Emmanuel Trélat’s research while affiliated with Université Paris Cité and other places

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Publications (285)


Multi-agent systems with multiple-wise interaction: Propagation of chaos and macroscopic limit
  • Preprint

February 2025

Thierry Paul

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Stefano Rossi

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Emmanuel Trélat

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Eth Zurich

We consider interacting multi-agent systems where the interaction is not only pairwise but involves simultaneous interactions among multiple agents (multiple-wise interaction). By passing through the mesoscopic and macroscopic limits with a fixed multiple-wise interaction of order m, we derive a macroscopic equation in the limit m \rightarrow \infty, capturing the dominant effects in large-size multiple-wise order.


Large-time optimal observation domain for linear parabolic systems

January 2025

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14 Reads

Annales de l Institut Henri Poincaré C Analyse Non Linéaire

Given a well-posed linear evolution system settled on a domain \Omega of \mathbb{R}^{d} , an observation subset \omega\subset\Omega and a time horizon T , the observability constant is defined as the largest possible nonnegative constant such that the observability inequality holds for the pair (\omega,T) . In this article we investigate the large-time behavior of the observation domain that maximizes the observability constant over all possible measurable subsets of a given Lebesgue measure. We prove that it converges exponentially, as the time horizon goes to infinity, to a limit set that we characterize. The mathematical technique is new and relies on a quantitative version of the bathtub principle.



Convergence in Nonlinear Optimal Sampled-Data Control Problems

October 2024

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7 Reads

IEEE Transactions on Automatic Control

Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution xx^* . On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem converges uniformly to xx^* as the norm of the partition tends to zero. Moreover, applying the Pontryagin Maximum Principle (PMP) to both problems, we prove that, if xx^* has a unique weak extremal lift with a costate p that is normal, then the costate of the sampled-data problem converges uniformly to p . In other words, under a normality assumption, control sampling commutes, at the limit of small partitions, with the application of the PMP.


Semigroup Theory

July 2024

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7 Reads

The objective of this chapter is to establish that, in an appropriate functional setting, there is a unique solution of the Cauchy problem where A is a linear operator on a Banach space X, and where y(t) and f(t) evolve in X, which is given by where (S(t))t0(S(t))_{t\geqslant 0} is the semigroup generated by the operator A.


Optimal Control

July 2024

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6 Reads

In Chap. 1, we have provided controllability properties for general classes of control systems. Considering some control problem of trying to reach some final configuration for the control system (1.1), from some initial configuration, with an admissible control, it happens that, in general, there exists an infinite number of controls making the job (think of all possibilities of realizing a parallel parking, for instance). Among this infinite number of controls, we now would like to select (at least) one control, achieving the desired controllability problem, and moreover minimizing some cost criterion (for instance, one would like to realize the parallel parking by minimizing the time, or by minimizing the fuel consumption). This is then an optimal control problem.


Controllability

July 2024

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1 Read

Let n and m be two positive integers. In this chapter we consider a control system in Rn\mathbb {R}^n where f:R×Rn×RmRnf:\mathbb {R}\times \mathbb {R}^n\times \mathbb {R}^m\rightarrow \mathbb {R}^n is of class C1C^1 with respect to (x,u) and locally integrable with respect to t, and the controls are measurable essentially bounded functions of time taking their values in some measurable subset Ω\Omega of Rm\mathbb {R}^m (set of control constraints).


Linear Control Systems in Banach Spaces

July 2024

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6 Reads

Throughout the chapter, we consider the linear autonomous control system where the state y(t) belongs to a Banach space X, y0Xy_0\in X, the control u(t) belongs to a Banach space U, A:D(A)XA:D(A)\rightarrow X is the generator of a C0C_0 semigroup (S(t))t0G(M,ω)(S(t))_{t\geqslant 0}\in \mathcal {G}(M,\omega ) on X, and BL(U,X1)B\in L(U,X_{-1}). The space X1X_{-1} has been defined in the previous chapter.


Stabilization

July 2024

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2 Reads

In this chapter, our objective will be to stabilize a possibly unstable equilibrium point by means of a feedback control.


Weyl formulae for some singular metrics with application to acoustic modes in gas giants

June 2024

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2 Reads

This paper is motivated by recent works on inverse problems for acoustic wave propagation in the interior of gas giant planets. In such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with boundary whose metric blows up near the boundary. Here, the spectral analysis of the corresponding Laplace-Beltrami operator is presented and the Weyl law is derived. The involved exponents depend on the Hausdorff dimension which, in the supercritical case, is larger than the topological dimension.


Citations (50)


... Let us comment on few existing results on this issue. Significant insights have been achieved in [17] and [5], where the authors explore variations of the linear quadratic (LQ) problem. Assuming certain controllability conditions, they derive the large-time expansion of the value function at the order of one in 1{T . ...

Reference:

Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems
Linear quadratic optimal control turnpike in finite and infinite dimension: Two-term expansion of the value function
  • Citing Article
  • June 2024

Systems & Control Letters

... Numerical methods for diffusion-type equations with nonlocal terms and describing phenomena in mathematical biology may be found in [21,22]. For nonstandard approaches concerning some reaction-diffusion systems, we may refer to [23][24][25]. ...

Approximate control of parabolic equations with on-off shape controls by Fenchel duality
  • Citing Article
  • February 2024

Annales de l Institut Henri Poincaré C Analyse Non Linéaire

... Such additional information (which is crucially required in the context of state estimation, as will be clear in Sections IV and V) is provided by exponential (or polynomial) turnpike characterizations that involve an explicit time-dependent bound on the difference of optimal trajectories and the turnpike, cf. e.g., [19], [21], [28], [29]. To cover arbitrary decay rates, we propose the following unified turnpike property involving general KL-functions. ...

Linear turnpike theorem
  • Citing Article
  • Publisher preview available
  • April 2023

Mathematics of Control Signals and Systems

... They also use results linking the original control problem with the auxiliary problem to derive a set of sensitivity relations without needing any compatibility assumptions. A recent survey is available in [136], and [137] provides a review of the recent approaches and results. ...

An algorithmic guide for finite-dimensional optimal control problems
  • Citing Chapter
  • January 2022

Handbook of Numerical Analysis

... This first step is useful to determine the structure of the optimal trajectory (i.e. the ordered sequence of regions that the optimal trajectory visits) and, second, to initialize an indirect numerical method applied to the original problem and based on the PMP stated in Theorem 3.1. The originality here is to incorporate the averaged Hamiltonian gradient condition, as well as the discontinuity jumps of the adjoint vector, to define an appropriate shooting function in addition to the classical terms defining the shooting function for classical (nonhybrid) optimal control problems (see [24,29]). ...

An algorithmic guide for finite-dimensional optimal control problems
  • Citing Book
  • January 2022

... The discretization of the HJB equation is carried out using the form of (31), which enables convenient computation. The partial differentials are discretized using the monotone and consistent upwind method (e.g., [71][72][73] ...

An algorithmic guide for finite-dimensional optimal control problems
  • Citing Preprint
  • December 2022

... These methods have in common that they focus on special classes of stochastic systems under periodicity assumptions. And, finally, yet another option for analyzing infinite horizon optimal control problems is to work with turnpikes [5,41], which can be used to analyze the long-term local behavior of systems near optimal steady-states or periodic orbits. ...

On the asymptotic behavior of the value function in large time optimal control problems
  • Citing Article
  • September 2022

IFAC-PapersOnLine

... Although numerous collective behavior models have been introduced and investigated to analyze the emergent flocking of self-organized systems, starting with the pioneering work of Cucker-Smale [2,3,8], to the best of our knowledge, this is the first time in the literature that the problem of control around flocks for discrete partial differential equations has been addressed and studied. In recent years, there have been important advances in the control theory for kinetic models, particularly the control to flocking of the Cucker-Smale model [1,10], which serves as our source of inspiration for studying the problem of controlling the propagation of lattice/chains nonlinear waves. ...

Controlling Swarms toward Flocks and Mills

SIAM Journal on Control and Optimization

... We point the interested readers to the excellent survey [12] for a complete description of the foundations of the finite-dimensional theory, and to the work [13] by the authors where we generalise the underlying core results to infinite-dimensional graphon dynamics. We also quote [14], which keenly builds on the approach of [5] to establish the exponential decay of the L 2 -norm in very general timeindependent asymmetric graphon dynamics. ...

Exponential Convergence Towards Consensus for Non-Symmetric Linear First-Order Systems in Finite and Infinite Dimensions
  • Citing Article
  • June 2022

SIAM Journal on Mathematical Analysis