Departamento de MatemÃ¡ticas
http://hdl.handle.net/10486/129167
20190720T17:38:17Z

Controllability of evolution equations with memory
http://hdl.handle.net/10486/688105
Controllability of evolution equations with memory
ChavesSilva, Felipe W.; Zhang, Xu; Zuazua, Enrique
This article is devoted to studying the null controllability of evolution equations with memory terms. The problem is challenging not only because the state equation contains memory terms but also because the classical controllability requirement at the final time has to be reinforced, involving the contribution of the memory term, to ensure that the solution reaches the equilibrium. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the adjoint system. We first consider finitedimensional dynamical systems involving memory terms and derive rank conditions for controllability. Then the null controllability property is established for some parabolic equations with memory terms, by means of Carleman estimates
First Published in SIAM Journal on Control and Optimization in Volume 55, Issue 4, 2017, Pages 24372459, published by the Society for Industrial and Applied Mathematics (SIAM)
20170808T00:00:00Z

Actuator design for parabolic distributed parameter systems with the moment method
http://hdl.handle.net/10486/688103
Actuator design for parabolic distributed parameter systems with the moment method
Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique
In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset of Rn. We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in , over all possible such distributions of a given measure. Using the moment method, we formulate a spectral optimal design problem, which consists of maximizing a criterion corresponding to an average over random initial data of the largest L2energy of controllers. Since we choose the moment method to control the PDE, our study mainly covers onedimensional parabolic operators, but we also provide several examples in higher dimensions. We consider two types of controllers: Either internal controls, modeled by characteristic functions, or lumped controls, that are tensorized functions in time and space. Under appropriate spectral assumptions, we prove existence and uniqueness of an optimal actuator distribution, and we provide a simple computation procedure. Numerical simulations illustrate our results
First Published in SIAM Journal on Control and Optimization in Volume 55, Issue 2, 2017, Pages 11281152, published by the Society for Industrial and Applied Mathematics (SIAM)
20170406T00:00:00Z

Controllability under positivity constraints of semilinear heat equations
http://hdl.handle.net/10486/688073
Controllability under positivity constraints of semilinear heat equations
Pighin, Dario; Zuazua, Enrique
In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with C 1 nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory. We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time
This is a precopyediting, authorproduced PDF of an article accepted for publication in Mathematical Control and Related Fields following peer review. The definitive publisherauthenticated version Mathematical Control and Related Fields 8.34 (2018): 935964 is available online at: http://www.aimsciences.org/article/doi/10.3934/mcrf.2018041
20180901T00:00:00Z

Local Elliptic Regularity for the Dirichlet Fractional Laplacian
http://hdl.handle.net/10486/688069
Local Elliptic Regularity for the Dirichlet Fractional Laplacian
Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique
We prove the W loc2s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of R N . The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cutoff, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions
Incluye tambiÃ©n:
Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian.
Advanced Nonlinear Studies, Volume 17, Issue 4, 1 October 2017, Pages 837839. DOI: https://doi.org/10.1515/ans20176020
20170421T00:00:00Z