## Section: Research Program

### Applications to control theory and signal processing

#### Robust stability analysis and stabilization problems for functional systems

Our expertise in the computer algebra aspects to stability and stabilization problems for multidimensional systems and for differential constant/distributed/varying delay systems [5], [6], [48], [47] will further be developed.

##### Computation of Lyapunov functions for homogeneous dynamical systems

We shall investigate the possibility to develop a computer algebra
package for the design of Lyapunov functions for homogeneous dynamical
systems based on an effective study of the differential algebra of
*generalized forms*, i.e. of Puiseux polynomials in signed powers
[106], [107].

##### Robust stability analysis of differential time-delay systems

The symbolic-numeric study of the robust stability of a differential
constant time-delay system with respect to the delay $h$, via the
variation of the zero locus of the associated
quasipolynomial $p(s,{e}^{-h\phantom{\rule{0.166667em}{0ex}}s})$ [70] in the
stability (resp. unstability) region ${\u2102}_{-}=\{s\in \u2102\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}\Re \left(s\right)<0\}$ (resp. $\overline{{\u2102}_{-}}=\u2102\setminus {\u2102}_{+}$) of $\u2102$, initiated
in [5], will be further developed. This problem is
another motivation for the development of a fast numerical algorithm
for the computation of Netwon-Puiseux series and its implementation in
a `C` library. They will be used to study how the
different branches of a quasipolynomial at a critical pair (namely
$({h}_{\u2606},{\omega}_{\u2606})\in {\mathbb{R}}_{>0}\times \mathbb{R}$
such that $p(i\phantom{\rule{0.166667em}{0ex}}{\omega}_{\u2606},{e}^{-i\phantom{\rule{0.166667em}{0ex}}h\phantom{\rule{0.166667em}{0ex}}{\omega}_{\u2606}})=0$) vary in ${\u2102}_{-}$ and in ${\u2102}_{+}$
with respect to $h$. See [5] and the references
therein.

Moreover, in collaboration with Mouze (Centrale Lille, France), we want to develop an effective study of the ring $\mathcal{E}=\mathbb{R}\left(s\right)\left[{e}^{-h\phantom{\rule{0.166667em}{0ex}}s}\right]\cap E$, where $E$ is the ring of entire functions. The ring $\mathcal{E}$ plays an important role for differential time-delay systems [66], [78]. Effective computation of Smith normal forms for matrices with entries in $\mathcal{E}$ and its implementation in a symbolic-numeric package will have many applications for synthesis problems of differential time-delay systems.

##### Stabilization problems for functional systems

We want to use the results developed
on the module-theoretic aspects of the ring of multivariate rational
functions without poles in the closed unit polydisc of ${\u2102}^{n}$ to
effectively compute stabilizing controllers of multidimensional
systems, as well as the Youla-Ku$\stackrel{\u02c7}{\mathrm{c}}$era parametrization
of all the stabilizing controllers [90]. This last
parametrization can be used to transform standard ${H}_{\infty}$-optimal
control problems, which are nonlinear by nature, into affine, and thus
convex optimal problems. See [37], [90] and the references
therein. Applications addressed in the former ANR MSDOS (2014–2018)
will be developed in collaboration with Bachelier (U. Poitiers). The
algorithms obtained in this direction will be unified in a unique `Maple` package.

Finally, the *noncommutative geometric approach* to robust problems for infinite-dimensional linear systems (e.g. differential time-delay or PD systems) [54], initiated in [92], will be further studied based on the mathematical concepts and methods introduced by (the Fields medalist) Connes [53]. We particularly want to investigate generalizations of Nyquist's theorem to infinite-dimensional systems
based on index theory (pairing of $K$-theory and $K$-homology), model
reduction based on Connes' interpretation of infinitesimal operators,
robustness metrics particularly the $\nu $-gap metric, etc. The
quantized differential calculus [53], based on Hankel
operators, as well as the connections and
curvatures on stabilizable systems will be further studied [92]. We aim to exploit these noncommutative differential geometric structures on the systems to get new inside in both the topology and geometry aspects of the ${H}_{\infty}$-control theory for infinite-dimensional systems [54].

#### Parameter estimation for linear & nonlinear functional systems

##### Linear functional systems

Our expertise on algebraic parameter estimation problem, coming from the former Non-A project-team, will be further developed. Following [65], this problem consists in estimating a set $\theta $ of parameters of a signal $x(\theta ,t)$ $-$ which satisfies a certain dynamics $-$ when the signal
$y\left(t\right)=x(\theta ,t)+\gamma \left(t\right)+\varpi \left(t\right)$ is observed, where $\gamma $
denotes a structured perturbation and $\varpi $ a noise. For instance,
$x$ can be a multi-sinusoidal waveform signal and $\theta $ phases,
frequencies, or amplitudes [13]. Based on a combination of
algebraic analysis techniques (rings of differential operators),
differential elimination theory (computation of annihilators), and
*operational calculus* (Laplace transform, convolution),
[65] shows how $\theta $ can sometimes be explicitly
determined by means of closed-form expressions using iterated
integrals of $y$. These integrals usually help to filter the effect of
the noise $\varpi $ on the estimation of the parameters $\theta $.

A first aim in this direction is to develop to a greater extent our recent work [108] that shows how the above approach can cover wider classes of signals such as *holonomic
signals* (e.g. signals decomposed into orthogonal polynomial
bases, special functions, possibly wavelets).

Moreover, [94] explains how larger classes of structured
perturbations $\gamma $ can be considered when the approach
developed in [65], [108], based on computation of *annihilators*, is
replaced by a new approach based on the more general algebraic concept
of *syzygies* [103]. This general approach to
the algebraic parameter estimation problem will be
developed. Following the ideas of [94], an
effective version of this general approach will also be done based on differential elimination techniques, i.e. Gröbner basis techniques for rings of differential operators. It will be
implemented in a dedicated `Maple` package which will extend the current prototype `NonA` package [94].

Furthermore, as an alternative to passing forth and backwards from the time domain to the operational (Laplace/frequency) domain by means of Laplace transform and its inverse as done in the standard algebraic parameter estimation method [65], [108], we aim to develop a direct time domain approach based on calculus on rings of integro-differential operators as described by the following picture ($L$ denotes the Laplace transform):

The direct computation will be handled by means of the effective methods of rings of integro-differential operators described in the above sections.

##### Nonlinear functional systems

For nonlinear control systems, the approach to the parameter estimation problem, recently proposed in [3] and based on the computation of integro-differential input-output equations, will be further developed based on the integration of fractions [2]. Such a representation better suits a numerical estimation of the parameters as shown in [3].

In [29], we have recently initiated an extension of the results developed in [3] to handle integro-differential equations such as Volterra-Kostitzin's equation. This general approach, based on an extension of the input-output ideal method for ordinary differential equations to the integro-differential ones, will be further developed based on the effective elimination theory for systems of integro-differential equations. An important advantage of this approach is that not only it solves the identifiability theoretical question but it also prepares a further parameter estimation step [57].