A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.

This paper presents coordination algorithms for groups of mobile agents performing deployment and coverage tasks. As an important modeling constraint, we assume that each mobile agent has a limited sensing or communication radius.
Based on the geometry of Voronoi partitions and proximity graphs, we analyze a class of aggregate objective functions and propose coverage algorithms in continuous and discrete time.
These algorithms have convergence guarantees and are spatially distributed with respect to appropriate proximity graphs. Numerical simulations illustrate the results.

An approach for solving inverse problems involving obstacles is proposed. The approach uses a level-set method which has been shown to be effective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry.We develop two computational methods based on this idea.One method results in a nonlinear time-dependant partial differential equation for the level-set function whose evolution minimizes the residual in the data fit. The second method is an optimization that generates a sequence of level-set functions that reduces the residual. The methods are illustrated in two applications : a deconvolution problem and a diffraction screen reconstruction problem.

Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary $\mathbb{R}^M$-valued reference signal r of class W1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that ${\varphi}(t)\| e(t)\| < 1$ for all t ≥ 0, where φ a prescribed real-valued function of class W1,∞ with the property that φ(s) > 0 for all s > 0 and $\liminf_{s\rightarrow\infty}{\varphi}(s)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u(t)=- \alpha ({\varphi}(t)\|e(t)\|)e(t)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha ({\varphi}(\cdot )\|e(\cdot )\|)$.

We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset S of a reference domain, and the complement of S is made of two other “phases”, the “void” and a fictitious “liquid” that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure S, which is the total work of the pressure and internal forces at the equilibrium displacement. In order to prevent from homogenization we add a penalization on the perimeter of S. We propose an approximation of our problem in the framework of Γ-convergence, based on an approximation of our three phases by a smooth phase-field. We detail the numerical implementation of the approximate energies and show a few experiments.

We study stability and stabilizability properties of systems with discontinuous righthand side (with solutions intended in Filippov's sense) by means of locally Lipschitz continuous and regular Lyapunov functions. The stability result is obtained in the more general context of differential inclusions. Concerning stabilizability, we focus on systemsaffine with respect to the input: we give some sufficient conditions for a system to be stabilized by means of a feedback law of the Jurdjevic-Quinn type.


The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions is studied. When boundary conditions bear on spatial derivatives up to order 2, the exact controllability result by Russell-Zhang is directly proved by means of Hilbert Uniqueness Method. When only the first spatial derivative at the right endpoint is assumed to be controlled, a quite different analysis shows that exact controllability holds too. From this last result we derive the exact boundary controllability for nonlinear KdV equation on bounded domains, for sufficiently small initial and final states.

We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-${\mathcal{KL}}$ estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-${\mathcal{KL}}$ estimate, exists if and only if the class-${\mathcal{KL}}$ estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all class-${\mathcal{KL}}$ estimates are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition is that the two positive semidefinite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for differential equations and differential inclusions that have appeared in the literature.

In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu)$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop systemgenerates a C 0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors andgeneralized eigenvectors that forms a Riesz basis for the state Hilbert space.Furthermore, we show that when the weight $\mu>\frac{1}{2}$ , for any time delay,we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When $\mu=\frac{1}{2}$ , we show that the system is at most asymptotically stable. When $\mu<\frac{1}{2}$ , the system is always unstable.

For constant linear systems we are introducing integral reconstructors and generalized proportional-integral controllers, which permit to bypass the derivative term in the classic PID controllers and more generally the usual asymptotic observers. Our approach, which is mainly of algebraic flavour, is based on the module-theoretic framework for linear systems and on operational calculus in Mikusiński's setting. Several examples are discussed.

We prove that any bounded sequence in a Hilbert homogeneous Sobolev space has a subsequence which can be decomposed as analmost-orthogonal sum of a sequence going strongly to zero in the corresponding Lebesgue space, and of a superposition of termsobtained from fixed profiles by applying sequences of translations and dilations. This decomposition contains in particular the variousversions of the concentration-compactness principle.

We consider the wave equation damped with a boundary nonlinear velocity feedback p(u'). Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction of a special weight function (that depends on the behavior of the function ρ in zero), and on a new nonlinear integral inequality.

In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.

For boundary or distributed controls, we get an approximate controllability result for the Navier-Stokes equations in dimension 2 in the case where the fluid is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions.

The ill-posed problem of solving linear equations in the space of vector-valued finiteRadon measures with Hilbert space data is considered. Approximate solutions are obtainedby minimizing the Tikhonov functional with a total variation penalty. The well-posednessof this regularization method and further regularization properties are mentioned.Furthermore, a flexible numerical minimization algorithm is proposed which convergessubsequentially in the weak* sense and with rate 𝒪(n-1) in terms of the functional values. Finally, numerical results for sparsedeconvolution demonstrate the applicability for a finite-dimensional discrete data spaceand infinite-dimensional solution space.

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems.

This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. They mainly have two effects (at first order with respect to the period ε): first, they add a dispersive term to the homogenized equation, and second, they yield an effective Fourier boundary condition.

The purpose of this paper is to show that the method of controlledLagrangians and its Hamiltonian counterpart (based on the notionof passivity) are equivalent under rather general hypotheses. Westudy the particular case of simple mechanical control systems(where the underlying Lagrangian is kinetic minus potentialenergy) subject to controls and external forces in some detail.The equivalence makes use of almost Poisson structures (Poissonbrackets that may fail to satisfy the Jacobi identity) on theHamiltonian side, which is the Hamiltonian counterpart of a classof gyroscopic forces on the Lagrangian side.