Adjoint computational methods for 2D inverse design of linear transport equations on unstructured grids

M. Morales-Hernández, E. Zuazua Adjoint computational methods for 2D inverse design of linear transport equations on unstructured grids Abstract: We address the problem of inverse design of linear hyperbolic transport equations in 2D heterogeneous media. We develop numerical algorithms based in gradient-adjoint methodologies on unstructured grids. While the flow equation is compulsorily solved by means…

Sparse source identification of linear diffusion–advection equations by adjoint methods

A. Monge, E. Zuazua Sparse source identification of linear diffusion–advection equations by adjoint methods Abstract: We present an algorithm for the time-inversion of diffusion–advection equations, based on the adjoint methodology. Given a final state distribution our main aim is to recover sparse initial conditions, constituted by a finite combination of Dirac deltas, identifying their location…

Turnpike in optimal shape design

G. Lance, E. Trélat, E. Zuazua Turnpike in optimal shape design Abstract: We investigate the turnpike problem in optimal control, in the context of time-evolving shapes. We focus here on the heat equation model where the shape acts as a source term, and we search the optimal time-varying shape, minimizing a quadratic criterion. We first…

Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions

Y. Privat, E. Trélat & E. Zuazua. Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions, 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…

Dynamics and control for multi-agent networked systems: a finite difference approach

U. Biccari, D. Ko, E. Zuazua Dynamics and control for multi-agent networked systems: a finite difference approach 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…

Phase portrait control for 1D monostable and bistable reaction-diffusion equations

Pouchol C., Trélat E., Zuazua E. Phase portrait control for 1D monostable and bistable reaction-diffusion equations , DOI: 10.1088/1361-6544/aaf07e Abstract: We consider the problem of controlling parabolic semilinear equations arising in population dynamics, either in finite time or infinite time. These are the monostable and bistable equations on (0,L) for a density of individuals 0≤y(t,x)≤1,…

The turnpike propety in nonlinear optimal control – A geometric approach

N. Sakamoto, D. Pighin, E. Zuazua The turnpike propety in nonlinear optimal control – A geometric approach Abstract: This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply…

A parabolic approach to the control of opinion spreading

D. Ruiz, E. Zuazua A parabolic approach to the control of opinion spreading 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…

Internal observability for coupled systems of linear partial differential equations

Lissy, P., Zuazua, E. Internal observability for coupled systems of linear partial differential equations , DOI: 10.1137/17M1119160 Abstract: We deal with the internal observability for some coupled systems of partial differential equations with constant or time-dependent coupling terms by means of a reduced number of observed components. We prove new general observability inequalities under some…