Thulasi Mylvaganam

Constructive

In this paper, a class of infinite-horizon, nonzero-sum differential games and their Nash equilibria are studied and the notion of εα-Nash equilibrium strategies is introduced. Dynamic strategies satisfying partial differential inequalities in place of the Hamilton-Jacobi-Isaacs partial differential equations associated with the differential games are constructed. These strategies constitute (local) εα-Nash equilibrium strategies for the differential game. The proposed methods are illustrated on a differential game for which the Nash equilibrium strategies are known and on a Lotka-Volterra model, with two competing species. Simulations indicate that both dynamic strategies yield better performance than the strategies resulting from the solution of the linear-quadratic approximation of the problem.

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A team of agents consisting of one leader and N followers is considered. The problem in which the leader is steered towards a target position while the followers create and maintain a formation about the leader, while avoiding collisions, is posed as a nonlinear differential game. Dynamic strategies which approximate the solution to the differential game are constructed. It is shown that for the case in which collisions are not considered the problem simplifies to a linear-quadratic differential game, for which approximate solutions are identified. These approximate solutions are well-suited for situations in which the agents do not have full information regarding the positions of all members of the team. Simulations illustrate the results for both the linear-quadratic and the nonlinear problems.

-Nash Equilibria for Nonzero-Sum Differential Games

The problem of steering a team of agents from their initial positions to a predefined end-configuration while avoiding collisions is formulated as a differential game. A method for approximating the solution of the differential game is then presented, providing ε-Nash strategies. It is shown that approximate solutions are sufficient to guarantee that the task of reaching the final configuration while avoiding collisions is achieved. The theory is illustrated by simulations.

A differential game approach to formation control for a team of agents with one leader

In this paper an approximate solution of a differential game arising in optimal monitoring is considered. The solution relies on the construction of a sequence of optimal control problems, solved using their associated Hamilton-Jacobi-Bellman partial differential equations, along with the use of an instantaeous player that acts between successive optimisation periods. Preliminary simulation results illustrate the ideas and show that continuous monitoring is achieved.

A constructive differential game approach to collision avoidance in multi-agent systems

A class of nonzero-sum differential games is considered and a dynamic state feedback control law that approximates the solution of the differential game is proposed. The control law relies upon the solution of algebraic equations in place of partial differential equations or inequalities and makes use of dynamics shared by the players, thus relaxing the structural assumption required in [1]. The idea is firstly illustrated by the two-player case and then extended to the N-player case. A simple numerical example completes the paper.

Approximate optimal monitoring: Preliminary results

A microgrid is an example of a small-scale power system. Such a system is of particular interest when considering networked electrical power systems with renewable sources. Microgrids are inherently different from the traditional power grid and thus require new control techniques. In this paper DC microgrid systems in which there are several controllable loads, which we refer to as players, are considered. The problem of reaching and maintaining a nominal operating condition while minimising the control efforts of each player is formulated as a differential game. Since closed-form solutions to differential games are usually not readily obtained, a systematic method of constructing approximate solutions is presented.

Approximate solutions to a class of nonlinear differential games

We consider a population of dynamic agents, also referred to as players. The state of each player evolves according to a linear stochastic differential equation driven by a Brownian motion and under the influence of a control and an adversarial disturbance. Every player minimizes a cost functional which involves quadratic terms on state and control plus a cross-coupling mean-field term measuring the congestion resulting from the collective behavior, which motivates the term “crowd-averse”. For this game we first illustrate the paradigm of robust mean-field games. Second, we provide a new approximate solution approach based on the extension of the state space and prove the existence of equilibria and their stability properties.

Control of microgrids using a differential game theoretic framework

We consider a population of dynamic agents, also referred to as players. The state of each player evolves according to a linear stochastic differential equation driven by a Brownian motion and under the influence of a control and an adversarial disturbance. Every player minimizes a cost functional which involves quadratic terms on state and control plus a cross-coupling mean-field term measuring the congestion resulting from the collective behavior, which motivates the term “crowd-averse.” Motivations for this model are analyzed and discussed in three main contexts: a stock market application, a production engineering example, and a dynamic demand management problem in power systems. For the problem in its abstract formulation, we illustrate the paradigm of robust mean-field games. Main contributions involve first the formulation of the problem as a robust mean-field game; second, the development of a new approximate solution approach based on the extension of the state space; third, a relaxation method to minimize the approximation error. Further results are provided for the scalar case, for which we establish performance bounds, and analyze stochastic stability of both the microscopic and the macroscopic dynamics.

Approximate solutions for crowd-averse robust mean-field games

The problem of continuously monitoring a region using a team of agents, for example unmanned aerial vehicles equipped with sensors, is formulated as a differential game. This allows for the use of heterogeneous agents. The differential game is approximated as a sequence of optimal control problems, for which approximate solutions are found using dynamic feedback and the notion of algebraic P̅ solution. A general form for such a solution is presented assuming the agents have single-integrator dynamics. The results can be interpreted as a trajectory plan for agents with more general dynamics. The case in which the agents have unicycle dynamics and are to track the generated trajectory plan is presented.

Crowd-Averse Robust Mean-Field Games: Approximation via State Space Extension

A two-player nonlinear Stackelberg differential game with player 1 and player 2 as leader and follower, respectively, is considered. The feedback Stackelberg solutions to such games rely on the solution of two coupled partial differential equations (PDEs) for which closed-form solutions cannot in general be found. A method for constructing strategies satisfying partial differential inequalities in place of the PDEs is presented. It is shown that these constitute approximate solutions to the differential game. The theory is illustrated by a numerical example.