1,857 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
Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves
In this paper, we shall prove a Carleman estimate for the so-called Zaremba
problem. Using some techniques of interpolation and spectral estimates, we
deduce a result of stabilization for the wave equation by means of a linear
Neumann feedback on the boundary. This extends previous results from the
literature: indeed, our logarithmic decay result is obtained while the part
where the feedback is applied contacts the boundary zone driven by an
homogeneous Dirichlet condition. We also derive a controllability result for
the heat equation with the Zaremba boundary condition.Comment: 37 pages, 3 figures. Final version to be published in Amer. J. Mat
Finite dimensional backstepping controller design
We introduce a finite dimensional version of backstepping controller design
for stabilizing solutions of PDEs from boundary. Our controller uses only a
finite number of Fourier modes of the state of solution, as opposed to the
classical backstepping controller which uses all (infinitely many) modes. We
apply our method to the reaction-diffusion equation, which serves only as a
canonical example but the method is applicable also to other PDEs whose
solutions can be decomposed into a slow finite-dimensional part and a fast
tail, where the former dominates the evolution in large time. One of the main
goals is to estimate the sufficient number of modes needed to stabilize the
plant at a prescribed rate. In addition, we find the minimal number of modes
that guarantee the stabilization at a certain (unprescribed) decay rate.
Theoretical findings are supported with numerical solutions.Comment: 28 pages, 2 figure
Challenges in Optimal Control of Nonlinear PDE-Systems
The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered
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