An All-Against-One Game Approach for the Multi-Player Pursuit-Evasion Problem

The traditional pursuit-evasion game considers a situation where one pursuer tries to capture an evader, while the evader is trying to escape. A more general formulation of this problem is to consider multiple pursuers trying to capture one evader. This general multi-pursuer one-evader problem can also be used to model a system of systems in which one of the subsystems decides to dissent (evade) from the others while the others (the pursuer subsystems) try to pursue a strategy to prevent it from doing so. An important challenge in analyzing these types of problems is to develop strategies for the pursuers along with the advantages and disadvantages of each. In this thesis, we investigate three possible and conceptually different strategies for pursuers: (1) act non-cooperatively as independent pursuers, (2) act cooperatively as a unified team of pursuers, and (3) act individually as greedy pursuers. The evader, on the other hand, will consider strategies against all possible strategies by the pursuers. We assume complete uncertainty in the game i.e. no player knows which strategies the other players are implementing and none of them has information about any of the parameters in the objective functions of the other players. To treat the three pursuers strategies under one general framework, an all-against-one linear quadratic dynamic game is considered and the corresponding closed-loop Nash solution is discussed. Additionally, different necessary and sufficient conditions regarding the stability of the system, and existence and definiteness of the closed-loop Nash strategies under different strategy assumptions are derived. We deal with the uncertainties in the strategies by first developing the Nash strategies for each of the resulting games for all possible options available to both sides. Then we deal with the parameter uncertainties by performing a Monte Carlo analysis to determine probabilities of capture for the pursuers (or escape for the evader) for each resulting game. Results of the Monte Carlo simulation show that in general, pursuers do not always benefit from cooperating as a team and that acting as non-cooperating players may yield a higher probability of capturing of the evader

An Overview of New Developments in Shale Gas: Induced Seismicity Aspect

New advances in technology such as advances in horizontal drilling, the use of multi-well drilling pads, and multi-stage hydraulic fracturing allow for the economic consideration of recovering formerly uneconomic, yet proven resources. Hydraulic fracturing perturbs the local stress field and causes slip/shearing in naturally fractured shale formations. Monitoring this process using microseismic techniques provides a valuable tool helping to detect the progression of the treatment and understand the efficacy of the operation. This article provides basic definitions regarding shale gas and development of shale gas reservoirs along with results of many new developments in the field of monitoring induced seismicity associated with hydraulic fracturing operations and characterizing of the efficacy of such operations

Consensus on Lie groups for the Riemannian Center of Mass

In this paper, we develop a consensus algorithm for distributed computation of the Riemannian center of mass (RCM) on Lie Groups. The algorithm is built upon a distributed optimization reformulation that allows developing an intrinsic, distributed (without relying on a consensus subroutine), and a computationally efficient protocol for the RCM computation. The novel idea for developing this fast distributed algorithm is to utilize a Riemannian version of distributed gradient flow combined with a gradient tracking technique. We first guarantee that, under certain conditions, the limit point of our algorithm is the RCM point of interest. We then provide a proof of global convergence in the Euclidean setting, that can be viewed as a "geometric" dynamic consensus that converges to the average from arbitrary initial points. Finally, we proceed to showcase the superior convergence properties of the proposed approach as compared with other classes of consensus optimization-based algorithms for the RCM computation

Data-driven Optimal Filtering for Linear Systems with Unknown Noise Covariances

This paper examines learning the optimal filtering policy, known as the Kalman gain, for a linear system with unknown noise covariance matrices using noisy output data. The learning problem is formulated as a stochastic policy optimization problem, aiming to minimize the output prediction error. This formulation provides a direct bridge between data-driven optimal control and, its dual, optimal filtering. Our contributions are twofold. Firstly, we conduct a thorough convergence analysis of the stochastic gradient descent algorithm, adopted for the filtering problem, accounting for biased gradients and stability constraints. Secondly, we carefully leverage a combination of tools from linear system theory and high-dimensional statistics to derive bias-variance error bounds that scale logarithmically with problem dimension, and, in contrast to subspace methods, the length of output trajectories only affects the bias term.Comment: arXiv admin note: text overlap with arXiv:2210.1487

Constrained Policy Synthesis: Riemannian Flows, Online Regulation, and Distributed Games

Thesis (Ph.D.)--University of Washington, 2023This dissertation makes contributions to decision-making processes in both cooperative and non-cooperative environments, spanning several domains from constrained and large- scale dynamical systems to network games and learning for control and estimation prob- lems. First, we examine linearly constrained policy optimization over stabilizing controllers, utilizing a Riemannian metric inherent to optimal control problems. We propose a novel Newton-type algorithm that leverages the manifold’s second-order geometry to ensure local convergence, demonstrating promising results in Structured and Output Linear Quadratic Regulators (LQR) problems. Additionally, we present a distributed model-free policy it- eration tailored for large networks of homogeneous systems. This algorithm enables the development of stabilizing distributed feedback controllers through a data-driven approach and the use of a learned stability margin. Addressing online regulation of partially unknown unstable linear systems, we introduce the Data-Guided Regulation (DGR) synthesis proce- dure, revealing novel geometric and system-theoretic properties while effectively regulating the system’s states. Furthermore, we explore distributed learning in network games using dual averaging, achieving sublinear regret bounds by optimizing global objectives composed of local objective functions and considering network structures. Lastly, we investigate optimal filtering policies for linear systems with unknown noise covariance matrices using noisy output data, minimizing prediction error through stochastic policy optimization and ensuring theoretical guarantees for biased gradients and stability constraints

Policy Optimization over Submanifolds for Constrained Feedback Synthesis

In this paper, we study linearly constrained policy optimizations over the manifold of Schur stabilizing controllers, equipped with a Riemannian metric that emerges naturally in the context of optimal control problems. We provide extrinsic analysis of a generic constrained smooth cost function, that subsequently facilitates subsuming any such constrained problem into this framework. By studying the second order geometry of this manifold, we provide a Newton-type algorithm with local convergence guarantees that exploits this inherent geometry without relying on the exponential mapping nor a retraction. The algorithm hinges instead upon the developed stability certificate and the linear structure of the constraints. We then apply our methodology to two well-known constrained optimal control problems. Finally, several numerical examples showcase the performance of the proposed algorithm

Multi-Pursuer Pursuit-Evasion Games Under Parameters Uncertainty: A Monte Carlo Approach

The traditional pursuit-evasion game considers a situation where one or more pursuers try to catch an evader, while the evader is trying to escape. Instead of moving entities, a more general scope of the pursuit-evasion game could also consider systems of systems. In a system consisting of many subsystems, when one subsystem decides to dissent and operate in a manner that is inconsistent with the others, a game situation similar to the pursuit-evasion game occurs. The dissenting subsystem can be viewed as an evader and the remaining conforming subsystems can be viewed as pursuers who oppose and try to prevent the dissention from succeeding. This is a more general system of systems, approach to the pursuit-evasion problem where the players are now subsystems of a system rather than moving entities. Clearly, any control strategy by any one of the subsystems to separate needs to be assessed against a variety of options that the remaining subsystems may use. In this paper, we consider a game where the pursuer subsystems consider three possible strategies: (1) act independently as Nash players, (2) act optimally as a team, and (3) act individually as greedy pursuers. The evader subsystem, on the other hand, will consider strategies against all possible strategies by the pursuer subsystems. We assume that no subsystem knows which strategies the other subsystems are implementing and none of the subsystems has information about any of the parameters in the objective functions of the other subsystems. We deal with these uncertainties by first developing the Nash strategies for each of the resulting games for all possible options available to both sides. Given the prevailing parameter uncertainty in developing the strategies, we perform a Monte Carlo analysis to determine probabilities of success (or failure) for each of the strategies considered by each side. We illustrate the results using two simulation scenarios of a pursuit-evasion example consisting of three pursuers moving on a plane and chasing one evader