18 research outputs found
Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift
We prove a version of the stochastic maximum principle, in the sense of
Pontryagin, for the finite horizon optimal control of a stochastic partial
differential equation driven by an infinite dimensional additive noise. In
particular we treat the case in which the non-linear term is of Nemytskii type,
dissipative and with polynomial growth. The performance functional to be
optimized is fairly general and may depend on point evaluation of the
controlled equation. The results can be applied to a large class of non-linear
parabolic equations such as reaction-diffusion equations
Necessary stochastic maximum principle for dissipative systems on infinite time horizon
We develop a necessary stochastic maximum principle for a finite-dimensional
stochastic control problem in infinite horizon under a polynomial growth and
joint monotonicity assumption on the coefficients. The second assumption
generalizes the usual one in the sense that it is formulated as a joint
condition for the drift and the diffusion term. The main difficulties concern
the construction of the first and second order adjoint processes by solving
backward equations on an unbounded time interval. The first adjoint process is
characterized as a solution to a backward SDE, which is well-posed thanks to a
duality argument. The second one can be defined via another duality relation
written in terms of the Hamiltonian of the system and linearized state
equation. Some known models verifying the joint monotonicity assumption are
discussed as well
Stochastic maximum principle for SPDEs with delay.
In this paper we develop necessary conditions for optimality, in the form of
the Pontryagin maximum principle, for the optimal control problem of a class of
infinite dimensional evolution equations with delay in the state. In the cost
functional we allow the final cost to depend on the history of the state. To
treat such kind of cost functionals we introduce a new form of anticipated
backward stochastic differential equations which plays the role of dual
equation associated to the control problem
A variational approach to the mean field planning problem
We investigate a first-order mean field planning problem of the form
\begin{equation} \left\lbrace\begin{aligned} -\partial_t u + H(x,Du) &= f(x,m)
&&\text{in } (0,T)\times \mathbb{R}^d, \\ \partial_t m - \nabla\cdot
(m\,H_p(x,Du)) &= 0 &&\text{in }(0,T)\times \mathbb{R}^d,\\ m(0,\cdot) = m_0,
\; m(T,\cdot) &= m_T &&\text{in } \mathbb{R}^d, \end{aligned}\right.
\end{equation} associated to a convex Hamiltonian with quadratic growth and
a monotone interaction term with polynomial growth. We exploit the
variational structure of the system, which encodes the first order optimality
condition of a convex dynamic optimal entropy-transport problem with respect to
the unknown density and of its dual, involving the maximization of an
integral functional among all the subsolutions of an Hamilton-Jacobi
equation. Combining ideas from optimal transport, convex analysis and
renormalized solutions to the continuity equation, we will prove existence and
(at least partial) uniqueness of a weak solution . A crucial step of our
approach relies on a careful analysis of distributional subsolutions to
Hamilton-Jacobi equations of the form ,
under minimal summability conditions on , and to a measure-theoretic
description of the optimality via a suitable contact-defect measure. Finally,
using the superposition principle, we are able to describe the solution to the
system by means of a measure on the path space encoding the local behavior of
the players
Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below
Given a complete, connected Riemannian manifold with Ricci
curvature bounded from below, we discuss the stability of the solutions of a
porous medium-type equation with respect to the 2-Wasserstein distance. We
produce (sharp) stability estimates under negative curvature bounds, which to
some extent generalize well-known results by Sturm and Otto-Westdickenberg. The
strategy of the proof mainly relies on a quantitative smoothing
property of the equation considered, combined with the Hamiltonian approach
developed by Ambrosio, Mondino and Savar\'e in a metric-measure setting
Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: equivalence and Gamma-convergence
This paper is devoted to the study of multi-agent deterministic optimal control problems. We initially provide a thorough analysis of the Lagrangian, Eulerian and Kantorovich formulations of the problems, as well as of their relaxations. Then we exhibit some equivalence results among the various representations and compare the respective value functions. To do it, we combine techniques and ideas from optimal transportation, control theory, Young measures and evolution equations in Banach spaces. We further exploit the connections among Lagrangian and Eulerian descriptions to derive consistency results as the number of particles/agents tends to infinity. To that purpose we prove an empirical version of the Superposition Principle and obtain suitable Gamma-convergence results for the controlled systems
Large deviations for Kac-like walks
We introduce a Kac's type walk whose rate of binary collisions preserves the
total momentum but not the kinetic energy. In the limit of large number of
particles we describe the dynamics in terms of empirical measure and flow,
proving the corresponding large deviation principle. The associated rate
function has an explicit expression. As a byproduct of this analysis, we
provide a gradient flow formulation of the Boltzmann-Kac equation