In two works, we have have developed a new avaraging technique for fast-varying continuous-time systems.

In 54, we studied linear systems with fast-varying almost periodic coefficients. We presented a novel transformation of the fast varying coefficients. This transformation enabled us to perform averaging over multiple time-scales for systems with constant delays, where the value of delay is not small (i.e. arbitrarily large with respect to the small parameter). We carried out stability analysis by employing time-varying Lyapunov functions (or functionals for the delayed case). The analysis leads to LMI conditions that are always feasible for small enough parameters.

The paper 53 is devoted to the problem of establishing input-to-state stability (ISS) for linear systems with multiple time-scales. The considered systems contain rapidly-varying, piecewise continuous and almost periodic coefficients with small parameters. The stability analysis relies on a novel system transformation, leading to a new system whose ISS guarantees the ISS of the original one. In this work, we unified this transformation with a new superposition-based system presentation. We employed time-varying Lyapunov functions for ISS analysis, where the novel system presentation plays a crucial role in deriving essentially less conservative compensating upper bounds. The analysis yields conditions of LMI type for ISS, leading to explicit bounds on the small parameters, decay rate and ISS gains. The LMIs are accompanied by suitable feasibility guarantees.

Event-triggered control has the advantage that it can reduce computational burdens of implementing feedback controls, by only changing control values when a significant enough event occurs. In order to decrease the number of needed switches of the control laws, we developed results relying on the theory of the positive systems and comparison systems called interval observers. In the papers 37 and 56, we provided a new input-to-state stabilizing event-triggered feedback design for linear systems with unknown input delays, unknown measurement delays, and unknown additive disturbances. We used the theory of positive systems, interval observers, and a vector version of Halanay’s inequality. We illustrated our method using a marine robotic model.

In contribution 39, to help the analysis of the stability properties of discrete-time systems, we provided a discrete-time vector valued analog of recently developed continuous-time trajectory-based estimates. We used it to provide a discrete-time version of the celebrate Halanay’s inequality. We combined the results with interval observers to prove exponential stability properties for discrete-time linear systems with uncertainties whose arbitrarily long input delays are compensated for by reduction model controls, and a robust global exponential stability result for observers for discrete-time linear systems.

In the papers 38 and 57, we provided new observer designs to simultaneously identify parameters and states of systems whose non-linearities have order two near the origin, which include cubic terms arising in the study of jump phenomena, process control, and bistable models of aerospace systems. This yields local exponential convergence of the state estimation error to zero, basin of attraction estimates, and fixed time parameter identification. We illustrated our result using Duffing’s equation, whose cubic term puts it outside the scope of prior methods

Piece-wise affine systems appear when linear dynamics are defined in different partitions of the state space. This type of systems naturally appears whenever actuators have different stages or saturate or whenever non-linear control laws are obtained as the solution to a parameterised optimization problem as, for instance for systems with feedback laws based on the so-called explicit Model Predictive Control. Even though the dynamics is simple to describe, the stability analysis, performance assessment and robustness analysis are difficult to perform since, due to the often used explicit representation, the Lyapunov stability and dissipation tests are often described in terms of a number of inequalities that increases exponentially on the number of sets in the partition since they are based on the enumeration of the partition transitions. Moreover regional stability and uncertainties corresponding to modification on the partition are difficult to study in this scenario.

To overcome these difficulties we have proposed an implicit representation for this class of systems in terms of ramp functions. The main advantage of such a representation lies on the fact that the ramp function can be exactly characterized in terms of linear inequalities and a quadratic equation, namely a linear complementarity condition. Thanks to the characterization of the ramp function and the implicit description of the PWA system, the verification of Lyapunov inequalities related to piecewise quadratic functions can be formulated as a semidefine programming whenever some co-positivity constraints are relaxed.

We have applied the results to the local analysis and synthesis of PWA control laws. Such a local formulation is based on local conditions for co-positivity of matrices. The proposed results encompass regional stability analysis formulations in the literature.

The stability conditions rely on the solution via convex optimization of piecewise quadratic inequalities in an implicit form, which can also be used to compute lower bounds to the minimum of non-convex and discontinuous functions.

A novel method is proposed for solving quadratic programming problems arising in model predictive control. The method is based on an implicit representation of the Karush–Kuhn–Tucker conditions using ramp functions. The method is shown to be highly efficient on both small and fairly large Quadratic Program problems, can be implemented using simple computer code, and has modest memory requirements. The proposed algorithm shows a performance improvement mainly for large dimension optimization problems whenever few constraints are active in the optimal solution.

One-dimensional hyperbolic systems are useful models for a wide range of natural phenomena, in particular those involving the propagation of some physical quantities, such as the propagation of electricity along a transmission line, of laser signals in optical fibers, of water in open channels, of water or gaz in ridig pipes, of vehicles in a road system, of vibrations on mechanical structures, of living organisms in the presence of certain chemicals, among many others 84. These numerous applications have motivated many recent works on the analysis of these kinds of systems, such as 83, 84, which address questions such as their stability, stabilizability, and controllability.

In the recent work 74, we have obtained criteria for exact and approximate controllability of linear one-dimensional hyperbolic systems in $L^p$ spaces, $p\in [1,+\infty )$, under the form of Hautus-type conditions expressed in the frequency domain. The strategy of this work relies on classical transformations of hyperbolic systems into difference equations (see, e.g., 84, 88), whose solutions can be expressed under the form of a suitable representation formula, the latter being useful for providing upper bounds on the minimal controllability times. The main controllability results of 74 are obtained by expressing the difference equations under consideration as input/output infinite-dimensional systems, which allows one to make use of the formalism developped by Y. Yamamoto in 94, 95, 93, similarly to what was previously done in 27 for the approximate and exact controllability of difference equations.

Finally, 74 considers the special case of flows in networks, identifying a topological obstruction for controllability and deducing Hautus-type controllability criteria for these systems, which reduce to Kalman criteria in the case where the delays of the corresponding difference equation are commensurate.

The dynamics of systems in which phenomena of different nature are present may contain different time scales. This is the case, for instance, of electric motors, in which the electrical time scale is typically much faster than the mechanical one. A natural question in these situations, which is the basis of the singular perturbation theory, is whether one can approximate the fastest time scale by an instantaneous process. For finite-dimensional systems, its answer is given by Tikhonov's theorem, which states roughly that such an approximation is valid as soon as the dynamics of the fastest time scale is stable 90.

However, many systems in practice involving propagative, diffusive, or reactive phenomena are naturally modeled as infinite-dimensional systems. Singular perturbation in infinite dimension has attracted much interest from researchers in recent years, and it was highlighted in particular that classical results for finite-dimensional systems may fail to hold true in general in such a context, according to which system evolves in a fast time scale (see, e.g., 92). Most works in the literature on singular perturbation for infinite-dimensional systms deal only with specific systems, showing approximation properties, when they hold, through the use of Lyapunov functions, as done in 92.

In order to get a better understanding of when singular perturbation methods work well for infinite-dimensional systems, we have revisited, in the recent work 50, the system considered in 92 made of a slow ODE coupled with a fast one-dimensional transport PDE. By adopting a spectral point of view of the problem, we have obtained sharper conditions under which approximation properties hold true, improving the results of 92. In addition, the spectral methods used in 50 have provided valuable insight on the behavior of singularly perturbed systems in infinite dimension, which might be used to address more general systems than the one from that reference.

Since the seminal works 86, 87, it has been known that spectral values of some families of time-delay systems of large multiplicity are often dominant (i.e., they attain the spectral abcissa of the system, and determine thus its asymptotic behavior), a property which came to be known as the multiplicity-induced-dominancy (MID) property. The validity of this property has an important impact in the stabilization of time-delay systems, since, when it holds, one may stabilize a system by selecting its free parameters in order to ensure that it admits a spectral value of large multiplicity which is dominant and has negative real part, ensuring stability.

In 72, we have studied the MID property for a family of delay-differential equations with a single delay when considering spectral values of multiplicity equal to $n+1$, where $n$ is the largest order of derivation appearing in the equation. In addition to several characterizations of spectral values with this multiplicity, 72 provides sufficient conditons for dominance and studies in more details the case of a chain of $n$ integrators, for which more precise results can be provided, together with links between multiple spectral values and the optimization of the spectral abscissa. The derived results show how the delay can be further exploited as a control parameter and are applied to some problems of stabilization of standard benchmarks with prescribed exponential decay.

The work 73 extends the results of 72 to roots of multiplicity larger than or equal to $n+1$, providing extensive characterizations of such roots. Many configurations are considered according to which coefficients of the system are assumed to be fixed and which ones are assumed to be available for choice, with a special attention paid to the so-called control-oriented configuration, in which the coefficients of highest order of the non-delayed part of the system are fixed, and all others are free. Sufficient conditions for dominance, similar to those of 72, are also provided in 73, together with a detailed analysis of two examples.

For instance, in 19, we characterize the MID property in the scalar neutral case with respect to the system parameters. Particular attention is paid to the so-called over-order multiplicities corresponding to real double and triple characteristic roots.

While most results on the MID property consider only systems with a single delay, the work 31 has addressed a first-order system with two delays. By considering the ratio between the smallest and the largest delay as a parameter, 31 provides a careful analysis of the behavior of the spectrum of the system with respect to this parameter, which is used to establish the MID property for roots of maximal multiplicity of such a class of systems. As a consequence, 31 also establishes that the inclusion of a second delay may help in stabilizing time-delay systems with constraints on their coefficients, with respect to a classical proportional-delayed controller. As an extension of the methodology of the latter, 59 characterizes the MID property for second-order systems controlled by a two-delay "block". As an application of which, the problem of stabilization of the classical pendulum with exclusive access to the delayed position is treated.

In recent years, the Team highligted an extension of the MID property called coexistent-real-roots-induced-dominancy (CRRID), see for instance 85, 82 applying for LTI functional differential equations of retarded type. The CRRID property consists in conditions on an LTI functional dynamical system's parameters guaranteeing the dominancy of coexistence of real spectral values. In 45, we extend such a property to a class of neutral systems, and exploit it in the boundary control of the standard transport equation. Namely, by using the CRRID property, we show that one can arbitrarily and robustly prescribe the exponential decay of the closed-loop transport solution, yielding the prospect of applying the CRRID partial poles placement methodology to hyperbolic PDE's.

We deal in 18 with a control-affine problem with scalar control subject to bounds, a scalar state con- straint and endpoint constraints of equality type. For the numerical solution of this problem, we propose a shooting algorithm and provide a sufficient condition for its local convergence. We exhibit an example that illustrates the theory.

We investigate in 75 the convergence of the Generalized Frank-Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. In a previous work 34, we had established some rates of convergence for this method, at the continuous level. We analyze here the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm. The article focuses on the theta-scheme, which we introduced in 25. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.

In the preprint 76, we formulate and investigate a mean field optimization (MFO) problem over a set of probability distributions $\mu$ with a prescribed marginal $m$. The cost function depends on an aggregate term, which is the expectation of $\mu$ with respect to a contribution function. This problem is of particular interest in the context of Lagrangian potential mean field games (MFGs) and their discretization. We provide a first-order optimality condition and prove strong duality. We investigate stability properties of the MFO problem with respect to the prescribed marginal, from both primal and dual perspectives. In our stability analysis, we propose a method for recovering an approximate solution to an MFO problem with the help of an approximate solution to an MFO with a different marginal $m$, typically an empirical distribution. We combine this method with the stochastic Frank-Wolfe algorithm, which we had introduced in 24, to derive a complete resolution method.

In the paper 32, we derived feedback control laws for isolation, contact regulation, and vaccination for infectious diseases, using a strict Lyapunov function. We use an SIQR (Susceptible, Infected, Quarantined, Recovered individuals) epidemic model describing transmission, isolation via quarantine, and vaccination for diseases to which immunity is long-lasting. Assuming that mass vaccination is not available to completely eliminate the disease in a time horizon of interest, we provided feedback control laws that drive the disease to a small endemic equilibrium. We prove the ISS robustness property on the entire state space, when the immigration perturbation is viewed as the uncertainty. We use an ISS Lyapunov function to construct the feedback control laws. A key ingredient in our analysis is that all compartment variables are present not only in the Lyapunov function, but also in a negative definite upper bound on its time derivative. We illustrate the efficacy of our method through simulations. Since the control laws are feedback, their values are updated based on data acquired in real time. We also discussed the degradation caused by the delayed data acquisition occurring in practical implementations, and we derive bounds on the delays under which the ISS property is maintained when delays are present.

An internal model control scheme is proposed in 49 to compensate both a long dead-time of a system and a harmonic disturbance. The controller is based on an inversion of the first-order model used to approximate the system dynamics together with an input delay. Two other components of the controller consist of a filter and an additional delay by which the harmonic modes are targeted via adjusting the control loop gain and phase shift. The design of the filter-delay pair is fully analytical and the implementation of the scheme is straightforward. The main attention is paid to the complete compensation of a single harmonic disturbance. Besides, an extension of the scheme is proposed to target a double harmonic disturbance. Increased attention is paid to the robustness aspects of the schemes. Outstanding performance in terms of harmonic disturbance compensation of the proposed schemes is demonstrated on a series of laboratory experiments.

The paper 48 presents a controller design for systems suffering from multi-harmonic periodic disturbance and substantial input time-delay. It forms an alternative approach to Repetitive Control where the goal is to stabilize a closed-loop that encapsulates an explicit time-delay model of the periodic signal. The proposed controller design is based on the Internal Model Control (IMC) framework, and it consists of the inverse system model and an appropriate distributed delay with an overall length related to the period of the disturbance. The properness of the controller can be ensured by utilizing a low-pass filter, however, such a component is shown to be unnecessary when the relative order of the system model is one. This fact makes the alternative approach especially suitable for systems approximated by a first-order model with input time-delay, leading to a straightforward controller design thanks to its simple structure and attainable conditions. Stability of the configuration is guaranteed by an ideal IMC framework. For further performance and robustness requirements for the non-ideal case the tuning of the controller is posed as a weighted-$H_{\infty }$ optimization problem where frequency-, spectral-and time-domain requirements are formulated as constraints. The overall control design is experimentally verified on a laboratory setup that has high-order dynamics approximated by a first-order model with input delay.

Following our project on the modeling and analysis of healthy and unhealthy cell population dynamics in leukemia, we have considered a nonlinear system with distributed delays where the parameters depend on growth-factor concentrations. Here, a change in one of the growth factor concentrations may lead to a switch in the corresponding model parameter. We have achieved a network representation of the switching system involving nodes and edges. Each node stands for a full-fledged nonlinear system with distributed delays where the parameters are constant. For each node, a stable positive steady state may exist. In this network framework, a change in the growth-factor concentration is interpreted as a transition from one node to another. We have proposed a method which provides a (sub)optimal therapeutic strategy, guiding the density of cells from an abnormal state towards a healthy one, through multiple drug infusions.