arXiv subjects

Dante Kalise

Publications and source records attributed to Dante Kalise.

At least 19 recordsLinked to original sources

arXiv2026

Control Strategies for Multi-Species Wasserstein Gradient Flows

We stabilize stationary states of coupled multi-species McKean--Vlasov equations by feedback control, extending a spectral approach based on the Wasserstein Hessian and a Riccati equation from a single density to a product of Wasserstein spaces. We give necessary and sufficient conditions for local exponential stabilization at a prescribed rate when only one species is actuated, together with the minimum number of control inputs required. For two species, the condition reduces to a rank condition on the Fourier coefficients of the cross-interaction kernel over the eigenspaces of the unactuated species' linearized dynamics. A two-community noisy Kuramoto model and a cell-sorting model illustrate the results.

math.OC↗

arXiv2026

Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

math.OC↗

arXiv2026

Nonlocal Mean Field Schr\"{o}dinger Bridge with Learned Interactions

The Schr\"odinger Bridge Problem connects an initial distribution to a terminal one along a minimum-energy stochastic process. Its mean-field extension, the Mean-Field Schr\"odinger Bridge, governs interacting populations whose dynamics and costs depend on the collective distribution. When these interactions are nonlocal, their direct evaluation scales quadratically with the population size, making large ensembles intractable within FBSDE-based solvers. We replace these terms with neural surrogates in state and time, trained on empirical interaction values along sampled trajectories and embedded in a four-stage alternating scheme that updates the forward and backward potentials and the surrogates in turn, while preserving forward--backward consistency and the prescribed endpoint marginals. We derive Gr\"onwall-type stability bounds quantifying how surrogate errors propagate to the generated trajectories under a small-gain condition. On crowd-navigation and high-dimensional opinion-dynamics benchmarks, the surrogates reproduce the trajectories obtained with exact evaluation at reduced training cost. The advantage is most significant when the interaction is a nonlinear functional of the measure, such as the normalized bounded-confidence drift, for which random-batch subsampling is biased and unstable whereas the learned surrogate remains accurate.

math.OC↗

arXiv2026

A Proximal Primal-Dual Approach to Generalized JKO Schemes for Doubly Nonlinear Parabolic Equations

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the $p$-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by computing explicit formulas for proximal operators with general costs amenable to efficient numerical approximation. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.

math.NA↗

arXiv2026

Feedback Control and Local Convexification of Wasserstein Gradient Flows

Building on the feedback stabilization framework that we developed for McKean-Vlasov PDEs, we treat a class of entropy-regularized free energies $F = E + \sigma\mathrm{Ent}$ on the flat torus, with $E$ possibly nonconvex and nonlocal, and show that the resulting controlled system admits a gradient flow formulation in Wasserstein space. Specifically, we realize the Wasserstein Hessian at a target stationary measure $\bar\mu$ as a self-adjoint operator with compact resolvent and show that its negative is unitarily equivalent to the generator of the linearized dynamics. The resulting feedback stabilization can then be interpreted as a local convexification of the free energy landscape: for any prescribed threshold $\delta > 0$, the control, obtained from an algebraic Riccati equation for the linearized problem, induces a finite-rank perturbation of the Wasserstein Hessian that lifts its spectrum above $\delta$, yielding local displacement convexity on a H\"older neighborhood of $\bar\mu$ in regimes where the uncontrolled energy is nonconvex or slowly contracting. Under a local well-posedness assumption, $\bar\mu$ is moreover a locally exponentially stable equilibrium of the closed-loop dynamics, with rate at least $\delta$. The framework covers McKean-Vlasov models and extends to moment-constrained Fokker-Planck dynamics and Fokker-Planck equations on closed Riemannian manifolds.

math.OC↗

arXiv2025

A PDE-constrained Optimization Approach to Optimal Trajectory Planning under Uncertainty via Reflected Schr\"odinger Bridges

A computational PDE-constrained optimization approach is proposed for optimal trajectory planning under uncertainty by means of an associated Schroedinger Bridge Problem (SBP). The proposed SBP formulation is interpreted as the mean-field limit associated with the energy-optimal evolution of a particle governed by a stochastic differential equation (SDE) with nonlinear drift and reflecting boundary conditions, constrained to prescribed initial and terminal densities. The resulting mean-field system consists of a nonlinear Fokker-Planck equation coupled with a Hamilton-Jacobi-Bellman equation, subject to two-point boundary conditions in time and Neumann boundary conditions in space. Through the Hopf-Cole transformation, this nonlinear system is recast as a pair of forward-backward advection-diffusion equations, which are amenable to efficient numerical solution via a standard finite element discretization. The weak formulation naturally enforces reflecting boundary conditions without requiring explicit particle-boundary collision detection, thus circumventing the computational difficulties inherent to particle-based methods in complex geometries. Numerical experiments on challenging 3D maze configurations demonstrate fast convergence, mass conservation, and validate the optimal controls computed through reflected SDE simulations.

math.OC↗

arXiv2025

Sparse stabilization of mean-field agent dynamics through a three-operator splitting method

We study the sparse stabilization of nonlinear multi-agent systems within a mean-field optimal control framework. The goal is to drive large populations of interacting agents toward consensus with minimal control effort. In the mean-field limit, the dynamics are described by a Vlasov-type kinetic equation, and sparsity is enforced through an l1-l2 penalization in the cost functional. The resulting nonsmooth optimization problem is solved via a three-operator splitting (TOS) method that separately handles smooth, nonsmooth, and constraint components through gradient, shrinkage, and projection steps. A particle-based Monte Carlo discretization with random batch interactions enables scalable computation while preserving the mean-field structure. Numerical experiments on the Cucker-Smale model demonstrate effective consensus formation with sparse, localized control actions, confirming the efficiency and robustness of the proposed approach.

math.OC↗

arXiv2025

Non-Convex Global Optimization as an Optimal Stabilization Problem: Convergence Rates

We develop a rigorous framework for global non-convex optimization by reformulating the minimization problem as a discounted infinite-horizon optimal control problem. For non-convex, continuous, and possibly non-smooth objective functions with multiple global minimizers, where classical gradient-based methods lack global convergence guarantees, we establish explicit exponential convergence rates with computable constants. Our analysis proves (i) variational convergence of the value function of the optimal control problem, (ii) convergence in the objective function for the original problem, as well as (iii) pathwise convergence of optimal trajectories to the minimizer set under minimal structural assumptions that require neither convexity, differentiability, nor {\L}ojasiewicz-type conditions on the objective. These quantitative results significantly strengthen the asymptotic theory developed in our previous work (arXiv:2511.10815). Numerical experiments demonstrate the practical effectiveness of the approach on challenging non-convex problems.

math.OC↗

arXiv2025

Non-Convex Global Optimization as an Optimal Stabilization Problem: Dynamical Properties

We study global optimization of non-convex functions through optimal control theory. Our main result establishes that (quasi-)optimal trajectories of a discounted control problem converge globally and practically asymptotically to the set of global minimizers. Specifically, for any tolerance $\eta > 0$, there exist parameters $\lambda$ (discount rate) and $t$ (time horizon) such that trajectories remain within an $\eta$-neighborhood of the global minimizers after some finite time $\tau$. This convergence is achieved directly, without solving ergodic Hamilton-Jacobi-Bellman equations. We prove parallel results for three problem formulations: evolutive discounted, stationary discounted, and evolutive non-discounted cases. The analysis relies on occupation measures to quantify the fraction of time trajectories spend away from the minimizer set, establishing both reachability and stability properties.

math.OC↗

arXiv2025

An Inexact Tensor-Train Primal-Dual Interior-Point Method for Semidefinite Programs

In this work, we introduce an interior-point method that employs tensor decompositions to efficiently represent and manipulate the variables and constraints of semidefinite programs, targeting problems where the solutions may not be low-rank but admit low-tensor-train rank approximations. Our method leverages a primal-dual infeasible interior-point framework with global and local convergence estimates. In experiments on Maximum Cut, Maximum Stable Set, and Correlation Clustering, the tensor-train interior-point method handles problems up to size $2^{12}$ with certified primal-dual accuracy, reported gaps of approximate order $10^{-4}$, and remains the only primal-dual SDP solver reported at the largest dimensions across all three benchmarks. Moreover, numerical evidence indicates that tensor-train ranks of the iterates typically remain structured and moderate along the interior-point trajectory, with instance-dependent growth for harder high-rank cases, supporting the scalability of the approach. Tensor-train interior-point methods offer a promising avenue for problems that lack traditional sparsity or low-rank structure, exploiting tensor-train structures instead.

math.OC↗

arXiv2025

Hierarchical clustering and dimensional reduction for optimal control of large-scale agent-based models

Agent-based models (ABMs) provide a powerful framework to describe complex systems composed of interacting entities, capable of producing emergent collective behaviours such as consensus formation or clustering. However, the increasing dimensionality of these models -- in terms of both the number of agents and the size of their state space -- poses significant computational challenges, particularly in the context of optimal control. In this work, we propose a scalable control frame work for large-scale ABMs based on a twofold model order reduction strategy: agent clustering and projection-based reduction via Proper Orthogonal Decomposition (POD). These techniques are integrated into a feedback loop that enables the design and application of optimal control laws over a reduced-order representation of the system. To illustrate the effectiveness of the approach, we consider the opinion dynamics model, a prototyp ical first-order ABM where agents interact through state-dependent influence functions. We show that our method significantly improves control efficiency, even in scenarios where direct control fails due to model complexity. Beyond its methodological contributions, this work also highlights the rel evance of opinion dynamics models in environmental contexts -- for example, modeling the diffusion of pro-environmental attitudes or decision-making processes in sustainable policy adoption -- where controlling consensus formation plays a crucial role.

math.OC↗

arXiv2025

Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

We develop a feedback control framework for stabilizing the McKean-Vlasov PDE on the torus. Our goal is to steer the dynamics toward a prescribed stationary distribution or accelerate convergence to it using a time-dependent control potential. We reformulate the controlled PDE in a weighted, zero-mean space and apply the ground-state transform to obtain a Schrodinger-type operator. The resulting operator framework enables spectral analysis, verification of the infinite-dimensional Hautus test, and construction of a Riccati-based feedback law derived from the linearized dynamics, yielding local exponential stabilization with a chosen convergence rate. We rigorously prove local exponential stabilization via maximal regularity arguments and nonlinear estimates. Numerical experiments on well-studied models in one and two dimensions (the noisy Kuramoto model for synchronization, the O(2) spin model in a magnetic field, and the von Mises attractive interaction potential) showcase the effectiveness of our control strategy, demonstrating convergence acceleration and stabilization of unstable equilibria.

math.OC↗

arXiv2025

Parallel-in-Time Preconditioning for Time-Dependent Variational Mean Field Games

We study the numerical approximation of a time-dependent variational mean field game system with local couplings and either periodic or Neumann boundary conditions. Following a variational approach, we employ a finite difference discretization and solve the resulting finite-dimensional optimization problem using the Chambolle--Pock primal--dual algorithm. As this involves computing proximal operators and solving ill-conditioned linear systems at each iteration, we embed within our solver a general class of parallel-in-time preconditioners based on suitably-chosen diagonalization techniques, applied using discrete Fourier transforms. These enable efficient, scalable iterative solvers for each linear system, with robustness across a wide range of viscosities. We further develop fast solvers for the resulting ill-conditioned systems arising at each time step, using exact recursive schemes for structured grids while allowing for other geometries. Numerical experiments confirm the improved performance and parallel scalability of our approach.

math.NA↗

arXiv2025

Time-optimal neural feedback control of nilpotent systems as a binary classification problem

A computational method for the synthesis of time-optimal feedback control laws for linear nilpotent systems is proposed. The method is based on the use of the bang-bang theorem, which leads to a characterization of the time-optimal trajectory as a parameter-dependent polynomial system for the control switching sequence. A deflated Newton's method is then applied to exhaust all the real roots of the polynomial system. The root-finding procedure is informed by the Hermite quadratic form, which provides a sharp estimate on the number of real roots to be found. In the second part of the paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network -- interpreted as a binary classifier -- via supervised learning. Numerical tests in integrators of increasing dimension assess the accuracy, robustness, and real-time-control capabilities of the approximate control law.

math.OC↗

arXiv2025

A Spectral Approach to Optimal Control of the Fokker-Planck Equation

In this paper, we present a spectral optimal control framework for Fokker-Planck equations based on the standard ground state transformation that maps the Fokker-Planck operator to a Schrodinger operator. Our primary objective is to accelerate convergence toward the (unique) steady state. To fulfill this objective, a gradient-based iterative algorithm with Pontryagin's maximum principle and the Barzilai-Borwein update is developed to compute time-dependent controls. Numerical experiments on two-dimensional ill-conditioned normal distributions and double-well potentials demonstrate that our approach effectively targets slow-decaying modes, thus increasing the spectral gap.

math.OC↗

arXiv2025

Separable Approximations of Optimal Value Functions and Their Representation by Neural Networks

The use of separable approximations is proposed to mitigate the curse of dimensionality related to the approximation of high-dimensional value functions in optimal control. The separable approximation exploits intrinsic decaying sensitivity properties of the system, where the influence of a state variable on another diminishes as their spatial, temporal, or graph-based distance grows. This property allows the efficient representation of global functions as a sum of localized contributions. A theoretical framework for constructing separable approximations in the context of optimal control is proposed by leveraging decaying sensitivity in both discrete and continuous time. Results extend prior work on decay properties of solutions to Lyapunov and Riccati equations, offering new insights into polynomial and exponential decay regimes. Connections to neural networks are explored, demonstrating how separable structures enable scalable representations of high-dimensional value functions while preserving computational efficiency.

math.OC↗

arXiv2024

A polynomial approximation scheme for nonlinear model reduction by moment matching

We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed to find an approximate solution to the invariance equation using a Newton iteration on the coefficients of a monomial basis expansion of the solution. These solutions to the invariance equations can then be used to construct reduced-order models. We assess the ability of the method to solve the invariance PDE system as well as to achieve moment matching and recover the steady-state behaviour of nonlinear systems with state dimension of order 1000 driven by linear and nonlinear signal generators.

math.OC↗

arXiv2024

A total variation flow scheme for ergodic mean field games

Motivated by recent developments in mean field games in ecology, in this paper we introduce a connection between the best response dynamics in evolutionary game theory, the minimization of the highest income of a game, and minimizing movement schemes. The aim of this work is to develop a variational approach to compute solutions of first order ergodic mean field games that may not possess a priori a variational structure. The study is complemented by a discussion and successful implementation of numerical algorithms, and comparisons between them in a variety of cases.

math.OC↗