arXiv subjects

Gaspard Ohlmann

Publications and source records attributed to Gaspard Ohlmann.

7 recordsLinked to original sources

arXiv2025

Synthesis of Safety Specifications for Probabilistic Systems

Ensuring that agents satisfy safety specifications can be crucial in safety-critical environments. While methods exist for controller synthesis with safe temporal specifications, most existing methods restrict safe temporal specifications to probabilistic-avoidance constraints. Formal methods typically offer more expressive ways to express safety in probabilistic systems, such as Probabilistic Computation Tree Logic (PCTL) formulas. Thus, in this paper, we develop a new approach that supports more general temporal properties expressed in PCTL. Our contribution is twofold. First, we develop a theoretical framework for the Synthesis of safe-PCTL specifications. We show how the reducing global specification satisfaction to local constraints, and define CPCTL, a fragment of safe-PCTL. We demonstrate how the expressiveness of CPCTL makes it a relevant fragment for the Synthesis Problem. Second, we leverage these results and propose a new Value Iteration-based algorithm to solve the synthesis problem for these more general temporal properties, and we prove the soundness and completeness of our method.

cs.LO↗

arXiv2025

ProSh: Probabilistic Shielding for Model-free Reinforcement Learning

Safety is a major concern in reinforcement learning (RL): we aim at developing RL systems that not only perform optimally, but are also safe to deploy by providing formal guarantees about their safety. To this end, we introduce Probabilistic Shielding via Risk Augmentation (ProSh), a model-free algorithm for safe reinforcement learning under cost constraints. ProSh augments the Constrained MDP state space with a risk budget and enforces safety by applying a shield to the agent's policy distribution using a learned cost critic. The shield ensures that all sampled actions remain safe in expectation. We also show that optimality is preserved when the environment is deterministic. Since ProSh is model-free, safety during training depends on the knowledge we have acquired about the environment. We provide a tight upper-bound on the cost in expectation, depending only on the backup-critic accuracy, that is always satisfied during training. Under mild, practically achievable assumptions, ProSh guarantees safety even at training time, as shown in the experiments.

cs.LG↗

arXiv2025

Scattering of Rational Solutions to the Half-Wave Maps Equation

This article studies the rational solutions of the Half-Wave Maps equation (HWM) in the non-singular spectrum case. We first provide characterizations to what we call \emph{scattering behavior}, and show that they imply scattering in Sobolev norm. We then provide a local condition implying \emph{scattering behavior}. Building on this, we show that any solution with non-singular spectrum scatters and give an explicit formula for the function to which the solution is scattering. This allows us to show that the scattering map is the identity. Additionally, we create, for any given number of spins and any target non-singular spectrum, global solutions of (HWM) with a spectrum arbitrarily close to the target. Finally, using a diagonal characterization of traveling waves, we show that if a wave scatters to a traveling wave, it is a scattering wave.

math.AP↗

arXiv2024

Half-Wave Maps: Explicit Formulas for Rational Functions with Simple Poles

We establish an explicit formula for the Half-Wave maps equation for rational functions with simple poles. The Lax pair provides a description of the evolution of the poles. By considering a half-spin formulation, we use linear algebra to derive a time evolution equation followed by the half-spins, in the moving frame provided by the Lax pair. We then rewrite this formula using a Toeplitz operator and $G$, the adjoint of the operator of multiplication by $x$ on the Hardy space $L_+^2(\mathbb{R})$.

math.AP↗

arXiv2024

On the stability of the ill-posedness of a quasi-linear wave equation in two dimensions with initial data in $H^{7/4} (\ln H^{-β})$

This article is the continuation of \cite{ohlmann2021illposedness} where we exhibited the ill-posedness of a quasi-linear wave equation in dimension $2$ for initial data in $H^{7/4}(\ln H^{7/4})$. Here, we look at modifications of the equation and show that the blow-up phenomenon still occurs. First, we study another equation with the same characteristics but a different underlying ODE. Later, we study an equation where a $x_2$ dependency is introduced. The latter case constitutes the main contribution of this paper, as we are able to show that the solution still behaves pathologically without having an explicit formula for either the characteristics or the values of the solution. Finally, we study the case where a perturbation of the initial data is introduced.

math.AP↗

arXiv2024

A Study on the Well-Posedness of 1D Energy-Critical Half-Wave Maps Equations

In this article, we study the well-posedness of the energy-critical half-wave maps equation (HWM) in dimension $1$. The half-wave maps equation emerges from the continuum limit of the Haldane Shastry spin chains and has been shown to arise as the continuum limit of Calogero-Moser classical spin systems. In higher dimension $d\geq 5$, it has been shown that (HWM) is well-posed by Krieger and Sire. This result has been improved by Krieger and Kiesenhofer to $d = 4$ but the Strichartz estimate on which the argument is built no longer holds for smaller dimensions. A Lax-pair structure has been revealed for this equation by Lenzmann and Gérard, indicating complete integrability and the fact that rational solutions stay rational for all time. The well-posedness of the (HWM) equation in lower dimensions remains an open problem. Here, we show the well-posedness of (HWM) in the rational case for finite times with separated poles, and for large and infinite times with distinct speeds of propagation.

math.AP↗

arXiv2022

Ill-posedness of a quasilinear wave equation in two dimensions for data in $H^{7/4}$

In this article, we study the ill-posedness of a quasilinear wave equation. It was shown by Tataru and Smith in 2005 that for any $s>7/4$ (or $11/4$ in our situation), the equation is well-posed in $H^{s}\times H^{s-1}$. We show a sharpness result by exhibiting a quasilinear wave equation and an initial data such that the Cauchy problem is ill-posed for in $H^{11/4} (\ln H)^{-β}\times H^{7/4} (\ln H^{-β})$.

math.AP↗