Sign Stability of Dual Switching Linear Continuous-Time Positive Systems

This paper investigates 1-moment exponential stability and exponential mean-square stability (EMS stability) under average dwell time (ADT) and the preset deterministic switching mechanism of dual switching linear continuous-time positive systems when a numerical realization does not exist. The signs of subsystem matrices, but not their structures of magnitude, are key information that causes a qualitative concept of stability called sign stability. Both 1-moment exponential stability and EMS stability, which are the traditional stability concepts, are generalized intrinsically. Hence, both 1-moment exponential sign stability and EMS sign stability are introduced and are proven based on sign equivalency. It is shown that they are symmetrically and qualitatively stable. Notably, the notion of stability can be checked quantitatively using some examples.

Stability concepts in bargaining games

In this thesis we discuss some novel concepts of stability in bargaining games, over a network setting. So far, the studies on bargaining games were done as profit sharing problems, whose underlying combinatorial optimization problems are of packing type. In our work, we study bargaining games from a cost sharing perspective, where the underlying combinatorial optimization problems are covering type problems. Unlike previous studies, where bargaining processes are restricted to only two players, we study bargaining games over a more generic hypergraph setting, which allows any bargaining process to be formed among any number of players. In previous studies of bargaining games, the objects that are being negotiated are assumed to be uniform and only the outcomes of the negotiations are allowed to be different. However, in our study, we accommodate possibilities of non-uniform weights of the objects that are being negotiated, which is closer to any real life scenario. Finally we extend our study to incorporate socially aware players by introducing a relaxed and innovative definition of stability.

Stability concepts in bargaining games

In this thesis we discuss some novel concepts of stability in bargaining games, over a network setting. So far, the studies on bargaining games were done as profit sharing problems, whose underlying combinatorial optimization problems are of packing type. In our work, we study bargaining games from a cost sharing perspective, where the underlying combinatorial optimization problems are covering type problems. Unlike previous studies, where bargaining processes are restricted to only two players, we study bargaining games over a more generic hypergraph setting, which allows any bargaining process to be formed among any number of players. In previous studies of bargaining games, the objects that are being negotiated are assumed to be uniform and only the outcomes of the negotiations are allowed to be different. However, in our study, we accommodate possibilities of non-uniform weights of the objects that are being negotiated, which is closer to any real life scenario. Finally we extend our study to incorporate socially aware players by introducing a relaxed and innovative definition of stability.

Stability Concepts of Riemann-Liouville Fractional-Order Delay Nonlinear Systems

First, we set up in an appropriate way the initial value problem for nonlinear delay differential equations with a Riemann-Liouville (RL) fractional derivative. We define stability in time and generalize Mittag-Leffler stability for RL fractional differential equations and we study stability properties by an appropriate modification of the Razumikhin method. Two different types of derivatives of Lyapunov functions are studied: the RL fractional derivative when the argument of the Lyapunov function is any solution of the studied problem and a special type of Dini fractional derivative among the studied problem.

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stability considerations play a central role in structural dynamics to determine states that are robust against perturbations during the operation. Linear stability concepts, such as the complex eigenvalue analysis, constitute the core of analysis approaches in engineering reality. However, most stability concepts are limited to local perturbations, i.e., they can only measure a state’s stability against small perturbations. Recently, the concept of basin stability was proposed as a global stability concept for multi-stable systems. As multi-stability is a well-known property of a range of nonlinear dynamical systems, this work studies the basin stability of bi-stable mechanical oscillators that are affected and self-excited by dry friction. The results indicate how the basin stability complements the classical binary stability concepts for quantifying how stable a state is given a set of permissible perturbations.

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stability considerations play a central role in structural dynamics to determine states that are robust against perturbations during the operation. Linear stability concepts, such as the complex eigenvalue analysis, constitute the core of analysis approaches in engineering reality. However, most stability concepts are limited to local perturbations, i.e. they can only measure a state’s stability against small perturbations. Recently, the concept of basin stability has been proposed as a global stability concept for multi-stable systems. As multi-stability is a well-known property of a range of nonlinear dynamical systems, this work studies the basin stability of bi-stable mechanical oscillators that are affected and self-excited by dry friction. The results indicate how the basin stability complements the classical binary stability concepts for quantifying how stable a state is given a set of permissible perturbations.

Finite-Time Stability Analysis: A Tutorial Survey

In the past decades, there has been a growing research interest in the field of finite-time stability and stabilization. This paper aims to provide a self-contained tutorial review in the field. After a brief introduction to notations and two distinct finite-time stability concepts, dynamical system models, particularly in the form of linear time-varying systems and impulsive linear systems, are studied. The finite-time stability analysis in a quantitative sense is reviewed, and a variety of stability results including state transition matrix conditions, the piecewise continuous Lyapunov-like function theory, and the converse Lyapunov-like theorem are investigated. Then, robustness and time delay issues are studied. Finally, fundamental finite-time stability results in a qualitative sense are briefly reviewed.

Stability Concepts: Beyond Nash Equilibria

The concept of an Evolutionarily Stable Strategy (ESS), which is a stronger stability condition than that of a Nash equilibrium, is introduced. A simple evolutionary dynamic, adaptive dynamics, is also introduced. This leads to the concept of convergence stability under adaptive dynamics. It is shown that these two stability criteria are independent for general games: a strategy can be an ESS but not be reachable under adaptive dynamics and a strategy may be an attractor under adaptive dynamics but a fitness minimum and so not an ESS. The latter situation leads to the possibility of evolutionary branching, a phenomenon in which the population splits into a mixture of two or more distinct morphs. Replicator dynamics provide another evolutionary dynamic, although it is argued that it is of limited relevance to biology. In some games, individuals interact with relatives. The effects of kin assortment, and the direct fitness and gene-centred approaches to games between kin are described and illustrated.

Collective Decision Making under Incomplete Knowledge: Possible and Necessary Solutions

Most solution concepts in collective decision making are defined assuming complete knowledge of individuals' preferences and of the mechanism used for aggregating them. This is often unpractical or unrealistic. Under incomplete knowledge, a solution advocated by many consists in quanrtifying over all completions of the incomplete preference profile (or all instantiations of the incompletely specified mechanism). Voting rules can be `modalized' this way (leading to the notions of possible and necessary winners), and also efficiency and fairness notions in fair division, stability concepts in coalition formation, and more. I give here a survey of works along this line.