Julie Valein

Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks

In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.

Stabilization of the Wave Equation on 1-d Networks

In this paper we study the stabilization of the wave equation on general 1-d networks. For that, we transfer known observability results in the context of control problems of conservative systems (see [R. Dager and E. Zuazua, Wave Propagation, Observation, and Control in 1-d Flexible Multi-structures, Math. Appl. 50, Springer-Verlag, Berlin, 2006]) into a weighted observability estimate for dissipative systems. Then we use an interpolation inequality similar to the one proved in [P. Begout and F. Soria, J. Differential Equations, 240 (2007), pp. 324-356] to obtain the explicit decay estimates of the energy for smooth initial data. The obtained decay rate depends on the geometric and topological properties of the network. We also give some examples of particular networks in which our results apply, yielding different decay rates.

Stabilization of Second Order Evolution Equations with Unbounded Feedback with Time-Dependent Delay

We consider abstract second order evolution equations with unbounded feedback with time-varying delay. Existence results are obtained under some realistic assumptions. We prove the exponential decay under some conditions by introducing an abstract Lyapunov functional. Our abstract framework is applied to the wave, to the beam, and to the plate equations with boundary delays.

Nonlinear control of a coupled PDE/ODE system modeling a switched power converter with a transmission line

We consider an infinite dimensional system modelling a boost converter connected to a load via a transmission line. The governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations model-ing the converter. The coupling is given by the boundary conditions and the nonlinear controller we introduce. We design a nonlinear saturating control law using a Lya-punov function for the averaged model of the system. The main results give the well-posedness and stability properties of the obtained closed loop system.

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

We consider the wave equation on an interval of length 1 with an interior damping at ξ. It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3–4):215–240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction

Finite dimensional approximations for a class of infinite dimensional time optimal control problems

\xi=\frac{p}{q},

Numerical approximation of some time optimal control problems

where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197–243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337–365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.

Wellposedness and stabilization of a class of infinite dimensional bilinear control systems

ABSTRACT In this work, we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge (in appropriate norms) to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems.

Stability of second order evolution equations with time-varying delays

In this work we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems.

STABILITY OF THE HEAT AND OF THE WAVE EQUATIONS WITH BOUNDARY TIME-VARYING DELAYS

We consider a class of infinite dimensional systems involving a control function u taking values in [0; 1]. This class contains, in particular, the average models of some infinite dimensional switched systems. We prove that the system is well-posed and obtain some regularity properties. Moreover, when u is given in an appropriate feedback form and the system satisfies appropriate observability assumptions, we show that the system is weakly stable. The main example concerns the analysis and stabilization of a model of Boost converter connected to a load via a transmission line. The main novelty consists in the fact that we give a rigorous wellposedness and stability analysis of coupled systems, in the presence of duty cycles.