In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constantand prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higherdimensions (cf. [15]). 
Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if then-dimensional problem always reduces to the three-dimensional case. In this paper we will prove that this is true in the euclidean as well as inthe noneuclidean case. At first glance one might consider this a trivial problem, but we will also give an example showing that this is not thecase. 


The problem of partial exact boundary controllability and exponential stability for the higher-dimensional linear system of thermoelasticity isconsidered. By introducing a velocity feedback on part of the boundary of the thermoelastic body, which is clamped along the rest of its boundary, toincrease the loss of energy, we prove that the energy in the system of thermoelasticity decays to zero exponentially. We also give a positive answer toa related open question raised by Alabau and Komornik for the Lamé system. Via Russell's “Controllability via Stabilizability" principle, we thenprove that the thermoelastic system is partially controllable with boundary controls without smallness restrictions on the coupling parameters. 


In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differentialequation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surroundingmanifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold,defined by Rabier and Rheinboldt for determined implicit differential equations, to underdetermined implicit differential equations. With thisgeometric framework we define a class of well-posed implicit differential equations for which we locally obtain, by means of a reductionprocedure, a controlled vector field on a submanifold W of the surrounding manifold X. We then show that the implicit Lagrange problem leadsto, locally, an explicit optimal control problem on the submanifold W for which the Pontryagin maximum principle is naturally used.

We present here a return method to describe some attainable sets on an interval of the classical Burger equation by means of the variation of the domain.

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in abounded domain with control distributed in an arbitrary fixed subdomain. The result that we obtain in this paper is as follows.Suppose that we have a given stationary point of the Navier-Stokes equations and our initial condition is sufficiently close to it. Thenthere exists a locally distributed control such that in a given moment of time the solution of the Navier-Stokes equations coincideswith this stationary point.

In this paper a model reference-based adaptive parameter estimator for a wide class of hyperbolic distributed parameter systems isconsidered. The proposed state and parameter estimator can handle hyperbolic systems in which the damping sesquilinear form maynot be symmetric (or even present) and a modification to the standard adaptive law is introduced to account for this lack of symmetry(or absence) in the damping form. In addition, the proposed scheme is modified for systems in which the input operator, bounded orunbounded, is also unknown. Parameters that are slowly time varying are also considered in this scheme via an extension of finitedimensional results. Using a Lyapunov type argument, state convergence is established and with the additional assumption ofpersistence of excitation, parameter convergence is shown. An approximation theory necessary for numerical implementation isestablished and numerical results are presented to demonstrate the applicability of the above parameter estimators.

This essentially numerical study, sets out to investigate various geometrical properties of exact boundary controllability of the waveequation when the control is applied on a part of the boundary. Relationships between the geometry of the domain, the geometry ofthe controlled boundary, the time needed to control and the energy of the control are dealt with. A new norm of the control and anenergetic cost factor are introduced. These quantities enable a detailed appraisal of the numerical solutions obtained and the detectionof trapped rays.

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.

The regularity of Lagrange multipliers for state-constrained optimal control problems belongs to the basic questions of controltheory. Here, we investigate bottleneck problems arising from optimal control problems for PDEs with certain mixed control-stateinequality constraints. We show how to obtain Lagrange multipliers in Lp spaces for linear problems and give an application to linearparabolic optimal control problems.

In domain optimization problems, normal variations of a reference domain are frequently used. We prove that such variations do notpreserve the regularity of the domain. More precisely, we give a bounded domain which boundary is m times differentiable and ascalar variation which is infinitely differentiable such that the deformed boundary is only m-1 times differentiable. We prove inaddition that the only normal variations which preserve the regularity are those with constant magnitude. This shows that the use of normal variations in an iterative approximation method for domain optimization generates a loss ofregularity at each iteration, and thus it is better to use transverse variations which preserve the regularity of the domain.

In the first part of this paper, we study the best constant involving the L2 norm in Wente's inequality. We prove that this best constantis universal for any Riemannian surface with boundary, or respectively, for any Riemannian surface without boundary. The secondpart concerns the study of critical points of the associate energy functional, whose Euler equation corresponds to H-surfaces. We willestablish the existence of a non-trivial critical point for a plan domain with small holes.

Interpretations of most existing controllability and observability notions for linear delay systems are given. Module theoreticcharacterizations are presented. This setting enables a clear and precise comparison of the various examined notions. A new notion ofcontrollability is introduced, which is called pi-freeness.

A vibrating string, modelled by the wave equation, with an interior mass is considered. It is viewed as a linear delay system. A trajectory tracking problem is solved using a new type of controllability.

We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derivenew duality results for the minimization of the max of two quasiconvex functions. Following Barron and al., we show that the levelsum provides quasiconvex viscosity solutions for Hamilton-Jacobi equations in which the initial condition is a general continuousquasiconvex function which is not necessarily Lipschitz or bounded.

For smooth or real-analytic single-input, control-affine, non-linear systems, with at least two ouputs, observability for any input ofa given class is generic. This class can be either the class of inputs bounded with their derivatives up to a certain order, or the class ofpolynomial inputs with bounded degree.

We study here an optimal control problem for a semilinear elliptic equation with an exponential nonlinearity, such that we cannotexpect to have a solution of the state equation for any given control. We then have to speak of pairs (control, state). After havingdefined a suitable functional class in which we look for solutions, we prove existence of an optimal pair for a large class of costfunctions using a non standard compactness argument. Then, we derive a first order optimality system assuming the optimal pair isslightly more regular.

In this paper we provide a new sufficiency theorem for regular syntheses. The concept of regular synthesis is discussed in [12], wherea sufficiency theorem for finite time syntheses is proved. There are interesting examples of optimal syntheses that are very regular, but whose trajectories have time domains not necessarilybounded. The regularity assumptions of the main theorem in [12] are verified by every piecewise smooth feedback control generatingextremal trajectories that reach the target in finite time with a finite number of switchings. In the case of this paper the situation iseven more complicate, since we admit both trajectories with finite and infinite time. We use weak differentiability assumptions on the synthesis and weak continuity assumptions on the associated value function.


In this article we consider a control system modelling a batch reactor in whichthree species X1, X2, X3 are reacting according to thescheme X1 → X2 → X3, each reaction being irreversible. The control is the temperature T of the reactions or the derivative of thistemperature with respect to time. The terminal constraint is to obtain a given concentration of the product X2 at the end of the batch.The objective of our study is to introduce and to apply all the mathematical tools to compute the time optimal control as aclosed-loop function. This work can be used to optimize the yield of chemical batch reactors.