],[bibcite key=AQ,AQ15] by studying effective versions of module theory and homological algebra which are important for the development of a mathematical systems theory of linear systems of integro-differential (constant or varying) delay equations and its applications in control theory. The rings of integro-differential operators are indeed highly more complicated than the purely differential case, due  %for instance
to the existence of zero-divisors, or the fact of having a coherent ring instead of a Noetherian ring \cite{Bavula}. Certain results effectively developed in [bibcite key=QR07,QR14] are known to still hold for rings of integro-differential operators [bibcite key=Bavula]. Our implementations (OreModules, OreMorphisms)  will be extended accordingly.

Integro-differential algebras are an extension of differential algebra \cite{Ritt,Kolchin} that include integral operators. These algebras will be algorithmically studied for nonlinear systems and it will allow constructing
integro-differential ideals and varieties and other fundamental structures in integro-differential elimination theory.

Furthermore, the works on the stability and stabilization problems of multidimensional systems, developed in the ANR MSDOS, 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 $\mathbb{C}^n$ [bibcite key=BQR15], [bibcite key=BCMQ17].
The stabilizability (resp. the existence of a doubly coprime factorization) of a multidimensional system is closely
related to a module property that has to be algorithmically verified prior to compute stabilizing controllers (resp. the family of all stabilizing controllers). Based on the works [bibcite key=AQ03,AQ06], we have recently proved in [bibcite key=BCMQ17] that the stabilizability condition is related to the development of an algorithmic proof of the so-called Polydisc Nullstellensatz [bibcite key=Bridges]. In addition, the existence of a doubly coprime factorization is related to a theorem obtained by the field medalist Deligne with a non-constructive proof. [bibcite key=AQ06]. This result can be seen as an
extension of the Quillen-Suslin theorem (Serre’s conjecture).
Based on our experience of the first implementation of the Quillen-Suslin theorem in the computer algebra system (Maple) [bibcite key=FQ07], we aim to develop this effective framework, a dedicated implementation and its  applications.

3. Focused applications:

GAIA will develop focused applications of the above two axes (effective algebra and computer algebra) to bring new methodologies to problems studied in control theory, signal processing and multidisciplinary domains.

Two ranges of applications will be tackled in the GAIA team:

Parameter estimation problem

Our leading expertise on algebraic methods for the parameter estimation problem developed in the {\sc Non-A} project-team will be further developed. A first aim will be to develop to a greater extent our recent work [bibcite key=UQ16] that shows how the Non-A approach can cover wider classes of signals, such as holonomic
signals (e.g. signals decomposed into orthogonal polynomial bases, special functions, possibly wavelets). Moreover, following [bibcite key=AQ17], an effective approach will be developed and implemented in a dedicated symbolic-numeric package. In particular, to consider larger classes of structured perturbations corrupting the observed signal, the so-called “syzygy approach”, which generalizes the “annihilator approach” developed by Non-A
for biased signals, will be algorithmically developed. Further, as an alternative to passing forth and backwards from the time domain to the operational domain by means of Laplace transform and its inverse as developed in the Non-A algebraic parameter estimation method, we aim to develop a direct time domain approach based on integro-differential operators. The closed-form expressions obtained within this extended framework will possibly continue to provide robust estimates in our multidisciplinary collaborations, in marine biology and human-machine interactions, as it is already the case of existing Non-A results [bibcite key=Ahmed] and [bibcite key=Uecc16] (in collaboration with Mjolnir, Inria).

For nonlinear control systems, the approach to the parameter estimation problem recently proposed in [bibcite key=Boulier3], based on the computation of integro-differential input-output equations, will be further developed based on effective integro-differential algebra results obtained in the “effective algebra” axis. Such a
representation better suits a numerical estimation of the parameters. A package dedicated to this approach will be developed and used in the collaboration with Verdi\`ere (Le Havre) and Castel-Gandolfo’s INSERM team (Rouen) for the model design of biological systems (e.g. astrocytes, neuron death) based on integro-differential systems. Finally, an extension of [bibcite key=Boulier3] will also be initiated for partial differential equations.

Further development of our recent approach for the anchor position self calibration problem and for the metric
multidimensional unfolding in [bibcite key=DQZ17] will be studied.

Stabilization problems

In collaboration with the Safran Electronics \& Defense company and Rouillier (Ouragan, Inria), we have recently initiated a parametric robust control theory based on computer algebra methods, as well as its applications to the stabilization of the line of sight of gyrostabilized platforms [bibcite key=AQAQ14,Rance]. For low-dimensional systems, this approach aims to determine closed-form solutions for robust controllers and for the robustness margins in terms of the unfixed system parameters (e.g. mass, length, inertia, mode). To achieve this objective,
we will analyze positive definite solutions of algebraic or differential Riccati equations (i.e. polynomial systems or
nonlinear differential systems with parameters) with computer algebra methods. The main applications of these results are twofold: the feasibility of an industrial project can be simplified by speeding up the computation of robust controllers and robust margins for systems with rapidly changing architecture parameters, and avoiding usual time-consuming optimization techniques. Secondly, adaptive and embeddable schemes for robust controllers can be proposed while coupling our approach with real-time parameter estimation methods (e.g. based on methods defined in the above point) [bibcite key=Rance]. Preliminary works in the direction have opened a great variety of questions such as the search for positive definite solutions of the polynomial and the differential systems defined by Riccati equations with parameters, the reduction of these equations and of the parameters based on symmetries, the development, with Rouillier (Ouragan), of efficient tools for drawing high degree curves and surfaces with the robustness  margins in terms of the system parameters, and the use of a certified numeric Newton-Puiseux algorithm for the design of robust controller.

Our leading expertise on stability and stabilization problems for multidimensional systems and time-delay systems  based on computer algebra methods will be further developed.

4. Transfer:

Our algorithmic results will yield dedicated implementations in computer algebra systems and target explicit  applications in engineering sciences (see above).



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