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Mathematical Control & Related Fields

2014 , Volume 4 , Issue 1

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Null controllability of retarded parabolic equations
Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix and Lahcen Maniar
2014, 4(1): 1-15 doi: 10.3934/mcrf.2014.4.1 +[Abstract](84) +[PDF](396.3KB)
We address in this work the null controllability problem for a linear heat equation with delay parameters. The control is exerted on a subdomain and we show how the global Carleman estimate due to Fursikov and Imanuvilov can be applied to derive results in this direction.
Controllability to trajectories for some parabolic systems of three and two equations by one control force
Assia Benabdallah, Michel Cristofol, Patricia Gaitan and Luz de Teresa
2014, 4(1): 17-44 doi: 10.3934/mcrf.2014.4.17 +[Abstract](87) +[PDF](579.4KB)
We present a controllability result for a class of linear parabolic systems of $3$ equations. We establish a global Carleman estimate for the solutions of systems of $2$ parabolic equations coupled with first order terms. Stability results for inverse coefficients problems are deduced.
Control of a Korteweg-de Vries equation: A tutorial
Eduardo Cerpa
2014, 4(1): 45-99 doi: 10.3934/mcrf.2014.4.45 +[Abstract](147) +[PDF](701.7KB)
These notes are intended to be a tutorial material revisiting in an almost self-contained way, some control results for the Korteweg-de Vries (KdV) equation posed on a bounded interval. We address the topics of boundary controllability and internal stabilization for this nonlinear control system. Concerning controllability, homogeneous Dirichlet boundary conditions are considered and a control is put on the Neumann boundary condition at the right end-point of the interval. We show the existence of some critical domains for which the linear KdV equation is not controllable. In despite of that, we prove that in these cases the nonlinearity gives the exact controllability. Regarding stabilization, we study the problem where all the boundary conditions are homogeneous. We add an internal damping mechanism in order to force the solutions of the KdV equation to decay exponentially to the origin in $L^2$-norm.
Almost periodic solutions for a weakly dissipated hybrid system
Sorin Micu and Ademir F. Pazoto
2014, 4(1): 101-113 doi: 10.3934/mcrf.2014.4.101 +[Abstract](82) +[PDF](377.8KB)
We consider a hybrid system coupling an elastic string with a rigid body at one end and we study the existence of an almost periodic solution when an almost periodic force $f$ acts on the body. The weak dissipation of the system does not allow to show the relative compactness of the trajectories which generally implies the existence of such solutions. Instead, we use Fourier analysis to show that the existence or not of the almost periodic solutions depends on the regularity and the exponents of the almost periodic nonhomogeneous term $f$.
Algebraic characterization of autonomy and controllability of behaviours of spatially invariant systems
Amol Sasane
2014, 4(1): 115-124 doi: 10.3934/mcrf.2014.4.115 +[Abstract](67) +[PDF](392.7KB)
We give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems, consisting of distributional solutions $w$, that are periodic in the spatial variables, to a system of partial differential equations $$ M\left(\frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_d} , \frac{\partial}{\partial t}\right) w=0, $$ corresponding to a polynomial matrix $M\in ({\mathbb{C}}[\xi_1,\dots, \xi_d, \tau])^{m\times n}$.

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