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Mathematical Control & Related Fields

2014 , Volume 4 , Issue 4

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Exact controllability of scalar conservation laws with strict convex flux
Adimurthi , Shyam Sundar Ghoshal and G. D. Veerappa Gowda
2014, 4(4): 401-449 doi: 10.3934/mcrf.2014.4.401 +[Abstract](149) +[PDF](657.7KB)
We consider the scalar conservation law with strict convex flux in one space dimension. In this paper we study the exact controllability of entropy solution by using initial or boundary data control. Some partial results have been obtained in [5],[23]. Here we investigate the precise conditions under which, the exact controllability problem admits a solution. The basic ingredients in the proof of these results are, Lax-Oleinik [15] explicit formula and finer properties of the characteristics curves.
Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain
Ahmed Bchatnia and Aissa Guesmia
2014, 4(4): 451-463 doi: 10.3934/mcrf.2014.4.451 +[Abstract](69) +[PDF](426.4KB)
In this work, we consider the Lamé system in 3-dimension bounded domain with infinite memories. We prove, under some appropriate assumptions, that this system is well-posed and stable, and we get a general and precise estimate on the convergence of solutions to zero at infinity in terms of the growth of the infinite memories.
Controllability of fast diffusion coupled parabolic systems
Felipe Wallison Chaves-Silva, Sergio Guerrero and Jean Pierre Puel
2014, 4(4): 465-479 doi: 10.3934/mcrf.2014.4.465 +[Abstract](68) +[PDF](342.7KB)
In this work we are concerned with the null controllability of coupled parabolic systems depending on a parameter and converging to a parabolic-elliptic system. We show the uniform null controllability of the family of coupled parabolic systems with respect to the degenerating parameter.
Asymptotic stability of Webster-Lokshin equation
Denis Matignon and Christophe Prieur
2014, 4(4): 481-500 doi: 10.3934/mcrf.2014.4.481 +[Abstract](56) +[PDF](530.2KB)
The Webster-Lokshin equation is a partial differential equation considered in this paper. It models the sound velocity in an acoustic domain. The dynamics contains linear fractional derivatives which can admit an infinite dimensional representation of diffusive type. The boundary conditions are described by impedance condition, which can be represented by two finite dimensional systems. Under the physical assumptions, there is a natural energy inequality. However, due to a lack of precompactness of the solutions, the LaSalle invariance principle can not be applied. The asymptotic stability of the system is proved by studying the resolvent equation, and by using the Arendt-Batty stability condition.
Multivariable boundary PI control and regulation of a fluid flow system
Cheng-Zhong Xu and Gauthier Sallet
2014, 4(4): 501-520 doi: 10.3934/mcrf.2014.4.501 +[Abstract](125) +[PDF](458.1KB)
The paper is concerned with the control of a fluid flow system governed by nonlinear hyperbolic partial differential equations. The control and the output observation are located on the boundary. We study local stability of spatially heterogeneous equilibrium states by using Lyapunov approach. We prove that the linearized system is exponentially stable around each subcritical equilibrium state. A systematic design of proportional and integral controllers is proposed for the system based on the linearized model. Robust stabilization of the closed-loop system is proved by using a spectrum method.
Observability and controllability analysis of blood flow network
Chun Zong and Gen Qi Xu
2014, 4(4): 521-554 doi: 10.3934/mcrf.2014.4.521 +[Abstract](101) +[PDF](588.7KB)
In this paper, we consider the initial-boundary value problem of a binary bifurcation model of the human arterial system. Firstly, we obtain a new pressure coupling condition at the junction based on the mass and energy conservation law. Then, we prove that the linearization system is interior well-posed and $L^2$ well-posed by using the semigroup theory of bounded linear operators. Further, by a complete spectral analysis for the system operator, we prove the completeness and Riesz basis property of the (generalized) eigenvectors of the system operator. Finally, we present some results on the boundary exact controllability and the boundary exact observability for the system.

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