Kooktae Lee

Kooktae Lee, Raktim Bhattacharya, Vijay Gupta

In the near future, massively parallel computing systems will be necessary to solve computation intensive applications. The key bottleneck in massively parallel implementation of numerical algorithms is the synchronization of data across processing elements (PEs) after each iteration, which results in significant idle time. Thus, there is a trend towards relaxing the synchronization and adopting an asynchronous model of computation to reduce idle time. However, it is not clear what is the effect of this relaxation on the stability and accuracy of the numerical algorithm. In this paper we present a new framework to analyze such algorithms. We treat the computation in each PE as a dynamical system and model the asynchrony as stochastic switching. The overall system is then analyzed as a switched dynamical system. However, modeling of massively parallel numerical algorithms as switched dynamical systems results in a very large number of modes, which makes current analysis tools available for such systems computationally intractable. We develop new techniques that circumvent this scalability issue. The framework is presented on a one-dimensional heat equation as a case study and the proposed analysis framework is verified by solving the partial differential equation (PDE) in a $\mathtt{nVIDIA\: Tesla^{\scriptsize{TM}}}$ GPU machine, with asynchronous communication between cores.

Kooktae Lee, Raktim Bhattacharya

In this paper, we consider the stability analysis of large-scale distributed networked control systems with random communication delays between linearly interconnected subsystems. The stability analysis is performed in the Markov jump linear system framework. There have been considerable researches on stability analysis of Markov jump systems, however, these methods are not applicable to large-scale systems because large numbers of subsystems result in an extremely large number of the switching modes. To avoid this scalability issue, we propose a new reduced mode model for stability analysis, which is computationally efficient. We also consider the case in which the transition probabilities for the Markov jump process contain uncertainties. We provide a new method that estimates bounds for uncertain Markov transition probability matrix to guarantee the system stability. The efficiency and the usefulness of the proposed methods are verified through examples.

Kooktae Lee, Raktim Bhattacharya

This paper provides a method to design an optimal switching sequence for jump linear systems with given Gaussian initial state uncertainty. In the practical perspective, the initial state contains some uncertainties that come from measurement errors or sensor inaccuracies and we assume that the type of this uncertainty has the form of Gaussian distribution. In order to cope with Gaussian initial state uncertainty and to measure the system performance, Wasserstein metric that defines the distance between probability density functions is used. Combining with the receding horizon framework, an optimal switching sequence for jump linear systems can be obtained by minimizing the objective function that is expressed in terms of Wasserstein distance. The proposed optimal switching synthesis also guarantees the mean square stability for jump linear systems. The validations of the proposed methods are verified by examples.

Kooktae Lee, Raktim Bhattacharya

In this paper, we study a convergence condition for asynchronous consensus problems in multi-agent systems. The convergence in this context implies the asynchronous consensus value converges to the synchronous one and is unique. Although it is reported in the literature that the consensus value under asynchronous communications may not coincide with the synchronous consensus value, it has not received much attention. In some applications, the discrepancy between them may result in serious consequences. For such applications it is critical to determine under what conditions the asynchronous consensus value is the same as the synchronous consensus value. We illustrate these issues with a few examples and then provide a condition, which guarantees that the asynchronous consensus value converges to the synchronous one. The validity of the proposed result is verified with simulations.

Kooktae Lee

This paper addresses the decentralized non-uniform area coverage problem for multi-agent systems, a critical task in missions with high spatial priority and resource constraints. While existing density-based methods often rely on computationally heavy Eulerian PDE solvers or heuristic planning, we propose Stochastic Density-Driven Optimal Control (D$^2$OC). This is a rigorous Lagrangian framework that bridges the gap between individual agent dynamics and collective distribution matching. By formulating a stochastic MPC-like problem that minimizes the Wasserstein distance as a running cost, our approach ensures that the time-averaged empirical distribution converges to a non-parametric target density under stochastic LTI dynamics. A key contribution is the formal convergence guarantee established via reachability analysis, providing a bounded tracking error even in the presence of process and measurement noise. Numerical results verify that Stochastic D$^2$OC achieves robust, decentralized coverage while outperforming previous heuristic methods in optimality and consistency.

Julian Martinez, Kooktae Lee

This paper addresses the spatiotemporal mismatch in multi-agent distribution tracking within time-varying environments. While recent advancements in Density-Driven Optimal Control (D$^2$OC) have enabled finite-time distribution matching using Optimal Transport theory, existing formulations primarily assume a stationary reference density. In dynamic scenarios, such as tracking evolving wildfires or moving plumes, this assumption leads to a structural tracking lag where the agent configuration inevitably falls behind the shifting reference flow. To resolve this, we propose a feedforward-augmented D$^2$OC framework that explicitly incorporates the reference velocity field, modeled via the continuity equation, into the control law. We provide a formal mathematical quantification of the induced tracking lag and analytically prove that the proposed predictive mechanism effectively reduces the cumulative tracking error. Furthermore, an analytical ultimate bound for the local Wasserstein distance is established under discretization errors and transport jitter. Theoretical analysis and numerical results demonstrate that our approach significantly mitigates tracking latency, ensuring robust and high-fidelity tracking performance in rapidly changing environments.

Julian Martinez, Kooktae Lee

Efficient coordination for collective spatial distribution is a fundamental challenge in multi-agent systems. Prior research on Density-Driven Optimal Control (D2OC) established a framework to match agent trajectories to a desired spatial distribution. However, implementing this as a predictive controller requires solving a large-scale Karush-Kuhn-Tucker (KKT) system, whose computational complexity grows cubically with the prediction horizon. To resolve this, we propose an analytical structural reduction that transforms the T-horizon KKT system into a condensed quadratic program (QP). This formulation achieves O(T) linear scalability, significantly reducing the online computational burden compared to conventional O(T^3) approaches. Furthermore, to ensure rigorous convergence in dynamic environments, we incorporate a contractive Lyapunov constraint and prove the Input-to-State Stability (ISS) of the closed-loop system against reference propagation drift. Numerical simulations verify that the proposed method facilitates rapid density coverage with substantial computational speed-up, enabling long-horizon predictive control for large-scale multi-agent swarms.

Rabiul Hasan Kabir, Kooktae Lee

An Optimal Transport (OT)-based decentralized collaborative multi-robot exploration strategy is proposed in this paper. This method is to achieve an efficient exploration with a predefined priority in the given domain. In this context, the efficiency indicates how a team of robots (agents) cover the domain reflecting the corresponding priority map (or degrees of importance) in the domain. The decentralized exploration implies that each agent carries out their exploration task independently in the absence of any supervisory agent/computer. When an agent encounters another agent within a communication range, each agent receives the information about which areas are already covered by other agents, yielding a collaborative exploration. The OT theory is employed to quantify the difference between the distribution formed by the robot trajectories and the given reference spatial distribution indicating the priority. A computationally feasible way is developed to measure the performance of the proposed exploration scheme. Further, the formal algorithm is provided for the efficient, decentralized, and collaborative exploration plan. Simulation results are presented to validate the proposed methods.

Rabiul Hasan Kabir, Kooktae Lee, Geronimo Macias

In this study, an ergodic environment exploration problem is introduced for a centralized multi-agent system. Given the reference distribution represented by the Mixture of Gaussian (MoG), the ergodicity is achieved when the time-averaged robot distribution is identical to the given reference distribution. The major challenge associated with this problem is to determine proper timing for a team of agents (robots) to visit each Gaussian component in the reference MoG for ergodicity. The ergodic function is defined as a measure of ergodicity and the condition for convergence is derived based on timing analysis. The proposed control strategy provides relatively reasonable performance to achieve the ergodicity. We provide the formal algorithm for centralized multi-agent control to achieve the ergodicity and simulation results are presented for the validation of the proposed algorithm.

Rabiul Hasan Kabir, Kooktae Lee

This paper addresses an Optimal Transport (OT)-based efficient multi-robot exploration problem, considering the energy constraints of a multi-robot system. The efficiency in this problem implies how a team of robots (agents) covers a given domain, reflecting a priority of areas of interest represented by a density distribution, rather than simply following a preset of uniform patterns. To achieve an efficient multi-robot exploration, the optimal transport theory that quantifies a distance between two density distributions is employed as a tool, which also serves as a means of performance measure. The energy constraints for the multi-robot system is then incorporated into the OT-based multi-robot exploration scheme. The proposed scheme is decoupled from robot dynamics, broadening the applicability of the multi-robot exploration plan to heterogeneous robot platforms. Not only the centralized but also decentralized algorithms are provided to cope with more realistic scenarios such as communication range limits between agents. To measure the exploration efficiency, the upper bound of the performance is developed for both the centralized and decentralized cases based on the optimal transport theory, which is computationally tractable as well as efficient. The proposed multi-robot exploration scheme is also applicable to a time-varying distribution, where the spatio-temporal evolution of the given reference distribution is desired. To validate the proposed method, multiple simulation results are provided.

Kooktae Lee, Raktim Bhattacharya

In this paper, we analyze the convergence as well as the rate of convergence of asynchronous distributed quadratic programming (QP) with dual decomposition technique. In general, distributed optimization requires synchronization of data at each iteration step due to the interdependency of data. This synchronization latency may incur a large amount of waiting time caused by an idle process during computation. We aim to attack this synchronization penalty in distributed QP problems by implementing asynchronous update of dual variable. The price to pay for adopting asynchronous computing algorithms is unpredictability of the solution, resulting in a tradeoff between speedup and accuracy. Thus, the convergence to an optimal solution is not guaranteed owing to the stochastic behavior of asynchrony. In this paper, we employ the switched system framework as an analysis tool to investigate the convergence of asynchronous distributed QP. This switched system will facilitate analysis on asynchronous distributed QP with dual decomposition, providing necessary and sufficient conditions for the mean square convergence. Also, we provide an analytic expression for the rate of convergence through the switched system, which enables performance analysis of asynchronous algorithms as compared with synchronous case. To verify the validity of the proposed methods, numerical examples are presented with an implementation of asynchronous parallel QP using OpenMP.