Y. Privat, E. Trélat & E. Zuazua. Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions. CALC VAR PARTIAL DIF. Springer Verlag, Vol. 58, No. 2 (2019) pp. 64. DOI: 10.1007/s00526-019-1522-3
Abstract: We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset Ω of IRn. The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a L1 constraint on densities, the so-called Rellich functions maximize this functional. Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that the L∞-norm of Rellich functions may be large, depending on the shape of Ω, we analyze the effect of adding pointwise constraints when maximizing the same functional. We investigate the optimality of bang-bang functions and Rellich densitiesfor this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximations. Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations.
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U. Biccari, D. Ko, E. Zuazua Dynamics and control for multi-agent networked systems: a finite difference approach. Math. Models Methods Appl. Sci., Vol. 29, No. 4 (2019), pp. 755–790. DOI: 10.1142/S0218202519400050
Abstract: We analyze the dynamics of multi-agent collective behavior models and their control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.
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D. Ruiz-Balet, E. Zuazua A parabolic approach to the control of opinion spreading. In: Berezovski A., Soomere T. (eds) Applied Wave Mathematics II. Mathematics of Planet Earth, Vol 6. Springer, Cham (2019), pp. 343-363. DOI: 10.1007/978-3-030-29951-4_15
Abstract: We analyze the problem of controlling to consensus a nonlinear system modeling opinion spreading. Our strategy makes use of known results on the controllability of spatially discretized semilinear parabolic equations. Both systems can be linked through time-rescaling. This allows us to derive explicit exponential estimates on the cost of approximately controlling these systems to consensus, as a function of the number of agents N entering in the system and the control time-horizon T.
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