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Alarcon T.,Basque Center for Applied Mathematics | Jensen H.J.,Imperial College London
Journal of the Royal Society Interface | Year: 2011

We point out that a simple and generic strategy in order to lower the risk for extinction consists in developing a dormant stage in which the organism is unable to multiply but may die. The dormant organism is protected against the poisonous environment. The result is to increase the survival probability of the entire population by introducing a type of zero reproductive fitness. This is possible, because the reservoir of dormant individuals act as a buffer that can cushion fatal fluctuations in the number of births and deaths, which without the dormant population would have driven the entire population to extinction. © 2010 The Royal Society.

Munch A.,University Blaise Pascal | Zuazua E.,Basque Center for Applied Mathematics
Inverse Problems | Year: 2010

The numerical approximation of exact or trajectory controls for the wave equation is known to be a delicate issue, since the pioneering work of Glowinski-Lions in the nineties, because of the anomalous behavior of the high-frequency spurious numerical waves. Various efficient remedies have been developed and analyzed in the last decade to filter out these highfrequency components: Fourier filtering, Tychonoff's regularization, mixed finite-element methods, multi-grid strategies, etc. Recently convergence rate results have also been obtained. This work is devoted to analyzing this issue for the heat equation, which is the opposite paradigm because of its strong dissipativity and smoothing properties. The existing analytical results guarantee that, at least in some simple situations, as in the finite-difference scheme in 1 - d, the null or trajectory controls for numerical approximation schemes converge. This is due to the intrinsic high-frequency damping of the heat equation that is inherited by its numerical approximation schemes. But when developing numerical simulations the topic appears to be much more subtle and difficult. In fact, efficiently computing the null control for a numerical approximation scheme of the heat equation is a difficult problem in itself. The difficulty is strongly related to the regularizing effect of the heat kernel. The controls of minimal L2-norm are characterized as minima of quadratic functionals on the solutions of the adjoint heat equation, or its numerical versions. These functionals are shown to be coercive in very large spaces of solutions, sufficient to guarantee the L2 character of controls, but very far from being identifiable as energy spaces for the adjoint system. The very weak coercivity of the functionals under considerationmakes the approximation problem exponentially ill-posed and the functional framework far from being well adapted to standard techniques in numerical analysis. In practice, the controls of the minimal L2-norm exhibit a singular highly oscillatory behavior near the final controllability time, which cannot be captured numerically. Standard techniques, such as Tychonoff's regularization or quasi-reversibility methods, allow a slight smoothing of the singularities but significantly reduce the quality of the approximation. In this paper we develop some more involved and less-standard approaches which turn out to be more efficient. We first discuss the advantages of using controls with compact support with respect to the time variable or the effect of adding numerical dissipative singular terms. We also develop the numerical version of the so-called transmutation method that allows writing the control of a heat process in terms of the corresponding control of the associated wave process, by means of a 'time convolution' with a one-dimensional controlled fundamental heat solution. This method, although it can be proved to converge, is also subtle in its computational implementation. Indeed, it requires using convergent numerical schemes for the control of the wave equation, a problem that, as mentioned above, is delicate in itself. But we also need to compute an accurate approximation of a controlled fundamental heat solution, an issue that requires its own analysis and significant numerical and computational new developments. These methods are thoroughly illustrated and discussed in the paper, accompanied by some numerical experiments in one space dimension that show the subtlety of the issue. These experiments allow one to compare the efficiency of the various methods. This is done in the case where the control is distributed in some subdomain of the domain where the heat process evolves but similar results and numerical experiments could be derived for other cases, such as the one in which the control acts on the boundary. The techniques we employ here can also be adapted to the multi-dimensional case. © 2010 IOP Publishing Ltd.

Lu Q.,Basque Center for Applied Mathematics | Lu Q.,University of Electronic Science and Technology of China
Inverse Problems | Year: 2012

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we study two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time T for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source. © 2012 IOP Publishing Ltd.

Beauchard K.,Ecole Normale Superieure de Cachan | Zuazua E.,Basque Center for Applied Mathematics
Archive for Rational Mechanics and Analysis | Year: 2011

This work is concerned with (n-component) hyperbolic systems of balance laws in m space dimensions. First, we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t → ∞. Using the Fourier transform, we examine the role that control theoretical tools, such as the classical Kalman rank condition, play. We build Lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable. In this way we extend the previous analysis by Shizuta and Kawashima under the so-called algebraic condition (SK). In particular, we show the existence of systems exhibiting more complex behavior than the one that the (SK) condition allows. We also discuss links between this analysis and previous literature in the context of damped wave equations, hypoellipticity and hypocoercivity. To conclude, we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws. Our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover. © 2010 Springer-Verlag.

Nakata Y.,Basque Center for Applied Mathematics
Nonlinear Analysis: Real World Applications | Year: 2011

We study the global dynamics of a time delayed epidemic model proposed by Liu et al. (2008) [J. Liu, J. Wu, Y. Zhou, Modeling disease spread via transport-related infection by a delay differential equation, Rocky Mountain J. Math. 38 (5) (2008) 15251540] describing disease transmission dynamics among two regions due to transport-related infection. We prove that if an endemic equilibrium exists then it is globally asymptotically stable for any length of time delay by constructing a Lyapunov functional. This suggests that the endemic steady state for both regions is globally asymptotically stable regardless of the length of the travel time when the disease is transferred between two regions by human transport. © 2011 Elsevier Ltd. All rights reserved.

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