1,530 research outputs found
Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states
In this paper we study the global approximate multiplicative controllability
for nonlinear degenerate parabolic Cauchy problems. In particular, we consider
a one-dimensional semilinear degenerate reaction-diffusion equation in
divergence form governed via the coefficient of the \-reaction term (bilinear
or multiplicative control). The above one-dimensional equation is degenerate
since the diffusion coefficient is positive on the interior of the spatial
domain and vanishes at the boundary points. Furthermore, two different kinds of
degenerate diffusion coefficient are distinguished and studied in this paper:
the weakly degenerate case, that is, if the reciprocal of the diffusion
coefficient is summable, and the strongly degenerate case, that is, if that
reciprocal isn't summable. In our main result we show that the above systems
can be steered from an initial continuous state that admits a finite number of
points of sign change to a target state with the same number of changes of sign
in the same order. Our method uses a recent technique introduced for uniformly
parabolic equations employing the shifting of the points of sign change by
making use of a finite sequence of initial-value pure diffusion pro\-blems. Our
interest in degenerate reaction-diffusion equations is motivated by the study
of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers
model) and some models in population genetics (see, e.g., the Fleming-Viot
model).Comment: arXiv admin note: text overlap with arXiv:1510.0420
Numerical controllability of the wave equation through primal methods and Carleman estimates
This paper deals with the numerical computation of boundary null controls for
the 1D wave equation with a potential. The goal is to compute an approximation
of controls that drive the solution from a prescribed initial state to zero at
a large enough controllability time. We do not use in this work duality
arguments but explore instead a direct approach in the framework of global
Carleman estimates. More precisely, we consider the control that minimizes over
the class of admissible null controls a functional involving weighted integrals
of the state and of the control. The optimality conditions show that both the
optimal control and the associated state are expressed in terms of a new
variable, the solution of a fourth-order elliptic problem defined in the
space-time domain. We first prove that, for some specific weights determined by
the global Carleman inequalities for the wave equation, this problem is
well-posed. Then, in the framework of the finite element method, we introduce a
family of finite-dimensional approximate control problems and we prove a strong
convergence result. Numerical experiments confirm the analysis. We complete our
study with several comments
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
Model reduction methods for bilinear control systems are compared by means of
practical examples of Liouville-von Neumann and Fokker--Planck type. Methods
based on balancing generalized system Gramians and on minimizing an H2-type
cost functional are considered. The focus is on the numerical implementation
and a thorough comparison of the methods. Structure and stability preservation
are investigated, and the competitiveness of the approaches is shown for
practically relevant, large-scale examples
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