arXiv subjects

Vishaal Krishnan

Publications and source records attributed to Vishaal Krishnan.

18 recordsLinked to original sources

arXiv2026

Generative optimal transport via forward-backward HJB matching

Controlling the evolution of a many-body stochastic system from a disordered reference state to a structured target ensemble, characterized empirically through samples, arises naturally in non-equilibrium statistical mechanics and stochastic control. The natural relaxation of such a system - driven by diffusion - runs from the structured target toward the disordered reference. The natural question is then: what is the minimum-work stochastic process that reverses this relaxation, given a pathwise cost functional combining spatial penalties and control effort? Computing this optimal process requires knowledge of trajectories that already sample the target ensemble - precisely the object one is trying to construct. We resolve this by establishing a time-reversal duality: the value function governing the hard backward dynamics satisfies an equivalent forward-in-time HJB equation, whose solution can be read off directly from the tractable forward relaxation trajectories. Via the Cole-Hopf transformation and its associated Feynman-Kac representation, this forward potential is computed as a path-space free energy averaged over these forward trajectories - the same relaxation paths that are easy to simulate - without any backward simulation or knowledge of the target beyond samples. The resulting framework provides a physically interpretable description of stochastic transport in terms of path-space free energy, risk-sensitive control, and spatial cost geometry. We illustrate the theory with numerical examples that visualize the learned value function and the induced controlled diffusions, demonstrating how spatial cost fields shape transport geometry analogously to Fermat's Principle in inhomogeneous media. Our results establish a unifying connection between stochastic optimal control, Schr\"odinger bridge theory, and non-equilibrium statistical mechanics.

cond-mat.stat-mech↗

arXiv2026

Stigmergic optimal transport

Efficient navigation in swarms often relies on the emergence of decentralized approaches that minimize traversal time or energy. Stigmergy, where agents modify a shared environment that then modifies their behavior, is a classic mechanism that can encode this strategy. We develop a theoretical framework for stigmergic transport by casting it as a stochastic optimal control problem: agents (collectively) lay and (individually) follow trails while minimizing expected traversal time. Simulations and analysis reveal two emergent behaviors: path straightening in homogeneous environments and path refraction at material interfaces, both consistent with experimental observations of insect trails. While reminiscent of Fermat's principle, our results show how local, noisy agent+field interactions can give rise to geodesic trajectories in heterogeneous environments, without centralized coordination or global knowledge, relying instead on an embodied slow fast dynamical mechanism.

q-bio.NC↗

arXiv2024

Hamiltonian bridge: A physics-driven generative framework for targeted pattern control

Patterns arise spontaneously in a range of systems spanning the sciences, and their study typically focuses on mechanisms to understand their evolution in space-time. Increasingly, there has been a transition towards controlling these patterns in various functional settings, with implications for engineering. Here, we combine our knowledge of a general class of dynamical laws for pattern formation in non-equilibrium systems, and the power of stochastic optimal control approaches to present a framework that allows us to control patterns at multiple scales, which we dub the "Hamiltonian bridge". We use a mapping between stochastic many-body Lagrangian physics and deterministic Eulerian pattern forming PDEs to leverage our recent approach utilizing the Feynman-Kac-based adjoint path integral formulation for the control of interacting particles and generalize this to the active control of patterning fields. We demonstrate the applicability of our computational framework via numerical experiments on the control of phase separation with and without a conserved order parameter, self-assembly of fluid droplets, coupled reaction-diffusion equations and finally a phenomenological model for spatio-temporal tissue differentiation. We interpret our numerical experiments in terms of a theoretical understanding of how the underlying physics shapes the geometry of the pattern manifold, altering the transport paths of patterns and the nature of pattern interpolation. We finally conclude by showing how optimal control can be utilized to generate complex patterns via an iterative control protocol over pattern forming pdes which can be casted as gradient flows. All together, our study shows how we can systematically build in physical priors into a generative framework for pattern control in non-equilibrium systems across multiple length and time scales.

cond-mat.soft↗

arXiv2023

Optimal control of interacting active particles on complex landscapes

Active many-body systems composed of many interacting degrees of freedom often operate out of equilibrium, giving rise to non-trivial emergent behaviors which can be functional in both evolved and engineered contexts. This naturally suggests the question of control to optimize function. Using navigation as a paradigm for function, we deploy the language of stochastic optimal control theory to formulate the inverse problem of shepherding a system of interacting active particles across a complex landscape. We implement a solution to this high-dimensional problem using an Adjoint-based Path Integral Control (APIC) algorithm that combines the power of recently introduced continuous-time back-propagation methods and automatic differentiation with the classical Feynman-Kac path integral formulation in statistical mechanics. Numerical experiments for controlling individual and interacting particles in complex landscapes show different classes of successful navigation strategies as a function of landscape complexity, as well as the intrinsic noise and drive of the active particles. However, in all cases, we see the emergence of paths that correspond to traversal along the edges of ridges and ravines, which we can understand using a variational analysis. We also show that the work associated with optimal strategies is inversely proportional to the length of the time horizon of optimal control, a result that follows from scaling considerations. All together, our approach serves as a foundational framework to control active non-equilibrium systems optimally to achieve functionality, embodied as a path on a high-dimensional manifold.

cond-mat.soft↗

arXiv2023

Data-Driven Feedback Linearization using the Koopman Generator

This paper contributes a theoretical framework for data-driven feedback linearization of nonlinear control-affine systems. We unify the traditional geometric perspective on feedback linearization with an operator-theoretic perspective involving the Koopman operator. We first show that if the distribution of the control vector field and its repeated Lie brackets with the drift vector field is involutive, then there exists an output and a feedback control law for which the Koopman generator is finite-dimensional and locally nilpotent. We use this connection to propose a data-driven algorithm Koopman Generator-based Feedback Linearization (KGFL) for feedback linearization. Particularly, we use experimental data to identify the state transformation and control feedback from a dictionary of functions for which feedback linearization is achieved in a least-squares sense. We also propose a single-step data-driven formula which can be used to compute the linearizing transformations. When the system is feedback linearizable and the chosen dictionary is complete, our data-driven algorithm provides the same solution as model-based feedback linearization. Finally, we provide numerical examples for the data-driven algorithm and compare it with model-based feedback linearization. We also numerically study the effect of the richness of the dictionary and the size of the data set on the effectiveness of feedback linearization.

math.OC↗

arXiv2023

Imitation and Transfer Learning for LQG Control

In this paper we study an imitation and transfer learning setting for Linear Quadratic Gaussian (LQG) control, where (i) the system dynamics, noise statistics and cost function are unknown and expert data is provided (that is, sequences of optimal inputs and outputs) to learn the LQG controller, and (ii) multiple control tasks are performed for the same system but with different LQG costs. We show that the LQG controller can be learned from a set of expert trajectories of length $n(l+2)-1$, with $n$ and $l$ the dimension of the system state and output, respectively. Further, the controller can be decomposed as the product of an estimation matrix, which depends only on the system dynamics, and a control matrix, which depends on the LQG cost. This data-based separation principle allows us to transfer the estimation matrix across different LQG tasks, and to reduce the length of the expert trajectories needed to learn the LQG controller to~$2n+m-1$ with $m$ the dimension of the inputs (for single-input systems with $l=2$, this yields approximately a $50\%$ reduction of the required expert data).

eess.SY↗

arXiv2022

Behavioral Feedback for Optimal LQG Control

In this work, we revisit the Linear Quadratic Gaussian (LQG) optimal control problem from a behavioral perspective. Motivated by the suitability of behavioral models for data-driven control, we begin with a reformulation of the LQG problem in the space of input-output behaviors and obtain a complete characterization of the optimal solutions. In particular, we show that the optimal LQG controller can be expressed as a static behavioral-feedback gain, thereby eliminating the need for dynamic state estimation characteristic of state space methods. The static form of the optimal LQG gain also makes it amenable to its computation by gradient descent, which we investigate via numerical experiments. Furthermore, we highlight the advantage of this approach in the data-driven control setting of learning the optimal LQG controller from expert demonstrations.

eess.SY↗

arXiv2022

Direct vs Indirect Methods for Behavior-based Attack Detection

We study the problem of data-driven attack detection for unknown LTI systems using only input-output behavioral data. In contrast with model-based detectors that use errors from an output predictor to detect attacks, we study behavior-based data-driven detectors. We construct a behavior-based chi-squared detector that uses a sequence of inputs and outputs and their covariance. The covariance of the behaviors is estimated using data by two methods. The first (direct) method employs the sample covariance as an estimate of the covariance of behaviors. The second (indirect) method uses a lower dimensional generative model identified from data to estimate the covariance of behaviors. We prove the consistency of the two methods of estimation and provide finite sample error bounds. Finally, we numerically compare the performance and establish a tradeoff between the methods at different regimes of the size of the data set and the length of the detection horizon. Our numerical study indicates that neither method is invariable superior, and reveals the existence of two regimes for the performance of the two methods, wherein the direct method is superior in cases with large data sets relative to the length of the detection horizon, while the indirect method is superior in cases with small data sets.

math.OC↗

arXiv2022

A Multiscale Analysis of Multi-Agent Coverage Control Algorithms

This paper presents a theoretical framework for the design and analysis of gradient descent-based algorithms for coverage control tasks involving robot swarms. We adopt a multiscale approach to analysis and design to ensure consistency of the algorithms in the large-scale limit. First, we represent the macroscopic configuration of the swarm as a probability measure and formulate the macroscopic coverage task as the minimization of a convex objective function over probability measures. We then construct a macroscopic dynamics for swarm coverage, which takes the form of a proximal descent scheme in the $L^2$-Wasserstein space. Our analysis exploits the generalized geodesic convexity of the coverage objective function, proving convergence in the $L^2$-Wasserstein sense to the target probability measure. We then obtain a consistent gradient descent algorithm in the Euclidean space that is implementable by a finite collection of agents, via a "variational" discretization of the macroscopic coverage objective function. We establish the convergence properties of the gradient descent and its behavior in the continuous-time and large-scale limits. Furthermore, we establish a connection with well-known Lloyd-based algorithms, seen as a particular class of algorithms within our framework, and demonstrate our results via numerical experiments.

math.OC↗

arXiv2022

Learning Lipschitz Feedback Policies from Expert Demonstrations: Closed-Loop Guarantees, Generalization and Robustness

In this work, we propose a framework to learn feedback control policies with guarantees on closed-loop generalization and adversarial robustness. These policies are learned directly from expert demonstrations, contained in a dataset of state-control input pairs, without any prior knowledge of the task and system model. We use a Lipschitz-constrained loss minimization scheme to learn feedback policies with certified closed-loop robustness, wherein the Lipschitz constraint serves as a mechanism to tune the generalization performance and robustness to adversarial disturbances. Our analysis exploits the Lipschitz property to obtain closed-loop guarantees on generalization and robustness of the learned policies. In particular, we derive a finite sample bound on the policy learning error and establish robust closed-loop stability under the learned control policy. We also derive bounds on the closed-loop regret with respect to the expert policy and the deterioration of closed-loop performance under bounded (adversarial) disturbances to the state measurements. Numerical results validate our analysis and demonstrate the effectiveness of our robust feedback policy learning framework. Finally, our results suggest the existence of a potential tradeoff between nominal closed-loop performance and adversarial robustness, and that improvements in nominal closed-loop performance can only be made at the expense of robustness to adversarial perturbations.

cs.LG↗

arXiv2021

On Direct vs Indirect Data-Driven Predictive Control

In this work, we compare the direct and indirect approaches to data-driven predictive control of stochastic linear time-invariant systems. The distinction between the two approaches lies in the fact that the indirect approach involves identifying a lower dimensional model from data which is then used in a certainty-equivalent control design, while the direct approach avoids this intermediate step altogether. Working within an optimization-based framework, we find that the suboptimality gap measuring the control performance w.r.t. the optimal model-based control design vanishes with the size of the dataset only with the direct approach. The indirect approach has a higher rate of convergence, but its suboptimality gap does not vanish as the size of the dataset increases. This reveals the existence of two distinct regimes of performance as the size of the dataset of input-output behaviors is increased. We show that the indirect approach, by relying on the identification of a lower dimensional model, has lower variance and outperforms the direct approach for smaller datasets, while it incurs an asymptotic bias as a result of the process noise and a (possibly) incorrect assumption on the order of the identified model. The direct approach, however, does not incur an asymptotic bias, and outperforms the indirect approach for larger datasets. Ultimately, by revealing the existence of two non-asymptotic regimes for the performance of direct and indirect data-driven predictive control designs, our study suggests that neither approach is invariably superior and that the choice of design must, in practice, be informed by the available dataset.

math.OC↗

arXiv2020

Lipschitz Bounds and Provably Robust Training by Laplacian Smoothing

In this work we propose a graph-based learning framework to train models with provable robustness to adversarial perturbations. In contrast to regularization-based approaches, we formulate the adversarially robust learning problem as one of loss minimization with a Lipschitz constraint, and show that the saddle point of the associated Lagrangian is characterized by a Poisson equation with weighted Laplace operator. Further, the weighting for the Laplace operator is given by the Lagrange multiplier for the Lipschitz constraint, which modulates the sensitivity of the minimizer to perturbations. We then design a provably robust training scheme using graph-based discretization of the input space and a primal-dual algorithm to converge to the Lagrangian's saddle point. Our analysis establishes a novel connection between elliptic operators with constraint-enforced weighting and adversarial learning. We also study the complementary problem of improving the robustness of minimizers with a margin on their loss, formulated as a loss-constrained minimization problem of the Lipschitz constant. We propose a technique to obtain robustified minimizers, and evaluate fundamental Lipschitz lower bounds by approaching Lipschitz constant minimization via a sequence of gradient $p$-norm minimization problems. Ultimately, our results show that, for a desired nominal performance, there exists a fundamental lower bound on the sensitivity to adversarial perturbations that depends only on the loss function and the data distribution, and that improvements in robustness beyond this bound can only be made at the expense of nominal performance. Our training schemes provably achieve these bounds both under constraints on performance and~robustness.

cs.LG↗

arXiv2020

Data-Driven Attack Detection for Linear Systems

This paper studies the attack detection problem in a data-driven and model-free setting, for deterministic systems with linear and time-invariant dynamics. Differently from existing studies that leverage knowledge of the system dynamics to derive security bounds and monitoring schemes, we focus on the case where the system dynamics, as well as the attack strategy and attack location, are unknown. We derive fundamental security limitations as a function of only the observed data and without estimating the system dynamics (in fact, no assumption is made on the identifiability of the system). In particular, (i) we derive detection limitations as a function of the informativity and length of the observed data, (ii) provide a data-driven characterization of undetectable attacks, and (iii) construct a data-driven detection monitor. Surprisingly, and in accordance with recent studies on data-driven control, our results show that model-based and data-driven security techniques share the same fundamental limitations, provided that the collected data remains sufficiently informative.

eess.SY↗

arXiv2019

A Probabilistic Framework for Moving-Horizon Estimation: Stability and Privacy Guarantees

This work proposes a unifying probabilistic framework for the design of robustly asymptotically stable moving-horizon estimators (MHE) for discrete-time nonlinear systems, and a mechanism to incorporate differential privacy in moving-horizon estimation. We begin with an investigation of the classical notion of strong local observability of nonlinear systems and its relationship to optimization-based state estimation. We then present a general moving-horizon estimation framework for strongly locally observable systems, as an iterative minimization scheme in the space of probability measures. This framework allows for the minimization of the estimation cost with respect to different metrics. In particular, we consider two variants, which we name $W_2$-MHE and KL-MHE, where the minimization scheme uses the 2-Wasserstein distance and the KL-divergence, respectively. The $W_2$-MHE yields a gradient-based estimator whereas the KL-MHE yields a particle filter, for which we investigate asymptotic stability and robustness properties. Stability results for these moving-horizon estimators are derived in the probabilistic setting, against the backdrop of the classical notion of strong local observability which, to the best of our knowledge, differentiates it from other previous works. We then propose a mechanism to encode differential privacy of the measurements used by the estimator via an entropy regularization of the MHE objective functional. In particular, we find sufficient bounds on the regularization parameter to achieve the desired level of differential privacy. Numerical simulations demonstrate the performance of these estimators.

math.OC↗

arXiv2019

Identification of critical nodes in large-scale spatial networks

The notion of network connectivity is used to characterize the robustness and failure tolerance of networks, with high connectivity being a desirable feature. In this paper, we develop a novel approach to the problem of identifying critical nodes in large-scale networks, with algebraic connectivity (the second smallest eigenvalue of the graph Laplacian) as the chosen metric. Employing a graph-embedding technique, we reduce the class of considered weight-balanced graphs to spatial networks with uniformly distributed nodes and nearest-neighbors communication topologies. Through a continuum approximation, we consider the Laplace operator on a manifold (with the Neumann boundary condition) as the limiting case of the graph Laplacian. We then reduce the critical node set identification problem to that of finding a ball of fixed radius, whose removal minimizes the second (Neumann) eigenvalue of the Laplace operator on the residual domain. This leads us to consider two functional and nested optimization problems. Resorting to the min-max theorem, we first treat the problem of determining the second smallest eigenvalue for a fixed domain by minimizing an energy functional. We then obtain a projected gradient flow that converges to the set of points satisfying the KKT conditions and prove that the only locally asymptotically stable critical point is the second eigenfunction of the Laplace operator. Building on these results, we consider the critical ball identification problem and construct a dynamics to converge asymptotically to these points. Finally, we provide a characterization of the location of critical nodes (for infinitesimally-small balls) as those points which belong to the nodal set of the second eigenfunction of the Laplace operator.

math.OC↗

arXiv2018

Distributed Control for Spatial Self-Organization of Multi-Agent Swarms

In this work, we design distributed control laws for spatial self-organization of multi-agent swarms in 1D and 2D spatial domains. The objective is to achieve a desired density distribution over a simply-connected spatial domain. Since individual agents in a swarm are not themselves of interest and we are concerned only with the macroscopic objective, we view the network of agents in the swarm as a discrete approximation of a continuous medium and design control laws to shape the density distribution of the continuous medium. The key feature of this work is that the agents in the swarm do not have access to position information. Each individual agent is capable of measuring the current local density of agents and can communicate with its spatial neighbors. The network of agents implement a Laplacian-based distributed algorithm, which we call pseudo-localization, to localize themselves in a new coordinate frame, and a distributed control law to converge to the desired spatial density distribution. We start by studying self-organization in one-dimension, which is then followed by the two-dimensional case.

math.OC↗

arXiv2018

Distributed Online Optimization for Multi-Agent Optimal Transport

We propose a scalable, distributed algorithm for the optimal transport of large-scale multi-agent systems. We formulate the problem as one of steering the collective towards a target probability measure while minimizing the total cost of transport, with the additional constraint of distributed implementation. Using optimal transport theory, we realize the solution as an iterative transport based on a proximal descent scheme. At each stage of the transport, the agents implement an online, distributed primal-dual algorithm to obtain local estimates of the Kantorovich potential for optimal transport from the current distribution of the collective to the target distribution. Using these estimates as their local objective functions, the agents then implement the transport by proximal descent. This two-step process is carried out recursively by the agents to converge asymptotically to the target distribution. We rigorously establish the underlying theoretical framework for the algorithm and test its behavior via numerical experiments.

math.OC↗

arXiv2013

Synchronization and semistability analysis of the Kuramoto model of coupled nonlinear oscillators

An interesting problem in synchronization is the study of coupled oscillators, wherein oscillators with different natural frequencies synchronize to a common frequency and equilibrium phase difference. In this paper, we investigate the stability and convergence in a network of coupled oscillators described by the Kuramoto model. We consider networks with finite number of oscillators, arbitrary interconnection topology, non-uniform coupling gains and non-identical natural frequencies. We show that such a network synchronizes provided the underlying graph is connected and certain conditions on the coupling gains are satisfied. In the analysis, we consider as states the phase and angular frequency differences between the oscillators, and the resulting dynamics possesses a continuum of equilibria. The synchronization problem involves establishing the Lyapunov stability of the fixed points and showing convergence of trajectories to these points. The synchronization result is established in the framework of semistability theory.

math.DS↗