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…
Controllability of the one-dimensional fractional heat equation under positivity constraints
U. Biccari, M. Warna, E. Zuazua Internal observability for coupled systems of linear partial differential equations Abstract: In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $(-\Delta)^s$ ($0
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…
A Two-Dimensional “Flea on the Elephant” Phenomenon and its Numerical Visualization
Bianchini, R., Gosse, L., Zuazua, E. A two-dimensional “flea on the elephant” phenomenon and its numerical visualization , Multiscale Modeling & Simulation, Vol. 17, No. 1 (2019), pages 137–166, DOI: 10.1137/18M1179985. Abstract: Localization phenomena (sometimes called “flea on the elephant”) for the operator $L ε = −ε 2 ∆u + p(x)u, p(x)$ being an unbounded…
Greedy optimal control for elliptic problems and its application to turnpike problems
Hernández-Santamaría V., Lazar M., Zuazua E. Greedy optimal control for elliptic problems and itsapplication to turnpike problems DOI: 10.1007/s00211-018-1005-z Abstract: We adapt and apply greedy methods to approximate in an efficient way the optimalcontrols for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling theparameter-space…
Turnpike in optimal shape design for heat equation
G. Lance, E. Trélat, E. Zuazua Turnpike in optimal shape design for heat equation Abstract: After a short presentation of the turnpike’s problem in optimal control, we focus on the example of the heat equation with a source term as a shape. We search the path and the shape of an optimal source which let…
