## Section: New Results

### Analysis of infinite-dimensional and multidimensional systems

#### - stability of (possibly fractional) neutral delay systems

Participants : Catherine Bonnet, André Fioravanti, Hitay Özbay, Jonathan R. Partington.

Both in the time-domain and frequency-domain approaches to stability of delay systems, neutral type systems are the most difficult systems to analyze since they may have chains of poles asymptotic to the imaginary axis in the complex plane. In [12] , we provide the location of all asymptotic poles for classical systems, and for the fractional case in [46] . In both cases, necessary and sufficient conditions for -stability are derived. Preliminary results about the robustness relative to a change in the delay or in the parameters are given.

#### Numerical methods for (possibly fractional) time-delay systems

Participants : Catherine Bonnet, André Fioravanti, Silviu-Iulian Niculescu, Hitay Özbay.

Many important properties of time-delay systems cannot be obtained in a pure algebraic form. In those cases, numerical methods have proved to be extremely efficient and precise. But most of the time, only classical retarded systems are considered. We studied how those algorithms can be adapted to the cases of neutral and fractional systems, and if not, how to obtain new algorithms for this class of systems [47] , [87] . Those results will be implemented in the YALTA toolbox described in Section 5.1 .

#### Differential flatness of analytic linear ordinary differential systems

Participants : Alban Quadrat, Daniel Robertz.

Based on an extension of Stafford's classical theorem in noncommutative algebra [107] developed in [85] , we prove in [64] that a controllable linear ordinary differential system with convergent power series coefficients (i.e., germs of real analytic functions) and at least two inputs is differentially flat [88] . This result extends a result obtained in [104] for linear ordinary differential systems with polynomial coefficients. The algorithm developed in [104] for the computation of injective parametrizations and bases of free differential modules with polynomial coefficients can be used to compute injective parametrizations and flat outputs of these classes of differentially flat systems. This algorithm allows us to remove artificial singularities which naturally appear in the computation of injective parametrizations and flat outputs obtained by means of Jacobson normal form computations.

#### Serre's reduction of multidimensional systems

Participants : Alban Quadrat, Thomas Cluzeau, Mohamed S. Boudellioua.

Given a linear multidimensional system (e.g., ordinary/partial differential systems, differential time-delay systems, difference systems), Serre's reduction aims at finding an equivalent linear multidimensional system which contains fewer equations and fewer unknowns. Finding Serre's reduction of a linear multidimensional system can generally simplify the study of structural properties and of different numerical analysis issues, and it can sometimes help solving the linear multidimensional system in closed form. In [14] , [74] , [42] , a constructive approach to Serre's reduction is developed for determined and underdetermined linear systems. In particular, an algorithm is given for the class of controllable 2D systems, which is used to find explicit Serre's reduction of many classical control differential time-delay systems. Serre's reduction of these systems generally simplifies their analysis and their synthesis.

In [44] and [83] , Serre's reduction problem is studied for underdetermined linear systems of partial differential equations with either polynomial, formal power series or analytic coefficients and with holonomic adjoints in the sense of algebraic analysis [77] , [78] . These linear partial differential systems are proved to be equivalent to a linear partial differential equation. In particular, an analytic linear ordinary differential system with at least one input is equivalent to a single ordinary differential equation. In the case of polynomial coefficients, we give an algorithm which computes the corresponding linear partial differential equation.

The algorithms obtained in [14] , [74] , [42] , [44] and [83] are implemented in the Serre package in development.

#### Purity filtration of multidimensional systems

Participant : Alban Quadrat.

In [63] , [72] , it is shown that every linear system of partial differential equations in n independent variables is equivalent to a linear system of partial differential defined by an upper block-triangular matrix of partial differential operators: each diagonal block is respectively formed by the elements of the system satisfying an i -dimensional (resp., (n + i) -dimensional) dynamics if the coefficients of the system are either constant or rational functions (resp., the coefficients are either polynomial, formal power series or convergent power series). Hence, the system can be integrated in cascade by successively solving (inhomogeneous) linear j -dimensional systems of partial differential equations to get a Monge parametrization of its solution space [105] . The results are based on an explicit construction of the purity filtration of the differential module associated with the linear systems of partial differential equations, which does not use spectral sequences [77] , [78] . These results can be extended to other classes of linear multidimensional systems such as differential time-delay systems or difference systems.

#### Extendability of multidimensional linear systems

Participant : Alban Quadrat.

Within the algebraic analysis approach to multidimensional linear systems defined by linear systems of partial differential equations with constant coefficients [72] , [79] , [102] , we show in [62] how to use different mathematical results developed in the literature of algebraic analysis [91] , [95] to obtain new characterizations of the concepts of controllability in the sense of Willems [101] and Pillai-Shankar [100] , observability, flatness and autonomous systems in terms of the possibility to extend (smooth or distribution) solutions of the multidimensional system and of its formal adjoint. Each characterization is equivalent to a module-theoretic property that can be constructively checked by means of the OreModules and QuillenSuslin packages.

#### Symmetries and parametrizations of multidimensional systems

Participants : Thomas Cluzeau, Alban Quadrat.

Within the algebraic analysis approach to linear systems theory [72] , [79] , [102] , [45] shows how the behaviour homomorphisms (namely, transformations which send the solutions of a linear multidimensional system to the solutions of another linear system [82] ) induce natural transformations on the autonomous elements of these systems (e.g., on the obstructions to controllability) and on the potentials (e.g., flat outputs) of their parametrizable parts (e.g., controllable parts). Extension of these results are then considered for linear systems inducing a chain of successive parametrizations (e.g., the divergence operator, the first group of Maxwell equations, the stress tensor in linear elasticity).

#### Factorization and decomposition problems for 2D Stokes and Oseen equations

Participants : Thomas Cluzeau, Alban Quadrat.

In [43] , within the constructive algebraic analysis approach to linear systems [72] , [79] , [102] , we study classical linear systems of partial differential equations in two independent variables with constant coefficients appearing in mathematical physics and engineering sciences such as the Stokes and Oseen equations studied in hydrodynamics and the Navier-Cauchy equations in linear elasticity. In particular, a precise description of the endomorphism ring of the differential module associated with the Stokes and Oseen, Navier-Cauchy equations is given. Using the fact that the endomorphism ring of the Stokes and Oseen equations in defines a cyclic differential module, we decide about the existence of factorizations of the matrices of differential operators defined by these systems and the decomposition of their solution spaces in direct sums.

#### New domain decomposition methods for linear PD systems

Participants : Thomas Cluzeau, Victorita Dolean, Frédéric Nataf, Alban Quadrat.

Within the framework of the PEPS Maths-ST2I SADDLES (Symbolic Algebra, Domain Decomposition, Linear Equations and Systems), the purpose of this work is to use algebraic and symbolic computation techniques such as Smith normal forms and Gröbner basis techniques to develop new Schwarz algorithms for domain decompositions and for preconditionners of linear systems of partial differential equations, especially for the Stokes and Oseen equations studied in hydrodynamics and the Navier-Cauchy equations in linear elasticity. New algorithms are developed to reduce the interface conditions and to solve the completion problem built on physical Smith variables. They are implemented in the Schwarz package (to appear) built upon the OreModules package. For more details, see http://www-math.unice.fr/~dolean/saddles/ .