Collision avoidance as a differential game: real-time approximation of optimal strategies using higher derivatives of the value function

2002 ◽  
Author(s):  
R. Lachner
Keyword(s):  
Real Time ◽  
Collision Avoidance ◽  
Differential Game ◽  
Value Function ◽  
Optimal Strategies ◽  
Time Approximation ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
The Value Function

Nested variational inequalities and related optimal starting–stopping problems

1992 ◽  
Vol 29 (01) ◽  
pp. 104-115 ◽  
Author(s):  
M. Sun
Keyword(s):  
Differential Equation ◽  
Variational Inequalities ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
Diffusion Model ◽  
Value Function ◽  
Optimal Strategies ◽  
Stochastic Differential Game ◽  
The Value Function

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.

Nested variational inequalities and related optimal starting–stopping problems

1992 ◽  
Vol 29 (1) ◽  
pp. 104-115 ◽  
Author(s):  
M. Sun
Keyword(s):  
Differential Equation ◽  
Variational Inequalities ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
Diffusion Model ◽  
Value Function ◽  
Optimal Strategies ◽  
Stochastic Differential Game ◽  
The Value Function

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.

THE VALUE FUNCTION OF A DIFFERENTIAL GAME WITH SIMPLE MOTIONS AND AN INTEGRO-TERMINAL COST

2018 ◽  
pp. 877-890
Author(s):  
Lyubov Gennad’evna Shagalova
Keyword(s):  
Differential Equation ◽  
State Space ◽  
Differential Game ◽  
Value Function ◽  
Piecewise Linear ◽  
The State ◽  
The Value Function

An antagonistic positional differential game of two persons is considered. The dynamics of the system is described by a differential equation with simple motions, and the payoff functional is integro-terminal. For the case when the terminal function and the Hamiltonian are piecewise linear, and the dimension of the state space is two, a finite algorithm for the exact construction of the value function is proposed.

Verification Theorems for Optimal Feedback Strategies in Differential Games

2003 ◽  
Vol 05 (02) ◽  
pp. 167-189 ◽  
Author(s):  
Ştefan Mirică
Keyword(s):  
Differential Games ◽  
Differential Game ◽  
Value Function ◽  
Generalized Derivatives ◽  
Differential Inequalities ◽  
Value Functions ◽  
Optimal Solutions ◽  
Optimal Feedback ◽  
Feedback Strategies ◽  
The Value Function

We give complete proofs to the verification theorems announced recently by the author for the "pairs of relatively optimal feedback strategies" of an autonomous differential game. These concepts are considered to describe the possibly optimal solutions of a differential game while the corresponding value functions are used as "instruments" for proving the relative optimality and also as "auxiliary characteristics" of the differential game. The 6 verification theorems in the paper are proved under different regularity assumptions accompanied by suitable differential inequalities verified by the generalized derivatives, mainly of contingent type, of the value function.

scholarly journals Approximation of value function of differential game with minimal cost

2021 ◽  
Vol 31 (4) ◽  
pp. 536-561
Author(s):  
Yu.V. Averboukh
Keyword(s):  
Differential Game ◽  
Continuous Time ◽  
Bellman Equation ◽  
Value Function ◽  
Stochastic Game ◽  
Inequality Constraints ◽  
Minimal Cost ◽  
Parabolic Pde ◽  
Zero Sum ◽  
The Value Function

The paper is concerned with the approximation of the value function of the zero-sum differential game with the minimal cost, i.e., the differential game with the payoff functional determined by the minimization of some quantity along the trajectory by the solutions of continuous-time stochastic games with the stopping governed by one player. Notice that the value function of the auxiliary continuous-time stochastic game is described by the Isaacs–Bellman equation with additional inequality constraints. The Isaacs–Bellman equation is a parabolic PDE for the case of stochastic differential game and it takes a form of system of ODEs for the case of continuous-time Markov game. The approximation developed in the paper is based on the concept of the stochastic guide first proposed by Krasovskii and Kotelnikova.

Double stopping by two decision-makers

1993 ◽  
Vol 25 (2) ◽  
pp. 438-452 ◽  
Author(s):  
K. Szajowski
Keyword(s):  
Markov Process ◽  
Optimal Stopping ◽  
Value Function ◽  
Optimal Strategies ◽  
Decision Makers ◽  
Secretary Problem ◽  
Gain Function ◽  
Competitive Situation ◽  
Zero Sum ◽  
The Value Function

A problem of optimal stopping of the discrete-time Markov process by two decision-makers (Player 1 and Player 2) in a competitive situation is considered. The zero-sum game structure is adopted. The gain function depends on states chosen by both decision-makers. When both players want to accept the realization of the Markov process at the same moment, the priority is given to Player 1. The construction of the value function and the optimal strategies for the players are given. The Markov chain case is considered in detail. An example related to the generalized secretary problem is solved.

scholarly journals Revisiting a Three-Player Pursuit-Evasion Game

2021 ◽  
Author(s):  
János Szőts ◽  
Andrey V. Savkin ◽  
István Harmati
Keyword(s):  
Differential Games ◽  
Value Function ◽  
Optimal Trajectories ◽  
Computationally Efficient ◽  
Partial Derivatives ◽  
Pursuit Evasion ◽  
Detailed Explanation ◽  
Evasion Game ◽  
Derivatives Of ◽  
The Value Function

AbstractWe consider the game of a holonomic evader passing between two holonomic pursuers. The optimal trajectories of this game are known. We give a detailed explanation of the game of kind’s solution and present a computationally efficient way to obtain trajectories numerically by integrating the retrograde path equations. Additionally, we propose a method for calculating the partial derivatives of the Value function in the game of degree. This latter result applies to differential games with homogeneous Value.

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