Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to infinity, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with [Formula: see text]. As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.

Impulse control of a two-link manipulation robot

A nonlinear problem of controlling the movements of a two-link manipulation robot is considered. The free mechanical system has two first integrals in involution. Methods of classical mechanics are used for analytical integration of the system of nonlinear differential equations. A trajectory connecting the initial and final positions of the two-link manipulation robot in the configuration space is found. Impulse controls at the initial moment of time impart the necessary energy to the robot to enter this trajectory. Impulse controls are also used to damp the speeds of the robot at the end position. In a computer simulation of the proposed procedure for moving the robot, generalized impulse controls are approximated by rectangular impulses.

A Bank Salvage Model by Impulse Stochastic Controls

The present paper is devoted to the study of a bank salvage model with a finite time horizon that is subjected to stochastic impulse controls. In our model, the bank’s default time is a completely inaccessible random quantity generating its own filtration, then reflecting the unpredictability of the event itself. In this framework the main goal is to minimize the total cost of the central controller, which can inject capitals to save the bank from default. We address the latter task, showing that the corresponding quasi-variational inequality (QVI) admits a unique viscosity solution—Lipschitz continuous in space and Hölder continuous in time. Furthermore, under mild assumptions on the dynamics the smooth-fit W l o c ( 1 , 2 ) , p property is achieved for any 1 < p < + ∞ .

IMPULSE CONTROL OF THE MANIPULATION ROBOT

A nonlinear control problem for a manipulation robot is considered. The solvability conditions for the problem are obtained in the class of special impulse controls. To achieve the control goal, the kinetic energy of the manipulation robot is used. When finding analytical formulas for controls, the classical first integrals of Lagrangian mechanics were used. The effectiveness of the proposed algorithm is illustrated by computer simulation.