Departamento de Matemáticashttp://hdl.handle.net/10486/1291672019-10-14T20:49:30Z2019-10-14T20:49:30ZFiltered gradient algorithms for inverse design problems of one-dimensional burgers equationGosse, LaurentZuazua, Enriquehttp://hdl.handle.net/10486/6884022019-08-23T08:23:00Z2017-03-09T00:00:00ZFiltered gradient algorithms for inverse design problems of one-dimensional burgers equation
Gosse, Laurent; Zuazua, Enrique
Gosse, Laurent; Natalini, Roberto
Inverse design for hyperbolic conservation laws is exemplified through the 1D Burgers equation which is motivated by aircraft’s sonic-boom minimization issues. In particular, we prove that, as soon as the target function (usually a Nwave) isn’t continuous, there is a whole convex set of possible initial data, the backward entropy solution being possibly its centroid. Further, an iterative strategy based on a gradient algorithm involving “reversible solutions” solving the linear adjoint problem is set up. In order to be able to recover initial profiles different from the backward entropy solution, a filtering step of the backward adjoint solution is inserted, mostly relying on scale-limited (wavelet) subspaces. Numerical illustrations, along with profiles similar to F-functions, are presented
The final publication is available at Springer via https://doi.org/10.1007/978-3-319-49262-9_7
2017-03-09T00:00:00ZControllability of evolution equations with memoryChaves-Silva, Felipe W.Zhang, XuZuazua, Enriquehttp://hdl.handle.net/10486/6881052019-09-20T07:47:56Z2017-08-08T00:00:00ZControllability of evolution equations with memory
Chaves-Silva, 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 finite-dimensional 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 2437-2459, published by the Society for Industrial and Applied Mathematics (SIAM)
2017-08-08T00:00:00ZActuator design for parabolic distributed parameter systems with the moment methodPrivat, YannickTrelat, EmmanuelZuazua, Enriquehttp://hdl.handle.net/10486/6881032019-09-20T07:49:55Z2017-04-06T00:00:00ZActuator 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 L2-energy of controllers. Since we choose the moment method to control the PDE, our study mainly covers one-dimensional 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 1128-1152, published by the Society for Industrial and Applied Mathematics (SIAM)
2017-04-06T00:00:00ZControllability under positivity constraints of semilinear heat equationsPighin, DarioZuazua, Enriquehttp://hdl.handle.net/10486/6880732019-09-20T07:51:33Z2018-09-01T00:00:00ZControllability 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 pre-copy-editing, author-produced PDF of an article accepted for publication in Mathematical Control and Related Fields following peer review. The definitive publisher-authenticated version Mathematical Control and Related Fields 8.3-4 (2018): 935-964 is available online at: http://www.aimsciences.org/article/doi/10.3934/mcrf.2018041
2018-09-01T00:00:00Z