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Brian D. O. Anderson

Publications and source records attributed to Brian D. O. Anderson.

At least 19 recordsLinked to original sources
arXiv2026

On a Class of Decentralized Feedback Controllers for the Networked Bivirus SIS Model

Recent work has established many properties of systems modeling the spread of two competing viruses in a population, and for networks with multiple connected populations with different spreading properties (e.g. men and women). This work considers the introduction of a class of nonlinear decentralized feedback controls aimed at reducing the fractions of populations infected at an endemic equilibrium. One surprising conclusion is that in some circumstances, new types of equilibria can arise. They can be viewed as an outcome of a transcritical bifurcation which is never observed for uncontrolled systems. In addition, we show that the controlled system is strongly monotone, and an unstable boundary equilibrium of the uncontrolled system cannot be stabilized using the class of decentralized state feedback controllers considered in this paper.

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arXiv2026

Reverse-Time Diffusion Processes for Discrete Time Linear and Nonlinear Systems with non-Gaussian Noise

Generative AI relies on finding reverse time models for a discrete-time forward diffusion with non-Gaussian initial state, but uses indirect approaches as there is no theory for direct reversal in discrete time. This paper develops a theory for directly finding reverse diffusions for discrete time nonlinear processes with non-Gaussian states and process noise. We also give a necessary and sufficient condition for the reverse model to be input-affine when the forward process is linear and the process noise Gaussian, and show that for a wide variety of state densities an input-affine reverse diffusion does not exist. This is among several differences between the reversal of stochastic difference equations and their continuous time counterparts.

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arXiv2026

Optimum adaptation of a Steiner network

The Euclidean Steiner tree problem, normally posed in two dimensions, seeks to connect a set of prescribed terminal nodes by placing additional nodes, known as Steiner points, with edges connecting such nodes either to another Steiner point or a terminal node, and with the placements minimising the sum of all the edge lengths of the associated tree. We consider a problem in which we start with a known solution to a Steiner tree problem, and the terminal positions are then perturbed. A first-order approximation theorem is established for efficiently updating the Steiner point positions to recover a Steiner tree solution after the perturbations to terminal nodes. Numerical examples illustrate the effectiveness of our approach (including a stepwise application for large perturbations) as well as its limitations.

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arXiv2026

Assessing performance tradeoffs in hierarchical organizations using a diffusive coupling model

We study a continuous-time dynamical system of nodes diffusively coupled over a hierarchical network to examine the efficiency and performance tradeoffs that organizations, teams, and command and control units face while achieving coordination and sharing information across layers. Specifically, after defining a network structure that captures real-world features of hierarchical organizations, we use linear systems theory and perturbation theory to characterize the rate of convergence to a consensus state, and how effectively information can propagate through the network, depending on the breadth of the organization and the strength of inter-layer communication. Interestingly, our analytical insights highlight a fundamental performance tradeoff. Namely, networks that favor fast coordination will have decreased ability to share information that is generated in the lower layers of the organization and is to be passed up the hierarchy. Numerical results validate and extend our theoretical results.

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arXiv2025

Performance Index Shaping for Closed-loop Optimal Control

The design of the performance index, also referred to as cost or reward shaping, is central to both optimal control and reinforcement learning, as it directly determines the behaviors, trade-offs, and objectives that the resulting control laws seek to achieve. A commonly used approach for this inference task in recent years is differentiable trajectory optimization, which allows gradients to be computed with respect to cost parameters by differentiating through an optimal control solver. However, this method often requires repeated solving of the underlying optimal control problem at every iteration, making the method computationally expensive. In this work, assuming known dynamics, we propose a novel framework that analytically links the performance index to the resulting closed-loop optimal control law, thereby transforming a typically bi-level inverse problem into a tractable single-level formulation. Our approach is motivated by the question: given a closed-loop control law that solves an infinite-horizon optimal control problem, how does this law change when the performance index is modified with additional terms? This formulation yields closed-form characterizations for broad classes of systems and performance indices, which not only facilitate interpretation and stability analysis, but also provide insight into the robust stability and input-to-state stable behavior of the resulting nonlinear closed-loop system. Moreover, this analytical perspective enables the generalization of our approach to diverse design objectives, yielding a unifying framework for performance index shaping. Given specific design objectives, we propose a systematic methodology to guide the shaping of the performance index and thereby design the resulting optimal control law.

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arXiv2025

Consensus Seminorms and their Applications

Consensus is a well-studied problem in distributed sensing, computation and control, yet deriving useful and easily computable bounds on the rate of convergence to consensus remains a challenge. This paper discusses the use of seminorms for this goal. A previously suggested family of seminorms is revisited, and an error made in their original presentation is corrected, where it was claimed that the a certain seminorm is equal to the well-known coefficient of ergodicity. Next, a wider family of seminorms is introduced, and it is shown that contraction in any of these seminorms guarantees convergence at an exponential rate of infinite products of matrices, generalizing known results on stochastic matrices to the class of matrices whose row sums are all equal one. Finally, it is shown that such seminorms cannot be used to bound the rate of convergence of classes larger than the well-known class of scrambling matrices.

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arXiv2024

Mode Consensus Algorithms With Finite Convergence Time

This paper studies the distributed mode consensus problem in a multi-agent system, in which the agents each possess a certain attribute and they aim to agree upon the mode (the most frequent attribute owned by the agents) via distributed computation. Three algorithms are proposed. The first one directly calculates the frequency of all attributes at every agent, with protocols based on blended dynamics, and then returns the most frequent attribute as the mode. Assuming knowledge at each agent of a lower bound of the mode frequency as a priori information, the second algorithm is able to reduce the number of frequencies to be computed at every agent if the lower bound is large. The third algorithm further eliminates the need for this information by introducing an adaptive updating mechanism. The algorithms find the mode in finite time, and estimates of convergence time are provided. The proposed first and second algorithms enjoy the plug-and-play property with a dwell time.

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arXiv2023

Neighboring Extremal Optimal Control Theory for Parameter-Dependent Closed-loop Laws

This study introduces an approach to obtain a neighboring extremal optimal control (NEOC) solution for a closed-loop optimal control problem, applicable to a wide array of nonlinear systems and not necessarily quadratic performance indices. The approach involves investigating the variation incurred in the functional form of a known closed-loop optimal control law due to small, known parameter variations in the system equations or the performance index. The NEOC solution can formally be obtained by solving a linear partial differential equation, akin to those encountered in the iterative solution of a nonlinear Hamilton-Jacobi equation. Motivated by numerical procedures for solving these latter equations, we also propose a numerical algorithm based on the Galerkin algorithm, leveraging the use of basis functions to solve the underlying Hamilton-Jacobi equation of the original optimal control problem. The proposed approach simplifies the NEOC problem by reducing it to the solution of a simple set of linear equations, thereby eliminating the need for a full re-solution of the adjusted optimal control problem. Furthermore, the variation to the optimal performance index can be obtained as a function of both the system state and small changes in parameters, allowing the determination of the adjustment to an optimal control law given a small adjustment of parameters in the system or the performance index. Moreover, in order to handle large known parameter perturbations, we propose a homotopic approach that breaks down the single calculation of NEOC into a finite set of multiple steps. Finally, the validity of the claims and theory is supported by theoretical analysis and numerical simulations.

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arXiv2023

Competitive Networked Bivirus SIS spread over Hypergraphs

The paper deals with the spread of two competing viruses over a network of population nodes, accounting for pairwise interactions and higher-order interactions (HOI) within and between the population nodes. We study the competitive networked bivirus susceptible-infected-susceptible (SIS) model on a hypergraph introduced in Cui et al. [1]. We show that the system has, in a generic sense, a finite number of equilibria, and the Jacobian associated with each equilibrium point is nonsingular; the key tool is the Parametric Transversality Theorem of differential topology. Since the system is also monotone, it turns out that the typical behavior of the system is convergence to some equilibrium point. Thereafter, we exhibit a tri-stable domain with three locally exponentially stable equilibria. For different parameter regimes, we establish conditions for the existence of a coexistence equilibrium (both viruses infect separate fractions of each population node).

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arXiv2023

Equilibria analysis of a networked bivirus epidemic model using Poincaré--Hopf and Manifold Theory

This paper considers a deterministic Susceptible-Infected-Susceptible (SIS) networked bivirus epidemic model (termed the bivirus model for short), in which two competing viruses spread through a set of populations (nodes) connected by two graphs, which may be different if the two viruses have different transmission pathways. The networked dynamics can give rise to complex equilibria patterns, and most current results identify conditions on the model parameters for convergence to the healthy equilibrium (where both viruses are extinct) or a boundary equilibrium (where one virus is endemic and the other is extinct). However, there are only limited results on coexistence equilibria (where both viruses are endemic). This paper establishes a set of ``counting'' results which provide lower bounds on the number of coexistence equilibria, and perhaps more importantly, establish properties on the local stability/instability properties of these equilibria. In order to do this, we employ the Poincaré-Hopf Theorem but with significant modifications to overcome several challenges arising from the bivirus system model, such as the fact that the system dynamics do not evolve on a manifold in the typical sense required to apply Poincaré-Hopf Theory. Subsequently, Morse inequalities are used to tighten the counting results, under the reasonable assumption that the bivirus system is a Morse-Smale dynamical system. Numerical examples are provided which demonstrate the presence of multiple attractor equilibria, and multiple coexistence equilibria.

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arXiv2023

Decentralised adaptive-gain control for eliminating epidemic spreading on networks

This paper considers the classical Susceptible--Infected--Susceptible (SIS) network epidemic model, which describes a disease spreading through $n$ nodes, with the network links governing the possible transmission pathways of the disease between nodes. We consider feedback control to eliminate the disease in scenarios where the disease would otherwise persist in an uncontrolled network. We propose a family of decentralised adaptive-gain control algorithms, in which each node has a control gain that adaptively evolves according to a differential equation, independent of the gains of other nodes. The adaptive gain is applied multiplicatively to either decrease the infection rate or increase the recovery rate. To begin, we assume all nodes are controlled, and prove that both infection rate control and recovery rate control algorithms eliminate the disease with the limiting gains being positive and finite. Then, we consider the possibility of controlling a subset of the nodes, for both the infection rate control and recovery rate control. We first identify a necessary and sufficient condition for the existence of a subset of nodes, which if controlled would result in the elimination of the disease. For a given network, there may exist several such viable subsets, and we propose an iterative algorithm to identify such a subset. Simulations are provided to demonstrate the effectiveness of the various proposed controllers.

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arXiv2023

Towards Understanding the Endemic Behavior of a Competitive Tri-Virus SIS Networked Model

This paper studies the endemic behavior of a multi-competitive networked susceptible-infected-susceptible (SIS) model. Specifically, the paper deals with three competing virus systems (i.e., tri-virus systems). First, we show that a tri-virus system, unlike a bi-virus system, is not a monotone dynamical system. Using the Parametric Transversality Theorem, we show that, generically, a tri-virus system has a finite number of equilibria and that the Jacobian matrices associated with each equilibrium are nonsingular. The endemic equilibria of this system can be classified as follows: a) single-virus endemic equilibria (also referred to as the boundary equilibria), where precisely one of the three viruses is alive; b) 2-coexistence equilibria, where exactly two of the three viruses are alive; and c) 3-coexistence equilibria, where all three viruses survive in the network. We provide a necessary and sufficient condition that guarantees local exponential convergence to a boundary equilibrium. Further, we secure conditions for the nonexistence of 3-coexistence equilibria (resp. for various forms of 2-coexistence equilibria). We also identify sufficient conditions for the existence of a 2-coexistence (resp. 3-coexistence) equilibrium. We identify conditions on the model parameters that give rise to a continuum of coexistence equilibria. More specifically, we establish i) a scenario that admits the existence and local exponential attractivity of a line of coexistence equilibria; and ii) scenarios that admit the existence of, and, in the case of one such scenario, global convergence to, a plane of 3-coexistence equilibria.

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arXiv2023

Exponential Consensus of Multiple Agents over Dynamic Network Topology: Controllability, Connectivity, and Compactness

This paper investigates the problem of securing exponentially fast consensus (exponential consensus for short) for identical agents with finite-dimensional linear system dynamics over dynamic network topology. Our aim is to find the weakest possible conditions that guarantee exponentially fast consensus using a Lyapunov function consisting of a sum of terms of the same functional form. We first investigate necessary conditions, starting by examining the system (both agent and network) parameters. It is found that controllability of the linear agents is necessary for reaching consensus. Then, to work out necessary conditions incorporating the network topology, we construct a set of Laplacian matrix-valued functions. The precompactness of this set of functions is shown to be a significant generalization of existing assumptions on network topology, including the common assumption that the edge weights are bounded piecewise constant functions or continuous functions. With the aid of such a precompactness assumption and restricting the Lyapunov function to one consisting of a sum of terms of the same functional form, we prove that a joint $(δ, T)$-connectivity condition on the network topology is necessary for exponential consensus. Finally, we investigate how the above two ``necessities'' work together to guarantee exponential consensus. To partially address this problem, we define a synchronization index to characterize the interplay between agent parameters and network topology. Based on this notion, it is shown that by designing a proper feedback matrix and under the precompactness assumption, exponential consensus can be reached globally and uniformly if the joint $(δ,T)$-connectivity and controllability conditions are satisfied, and the synchronization index is not less than one.

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arXiv2023

The Domain of Attraction of the Desired Path in Vector-field Guided Path Following

In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the vector field vanishes creates a topological obstruction to global convergence to the desired path and some associated topological analysis has been conducted in our previous work. In this paper, we strengthen the result in our previous work by showing that the domain of attraction of the desired path, which is a compact asymptotically stable one-dimensional embedded submanifold of an $n$-dimensional ambient manifold $\mathcal{M}$, is homeomorphic to $\mathbb{R}^{n-1} \times \mathbb{S}^1$, and not just homotopy equivalent to $\mathbb{S}^1$. This result is extended for a $k$-dimensional compact manifold for $k \ge 2$.

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arXiv2023

Competitive epidemic networks with multiple survival-of-the-fittest outcomes

We use a deterministic model to study two competing viruses spreading over a two-layer network in the Susceptible--Infected--Susceptible (SIS) framework, and address a central problem of identifying the winning virus in a "survival-of-the-fittest" battle. Existing sufficient conditions ensure that the same virus always wins regardless of initial states. For networks with an arbitrary but finite number of nodes, there exists a necessary and sufficient condition that guarantees local exponential stability of the two equilibria corresponding to each virus winning the battle, meaning that either of the viruses can win, depending on the initial states. However, establishing existence and finding examples of networks with more than three nodes that satisfy such a condition has remained unaddressed. In this paper, we prove that, for any arbitrary number of nodes, such networks exist. We do this by proving that given almost any network layer of one virus, there exists a network layer for the other virus such that the resulting two-layer network satisfies the aforementioned condition. To operationalize our findings, a four-step procedure is developed to reliably and consistently design one of the network layers, when given the other layer. Conclusions from numerical case studies, including a real-world mobility network that captures the commuting patterns for people between $107$ provinces in Italy, extend on the theoretical result and its consequences.

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arXiv2022

Cooperative Tuning of Multi-Agent Optimal Control Systems

This paper investigates the problem of cooperative tuning of multi-agent optimal control systems, where a network of agents (i.e. multiple coupled optimal control systems) adjusts parameters in their dynamics, objective functions, or controllers in a coordinated way to minimize the sum of their loss functions. Different from classical techniques for tuning parameters in a controller, we allow tunable parameters appearing in both the system dynamics and the objective functions of each agent. A framework is developed to allow all agents to reach a consensus on the tunable parameter, which minimizes team loss. The key idea of the proposed algorithm rests on the integration of consensus-based distributed optimization for a multi-agent system and a gradient generator capturing the optimal performance as a function of the parameter in the feedback loop tuning the parameter for each agent. Both theoretical results and simulations for a synchronous multi-agent rendezvous problem are provided to validate the proposed method for cooperative tuning of multi-agent optimal control.

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arXiv2022

Guiding Vector Fields for Following Occluded Paths

Accurately following a geometric desired path in a two-dimensional space is a fundamental task for many engineering systems, in particular mobile robots. When the desired path is occluded by obstacles, it is necessary and crucial to temporarily deviate from the path for obstacle/collision avoidance. In this paper, we develop a composite guiding vector field via the use of smooth bump functions, and provide theoretical guarantees that the integral curves of the vector field can follow an arbitrary sufficiently smooth desired path and avoid collision with obstacles of arbitrary shapes. These two behaviors are reactive since path (re)-planning and global map construction are not involved. To deal with the common deadlock problem, we introduce a switching vector field, and the Zeno behavior is excluded. Simulations are conducted to support the theoretical results.

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