Differential Game Theory Research Articles

A saddle-point theorem for strongly and weakly convex functions

We prove a theorem on the existence, uniqueness, and continuous dependence on parameters for a saddle point in a type of minimax problem that arises, for example, in differential game theory. Our theorem on the existence of a saddle point does not follow from the well-known theorems of von Neumann, Ky Fan, Sion and others since the intersection of sublevel sets of the function considered may be disconnected and non-empty. The hypotheses of our theorem are stated in terms of the strong and weak convexity of functions defined on a Banach space. We study properties of strongly and weakly convex functions related to the operations of minimization and maximization. We obtain unimprovable estimates of convexity parameters for the infimal convolution (episum) and epidifference of functions. This results in the construction of a calculus of convexity parameters of functions with respect to epioperations. We give typical examples and show that the hypotheses of our theorems are essential.

Agile and Precise Attitude Switching Maneuver of Flexible Spacecraft based on Nonstationary Frequency-Shaped Robust Control

Flexible spacecrafts have been requested to perform agile and precise attitude switching maneuver recently. For this requirement, the controller of the flexible spacecrafts is required to be robust against high-frequency unmodelled uncertainty and to have high performance of motion and vibration control. In this paper, an attitude controller of the flexible spacecrafts is designed by nonstationary frequency-shaped robust control based on the differential game theory. The frequency weights of the proposed controller are designed to envelop high-frequency unmodelled uncertainty and enable the controller to suppress the spillover instability. Moreover, the frequency weights for the state vector are designed in the time domain. The proposed controller generates optimal input torques to perform agile and precise attitude switching maneuver of the spacecraft. Several simulation results show the effectiveness of the proposed controller.

ON NONLINEAR ELLIPTIC BELLMAN SYSTEMS FOR A CLASS OF STOCHASTIC DIFFERENTIAL GAMES IN ARBITRARY DIMENSION

We consider nonlinear elliptic Bellman systems which arise in the theory of stochastic differential games. The right-hand sides of the equations (which are called Hamiltonians) may have quadratic growth with respect to the gradient of the unknowns. Under certain assumptions on the Hamiltonians, that are satisfied for many types of stochastic games, we establish the existence of a regular solution. This partially generalizes recent works of Bensoussan, Frehse and Vogelgesang from two-dimensional setting onto arbitrary dimension. The main novelty of the paper consists of introducing a new (semi-continuity) method for obtaining the continuity of the solution and the corresponding estimates.

A Design of Entrapment Strategies for the Distributed Pursuit-Evasion Game⋆

AbstractIn this paper, the entrapment strategies are proposed for a multiple-pursuer single-evader game. The aimed game is nonzero-sum and under distributed information, where the global information is not available for all the players. Since the information is distributive, cooperative control is integrated with the the differential game theory to deal with the difficulty of lack of information. The issue of entrapment strategies is taken into account for the pursuers to chase the evader in a more intelligent manner. Simulation results are presented.

Risk-Sensitive Mean-Field Stochastic Differential Games

AbstractIn this paper, we study a class of risk-sensitive mean-field stochastic differential games. Under regularity assumptions, we use results from standard risk-sensitive differential game theory to show that the mean-field value of the exponentiated cost functional coincides with the value function of a Hamilton-Jacobi-Bellman-Fleming (HJBF) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations and HJBF equations.

Tracking a reference solution of a control system of phase field equations

We consider the problem of tracking a reference solution of a dynamical system described by a pair of distributed differential equations, the phase field equations. To solve this problem, we propose an algorithm based on Yu.S. Osipov’s theory of dynamic inversion and on N.N. Krasovskii’s extremal shift method developed in the theory of positional differential games.

Some Recent Aspects of Differential Game Theory

This survey paper presents some new advances in theoretical aspects of differential game theory. We particular focus on three topics: differential games with state constraints; backward stochastic differential equations approach to stochastic differential games; differential games with incomplete information. We also address some recent development in nonzero-sum differential games (analysis of systems of Hamilton–Jacobi equations by conservation laws methods; differential games with a large number of players, i.e., mean-field games) and long-time average of zero-sum differential games.

Complete Solution of a Differential Game with Linear Dynamics and Bounded Controls

A differential game is an appropriate mathematical model for real-life control problems, which either involve many decision-makers or contain a high degree of uncertainties. There is a rich literature devoted to the theory of differential games (see e.g. [1–5]). A zero-sum finite-horizon differential game with linear dynamics and bounded controls was studied extensively in the literature, because of its considerable meaning both in theory and applications (see e.g. [6–10] and the references therein). Important applications of this game are: a pursuit-evasion problem (see e.g. [11–14]), an airplane landing problem under windshear conditions (see [15] and references therein), and some others. Different versions of this game were analyzed in the literature. A simple example with the ideal dynamics of the players was considered in [6]. The game with a first-order

An approach-evasion differential game: Stochastic guide

An approach-evasion positional differential game is considered for a conflict-controlled motion and a target set within a given set. Use is made of a solution of the associated boundary-value problem for a parabolic equation degenerating as the diffusion term vanishes to a Hamilton-Jacobi type equation, which is typical for techniques in the theory of differential games. Based on this, a control scheme with a stochastic guide is developed.

Coordinating decisions by supply chain partners in a vendor-managed inventory relationship

We look into the linked decision making in the vendor-managed inventory (VMI) relationship. It is a supply chain management model, where the retailer decides the retail price while the vendor determines its capacity commitment. In this model, the retailer and the vendor should coordinate their decisions in order to maximize their individual profit or the total profit combining the two participants together. The vendor has to take into account the demand pattern throughout the product life cycle (PLC) when it decides its capacity commitment, which will affect its inventory management cost during the PLC, while the retailer should change the retail price over the PLC so as to maximize the revenues and minimize the inventory cost at the same time. Employing a system dynamics simulation approach based on differential game theory, which also takes into account the product characteristics such as the demand’s innovation and imitation effects, we analyze and confirm the dynamic coordination of key decision variables by the supply chain partners in the VMI relationship.

Numerical construction of Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game

Numerical methods are proposed for constructing Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game with terminal cost functionals and geometric constraints on the players’ controls. The formalization of the players’ strategies and of the motions generated by them is based on the formalization and results from the theory of positional zero-sum differential games developed by N.N. Krasovskii and his school. It is assumed that the game is reduced to a planar game and the constraints on the players’ controls are given in the form of convex polygons. The problem of finding solutions of the game may be reduced to solving nonstandard optimal control problems. Several computational geometry algorithms are used to construct approximate trajectories in these problems, in particular, algorithms for constructing the convex hull as well as the union, intersection, and algebraic sum of polygons.

On nontraditional problems of portfolio management

Some applications of portfolio investment theory, which goes back to papers by H. Markowitz and J. Tobin, are considered. First, problems of dynamic control of an investment portfolio are considered in the case when the behavior of the averaged characteristics of returns on assets is described not by stochastic differential equations but by deterministic differential inclusions. Such a description of the uncertainty allows one to apply methods of the theory of guaranteed control and the theory of positional differential games. Second, it is shown how the theory developed initially for portfolios of risky financial instruments (shares) can be used, after an appropriate modification, for studying objects of different nature as well. In particular, problems of constructing an effective project portfolio and effective portfolios of contractors and customers of an enterprise are considered.

Vertical passage of obstacles by an aircraft under wind disturbance

The adaptive control method is applied to the problem of the vertical passage of obstacles by an aircraft under wind disturbance. Constructions of the theory of differential games are used.

Modeling crowd dynamics by the mean-field limit approach

We propose to couple through optimal control theory, a dynamical crowd evolution models where pedestrians make decisions with a well-defined set of actions and strategies by minimizing their utility function. We explain how situations of this kind can be modeled by making use of differential game theory. By leveraging on some simple examples, we introduce the most fundamental concept of mean field games recently introduced by Lasry and Lions [J.-M. Lasry, P.-L. Lions, Mean field games, Jpn. J. Math. 2 (1) (2007) 229–260]. The main interest of mean field games in our work is to simplify the interactions between pedestrians. As a consequence, we derive a continuum formulation of the crowd dynamics and nonlinear systems involving partial differential equations of crowd models.

Aircraft landing control under wind disturbances

Results from differential game theory are applied to construct an adaptive control in linear systems with an unknown level of dynamic disturbances. The efficiency of the method is exemplified by a problem of aircraft landing under wind disturbance.

Singular boundary characteristics of the Hamilton–Jacobi equation

Situations exist in boundary value problems for first order partial differential equations arising in physics (the Hamilton–Jacobi equation), optimal control theory (the Bellman equation) and the theory of differential games (the Isaacs equation) when the value of the required function is not given on a part of the boundary or not at all, or it is not the limit of the (generalized) solution of the problem. Nevertheless, such conditions are required for constructing the solution (by the method of characteristics, for example). It is shown that the required boundary values can be exposed as a specific continuation of the conditions that are known in the boundary submanifolds of the given part of the boundary. This extension of the conditions is accomplished using the characteristic curves starting in a known submanifold of the boundary and running along the boundary. The characteristics are a generalization of the classical characteristics associated with a partial differential equation. They are called singular characteristics, and the theory of these has been developed in a number of the author's papers. After obtaining these “natural” boundary conditions, the solution is constructed using the conventional method of integrating the equations of the classical characteristics. Conditions of the Dirichlet and Neumann type are considered. The technique is illustrated using a numerical example from the theory of differential games containing a number of parameters.

Cooperative Guidance for Naval Area Defence

AbstractIn this paper, we proposed a cooperative guidance strategy to consider many on many engagements for maritime area air defence. The main idea is that a cooperative missile guidance law can defend the area as long as it guarantees that the keep out zone is always within the defended area defined by Earliest Intercept Geometry (EIG) or a No Escape Zone (NEZ). Since one of the most important issues for the cooperative guidance is to allocate attacking missiles to defending missiles, we focused on developing an allocation plan. The cooperative guidance algorithm proposed in this paper consists of two main parts: a pre allocation algorithm designed by using the EIG guidance law and Differential Geometry Guidance Law (DGGL); in flight target allocation proposed based on the differential game theory. Whilst the pre allocation plan is activated during the mid course and deactivated after the mid course, the in flight target allocation is activated during all in flight phases to refine the initial allocation. Mathematical analysis for the allocation algorithms is carried out to check their characteristics and the performance of the proposed strategy is verified by some numerical examples.

1B12 Agile and Precise Attitude Switching Maneuver of Flexible Spacecraft based on Nonstationary Frequency-Shaped Robust Control

Flexible spacecrafts have been required to agile and precise attitude switching maneuver recently. For this requirement, the controller of the flexible spacecrafts is required to be robust against high-frequency unmodelled uncertainty and to have high performances of motion and vibration control. In this paper, the attitude controller of the flexible spacecrafts is designed by nonstationary frequency-shaped robust control based on the differential game theory. The weightings of the proposed controller are designed to be replaced high-frequency unmodelled uncertainty and enable the controller to suppress the spillover instability. The proposed controller generates optimal input torques to control agile and precise attitude switching maneuver of the spacecraft by the weightings of the state vector which are designed in the time domain. Several simulation results show the effectiveness of the proposed controller.

On the convergence of singular perturbations of Hamilton-Jacobi equations

This paper is devoted to singular perturbation problems for first order equations.Under some coercivity and periodicity assumptions, we establish the uniform convergence and we provide an estimate of the rate of convergence, which we consider the main result of the paper.  &nbsp We shall also show that our results apply to the homogenization problem for coercive and periodic equations.Finally, some examples arising in optimal control and differential games theory will be discussed.

A differential game theoretical analysis of mechanistic models for territoriality

In this paper, elements of differential game theory are used to analyze a spatially explicit home range model for interacting wolf packs when movement behavior is uncertain. The model consists of a system of partial differential equations whose parameters reflect the movement behavior of individuals within each pack and whose steady-state solutions describe the patterns of space-use associated to each pack. By controlling the behavioral parameters in a spatially-dynamic fashion, packs adjust their patterns of movement so as to find a Nash-optimal balance between spreading their territory and avoiding conflict with hostile neighbors. On the mathematical side, we show that solving a nonzero-sum differential game corresponds to finding a non-invasible function-valued trait. From the ecological standpoint, when movement behavior is uncertain, the resulting evolutionarily stable equilibrium gives rise to a buffer-zone, or a no-wolf's land where deer are known to find refuge.