Keywords: Control of Distributed Parameter Systems
Abstract: The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.

Apart from four possible pathologies, if the monopolist's costs grow quadratically with the product type we show that a smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Based on works-in-progress including "The monopolist's free boundary problem in the plane" (https://arxiv.org/abs/2412.15505), with Lucas O'Brien (MIT), Cale Rankin (Monash University) and Kelvin Shuangjian Zhang (Fudan University) in various combinations.

Keywords: Nonlinear Systems and Control, Feedback Control Systems
Abstract: We introduce the notion of descriptor embedding for nonlinear systems and use it for the data-driven design of stabilizing controllers. Specifically, we provide sufficient data-dependent LMI conditions which, if feasible, return a stabilizing nonlinear controller of the form u = K(x)Z(x) where K(x) belongs to a polytope and Z is a user-defined function. A simulation example is used to illustrate the results and compare them to existing methods.

Keywords: Optimal Control, Nonlinear Systems and Control
Abstract: We study spatial decay properties of sensitivities in a nonlinear optimal control problem with a graph-structured interaction topology. For a problem with nonlinear decoupled dynamics and quadratic cost, we show that a perturbation of the zero initial condition at a single node induces an optimal trajectory whose node-wise norms decay exponentially with the graph distance from the perturbed node. The analysis, based on a nonlinear null-controllability condition, provides a first step toward extending known spatial decay results from linear–quadratic to nonlinear systems. A numerical example illustrates the theoretical findings.

Keywords: Machine Learning and Control
Abstract: We present a Koopman autoencoder-based least-squares policy iteration (KAELSPI) algorithm in reinforcement learning (RL). The KAE-LSPI algorithm is based on reformulating the so-called least-squares fixed-point approximation method in terms of the extended dynamic mode decomposition (EDMD) matrix, thereby enabling automatic features learning via the Koopman autoencoder (KAE) framework. The approach is motivated by the lack of a systematic method for choosing features or kernels in linear approximation RL techniques. We apply the KAE-LSPI algorithm to a toy example of the stochastic chain walk problem. Unlike previous works, no features or kernels need to be fixed a priori in our approach.

Keywords: Nonlinear Systems and Control, Linear Systems
Abstract: The idea of embedding or immersing a complex nonlinear dynamical system into a higher-dimensional yet simpler one has important applications in analysis, design, and learning of nonlinear systems. This extended abstract summarizes several recent results on when finite dimensional linear immersions exist for discrete-time and continuous-time dynamical systems. In particular, we show that the existence of countably many but more than one omega-limit sets constitutes an obstruction for linear immersibility with continuous mappings. We present several examples for existence and non-existence and discuss the implications of this obstruction in the context of learning dynamics.

Keywords: Optimization : Theory and Algorithms, Machine Learning and Control
Abstract: Fréchet regression, or conditional Barycenters, is a flexible framework for modeling relationships between covariates (usually Euclidean) and response variables on general metric spaces, e.g., probability distributions or positive definite matrices. However, in contrast to classical barycenter problems, computing conditional counterparts in many non-Euclidean spaces remains an open challenge, as they yield non-convex optimization problems with an affine structure. In this work, we study the existence and computation of conditional barycenters, specifically in the space of positive-definite matrices with the Bures-Wasserstein metric. We provide a sufficient condition for the existence of a minimizer of the conditional barycenter problem that characterizes the regression range of extrapolation. Moreover, we further characterize the optimization landscape, proving that under this condition, the objective is free of local maxima. Additionally, we develop a projection-free and provably correct algorithm for the approximate computation of first-order stationary points. Numerical experiments validate the performance of the proposed methods on regression problems of biological networks.

Keywords: Infinite Dimensional Systems Theory, Systems on Graphs, Stability
Abstract: Oscillator Ising Machines (OIMs) have emerged as promising computational architectures for approximating solutions to combinatorial optimization problems. We derive and analyze the mean-field limit of an OIM model and show that it inherits the gradient-flow structure of the finite-dimensional dynamics. We identify conditions under which this mean-field evolution admits an Eulerian formulation as a gradient flow on the Wasserstein space of probability measures, and contrast this with a general Lagrangian formulation that is always available. The gradient-flow structure strongly constrains the long-time dynamics and, in the totally symmetric case, enables a complete classification of limit solutions and their stability. In particular, all limit solutions are fixed points with a highly restricted phase structure. We also present numerical evidence that the mean-field model correctly predicts behavioral regimes in large Erdős–Rényi networks.

Keywords: Optimization : Theory and Algorithms, Nonlinear Filtering and Estimation, Large Scale Systems
Abstract: We study a discrete Schrödinger bridge problem with partial marginal information, where only a subset of the state space is observed and the total transported mass is not fixed. In this setting, the problem is inherently ill-posed, preventing the direct use of classical Sinkhorn-type iterations, and potentially leading to non-uniqueness of solutions. To address this issue, we propose an entropic proximal point scheme that alternates between Schrödinger bridge problems with a fixed marginal and updates of the unobserved mass components. Each proximal step reduces to a Schrödinger bridge problem that can be solved efficiently using Sinkhorn-type iterations. We further analyze the structure of the solution set and derive conditions under which uniqueness of the optimal solution is recovered, relating these conditions to an observability property of an associated linear system.

Keywords: System Identification, Optimization : Theory and Algorithms, Linear Systems
Abstract: We study the problem of identifying an optimal coupling between input–output distributional data generated by a causal dynamical system. The coupling is required to satisfy prescribed marginal distributions and a causality constraint reflecting the temporal structure of the system. We formulate this problem as a Schrödinger Bridge, which seeks the coupling closest—in Kullback–Leibler divergence—to a given prior while enforcing both marginal and causality constraints. For the case of Gaussian marginals and general time-dependent quadratic cost functions, we derive a fully tractable characterization of the Sinkhorn iterations that converges to the optimal solution. Beyond its theoretical contribution, the proposed framework provides a principled foundation for applying causal optimal transport methods to system identification from distributional data.

Keywords: Optimal Control, Stochastic Control and Estimation, Optimization : Theory and Algorithms
Abstract: We introduce and discuss a new type of transportation problems, where particles/agents in the ensemble are labeled and their relative position along their journey is of interest. Of particular importance in our program are control laws that steer ensembles along cycles ensuring that individual particles return to their original position. This feature is in contrast with the classical theory of optimal transport where the primary object of study is the path of probability densities, without any concern about particle labels. We focus on the case Gaussian distributions and linear dynamics, and explore a hitherto unstudied sub-Riemannian structure of Monge-Kantorovich transport where the relative position of particles along their journey is modeled by the holonomy of the transportation schedule.

Keywords: Port-Hamiltonian Systems, Dissipativity, Nonlinear Systems and Control
Abstract: The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work, port-Hamiltonian formulations for both the inviscid and the viscous Burgers' equations are proposed, enabling a representation that incorporates both convective and dissipative effects. Boundary control and observation are naturally handled within this framework. Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derived. The relationship between time step, spatial resolution, and viscosity required to achieve numerical stability is analyzed. Numerical experiments validate the effectiveness of the approach.

Keywords: Port-Hamiltonian Systems, Mathematical Theory of Networks and Circuits, Linear Systems
Abstract: Linear port-Hamiltonian systems (PHS) defined by Dirac and Lagrange structures, referred to as port-Hamiltonian differential-algebraic equation systems (PH DAEs), incorporate a class of algebraic constraints arising from energy deficiency in the Hamiltonian definition, known as Lagrange algebraic constraints. This paper presents two novel equivalent characterizations of linear PH DAEs with Lagrange constraints, including resistive and external ports. Relations to constrained input–output PHS formulations are established. The constraints are shown to be transferable between Dirac and Lagrange structures, enabling equivalent representations in terms of state or effort variables. The proposed formulations are illustrated using a micro-robotic contact scenario model.

Keywords: Dissipativity, Mathematical Theory of Networks and Circuits, Large Scale Systems
Abstract: We characterize a class of linear, time invariant systems whose state-space realizations form a Stieltjes matrix when stacked appropriately. These systems share properties of both passive and internally positive systems. The class is preserved under negative feedback, their inverses are internally positive and a diagonal storage function is necessary and sufficient for dissipativity. We show that Stieltjes systems arise naturally from a class of RC circuits.

Keywords: Physical Systems Theory, Mathematical Theory of Networks and Circuits
Abstract: Duality plays a fundamental role in circuit theory, where it arises both at the network level and at the component level. While component-level duality is well understood for resistive, capacitive, and inductive elements, its systematic treatment for memristive elements remains limited. In this extended abstract, we explore how duality emerges for memristive elements as an extension of the duality for resistive elements. Since memristive elements incorporate memory effects, the classical content-cocontent duality of resistive elements does not directly extend in a straightforward manner. To address this challenge, we adopt a Riemannian approach to modeling memristive elements and explore how notions of content, cocontent, and variational principles may be reformulated in this framework.

Keywords: Dissipativity, Networked Control Systems, Optimal Control
Abstract: Despite longstanding interest, stability analysis and controller synthesis remain challenging for large-scale interconnections of heterogenious, nonlinear systems. Moreover, improved computational efficiency and information privacy are increasingly important. This extended abstract reviews recent progress in dissipativity-based distributed stability analysis and introduces a distributed controller synthesis framework that integrates alternating direction methods of multipliers (ADMM), chordal decomposition, and iterative convex overbounding (ICO) to overcome current limitations in scaling and privacy.