Characterization of well-posedness of piecewise linear systems

One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. The paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Caratheodory. The concepts of jump solutions or of sliding modes are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multimodal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-pose

Dynamic Price Competition with Price Adjustment Costs and Product Differentiation

We study a discrete time dynamic game of price competition with spatially differentiated products and price adjustment costs. We characterise the Markov perfect and the open-loop equilibrium of our game. We find that in the steady state Markov perfect equilibrium, given the presence of adjustment costs, equilibrium prices are always higher than prices at the repeated static Nash solution, even though, adjustment costs are not paid in steady state. This is due to intertemporal strategic complementarity in the strategies of the firms and from the fact that the cost of adjusting prices adds credibility to high price equilibrium strategies. On the other hand, the stationary open-loop equilibrium coincides always with the static solution. Furthermore, in contrast to continuous time games, we show that the stationary Markov perfect equilibrium converges to the static Nash equilibrium when adjustment costs tend to zero. Moreover, we obtain the same convergence result when adjustment costs tend to infinity.Price adjustment costs, Difference game, Markov perfect equilibrium, Open-loop equilibrium

On Evolutionary Stability of Spiteful Preferences

The paper analyzes under what conditions spiteful preferences are evolutionarily stable applying the indirect evolution approach. With a quadratic material payo¤ function, spiteful preferences are evolutionarily stable for a large set of parameters. It is shown that strategic substitutability or complementarity is endogenous property of the game played with evolutionarily stable preferences. Its relation to properties of the material payoff function is analyzed. Finally, it is shown that with incomplete information only selfish preferences are evolutionarily stable.indirect evolution;spite;endogenous preferences

The Nature of the Chemical Process. 1. Symmetry Evolution - Revised Information Theory, Similarity Principle and Ugly Symmetry

Three laws of information theory have been proposed. Labeling by introducing nonsymmetry and formatting by introducing symmetry are defined. The function L (L=lnw, w is the number of microstates, or the sum of entropy and information, L=S+I) of the universe is a constant (the first law of information theory). The entropy S of the universe tends toward a maximum (the second law law of information theory). For a perfect symmetric static structure, the information is zero and the static entropy is the maximum (the third law law of information theory). Based on the Gibbs inequality and the second law of the revised information theory we have proved the similarity principle (a continuous higher similarity-higher entropy relation after the rejection of the Gibbs paradox) and proved the Curie-Rosen symmetry principle (a higher symmetry-higher stability relation) as a special case of the similarity principle. Some examples in chemical physics have been given. Spontaneous processes of all kinds of molecular interaction, phase separation and phase transition, including symmetry breaking and the densest molecular packing and crystallization, are all driven by information minimization or symmetry maximization. The evolution of the universe in general and evolution of life in particular can be quantitatively considered as a series of symmetry breaking processes. The two empirical rules - similarity rule and complementarity rule - have been given a theoretical foundation. All kinds of periodicity in space and time are symmetries and contribute to the stability. Symmetry is beautiful because it renders stability. However, symmetry is in principle ugly because it is associated with information loss.Comment: 29 pages, 14 figure

Complementarity in classical dynamical systems

The concept of complementarity, originally defined for non-commuting observables of quantum systems with states of non-vanishing dispersion, is extended to classical dynamical systems with a partitioned phase space. Interpreting partitions in terms of ensembles of epistemic states (symbols) with corresponding classical observables, it is shown that such observables are complementary to each other with respect to particular partitions unless those partitions are generating. This explains why symbolic descriptions based on an \emph{ad hoc} partition of an underlying phase space description should generally be expected to be incompatible. Related approaches with different background and different objectives are discussed.Comment: 18 pages, no figure

Electricity System Expansion Studies to Consider Uncertainties and Interactions in Restructured Markets

This dissertation concerns power system expansion planning under different market mechanisms. The thesis follows a three paper format, in which each paper emphasizes a different perspective. The first paper investigates the impact of market uncertainties on a long term centralized generation expansion planning problem. The problem is modeled as a two-stage stochastic program with uncertain fuel prices and demands, which are represented as probabilistic scenario paths in a multi-period tree. Two measurements, expected cost (EC) and Conditional Value-at-Risk (CVaR), are used to minimize, respectively, the total expected cost among scenarios and the risk of incurring high costs in unfavorable scenarios. We sample paths from the scenario tree to reduce the problem scale and determine the sufficient number of scenarios by computing confidence intervals on the objective values. The second paper studies an integrated electricity supply system including generation, transmission and fuel transportation with a restructured wholesale electricity market. This integrated system expansion problem is modeled as a bi-level program in which a centralized system expansion decision is made in the upper level and the operational decisions of multiple market participants are made in the lower level. The difficulty of solving a bi-level programming problem to global optimality is discussed and three problem relaxations obtained by reformulation are explored. The third paper solves a more realistic market-based generation and transmission expansion problem. It focuses on interactions among a centralized transmission expansion decision and decentralized generation expansion decisions. It allows each generator to make its own strategic investment and operational decisions both in response to a transmission expansion decision and in anticipation of a market price settled by an Independent System Operator (ISO) market clearing problem. The model poses a complicated tri-level structure including an equilibrium problem with equilibrium constraints (EPEC) sub-problem. A hybrid iterative algorithm is proposed to solve the problem efficiently and reliably