Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

A Strategic Analysis of Terrorist Activity and Counter-Terrorism Policies

This paper develops a model of terrorist activity and behavior. A terrorist organization chooses the size and the number of attacks. The defending state chooses the level of security-deterrence measures. The equilibrium sequence is such that the Organization moves first, followed by the State. A defensive policy such as an innovation in security-deterrence technology tends to reduce the size of attacks but increase their number, while an offensive policy, lowering the total "strike" or "output" potential of a terrorist organization, has opposite effects. Both policies reduce the expected damage from terror. An individual's decision to become a terrorist or a financier is also modeled, leading to endogenous supplies of terrorists and funds. The effects of terrorist-flushing measures, provisions to curb the flow of funds to terrorist organizations and income-enhancing policies are evaluated by taking into account their "supply-side" effects.

Magnetic Equilibria and Instabilities

Solar flares are understood as a process of explosive liberation of magnetic energy, coming after a slow phase of energy build-up. The slow evolution of magnetic equilibria may end up with (a) the termination of an equilibrium sequence, or (b) an instability. The distinction between the two can be made by drawing schematic potential curves. Case (a) has been extensively studied in two-dimensional models. The appearance of multiple solutions, or disappearance of a solution takes place as the system evolves away from the current-free configuration. Case (b) can be discussed in terms of ideal MHD or resistive MHD instabilities. A possible route to explosive energy release is suggested by combining these two cases.