In , , , it is shown that every linear functional system (e.g., PD systems, differential time-delay systems, difference systems) is equivalent to a linear functional system defined by an upper block-triangular matrix of functional operators: each diagonal block is respectively formed by a generating set of the elements of the system satisfying a purely $i$-codimensional system. Hence, the system can be integrated in cascade by successively solving (inhomogeneous) $i$-codimensional linear functional systems to get a Monge parametrization of its solution space . The results are based on an explicit construction of the grade/purity filtration of the module associated with the linear functional system. This new approach does not use involved spectral sequence arguments as is done in the literature of modern algebra , . To our knowledge, the algorithm obtained in , is the most efficient algorithm existing in the literature of non-commutative algebra. It was implemented in the PurityFiltrationpackage developed in Maple (see Section  ) and in the homalgpackage of GAP 4 (see Section  ).

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 , 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 , . 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.

In , we give a complete constructive form of the classical Fitting's lemma in module theory which studies the relation between equivalences of linear systems and isomorphisms of their associated finitely presented modules. The corresponding algorithms were implemented in the OreMorphismspackage (see Section  ).

Many partial differential systems appearing in mathematical physics, engineering sciences and mathematical biology are nonlinear. Unfortunately, algebraic analysis and $D$-module theory, which were successful for the algorithmic study of linear partial differential systems, cannot consider nonlinear PD systems. This project aims at developing a generalization of the algebraic analysis approach to certain classes of quasilinear partial differential systems appearing in mathematical physics and engineering sciences (e.g., Burgers' equation, shallow water, Euler equations for an incompressible fluid, traffic flow, gas flow). In , we have shown how constructive methods of differential algebra and algebraic analysis could be combined to extend results obtained in and for linear PD systems and how they allowed us to compute conservation laws, internal symmetries and decompositions of the solution space of certain classes of quasilinear PD systems. The algorithms have been implemented in the JanetMorphismspackage dedicated to this new approach (see Section  ).

In , , , it was shown how the fractional representation approach to analysis and synthesis problems developed by Vidyasagar, Desoer, Callier, Francis, Zames..., could be recast into a modern algebraic analysis approach based on module theory (e.g., fractional ideals, algebraic lattices) and the theory of Banach algebras. This new approach successfully solved open questions in the literature. Basing ourselves on this new approach, we explain in why the non-commutative geometry developed by Alain Connes is a natural framework for the study of stabilizing problems of infinite-dimensional systems. Using the 1-dimensional quantized calculus developed in non-commutative geometry and results obtained in , , , we show that every stabilizable system and their stabilizing controllers naturally admit geometric structures such as connections, curvatures, Chern classes...  These results are the first steps toward the use of the natural geometry of the stabilizable systems and their stabilizing controllers in the study of the important $H_{\infty }$and $H_2$-problems.

In , we study the stabilization problem of a linear system formed by a simple integrator and a time-delay. We show that the stabilizing controllers of such a system can be be rewritten as the closed-loop system defined by the stabilizing controllers of the simple integrator and a distributed delay. This result is used to study tracking problems appearing in the study of inertially stabilized platforms for optical imaging systems.

In , we present results on algorithmic methods for singular boundary problems for linear ordinary differential equations. Moreover, the implementation of integro-differential operators and the corresponding algorithms for boundary problems in the computer algebra system Maple is discussed. The operations implemented for regular boundary problems include computing Green's operators as well as composing and factoring boundary problems. For singular boundary problems, compatibility conditions and generalized Green's operators can be computed. In  , we give a survey and new results on our algebraic and symbolic approach to boundary problem developed over the last years. The construction of integro-differential operators and polynomials over an integro-differential algebra is described in detail along with a generic implementation of the corresponding canonical forms and algorithms.

In a joint work, Stefan Müller and G. Regensburger propose a notion of generalized mass action systems that could serve as a more realistic model for reaction networks in intracellular environments; classical mass action systems capture chemical reaction networks in homogeneous and dilute solutions, see e.g. and . We show that several results of Chemical Reaction Network Theory carry over to the case of generalized mass action kinetics. Our main result essentially states that, if the sign vectors of the stoichiometric and the kinetic-order subspace coincide, there exists a unique positive complex balancing equilibrium in every stoichiometric compatibility class.

In a cooperation with Hansjörg Albrecher, Corina Constantinescu (both University of Lausanne), Zbigniew Palmowski (University of Wroclaw), and Markus Rosenkranz (University of Kent), we developed a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients and the corresponding Green's operators (integral operators), generalizing the approach from . We obtain also closed-form solutions for more specific functions in the compound Poisson risk model.

A. Quadrat and G. Regensburger (in the frame of his grant) are working on a new approach for studying algebraic and algorithmic properties of systems of linear ordinary integro-differential equations with boundary conditions. In a recent series of papers, in particular, , , V. V. Bavula obtained numerous algebraic results for modules over the ring of integro-differential operators with polynomial coefficients using generalized Weyl algebras. We are interested in how far some of his approach can be made algorithmic and generalized to boundary problems. First results in this direction were presented at the Journées Nationales de Calcul Formel (JNCF 2011).

In order to yield the set of all stabilizing controllers of a large class of MIMO fractional time-delay systems, we may look for coprime factorizations of the transfer function and their corresponding Bézout factors. As primary results, in considering $H_{\infty }$stability, left coprime factorizations and left Bézout factors have been determined analytically from the transfer function. Then a particular stabilizing controller has been derived. We also proved the existence of right coprime factorizations.

The $H_{\infty }$-stability analysis of (fractional) delay systems of neutral type has been studied in . It was shown that the chains of poles are asymptotic to vertical axes which position depend on the roots of a certain polynomial (easily determined from the given transfer function). In the case of roots of multiplicity one was completely treated whereas the case of roots of multiplicity greater than one was considered only in the particularc case of neutral systems whose transfer function is a product of transfer functions of neutral systems with one delay. This year, we have studied more deeply the case of roots of multiplicity greater than one for both, standard and fractional delay systems.

In , we have treated the time-delay linear systems control design in the framework of complete and partial information. We were able to find linear controllers that increase the first stability window imposing at the same time that the delay-free system is stable using some properties about the norms of the state-space matrices. Our method treated the design problem by numeric routines based on Linear Matrix Inequalities (LMI) arisen from classical linear time invariant system theory coupled together with a unidimensional search. Both the state and output feedback design, were solved.

The technique, based on the notion of interval observer, is a recent state estimation technique, which offers the advantage of providing information on the current state of a system at any instant of time. The firts interval observers where relying on the assumption that the system was cooperative or, roughly speaking, "almost" cooperative.

In the contribution , we have proved that, for any time-invariant exponentially stable linear system with additive disturbances, time-varying exponentially stable interval observers can be constructed. The technique of construction relies on the Jordan canonical form that any real matrix admits and on time-varying changes of coordinates for elementary Jordan blocks which lead to cooperative linear systems. We applied our to the case of linear systems with input and output that are detectable.

The paper focused on the analysis and the design of families of interval observers for linear systems with a point-wise delay. First, we proved that classical interval observers for systems without delays are not robustwith respect to the presence of delays, no matter how small delays are. Next, we have shown that, in general, for linear systems with delay, the classical interval observers endowed with a point-wise delay are unstable. A new type of design of interval observers enabling to circumvent these obstacles is proposed. It provides with framers that incorporate distributed delayterms. The proposed interval observers are assessed through a non-linear biotechnological model.

In and , for families of partial differential equations (PDEs) with particular stabilizing boundary conditions, we have constructed strict Lyapunov functions. The PDEs under consideration were parabolic and, in addition to the diffusion term, might contain a nonlinear source term plus a convection term. The boundary conditions were the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions relied on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions were used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function made it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

In , we have considered a family of time-varying hyperbolic systems of balance laws. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides with a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach which gives a time-varying strict Lyapunov function which allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of Input-to-State Stability (ISS) type.

In the work , we propose a new approach for the stabilization of nonlinear time-varying forward-complete systems with delay in the input. This approach is a new reduction modelapproach, which relies on operators of a new type. It presents three advantages. First, the corresponding control laws do not include distributed terms. Second, it yields closed-loop systems with positive solutions that can be easily derived. Finally, the stabilized systems possess some robustness properties (for instance of ISS type) that can be estimated.

In the contributions and , we revisited the problem of constructing globally asymptotically stabilizing control laws for nonlinear systems in feedback form with a known pointwise delay in the input. The result we obtained covers a family of systems wider than those studied in the literature and endows with control laws with a single delay, in contrast to those given in previous works which include two distinct pointwise delays or distributed delays. The strategy of design is based on the construction of an appropriate Lyapunov-Krasovskii functional. An illustrative example ends the paper.

The usual adaptive control problem is to design a controller that forces all trajectories of the system to track a prescribed trajectory, while keeping the estimator of the unknown constant parameter vector bounded. We studied the important and more difficult adaptive tracking and estimation problem of simultaneously (1) forcing the trajectories of the system to track a given trajectory and (2) identifying the parameter vector. This problem was known to be solvable when the regressor satisfies a persistency of excitation condition, but the known results did not provide a strict Lyapunov function for the augmented error dynamics and so could not certify good performance such as ISS with respect to uncertainty added to the controller.

Our main result from covers adaptively controlled first-order nonlinear systems that satisfy the persistency of excitation condition and are affine in the unknown parameter vector. Our contribution consists in particular in constructing global strict Lyapunov functions for the augmented error dynamics In , we extended to adaptive tracking for nonlinear systems in feedback form with multiple inputs and unknown high-frequency gains multiplying the controllers. The control gains must be identified as part of the control objective. High-frequency gains are important for electric motors, flight dynamics, and robot manipulators. We used a persistency of excitation condition that again ensured tracking and parameter identification and led to the explicit construction of a global strict Lyapunov function for the closed loop augmented error dynamics. The strict Lyapunov function was key to proving integral ISS with respect to time varying uncertainties added to the unknown parameters.

New results have been obtained toward the computation of feasible sets for linear model predictive control techniques, based on set relations and not on the conventional orthogonal projection. Further, the problem of computing suitable inner approximations of the feasible sets was considered. Such approximations are characterized by simpler polytopic representations, and preserve essential properties as convexity, positive invariance, inclusion of the set of expected initial states.

Efficient dedicated methods have been developed for Hinfinity controller synthesis. However, such methods require translating the design objectives using weighting filters, whose tuning is not easy; in addition they lead to high order controllers which have to be reduced. A particle swarm optimization method is used to solve both problems successively: after having optimized the filters according to the design specifications using a full order controller but no reformulation of the constraints, a reduced-order one is computed using the obtained filters. Experimental tests for a pendulum in the cart exhibit much than satisfactory results , . In addition, the design of Hinfinity static output feedback has been done and tested using the Compleib librairy and exhibiting similar results to those obtained with the HIFOO solver .

The identification of systems is a key feature to get representative models and so to design efficient control laws. Numerous methods exist to identify the parameters of nonlinear systems when the global structure of the model is given. The problem appears to be much more difficult when this structure is unknown (symbolic regression). We introduces ant colony optimization (ACO) in solving the problem of non linear systems identification in the case of an unknown structure of the mode . Numerical results prove the viability of the approach.

Many studies have considered the solution of Unit Commitment problems for the management of energy networks. In this field, earlier work addressed the problem in determinist cases and in cases dealing with demand uncertainties. In this paper , the authors develop a method to deal with uncertainties related to the cost function. Indeed, such uncertainties often occur in energy networks (waste incinerator with a priori unknown waste amounts, cogeneration plant with uncertainty of the sold electricity price...). The corresponding optimization problems are large scale stochastic non-linear mixed integer problems. The developed solution method is a recourse based programming one. The main idea is to consider that amounts of energy to produce can be slightly adapted in real time, whereas the on/off statuses of units have to be decided very early in the management procedure. Results show that the proposed approach remains compatible with existing Unit Commitment programming methods and presents an obvious interest with reasonable computing loads.

In order to better take into account physiological phenomena as well as better understand the effect of the new anti-FLT3 therapy for AML, we have modified the model M. Adimy and F. Crauste in two ways :

- we have introduced a modeling of quick self-renewal of cells in each stage of maturation.

- we have modeled each phase of the proliferating compartment (that is $G_1$, $S$, $G_2$and $M$) separately. For the time being, only the $M$-phase is supposed to have a fixed time duration as it is well-known that the short time necessary to perform mitosis is hardly submitted to any variation.

In parallel to this modeling task, Faten Merhi and Annabelle Ballesta have performed experiments in order to identify parameters of the model. These experiments (which will continue in 2012) tend to show that we will converge to a patient-dependant model.

The paper focuses on the stability analysis of a delay-differential system encountered in modeling immune dynamics during imatinib treatment for chronic myelogenous leukemia (CML). A simple algorithm is proposed for the analysis of delay effects on the stability. Such an algorithm takes advantage of the particular structure of the dynamical interconnections of the model. The analysis shows that the model yields three fixed points, two of which are always unstable and one of which is sometimes stable. The stable fixed point corresponds to an equilibrium solution in which the leukemia population is kept below the cytogenetic remission level. This result implies that, during imatinib treatment, the resulting anti-leukemia immune response can serve to control the leukemia population. However, the rate of approach to the stable fixed point is very slow, indicating that the immune response is largely ineffective at driving the leukemia population towards the stable fixed point. To extend the stability analysis with respect to the delay parameter, we conduct a global nonlinear analysis to demonstrate the existence of unbounded solutions. We provide sufficient conditions based on initial cell concentrations that guarantee unbounded solutions and comment on how these conditions can serve to predict whether imatinib treatment will result in a sustained remission based on a patient's initial leukemia load and initial anti-leukemia $T$cell concentration.

We worked on the control of human heart rate during exercise , which is a problem that has implications for the development of protocols for athletics, assessing physical fitness, weight management, and the prevention of heart failure. We provided new stabilization techniques, based on the notion of tracking, for a recently-proposed nonlinear model for human heart rate response that describes the central and peripheral local responses during and after treadmill exercise. The control we proposed respect the sign constraint imposed by the model and we proposed observers to cope with the case (important from a practical point of view) where some of the variables are not measured.

The works , was inspired by the recent Deepwater Horizon oil spill disaster. The goal was to develop and implement robotic surveying methods to evaluate the immediate and longer term environmental impacts of the oil spills. It was joint with Michael Malisoff from the LSU and a Georgia Tech robotics team led by Fumin Zhang. Robotic surveying methods provide a low cost and convenient way to collect data in marsh areas that are difficult to access by human based methods. We designed strict Lyapunov functions that made it possible to use ISS to quantify the robustness of collision avoiding curve tracking controllers under controller uncertainty. The controllers are designed to keep the robot a fixed distance from, but moving parallel to, a two dimensional curve. Four challenges in applying ISS to curve tracking are (a) the need to restrict the magnitudes of the uncertainty to keep the state in the state space and build a strict Lyapunov function, (b) the likelihood of time delays in the controllers in real time applications, (c) possible parameter uncertainty such as unknown control gains, and (d) generalizations to three dimensional curve tracking. We overcame challenge (a) by finding maximum bounds on the perturbations that maintain forward invariance of a nested family of hexagons that fill the state space and transforming a nonstrict Lyapunov function into a strict Lyapunov function on the full state space. To address challenge (b), we used a Lyapunov-Krasovskii approach from to convert the strict Lyapunov function into a Lyapunov-Krasovskii functional. This led to an upper bound on the admissible controller delay that can be introduced into the controller while still maintaining ISS.