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Marc Quincampoix

Publications and source records attributed to Marc Quincampoix.

15 recordsLinked to original sources
arXiv2025

Generalized differentiation in Wasserstein space and application to multiagent control problem

Several concepts of generalized differentiation in Wasserstein space have been proposed in order to deal with the intrinsic nonsmoothness arising in the context of optimization problems in Wasserstein spaces. In this paper we introduce a concept of admissible variation encompassing some of the most popular definitions as special cases, and using it to derive a comparison principle for viscosity solutions of an Hamilton Jacobi Bellman equation following from an optimal control of a multiagent systems.

math.OC↗
arXiv2025

Strong bi-metric regularity in affine optimal control problems

The paper presents new sufficient conditions for the property of strong bi-metric regularity of the optimality map associated with an optimal control problem which is affine with respect to the control variable ({\em affine problem}). The optimality map represents the system of first order optimality conditions (Pontryagin maximum principle), and its regularity is of key importance for the qualitative and numerical analysis of optimal control problems. The case of affine problems is especially challenging due to the typical discontinuity of the optimal control functions. A remarkable feature of the obtained sufficient conditions is that they do not require convexity of the objective functional. As an application, the result is used for proving uniform convergence of the Euler discretization method for a family of affine optimal control problems.

math.OC↗
arXiv2025

Reachability for multiagent control systems via Lyapunov functions

This paper concerns the problem of reachability of a given state for a multiagent control system in $\mathbb{R}^d$. In such a system, at every time each agent can choose his/her velocity which depends both on his/her position and on the position of the whole crowd of agents (modeled by a probability measure on $ \mathbb{R}^d$). The main contribution of the paper is to study the above reachability problem with a given rate of attainability through a Lyapunov method adapted to the Wasserstein space of probability measures. As a byproduct we obtain a new comparison result for viscosity solutions of Hamilton Jacobi equations in the Wasserstein space.

math.OC↗
arXiv2024

Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time

A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $λ$ tends to zero. In the literature, this problem has been addressed under various controllability or ergodicity conditions ensuring that the rescaled value function converges uniformly to a constant limit. In this case the limit can be characterized as the unique constant such that a suitable Hamilton-Jacobi equation has at least one continuous viscosity solution. In this paper, we study this problem without such conditions, so that the aforementioned limit needs not be constant. Our main result characterizes the uniform limit (when it exists) as the maximal subsolution of a system of Hamilton-Jacobi equations. Moreover, when such a subsolution is a viscosity solution, we obtain the convergence of optimal values as well as a rate of convergence. This mirrors the analysis of the discrete time case, where we characterize the uniform limit as the supremum over a set of sub-invariant half-lines of the dynamic programming operator. The emerging structure in both discrete and continuous time models shows that the supremum over sub-invariato half-lines with respect to the Lax-Oleinik semigroup/dynamic programming operator, captures the behavior of the limit cost as discount vanishes.

math.OC↗
arXiv2022

Extending Rademacher Theorem to Set-Valued Maps

Rademacher theorem asserts that Lipschitz continuous functions between Euclidean spaces are differentiable almost everywhere. In this work we extend this result to set-valued maps using an adequate notion of set-valued differentiability relating to convex processes. Our approach uses Rademacher theorem but also recovers it as a special case.

math.CA↗
arXiv2020

Characterization of Filippov representable maps and Clarke subdifferentials

The ordinary differential equation $\dot{x}(t)=f(x(t)), \; t \geq 0 $, for $f$ measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the function $f$ with its Filippov regularization $F_{f}$ and consider the differential inclusion $\dot{x}(t)\in F_{f}(x(t))$ which always has a solution. It is interesting to know, inversely, when a set-valued map $Φ$ can be obtained as the Filippov regularization of a (single-valued, measurable) function. In this work we give a full characterization of such set-valued maps, hereby called Filippov representable. This characterization also yields an elegant description of those maps that are Clarke subdifferentials of a Lipschitz function.

math.OC↗
arXiv2018

A differential game on Wasserstein space. Application to weak approachability with partial monitoring

Studying continuous time counterpart of some discrete time dynamics is now a standard and fruitful technique, as some properties hold in both setups. In game theory, this is usually done by considering differential games on Euclidean spaces. This allows to infer properties on the convergence of values of a repeated game, to deal with the various concepts of approachability, etc. In this paper, we introduce a specific but quite abstract differential game defined on the Wasserstein space of probability distributions and we prove the existence of its value. Going back to the discrete time dynamics, we derive results on weak approachability with partial monitoring: we prove that any set satisfying a suitable compatibility condition is either weakly approachable or weakly excludable. We also obtain that the value for differential games with nonanticipative strategies is the same that those defined with a new concept of strategies very suitable to make links with repeated games.

math.OC↗
arXiv2015

Differential games with asymmetric information and without Isaacs condition

We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs condition. The dynamics is an ordinary differential equation parametrised by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. Moreover, the value function can be characterised in term of the unique viscosity solution in some dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the Isaacs condition which is usually assumed in differential games.

math.OC↗
arXiv2015

Stochastic differential equations of second order with a small parameter

We consider boundary value problems for stochastic differential equations of second order with a small parameter. For this case we prove a special existence and unicity theorem for strong solutions. The asymptotic behavior of these solutions as small parameter goes to zero is studied. The stochastic averaging theorem for such equations is shown. The limits in the explicit form for the solutions as a small parameter goes to zero are found.

math.PR↗
arXiv2015

Limit value for optimal control with general means

We consider optimal control problem with an integral cost which is a mean of a given function. As a particular case, the cost concerned is the Cesàro average. The limit of the value with Cesàro mean when the horizon tends to infinity is widely studied in the literature. We address the more general question of the existence of a limit when the averaging parameter converges, for values defined with means of general types. We consider a given function and a family of costs defined as the mean of the function with respect to a family of probability measures -- the evaluations -- on R_+. We provide conditions on the evaluations in order to obtain the uniform convergence of the associated value function (when the parameter of the family converges). Our main result gives a necessary and sufficient condition in term of the total variation of the family of probability measures on R_+. As a byproduct, we obtain the existence of a limit value (for general means) for control systems having a compact invariant set and satisfying suitable nonexpansive property.

math.OC↗
arXiv2014

Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition

In the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors' best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition $π$ of the time interval $[0,T]$. The underlying stochastic controls for the both players are randomized along $π$ by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point $t_{j-1}$ of the subintervals generated by $π$, the controls of Players 1 and 2 are conditionally independent over $[t_{j-1},t_j)$. It is shown that the associated lower and upper value functions $W^π$ and $U^π$ converge uniformly on compacts to a function $V$, the so-called value in mixed strategies, as the mesh of $π$ tends to zero. This function $V$ is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation.

math.PR↗
arXiv2013

On an unified framework for approachability in games with or without signals

We unify standard frameworks for approachability both in full or partial monitoring by defining a new abstract game, called the "purely informative game", where the outcome at each stage is the maximal information players can obtain, represented as some probability measure. Objectives of players can be rewritten as the convergence (to some given set) of sequences of averages of these probability measures. We obtain new results extending the approachability theory developed by Blackwell moreover this new abstract framework enables us to characterize approachable sets with, as usual, a remarkably simple and clear reformulation for convex sets. Translated into the original games, those results become the first necessary and sufficient condition under which an arbitrary set is approachable and they cover and extend previous known results for convex sets. We also investigate a specific class of games where, thanks to some unusual definition of averages and convexity, we again obtain a complete characterization of approachable sets along with rates of convergence.

cs.GT↗
arXiv2012

Value Function of Differential Games without Isaacs Conditions. An Approach with Non-Anticipative Mixed Strategies

In the present paper we investigate the problem of the existence of a value for differential games without Isaacs condition. For this we introduce a suitable concept of mixed strategies along a partition of the time interval, which are associated with classical nonanticipative strategies (with delay). Imposing on the underlying controls for both players a conditional independence property, we obtain the existence of the value in mixed strategies as the limit of the lower as well as of the upper value functions along a sequence of partitions which mesh tends to zero. Moreover, we characterize this value in mixed strategies as the unique viscosity solution of the corresponding Hamilton-Jacobi-Isaacs equation.

math.OC↗