Characterizing and inferring properties and behaviors of objects or phenomena from observations using models is common to many research fields. For dynamical systems encountered in the domains of engineering and physiology, this is of practical importance for monitoring, prediction, and control. For such purposes, we consider most frequently, the following model of dynamical systems:

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\displaystyle \frac{dx\left(t\right)}{dt}}& =& f\left(x\right(t),u(t),\theta ,w(t\left)\right)\hfill \\ \hfill y\left(t\right)& =& g\left(x\right(t),u(t),\theta ,v(t\left)\right)\hfill \end{array}\end{array}$$

where $x\left(t\right)$, $u\left(t\right)$ and $y\left(t\right)$ represent respectively the state, input and output of the system, $f$ and $g$ characterize the state and output equations, parameterized by $\theta$ and subject to modeling and measurement uncertainties $w\left(t\right)$ and $v\left(t\right)$. Modeling is usually based on physical knowledge or on empirical experiences, strongly depending on the nature of the system. Typically only the input $u\left(t\right)$ and output $y\left(t\right)$ are directly observed by sensors. Inferring the parameters $\theta$ from available observations is known as system identification and may be useful for system monitoring , whereas algorithms for tracking the state trajectory $x\left(t\right)$ are called observers. The members of SISYPHE have gained important experiences in the modeling of some engineering systems and biomedical systems. The identification and observation of such systems often remain challenging because of strong nonlinearities . Concerning control, robustness is an important issue, in particular to ensure various properties to all dynamical systems in some sets defined by uncertainties , . The particularities of ensembles of connected dynamical systems raise new challenging problems.

Examples of reduced order models:

- Reduced order modeling of the cardiovascular system for signal & image processing or control applications. See section .

- Excitable neuronal networks & control of the reproductive axis by the GnRH. See section .

- Modeling, Control, Monitoring and Diagnosis of Depollution Systems. See section .

Linear stationary waves. Our main example of classical system that is interesting to see as a quantum-like system is the Telegrapher Equation, a model of transmission lines, possibly connected into a network. This is the standard model for electrical networks, where $V$ and $I$ are the voltage and intensity functions of $z$ and $k$, the position and frequency and $R\left(z\right)$, $L\left(z\right)$, $C\left(z\right)$, $G\left(z\right)$ are the characteristics of the line:

$$\frac{\partial V(z,k)}{\partial z}=-\left(R\left(z\right)+jkL\left(z\right)\right)I(z,k),\frac{\partial I(z,k)}{\partial z}=-(G\left(z\right)+jkC\left(z\right))V(z,k)$$

Since the work of Noordergraaf , this model is also used for hemodynamic networks with $V$ and $I$ respectively the blood pressure and flow in vessels considered as 1D media, and with $R=\frac{8\pi \eta }{S^2}$, $L=\frac{\rho }{S}$, $C=\frac{3S(r+h)}{E(2r+h)}$ where $\rho$ and $\eta$ are the density and viscosity of the blood ; $r$, $h$ and $E$ are the inner radius, thickness and Young modulus of the vessel. $S=\pi r^2$. The conductivity $G$ is a small constant for blood flow.

Monitoring such networks is leading us to consider the following inverse problem: get information on the functions $R$, $L$, $C$, $G$ from the reflection coefficient $ℛ\left(k\right)$ (ratio of reflected over direct waves) measured in some location by Time or Frequency Domain Reflectometry.

To study this problem it is convenient to use a Liouville transform, setting $x\left(z\right)={\int }_0^z\sqrt{L\left(z^{\text{'}}\right)C\left(z^{\text{'}}\right)}d{z}^{\text{'}}$, to introduce auxiliary functions ${\displaystyle q^{\pm }\left(x\right)=\frac{1}{4}\frac{d}{x}\left(ln\frac{L\left(x\right)}{C\left(x\right)}\right)\pm \frac{1}{2}\left(\frac{R\left(x\right)}{L\left(x\right)}-\frac{G\left(x\right)}{C\left(x\right)}\right)}$ and ${\displaystyle q_p\left(x\right)=\frac{1}{2}\left(\frac{R\left(x\right)}{L\left(x\right)}+\frac{G\left(x\right)}{C\left(x\right)}\right)}$, so that () becomes a Zakharov-Shabat system that reduces to a Schrödinger equation in the lossless case ($R=G=0$):

$${\displaystyle \frac{\partial v_1}{\partial x}=\left(q_p-jk\right)v_1+q^{+}{v}_2,\frac{\partial v_2}{\partial x}=-\left(q_p-jk\right)v_2+q^{-}{v}_1}$$

and ${\displaystyle I(x,k)=\frac{1}{\sqrt{2}}{\left[\frac{C\left(x\right)}{L\left(x\right)}\right]}^{\frac{1}{4}}(v_1(x,k)+v_2(x,k))}$, ${\displaystyle V(x,k)=-\frac{1}{\sqrt{2}}{\left[\frac{L\left(x\right)}{C\left(x\right)}\right]}^{\frac{1}{4}}(v_1(x,k)-v_2(x,k))}$.

Our inverse problem becomes now an inverse scattering problem for a Zakharov-Shabat (or Schrödinger) equation: find the potentials $q^{\pm }$ and $q_p$ corresponding to $ℛ$. This classical problem of mathematical physics can be solved using e.g. the Gelfand-Levitan-Marchenko method.

Nonlinear traveling waves. In some recent publications  , , we use scattering theory to analyze a measured Arterial Blood Pressure (ABP) signal. Following a suggestion made in  , a Korteweg-de Vries equation (KdV) is used as a physical model of the arterial flow during the pulse transit time. The signal analysis is based on the use of the Lax formalism: the iso-spectral property of the KdV flow allows to associate a constant spectrum to the non stationary signal. Let the non-dimensionalized KdV equation be

$$\frac{\partial y}{\partial t}-6y\frac{\partial y}{\partial x}+\frac{{\partial }^{3}y}{\partial x^3}=0$$

In the Lax formalism, $y$ is associated to a Lax pair: a Schrödinger operator, ${\displaystyle L\left(y\right)=-\frac{{\partial }^2}{\partial x^2}+y}$ and an anti-Hermitian operator ${\displaystyle M\left(y\right)=-4\frac{{\partial }^3}{\partial x^3}+3y\frac{\partial }{\partial x}+3\frac{\partial }{\partial x}y}$. The signal $y$ is playing here the role of the potential of $L\left(y\right)$ and is given by an operator equation equivalent to ():

$$\frac{\partial L\left(y\right)}{\partial t}=[M\left(y\right),L\left(y\right)]$$

Scattering and inverse scattering transforms can be used to analyze $y$ in term of the spectrum of $L\left(y\right)$ and conversely. The “bound states” of $L\left(y\right)$ are of particular interest: if $L\left(y\right)$ is solution of () and $L\left(y\right(t\left)\right)$ has only bound states (no continuous spectrum), then this property is true at each time and $y$ is a soliton of KdV. For example the arterial pulse pressure is close to a soliton , .

Inverse scattering as a generalized Fourier transform. For “pulse-shaped” signals $y$, meaning that $y\in L^1(ℝ;(1+{\left|x\right|}^2\left)dx\right)$, the squared eigenfunctions of $L\left(y\right)$ and their space derivatives are a basis in $L^1(ℝ;dx)$ (see e.g. ) and we use this property to analyze signals. Remark that the Fourier transform corresponds to using the basis associated with $L\left(0\right)$. The expression of a signal $y$ in its associated basis is of particular interest. For a positive signal (as e.g. the arterial pressure), it is convenient to use $L(-y)$ as $-y$ is like a multi-well potential, and the Inverse scattering transform formula becomes:

$$y\left(x\right)=4\sum _{n=1}^{n=N}{\kappa }_n{\psi }_n^2\left(x\right)-\frac{2i}{\pi }{\int }_{-\infty }^{-\infty }kℛ\left(k\right)f^2(k,x)dk$$

where ${\psi }_n$ and $f(k,.)$ are solutions of $L(-y)f=k^{2}f$ with $k=i{\kappa }_n$, ${\kappa }_n>0$, for ${\psi }_n$ (bound states) and $k>0$ for $f(k,.)$ (Jost solutions). The discrete part of this expression is easy to compute and provides useful informations on $y$ in applications. The case $ℛ=0$ ($-y$ is a reflectionless potential) is then of particular interest as $2N$ parameters are sufficient to represent the signal. We investigate in particular approximation of pulse-shaped signals by such potentials corresponding to N-solitons.

Interesting applications for quantum control have motivated seminal studies in such wide-ranging fields as chemistry, metrology, optical networking and computer science. In chemistry, the ability of coherent light to manipulate molecular systems at the quantum scale has been demonstrated both theoretically and experimentally  . In computer science, first generations of quantum logical gates (restrictive in fidelity) has been constructed using trapped ions controlled by laser fields (see e.g. the “Quantum Optics and Spectroscopy Group, Univ. Innsbruck”). All these advances and demands for more faithful algorithms for manipulating the quantum particles are driving the theoretical and experimental research towards the development of new control techniques adapted to these particular systems. A very restrictive property, particular to the quantum systems, is due to the destructive behavior of the measurement concept. One can not measure a quantum system without interfering and perturbing the system in a non-negligible manner.

Quantum decoherence (environmentally induced dissipations) is the main obstacle for improving the existing algorithms  . Two approaches can be considered for this aim: first, to consider more resistant systems with respect to this quantum decoherence and developing faithful methods to manipulate the system in the time constants where the decoherence can not show up (in particular one can not consider the back-action of the measurement tool on the system); second, to consider dissipative models where the decoherence is also included and to develop control designs that best confronts the dissipative effects.

In the first direction, we consider the Schrödinger equation where $\Psi (t,x)$, $-\frac{1}{2}\Delta$, $V$, $\mu$ and $u\left(t\right)$ respectively represent the wavefunction, the kinetic energy operator, the internal potential, the dipole moment and the laser amplitude (control field):

$$\begin{array}{c}\hfill i\frac{d}{dt}\Psi (t,x)=(H_0+u\left(t\right)H_1)\Psi (t,x)=(-\frac{1}{2}\Delta +V\left(x\right)+u\left(t\right)\mu \left(x\right)){\Psi (t,x),\Psi |}_{t=0}={\Psi }_0,\end{array}$$

While the finite dimensional approximations ($\Psi \left(t\right)\in {ℂ}^N$) have been very well studied (see e.g. the works by H. Rabitz, G. Turinici, ...), the infinite dimensional case ($\Psi (t,.)\in L^2({ℝ}^N;ℂ)$) remains fairly open. Some partial results on the controllability and the control strategies for such kind of systems in particular test cases have already been provided  , , . As a first direction, in collaboration with K. Beauchard (CNRS, ENS Cachan) et J-M Coron (Paris-sud), we aim to extend the existing ideas to more general and interesting cases. We will consider in particular, the extension of the Lyapunov-based techniques developed in  , , . Some technical problems, like the pre-compactness of the trajectories in relevant functional spaces, seem to be the main obstacles in this direction.

In the second direction, one needs to consider dissipative models taking the decoherence phenomena into account. Such models can be presented in the density operator language. In fact, to the Schrödinger equation (), one can associate an equation in the density operator language where $\rho =\Psi {\Psi }^{*}$ represents the projection operator on the wavefunction $\Psi$ ($[A,B]=AB-BA$ is the commutator of the operators $A$ and $B$):

$$\frac{d}{dt}\rho =-i[H_0+u\left(t\right)H_1,\rho ],$$

Whenever, we consider a quantum system in its environment with the quantum jumps induced by the vacuum fluctuations, we need to add the dissipative effect due to these stochastic jumps. Note that at this level, one also can consider a measurement tool as a part of the environment. The outputs being partial and not giving complete information about the state of the system (Heisenberg uncertainty principle), we consider a so-called quantum filtering equation in order to model the conditional evolution of the system. Whenever the measurement tool composes the only (or the only non-negligible) source of decoherence, this filter equation admits the following form:

$$\begin{array}{c}d{\rho }_t=-i[H_0+u\left(t\right)H_1,{\rho }_t]dt+(L{\rho }_t{L}^{*}-\frac{1}{2}{L}^{*}L{\rho }_t-\frac{1}{2}{\rho }_t{L}^{*}L)dt\hfill \\ \hfill +\sqrt{\eta }(L{\rho }_t+{\rho }_t{L}^{*}-\text{Tr}\left[(L+L^{*}){\rho }_t\right]{\rho }_t)d{W}_t,\end{array}$$

where $L$ is the so-called Lindblad operator associated to the measurement, $0<\eta \le 1$ is the detector's efficiency and where the Wiener process $W_t$ corresponds to the system output $Y_t$ via the relation $d{W}_t=d{Y}_t-\text{Tr}\left[(L+L^{*}){\rho }_t\right]dt$. This filter equation, initially introduced by Belavkin  , is the quantum analogous of a Kushner-Stratonovic equation. In collaboration with H. Mabuchi and his co-workers (Physics department, Caltech), we would like to investigate the derivation and the stochastic control of such filtering equations for different settings coming from different experiments  .

Finally, as a dual to the control problem, physicists and chemists are also interested in the parameter identification for these quantum systems. Observing different physical observables for different choices of the input $u\left(t\right)$, they hope to derive more precise information about the unknown parameters of the system being parts of the internal Hamiltonian or the dipole moment. In collaboration with C. Le Bris (Ecole des ponts and Inria), G. Turinici (Paris Dauphine and Inria), P. Rouchon (Ecole des Mines) and H. Rabitz (Chemistry department, Princeton), we would like to propose new methods coming from the systems theory and well-adapted to this particular context. A first theoretical identifiability result has been proposed  . Moreover, a first observer-based identification algorithm is under study.

Understanding the complex mechanisms involved in the cardiac pathological processes requires fundamental researches in molecular and cell biology, together with rigorous clinical evaluation protocols on the whole organ or system scales. Our objective is to contribute to to these researches by developing low-order models of the cardiac mechano-energetics and control mechanisms, for applications in model-based cardiovascular signal or image processing.

We consider intrinsic heart control mechanisms, ranging from the Starling and Treppe effects on the cell scale to the excitability of the cardiac tissue and to the control by the autonomous nervous system. They all contribute to the function of the heart in a coordinated manner that we want to analyze and assess. For this purpose, we study reduced-order models of the electro-mechanical activity of cardiac cells designed to be coupled with measures available on the organ scale (e.g. ECG and pressure signals). We study also the possibility to gain insight on the cell scale by using model-based multiscale signal processing techniques of long records of cardiovascular signals.

Here are some questions of this kind, we are considering:

- Modeling the controlled contraction/relaxation from molecular to tissue and organ scales.

- Direct and inverse modeling the electro-mechanical activity of the heart on the cell scale.

- Nonlinear spectral analysis of arterial blood pressure waveforms and application to clinical indexes.

- Modeling short-term and long-term control dynamics on the cardiovascular-system scale. Application to a Total Artificial Heart.

The ovulatory success is the main limiting factor of the whole reproductive process, so that a better understanding of ovulation control is needed both for clinical and zootechnical applications. It is necessary to improve the treatment of anovulatory infertility in women, as it can be by instance encountered in the PolyCystic Ovarian Syndrome (PCOS), whose prevalence among reproductive-age women has been estimated at up to 10%. In farm domestic species, embryo production following FSH stimulation (and subsequent insemination) enables to amplify the lineage of chosen females (via embryo transfer) and to preserve the genetic diversity (via embryo storage in cryobanks). The large variability in the individual responses to ovarian stimulation treatment hampers both their therapeutic and farming applications. Improving the knowledge upon the mechanisms underlying FSH control will help to improve the success of assisted reproductive technologies, hence to prevent ovarian failure or hyperstimulation syndrome in women and to manage ovulation rate and ovarian cycle chronology in farm species.

To control ovarian cycle and ovulation, we have to deeply understand the selection process of ovulatory follicles, the determinism of the species-specific ovulation rate and of its intra- and between-species variability, as well as the triggering of the ovulatory GnRH surge from hypothalamic neurons.

Beyond the strict scope of Reproductive Physiology, this understanding raises biological questions of general interest, especially in the fields of

Molecular and Cellular Biology. The granulosa cell, which is the primary target of FSH in ovarian follicles, is a remarkable cellular model to study the dynamical control of the transitions between the cellular states of quiescence, proliferation, differentiation, and apoptosis, as well as the adaptability of the response to the same extra-cellular signal according to the maturity level of the target cell. Moreover, the FSH receptor belongs to the seven transmembrane spanning receptor family, which represent the most frequent target (over 50%) amongst the therapeutic agents currently available. The study of FSH receptor-mediated signaling is thus not only susceptible to allow the identification of relaying controls to the control exerted by FSH, but it is also interesting from a more generic pharmacological viewpoint.

Neuroendocrinology and Chronobiology. The mechanisms underlying the GnRH ovulatory surge involve plasticity phenomena of both neuronal cell bodies and synaptic endings comparable to those occurring in cognitive processes. Many time scales are interlinked in ovulation control from the fastest time constants of neuronal activation (millisecond) to the circannual variations in ovarian cyclicity. The influence of daylength on ovarian activity is an interesting instance of a circannual rhythm driven by a circadian rhythm (melatonin secretion from the pineal gland).

Simulation and control of a multiscale conservation law for follicular cells

In the past years, we have designed a multiscale model of the selection process of ovulatory follicles, including the cellular, follicular and ovarian levels , . The model results from the double structuration of the granulosa cell population according to the cell age (position within the cell cycle) and to the cell maturity (level of sensitivity towards hormonal control). In each ovarian follicle, the granulosa cell population is described by a density function whose changes are ruled by conservation laws. The multiscale structure arises from the formulation of a hierarchical control operating on the aging and maturation velocities as well on the source terms of the conservation law. The control is expressed from different momentums of the density leading to integro-differential expressions.

Future work will take place in the REGATE project and will consist in:

- predicting the selection outcome (mono-, poly-ovulation or anovulation / ovulation chronology) resulting from given combinations of parameters and corresponding to the subtle interplay between the different organs of the gonadotropic axis (hypothalamus, pituitary gland and ovaries). The systematic exploration of the situations engendered by the model calls for the improvement of the current implementation performances. The work will consist in improving the precision of the numerical scheme, in the framework of the finite volume method and to implement the improved scheme,

- solving the control problems associated with the model. Indeed, the physiological conditions for the triggering of ovulation, as well as the counting of ovulatory follicles amongst all follicles, define two nested and coupled reachability control problems. Such particularly awkward problems will first be tackled from a particle approximation of the density, in order to design appropriate control laws operating on the particles and allowing them to reach the target state sets.

Connectivity and dynamics of the FSH signaling network in granulosa cells

The project consists in analyzing the connectivity and dynamics of the FSH signaling network in the granulosa cells of ovarian follicles and embedding the network within the multiscale representation described above, from the molecular up to the organic level. We will examine the relative contributions of the G${\alpha }_s$ and $\beta$arrestin-dependent pathways in response to FSH signal, determine how each pathway controls downstream cascades and which mechanisms are involved in the transition between different cellular states (quiescence, proliferation, differentiation and apoptosis). On the experimental ground, we propose to develop an antibody microarray approach in order to simultaneously measure the phosphorylation levels of a large number of signaling intermediates in a single experiment. On the modeling ground, we will use the BIOCHAM (biochemical abstract machine) environment first at the boolean level, to formalize the network of interactions corresponding to the FSH-induced signaling events on the cellular scale. This network will then be enriched with kinetic information coming from experimental data, which will allow the use of the ordinary differential equation level of BIOCHAM. In order to find and fine-tune the structure of the network and the values of the kinetic parameters, model-checking techniques will permit a systematic comparison between the model behavior and the results of experiments. In the end, the cell-level model should be abstracted to a much simpler model that can be embedded into a multiscale one without losing its main characteristics.

Bifurcations in coupled neuronal oscillators.

We have proposed a mathematical model allowing for the alternating pulse and surge pattern of GnRH (Gonadotropin Releasing Hormone) secretion . The model is based on the coupling between two systems running on different time scales. The faster system corresponds to the average activity of GnRH neurons, while the slower one corresponds to the average activity of regulatory neurons. The analysis of the slow/fast dynamics exhibited within and between both systems allows to explain the different patterns (slow oscillations, fast oscillations and periodical surge) of GnRH secretion.

This model will be used as a basis to understand the control exerted by ovarian steroids on GnRH secretion, in terms of amplitude, frequency and plateau length of oscillations and to discriminate a direct action (on the GnrH network) from an indirect action (on the regulatory network) of steroids. From a mathematical viewpoint, we have to fully understand the sequences of bifurcations corresponding to the different phases of GnRH secretion. This study will be derived from a 3D reduction of the original model.