13,218 research outputs found
On the Fattorini Criterion for Approximate Controllability and Stabilizability of Parabolic Systems
In this paper, we consider the well-known Fattorini's criterion for
approximate controllability of infinite dimensional linear systems of type
. We precise the result proved by H. O. Fattorini in
\cite{Fattorini1966} for bounded input , in the case where can be
unbounded or in the case of finite-dimensional controls. More precisely, we
prove that if Fattorini's criterion is satisfied and if the set of geometric
multiplicities of is bounded then approximate controllability can be
achieved with finite dimensional controls. An important consequence of this
result consists in using the Fattorini's criterion to obtain the feedback
stabilizability of linear and nonlinear parabolic systems with feedback
controls in a finite dimensional space. In particular, for systems described by
partial differential equations, such a criterion reduces to a unique
continuation theorem for a stationary system. We illustrate such a method by
tackling some coupled Navier-Stokes type equations (MHD system and micropolar
fluid system) and we sketch a systematic procedure relying on Fattorini's
criterion for checking stabilizability of such nonlinear systems. In that case,
the unique continuation theorems rely on local Carleman inequalities for
stationary Stokes type systems
Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design
This paper develops a control and estimation design for the one-phase Stefan
problem. The Stefan problem represents a liquid-solid phase transition as time
evolution of a temperature profile in a liquid-solid material and its moving
interface. This physical process is mathematically formulated as a diffusion
partial differential equation (PDE) evolving on a time-varying spatial domain
described by an ordinary differential equation (ODE). The state-dependency of
the moving interface makes the coupled PDE-ODE system a nonlinear and
challenging problem. We propose a full-state feedback control law, an observer
design, and the associated output-feedback control law via the backstepping
method. The designed observer allows estimation of the temperature profile
based on the available measurement of solid phase length. The associated
output-feedback controller ensures the global exponential stability of the
estimation errors, the H1- norm of the distributed temperature, and the moving
interface to the desired setpoint under some explicitly given restrictions on
the setpoint and observer gain. The exponential stability results are
established considering Neumann and Dirichlet boundary actuations.Comment: 16 pages, 11 figures, submitted to IEEE Transactions on Automatic
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