Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 130-135).

This thesis is devoted to a theoretical and numerical investigation of methods to solve open-loop non zero-sum differential Nash games. These problems arise in many applications, e.g., biology, economics, physics, where competition between different agents appears. In this case, the goal of each agent is in contrast with those of the others, and a competition game can be interpreted as a coupled optimization problem for which, in general, an optimal solution does not exist. In fact, an optimal strategy for one player may be unsatisfactory for the others. For this reason, a solution of a game is sought as an equilibrium and among the solutions concepts proposed in the literature, that of Nash equilibrium (NE) is the focus of this thesis. The building blocks of the resulting differential Nash games are a dynamical model with different control functions associated with different players that pursue non-cooperative objectives. In particular, the aim of this thesis is on differential models having linear or bilinear state-strategy structures. In this framework, in the first chapter, some well-known results are recalled, especially for non-cooperative linear-quadratic differential Nash games. Then, a bilinear Nash game is formulated and analysed. The main achievement in this chapter is Theorem 1.4.2 concerning existence of Nash equilibria for non-cooperative differential bilinear games. This result is obtained assuming a sufficiently small time horizon T, and an estimate of T is provided in Lemma 1.4.8 using specific properties of the regularized Nikaido-Isoda function. In Chapter 2, in order to solve a bilinear Nash game, a semi-smooth Newton (SSN) scheme combined with a relaxation method is investigated, where the choice of a SSN scheme is motivated by the presence of constraints on the players’ actions that make the problem non-smooth. The resulting method is proved to be locally convergent in Theorem 2.1, and an estimate on the relaxation parameter is also obtained that relates the relaxation factor to the time horizon of a Nash equilibrium and to the other parameters of the game. For the bilinear Nash game, a Nash bargaining problem is also introduced and discussed, aiming at determining an improvement of all players’ objectives with respect to the Nash equilibrium. A characterization of a bargaining solution is given in Theorem 2.2.1 and a numerical scheme based on this result is presented that allows to compute this solution on the Pareto frontier. Results of numerical experiments based on a quantum model of two spin-particles and on a population dynamics model with two competing species are presented that successfully validate the proposed algorithms. In Chapter 3 a functional formulation of the classical homicidal chauffeur (HC) Nash game is introduced and a new numerical framework for its solution in a time-optimal formulation is discussed. This methodology combines a Hamiltonian based scheme, with proximal penalty to determine the time horizon where the game takes place, with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time. The resulting numerical optimization scheme has a bilevel structure, which aims at decoupling the computation of the end-time from the solution of the pursuit-evader game. Several numerical experiments are performed to show the ability of the proposed algorithm to solve the HC game. Focusing on the case where a collision may occur, the time for this event is determined. The last part of this thesis deals with the analysis of a novel sequential quadratic Hamiltonian (SQH) scheme for solving open-loop differential Nash games. This method is formulated in the framework of Pontryagin’s maximum principle and represents an efficient and robust extension of the successive approximations strategy in the realm of Nash games. In the SQH method, the Hamilton-Pontryagin functions are augmented by a quadratic penalty term and the Nikaido-Isoda function is used as a selection criterion. Based on this fact, the key idea of this SQH scheme is that the PMP characterization of Nash games leads to a finite-dimensional Nash game for any fixed time. A class of problems for which this finite-dimensional game admits a unique solution is identified and for this class of games theoretical results are presented that prove the well-posedness of the proposed scheme. In particular, Proposition 4.2.1 is proved to show that the selection criterion on the Nikaido-Isoda function is fulfilled. A comparison of the computational performances of the SQH scheme and the SSN-relaxation method previously discussed is shown. Applications to linear-quadratic Nash games and variants with control constraints, weighted L1 costs of the players’ actions and tracking objectives are presented that corroborate the theoretical statements. / Diese Dissertation handelt von eine theoretischen und numerischen Untersuchung von Methoden zur Lösung von Open-Loop-Nicht-Nullsummen-Differential-Nash-Spielen. Diese Probleme treten in vielen Anwendungen auf, z.B., Biologie, Wirtschaft, Physik, in denen die Konkurrenz zwischen verschiedenen Wirkstoffen bzw. Agenten auftritt. In diesem Fall steht das Ziel jedes Agenten im Gegensatz zu dem der anderen und ein Wettbewerbsspiel kann als gekoppeltes Optimierungsproblem interpretiert werden. Im Allgemeinen gibt es keine optimale Lösung für ein solches Spiel. Tatsächlich kann eine optimale Strategie für einen Spieler für den anderen unbefriedigend sein. Aus diesem Grund wird ein Gle- ichgewicht eines Spiels als Lösung gesucht, und unter den in der Literatur vorgeschlagenen Lösungskonzepten steht das Nash-Gleichgewicht (NE) im Mittelpunkt dieser Arbeit. ...

No description available.

No description available.

Thesis (Ph. D.)--Ohio State University, 1983. / Includes bibliographical references (leaves 128-132). Available online via OhioLINK's ETD Center.

The purpose of this dissertation is to apply behavior learning concepts to incomplete-information continuous time games. Realistic game scenarios are often incomplete-information games in which the players withhold information. A player may not know its opponent’s objectives and strategies prior to the start of the game. This lack of information can limit the player’s ability to play optimally. If the player can observe the opponent’s actions, it can better optimize its achievements by taking corrective actions. In this research, a framework to learn an opponent’s behavior and take corrective actions is developed. The framework will allow a player to observe the opponent’s actions and formulate behavior models. The developed behavior model can then be utilized to find the best actions for the player that optimizes the player’s objective function. In addition, the framework proposes that the player plays a safe strategy at the beginning of the game. A safe strategy is defined in this research as a strategy that guarantees a minimum pay-off to the player independent of the other player’s actions. During the initial part of the game, the player will play the safe strategy until it learns the opponent’s behavior. Two methods to develop behavior models that differ in the formulation of the behavior model are proposed. The first method is the Cost-Strategy Recognition (CSR) method in which the player formulates an objective function and a strategy for the opponent. The opponent is presumed to be rational and therefore will play to optimize its objective function. The strategy of the opponent is dependent on the information available to the opponent about other players in the game. A strategy formulation presumes a certain level of information available to the opponent. The previous observations of the opponent’s actions are used to estimate the parameters of the formulated behavior model. The estimated behavior model predicts the opponent’s future actions. The second method is the Direct Approximation of Value Function (DAVF) method. In this method, unlike the CSR method, the player formulates an objective function for the opponent but does not formulates a strategy directly; rather, indirectly the player assumes that the opponent is playing optimally. Thus, a value function satisfying the HJB equation corresponding to the opponent’s cost function exists. The DAVF method finds an approximate solution for the value function based on previous observations of the opponent’s control. The approximate solution to the value function is then used to predict the opponent’s future behavior. Game examples in which only a single player is learning its opponent’s behavior are simulated. Subsequently, examples in which both players in a two-player game are learning each other’s behavior are simulated. In the second part of this research, a reorientation control maneuver for a spinning spacecraft will be developed. This will aid the application of behavior learning and differential games concepts to the specific scenario involving multiple spinning spacecraft. An impulsive reorientation maneuver with coasting will be analytically designed to reorient the spin axis of the spacecraft using a single body fixed thruster. Cooperative maneuvers of multiple spacecraft optimizing fuel and relative orientation will be designed. Pareto optimality concepts will be used to arrive at mutually agreeable reorientation maneuvers for the cooperating spinning spacecraft.

No description available.

No description available.

No description available.

Pursuit and evasion games have garnered much research attention since the class of problems was first posed over a half century ago. With wide applicability to both civilian and military problems, the study of pursuit and evasion games showed much early promise. Early work generally focused on analytical solutions to games involving a single pursuer and a single evader. These solutions generally assumed simple system dynamics to facilitate convergence to a solution. More recently, numerical techniques have been utilized to solve more difficult problems. While many sophisticated numerical tools exist for standard optimization and optimal control problems, developing a more complete set of numerical tools for pursuit and evasion games is still a developing topic of research. This thesis extends the current body of numeric solution tools in two ways. First, an existing approach that modifies sophisticated optimization tools to solve two player pursuer and evasion games is extended to incorporate a class of state inequality constraints. Several classical problems are solved to illustrate the e±cacy of the new approach. Second, a new cooperation metric is introduced into the system objective function for multi-player pursuit and evasion games. This new cooperation metric encourages multiple pursuers to surround and then proceed to capture an evader. Several examples are provided to demonstrate this new cooperation metric.