Uniqueness of the optimal control is obtained by assuming certain conditions on the crowding effect of the species. Moreover, an approximation procedure for the unique optimal control isdeveloped.

If a smooth nonlinear affine control system has a controllable linear approximation, a standard technique for constructing a smooth (linear) asymptotically stabilizing feedbackcontrol is via theLQR (linear, quadratic, regulator) method. The nonlinear system may not have a controllable linear approximation, but instead may be shown to be small (or large) time locally controllable via a high order, homogeneous approximation. In this case one can attempt to construct an asymptotically stabilizing feedback control as the optimal control, relative to a cost functional with homogeneous integrand, for the approximating system. Necessary, and some sufficient, conditions for the existence of a smooth (real analytic), stabilizing feedback control of this form are given. For some systems which satisfy these necessary conditions, the specific form of a stabilizing control is established.

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.

The concepts of stability, attractivity and asymptotic stability for systems subject to restrictions of the input values are introduced and analyzed in terms of Lyapunov functions. A comparisonwith the well known input-to-state stability property introduced by Sontag is provided. We use these concepts in order to derive sufficient conditions for global stabilization for triangular andfeedforward systems by means of saturated bounded feedback controllers and also recover some recent results due to Teel.

This paper is concerned with the null controllability of systems governed by semilinear parabolic equations. The control is exerted either on a small subdomain or on a portion of the boundary. We prove that the system is null controllable when the nonlinear term f(s) grows slower than s . log|s| as |s| → ∞.

Dans cet article nous prouvons un nouveau résultat d'existence pour une classe de problèmes d'optimisation de forme assez générale. Les ouverts que nous considérons possèdent une contrainte de nature géométrique sur la normale intérieure. Ce travail est motivé par la formulation variationnelle d'un problème à frontière libre dont la solution possède cette propriété géométrique.

We consider a semilinear heat equation in an unboundeddomain Ω with partially known initial data. The insensitizing problemconsists in finding a control function such that some functional of thestate is locally insensitive to the perturbations of these initial data. Forbounded domains Bodart and Fabre proved the existence of insensitizing controls of the norm of the observation of the solution in an opensubset of the domain. In this paper we prove similar results when Ω isunbounded. We consider the problem in bounded domains of the formΩr = Ω ∩ Br where Br denotes the ball centered in zero of radius r Weshow that for r large enough the control proposed by Bodart and Fabrefor the problem in Ωr , provides an insensitizing control for our problemin Ω.

This paper considers control affine systems in $\lambda _{2}$ with two inputs, and gives necessary and sufficient conditions for dynamic feedback linearization of these systems with the restriction that the "linearizing outputs" must be some functions of the original state and inputs only. This also gives conditions for non-affine systems in $\lambda _{2}$ .

We consider a hybrid, one-dimensional, linear system consisting in two flexible strings connected by a point mass. It is known that this system presents two interesting features. First, it is wellposed in an asymmetric space in which solutions have one more degree of regularity to one side of the point mass. Second, that the spectral gap vanishes asymptotically. We prove that the first property is a consequence of the second one. We also consider a system in which the point mass is replaced by a string of length 2ε and density 1/2ε. We show that, as ε → 0, thesolutions of this system converge to those of the original one. We also analyze the convergence of the spectrum and obtain the well-posedness of the limit system in the asymmetric space as a consequence of non-standard uniform bounds of solutions of the approximate problems. Finally we consider the controllability problem. It is well known that the limit system with L-controls on one end is exactly controllable in an asymmetric space. We show how this result can be obtained as the limit when ε → 0 of partial controllability results for the approximate systems inwhich the number of controlled frequencies converges to infinity as ε → 0. This is done by means of some new results on non-harmonic Fourier series.

To find a zero of a maximal monotone operator, an extension of the Auxiliary Problem Principle to nonsymmetric auxiliary operators is proposed. The main convergence result supposes a relationship between the main operator and the nonsymmetric component of the auxiliary operator. When applied to the particular case of convex-concave functions, this result implies the convergence of the parallel version of the Arrow-Hurwicz algorithm under the assumptions of Lipschitz and partial Dunn properties of the main operator. The latter is systematically enforced by partial regularization. In the strongly monotone case, it is shown that the convergence is linear in the average. Moreover, if the symmetric part of the auxiliary operator is linear, the Lipschitz property of the inverse suffices to ensure a linear convergence rate in the average.

This work is devoted to numerical experiments for multidimensional Spectral Inverse Problems. We check the efficiency of the algorithm based on the BC-method, which exploits relations between Boundary Control Theory and Inverse Problems. As a test, the problem for an ellipse is considered. This case is of interest due to the fact that a field of normal geodesics loses regularity on a nontrivial separation set. The main result is that the BC-algorithm works quite successfully in spite of this complication. A theoretical introduction to the BC-method is included.

For linear control systems with coefficients determined by a dynamical system null controllability is discussed. If uniform local null controllability holds, and if the Lyapounov exponents of thehomogeneous equation are all non-positive, then the system is globally null controllable for almost all paths of the dynamical system. Even if some Lyapounov exponents are positive, anirreducibility assumption implies that, for a dense set of paths, the system is globally null controllable.

We consider transmission problems for general second order linear hyperbolic systems having piecewise constant coefficients in a bounded, open connected set with smooth boundary and controlled through the Dirichlet boundary condition. It is proved that such a system is exactly controllable in an appropriate function space provided the interfaces where the coefficients have a jump discontinuity are all star-shaped with respect to one and the same point and the coefficients satisfy a certain monotonicity condition.

We consider the problem : (P) Minimize $\lambda _{2}$ over u ∈ K ∩ X, where α≥ 0, β > 0, K is a closed convex subset of L 2(Ω), and the last additive term denotes the BV-seminorm of u, T is a linear operator from L 2 ∩ BV into the observation space Y. We formulate necessary optimality conditions for (P). Then we show that (P) admits, for givenregularization parameters α and β, solutions which depend in a stable manner on the data z. Finally we study the asymptotic behavior when α = β → 0. The regularized solutions ûβ of (P) converge to the L 2 ∩ BV minimal norm solution of the unregularized problem. The rate of convergence is β½ when the minimum-norm solution û is smooth enough.

This article deals with the local sub-Riemannian geometry on ℜ3, (D,g) where D is the distribution ker ω, ω being the Martinet one-form : dz - ½y2dxand g is a Riemannian metric on D. We prove that we can take g as a sum of squares adx2 + cd2 . Then we analyze the flat case where a = c = 1. We parametrize the set of geodesics using elliptic integrals. This allows to compute the exponential mapping, the wave front, the conjugate and cut loci and the sub-Riemannian sphere. A direct consequence of our computations is to show that the sphere and the distance function are not sub-analytic. Some of these computations are generalized to a one parameter deformation of the flat case.