arXiv subjects

Emmanuel Trélat

Publications and source records attributed to Emmanuel Trélat.

At least 19 recordsLinked to original sources

arXiv2026

Global exponential turnpike properties for optimal control of the viscous Burgers equation

We establish global exponential turnpike properties for quadratic optimal tracking problems governed by the one-dimensional viscous Burgers equation with localized internal control. For every initial datum, finite-horizon optimal solutions approach the unique optimal periodic regime when the periodic tracking target is sufficiently small; the zero-target case yields a global steady turnpike at the origin, with no smallness assumption on the initial datum. To our knowledge, these are the first global exponential turnpike results for the viscous Burgers equation. The proof combines a local exponential turnpike, obtained through strict convexity and periodic Riccati theory, with a parabolic dissipation argument that provides an absorbing time independent of the horizon.

math.OC↗

arXiv2026

Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces

Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-γ/t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set. Let $A$ be a nonnegative self-adjoint operator on sections of a finite-rank Euclidean vector bundle over a doubling metric measure space, satisfying ultracontractivity, Davies-Gaffney estimates (equivalently, finite speed of propagation for the wave equation) and a pointwise local Weyl law. If a weighted integrated observability inequality holds on a measurable set $ω$, for a fixed horizon $T\in(0,+\infty]$ and an admissible weight $h$, then, for every $0<κ<\frac{1}{2}$, $$ h(t)\leq A_{T,κ}\exp\left(-κ\frac{\mathcal{L}(ω)^2}{t}\right),\qquad 0<t<T, $$ where $\mathcal{L}(ω)$ is the essential maximal distance to $ω$, replaced by any finite radius when $\mathcal{L}(ω)=+\infty$. Thus, for $h(t)=e^{-γ/t}$, necessarily $γ\geq\mathcal{L}(ω)^2/2$. This settles, with the optimal threshold, the maximal-distance lower bound for the infinite-time constant left open in earlier work. In the control-norm convention, the fast-control rate is at least $\mathcal{L}(ω)^2/4$, recovering Miller's bound. The proof rests on the spectral packet $(\cosh(r\sqrt A)-1)e^{-tA}$. A pointwise Weyl law gives its sharp lower growth, while finite propagation speed and a weak-kernel Kannai transmutation formula make it exponentially small on $ω$. Without kernel continuity or compact resolvent, we develop pointwise spectral measures and weak wave kernels. The framework covers Laplace-type operators on compact Riemannian manifolds, coupled heat systems, Schrödinger operators on $\mathbb{R}^d$, equiregular sub-Laplacians and Grushin models, and $δ'$-coupled Laplacians on metric graphs.

math.AP↗

arXiv2026

Turnpike in optimal control and beyond: a survey

The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide class of long-horizon optimal control problems, the optimal trajectory spends most of its time near a steady-state solution (the ''turnpike'') rather than being influenced by the initial or final conditions. In this article, we provide a survey on the turnpike property in optimal control, adding several recent and novel considerations. After some historical insights, we present an elementary proof of the exponential turnpike property for linear-quadratic optimal control problems in finite dimension. Next, we show an extension to nonlinear optimal control problems, with a local exponential turnpike property. On simple but meaningful examples, we illustrate the local and global aspects of the turnpike theory, clarifying the global picture and raising new questions. We discuss key generalizations, in infinite dimension and other various settings, and review several applications of the turnpike theory across different fields.

math.OC↗

arXiv2026

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

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.

math.AP↗

arXiv2026

Boundary-compatible interacting approximations of quasilinear PDEs on bounded domains

We develop a general operator-theoretic route that turns Kato-type quasilinear evolution systems on a Banach scale $(Z,X)$ into finite-dimensional interacting approximations. The construction proceeds in two steps. First, one introduces a regularized family $(A_\varepsilon,f_\varepsilon)$ indexed by a scale parameter $\varepsilon>0$, for which the drift $A_\varepsilon[t,z]z+f_\varepsilon[t,z]$ takes values in an output space $Y$ suitable for discretization. Second, one discretizes this regularized dynamics by a sampling-reconstruction pair $(P_N,R_N)$ and obtains an interacting ODE on a finite-dimensional state space $V_N\simeq\R^{dN}$. Our main abstract theorem provides a quantitative estimate of the discrepancy $y_\varepsilon^N-y$ between the lifted discrete solution and the exact one, separating the regularization error $χ(\varepsilon)$ from the discretization error $(1+L_\varepsilon)N^{-γ}$, where $L_\varepsilon$ measures the size of the regularized drift in the output norm. This makes explicit the trade-off between the regularization scale $\varepsilon$, the discretization scale $N$, and the possible deterioration of $L_\varepsilon$ as $\varepsilon\to 0$. As a running example, we focus on quasilinear PDEs on bounded Lipschitz domains with boundary conditions. We show that Burenkov's variable-step mollifiers provide a boundary-compatible kernelization: they regularize differential operators into explicit integral-interaction operators supported inside the domain and preserve boundary traces of sufficiently regular fields. In this setting one can choose an output space $Y$ for which $L_\varepsilon$ remains uniformly bounded, leading to algebraic convergence rates in $N$ for quasi-uniform discretizations.

math.NA↗

arXiv2026

Optimal drift optimizer for non-convex optimization

We study a finite-horizon stochastic control criterion for non-convex optimization in which Brownian exploration is balanced against a quadratic control cost. Rather than emphasizing the classical Hopf--Cole representation, we isolate the exact drift selected by the criterion and reorganize it in a form adapted to optimization. The key object is the conditional terminal law of the optimal process. We show that this law is a Gibbs measure for a proximally penalized energy, yielding three exact representations of the drift: potential, averaged-gradient, and barycentric. We then analyze two asymptotic regimes relevant for optimization. As terminal time is approached, the drift recovers a scaled gradient-descent field. In the low-temperature regime, assuming a unique global minimizer, the conditional terminal law concentrates on it even in the presence of nonglobal local minima, and the drift converges to an affine attraction field toward it. In the nondegenerate case we also derive Laplace asymptotics for the drift, the value function, and the covariance of the conditional terminal law. Finally, we record a simple gradient-free discretization suggested by the barycentric formula.

math.OC↗

arXiv2026

Weyl asymptotics for singular metrics with a variable boundary degeneracy exponent

We consider a compact smooth manifold $X$ of dimension $n+1$ with boundary $M=\partial X$. In a collar neighborhood of $M$, we assume that the metric has the form $g=u^{-α}\bar g$, where $u$ is a boundary defining function, $α\in C^1(M;[0,2))$ and $\bar g$ is a $C^1$ Riemannian metric up to $M$. Since $α<2$, the boundary lies at finite $g$-distance and $(X,g)$ is a singular metric space. We study the Weyl asymptotics of the Friedrichs Laplacian $\triangle\_g$ when the degeneracy exponent $α$ varies along $M$. If the maximum $α\_{\mathrm{max}}$ of $α$ on $M$ is strictly larger than the critical value $α\_c=\frac{2}{n+1}$, then we prove that the points where $α$ is close to $α\_{\mathrm{max}}$ govern the leading term in the Weyl asymptotics. If $α\_{\mathrm{max}}\leqα\_c$, then the leading term is governed by the truncated volume $\vol\_g(\{\dist(\cdot,M)>λ^{-1/2}\})$. When the maximum set of $α$ is Morse-Bott, we compute the associated constants and the logarithmic corrections. To the best of our knowledge, this is the first Weyl law in this setting with a boundary-dependent degeneracy exponent. The results highlight a sharp transition at $α\_c$ between a boundary-dominated non-classical regime and a truncated-volume regime.

math.SP↗

arXiv2026

Mean field, hydrodynamic and graph limits for deterministic interacting particle systems: a survey with quantitative estimates

We present a unified framework, with quantitative estimates, for deterministic interacting particle systems whose pairwise interactions may depend on heterogeneous labels. Heterogeneity is kept at every level by adding a frozen label variable $x\inΩ$ to the state. Within this framework we compare several limiting procedures: the direct continuum / graph limit, the mean field limit yielding a Vlasov equation on the extended space of labels and states, the Liouville lift of the particle system together with propagation of chaos through marginals of arbitrary order, and the hydrodynamic moment closures. We give a common language for these limits and identify precisely where the various passages commute and where they do not; in particular, we separate the continuum / graph limit equation from the classical hydrodynamic Euler equations and characterize when the former arises as a moment closure of the latter (linearity in $(ξ,ξ')$ or monokinetic ansatz). Along the way, we prove quantitative convergence estimates for the graph limit and for the passages from particles or Liouville to Vlasov, and we discuss the limitations of the framework, in particular concerning singular kernels and stochastic dynamics. The paper is written as a survey with original contributions, with an emphasis on estimates, examples, and a clear delineation of scope.

math.AP↗

arXiv2026

Boundary observability for gas giant metrics

We study the observability of waves on gas giant manifolds which are a class of Riemannian manifolds whose metrics are singular at the boundary. Such manifolds arise naturally in modeling of acoustic wave propagation in gas giant planets.We establish an observability inequality using full boundary measurements given by a Neumann-type trace that is natural in the gas giant setting. The proof proceeds in two steps. First, observability for a general gas giant metric is reduced to the so-called separable case via a perturbation argument. In the separable case, we employ a uniform-in-tangential-frequency analysis combined with an Ingham inequality to prove observability.

math.OC↗

arXiv2026

Moving localized observations and Ces{à}ro asymptotic observability for conservative PDEs

We develop a deterministic large-time mechanism yielding Ces{à}ro asymptotic observability inequalities from moving localized observations for conservative evolutions. On each observation interval, exact convexification on a compact measured homogeneous space replaces full observation on the whole observation manifold by a finite convex combination of translates of one prototype subset. A switching realization theorem then turns that static design into a genuinely moving observer, while a Hilbertian tail-reduction proposition shows that interval estimates proved only on growing spectral windows still recover the full conserved energy after Ces{à}ro averaging. The resulting design-to-observability chain applies to interior observations for wave, Klein-Gordon, and Schr{ö}dinger equations on compact measured homogeneous manifolds, to moving boundary caps on the Euclidean ball, and to a singular almost-separated gas-giant boundary model. The framework is especially relevant when each instantaneous observation set is too small for one to expect a finite-time GCC or time-dependent GCC statement.

math.AP↗

arXiv2026

Exponential Turnpike Theorems for Nonlinear Deterministic Meanfield Optimal Control Problems

In this article, we establish exponential turnpike theorems for a class of nonlinear deterministic meanfield optimal control problems. We carry out our analysis simultaneously in the so-called Lagrangian and Eulerian frameworks. In the Lagrangian setting, the problem is lifted to a Hilbert space of random variables, and we prove an exponential turnpike theorem by combining first-order optimality conditions, a second-order expansion of the lifted Hamiltonian, and an operator Riccati diagonalization argument. In the Eulerian setting, we derive intrinsic KKT conditions for the static constrained problem, and show how the Eulerian second-order hypotheses split into a horizontal part, transferred by unitary conjugation to the lifted space, and a vertical part which reduces to uniform pointwise stabilizability and detectability conditions on multiplication operators. This yields an exponential turnpike theorem in the Wasserstein space for optimal Pontryagin triples. Along the way, we %provide explicit the link between Wasserstein Hessians and their Lagrangian lifts, and provide several remarks clarifying the role of occupation measures, local Eulerian minimizers, and control constraints in our results.

math.OC↗

arXiv2026

From PDEs on standard domains to self-similar particle systems on fractals

We construct transported PDEs on self-similar fractal domains from reference equations posed on the unit interval, and derive explicit self-similar interacting particle systems that approximate the resulting dynamics. The construction combines a measure-preserving isometry between $L^2$-spaces on $[0,1]$ and on the fractal \cite{Med2026}, a nonlocal-to-local approximation of differential operators \cite{PauTre2025}, and a Galerkin discretization on the canonical self-similar partitions. This yields a two-parameter approximation scheme whose error separates a nonlocal consistency term from a Galerkin network term. We work out the transport, Burgers, and heat equations, discuss the relation with intrinsic operators on fractals, and outline extensions to local charts and to pullbacks of nonlocal equations on fractal domains. Moreover, the reverse mapping transforms a nonlocal evolution equation on a fractal domains into the evolution equation on the unit interval, where the methods of classical numerical analysis can be applied. This suggests a promising direction for the development of numerical methods for nonlocal models on fractals, including fractional heat equation, fractal scattering, and related models.

math.AP↗

arXiv2026

A flatness proof of the exponential turnpike phenomenon for linear-quadratic optimal control problems

We revisit finite-dimensional linear-quadratic optimal control from the viewpoint of differential flatness. If the pair (A, B) is controllable, then the linear control system is flat, and every trajectory can be parametrized by a flat output and finitely many of its derivatives. Once this parametrization is inserted into the quadratic functional, the Euler-Lagrange condition becomes a linear differential equation with constant coefficients, or more generally a polynomial matrix differential equation. After reduction to Smith normal form, this equation decouples into scalar constant-coefficient equations, and its solutions are exponentialpolynomials. This yields a viewpoint on the turnpike phenomenon that is quite different from the classical Hamiltonian-Riccati analysis: the turnpike mechanism appears directly from the stable-unstable splitting of the reduced flat equation. In particular, when the reduced Euler-Lagrange operator has no purely imaginary characteristic roots and when the endpoint constraints act nondegenerately on the stable and unstable modes, the optimal trajectory consists of a left-boundary layer, a right-boundary layer, and a long interior arc exponentially close to the static optimum. The same viewpoint also clarifies what changes when some weights are only semidefinite: the order of the reduced equation may drop, some endpoint conditions may become incompatible, and polynomial or oscillatory modes may destroy the exponential turnpike. It also gives a natural meaning to certain endpoint constraints on the control and on finitely many derivatives of the control: such traces are not defined on the ambient L 2 control space, but they are meaningful on the smooth extremals selected by the reduced Euler-Lagrange equation. We formulate this principle as a general theorem and illustrate it in detail on the double integrator.

math.OC↗

arXiv2026

Probabilistic algorithm for computing all local minimizers of Morse functions on a compact domain

Let K be the unit-cube in Rn and f\,: K $\rightarrow$ R^n be a Morse function. We assume that the function f is given by an evaluation program $Γ$ in the noisy model, i.e., the evaluation program $Γ$ takes an extra parameter $η$ as input and returns an approximation that is $η$-close to the true value of f . In this article, we design an algorithm able to compute all local minimizers of f on K . Our algorithm takes as input $Γ$, $η$, a numerical accuracy parameter $ε$ as well as some extra regularity parameters which are made explicit. Under assumptions of probabilistic nature -- related to the choice of the evaluation points used to feed $Γ$ --, it returns finitely many rational points of K , such that the set of balls of radius $ε$ centered at these points contains and separates the set of all local minimizers of f . Our method is based on approximation theory, yielding polynomial approximants for f , combined with computer algebra techniques for solving systems of polynomial equations. We provide bit complexity estimates for our algorithm when all regularity parameters are known. Practical experiments show that our implementation of this algorithm in the Julia package Globtim can tackle examples that were not reachable until now.

cs.SC↗

arXiv2026

Generalisation of Farkas' lemma beyond closedness: a constructive approach via Fenchel-Rockafellar duality

Farkas' lemma is an ubiquitous tool in optimisation, as it provides necessary and sufficient conditions to have $b \in A(P)$, where $P$ is a closed convex cone, $A$ is a (continuous) linear mapping and $b$ is a fixed vector. The standard underlying hypothesis is the closedness of $A(P)$, which is not always satisfied and can be difficult to check. We devise a new method to generalise Farkas' lemma, based on a primal-dual pair of optimisation problems and Fenchel-Rockafellar duality theory. We work under the sole hypothesis that $P$ be generated by a closed bounded convex set. This hypothesis is weaker than in previous generalisations of Farkas' lemma, which almost all require that $A(P)$ be closed, or, in few cases, that only $P$ be closed. In our case, $P$ (and a fortiori $A(P)$) is not necessarily closed; we uncover necessary and sufficient conditions both for $b \in A(P)$ and $b \in \overline{A(P)}$. For a given $\e \geq 0$, we exhibit constructive characterisations of $x \in P$ such that $\|Ax-b\| \leq \e$ when it exists, by means of optimality conditions. For $\e = 0$, these strongly rely on whether the dual problem admits a solution, and we discuss conditions under which it does. Finally, we also explain how, upon relaxation, we may apply our method to a nonconvex cone.

math.OC↗

arXiv2026

Polynomial Turnpike Property for a Class of Infinite-Dimensional Oscillating Systems

We establish a polynomial turnpike estimate for an optimal control problem consisting of a system of infinitely many controlled oscillators, considered as an abstract differential equation in a Hilbert space, with a quadratic cost. Our proof relies on spectral considerations and on the construction of a Riesz basis. A concrete example is given, which involves a rotating bodybeam system. To our knowledge, this is the first example of a pointwise turnpike estimate around a steady-state that is polynomial but not exponential.

math.OC↗

arXiv2026

Controllability of wave-heat and heat-wave cascades

We study boundary controllability of one-dimensional coupled hyperbolic-parabolic cascades, focusing on the fine structure of reachable sets. The main model is a wave-heat cascade in which a boundary control acts on the wave equation and drives the heat equation through an internal coupling. We provide a sharp minimal time for the hyperbolic part (T > 2L) and a complete spectral characterization of exact controllability in weighted Hilbert spaces, whose definition depends explicitly on the coupling profile through a sequence of modal coefficients. In particular, internal couplings may generate nonstandard highly irregular controllability spaces and yield a generic (full measure) but non-robust controllability property. The analysis relies on Riesz basis decompositions and on an Ingham-M{ü}ntz inequality. We also prove that the exact controllability space is not invariant along Hilbert Uniqueness Method trajectories: even if both endpoints belong to the controllability space, the associated minimal-energy trajectory may leave it at intermediate times. Finally, we compare with the reversed (heat-wave) cascade and discuss how reversing the direction of the coupling transfers the loss of regularity between the parabolic and hyperbolic components.

math.OC↗

arXiv2026

Stabilization of a Wave-Heat Cascade System

We consider the output-feedback stabilization of a one-dimensional cascade coupling a reaction-diffusion equation and a wave equation through an internal term, with Neumann boundary control acting at the wave endpoint. Two measurements are available: the wave velocity at the controlled boundary and a temperature-type observation of the reaction-diffusion component, either distributed or pointwise. Under explicit, necessary and sufficient conditions on the coupling and observation profiles, we show that the generator of the open-loop system is a Riesz-spectral operator. Exploiting this structure, we design a finite-dimensional dynamic output-feedback law, based on a finite number of parabolic modes, which achieves arbitrary exponential decay in both the natural energy space and a stronger parabolic norm. The construction relies on a spectral reduction and a Lyapunov argument in Riesz bases. We also extend the design to pointwise temperature or heat-flux measurements.

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