74 research outputs found
How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability
Published versio
Simultaneous exact controllability and some applications
Published versio
Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations
A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani et al. (Automatica 46(10), 1616-1625, 2010 ). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed)
Controllability under positivity constraints of multi-d wave equations
We consider both the internal and boundary controllability problems for wave
equations under non-negativity constraints on the controls. First, we prove the
steady state controllability property with nonnegative controls for a general
class of wave equations with time-independent coefficients. According to it,
the system can be driven from a steady state generated by a strictly positive
control to another, by means of nonnegative controls, when the time of control
is long enough. Secondly, under the added assumption of conservation and
coercivity of the energy, controllability is proved between states lying on two
distinct trajectories. Our methods are described and developed in an abstract
setting, to be applicable to a wide variety of control systems
On Polynomial Stability of Coupled Partial Differential Equations in 1D
We study the well-posedness and asymptotic behaviour of selected PDE-PDE and
PDE-ODE systems on one-dimensional spatial domains, namely a boundary coupled
wave-heat system and a wave equation with a dynamic boundary condition. We
prove well-posedness of the models and derive rational decay rates for the
energy using an approach where the coupled systems are formulated as feedback
interconnections of impedance passive regular linear systems.Comment: 12 pages, 1 figure, accepted for publication in the Proceedings of
"Semigroups of Operators: Theory and Applications", Kazimierz Dolny, Poland,
October 201
Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator
We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state. We give necessary and sufficient condition to have an exponential rate of convergence. Numerical simulations are presented to illustratethe theoretical results
Controllability and Qualitative properties of the solutions to SPDEs driven by boundary L\'evy noise
Let be the solution to the following stochastic evolution equation (1)
du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking
values in an Hilbert space \HH, where is a \RR valued L\'evy process,
an infinitesimal generator of a strongly continuous semigroup,
\sigma:H\to \RR bounded from below and Lipschitz continuous, and B:\RR\to H
a possible unbounded operator. A typical example of such an equation is a
stochastic Partial differential equation with boundary L\'evy noise. Let
\CP=(\CP_t)_{t\ge 0} %{\CP_t:0\le t<\infty}T>0BAx\in H\CP_T^\star \delta_xH\HHLAB$ the solution of Equation [1] is
asymptotically strong Feller, respective, has a unique invariant measure. We
apply these results to the damped wave equation driven by L\'evy boundary
noise
Exact controllability of non-Lipschitz semilinear systems
We present sufficient conditions for exact controllability of a semilinear infinite-dimensional dynamical system. The system mild solution is formed by a noncompact semigroup and a nonlinear disturbance that does not need to be Lipschitz continuous. Our main result is based on a fixed point-type application of the Schmidt existence theorem and illustrated by a nonlinear transport partial differential equation
Locally distributed control for a model of fluid-structure interaction
International audienceWe consider the equations modeling the coupled vibrations of a fluid-solid system. The control acts in a subset of a domain occupied by the fluid. Our main result asserts that we have exact controllability and exponential stabilizability provided that the support of the control contains a neighborhood of the solid and a neighborhood of the exterior boundary. This improves the existing exact controllability results, which require a control which is active in the whole fluid domain. The proof is based on a frequency domain approach, combined with the use of appropriate multipliers. Moreover, we show that the strong stabilizability property holds for any open control region
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