Proportional navigation and the game of two cars: The case of a pursuer with variable speed
type: Item
instance of: scholarly article
title: Proportional navigation and the game of two cars: The case of a pursuer with variable speed
zbMATH Open document ID: 0688.90068
DOI: 10.1016/0898-1221(89)90125-9
author: Y. Yavin
author: R. de Villiers
publication date: 1989-01-01T00:00:00Z
review text: A stochastic version of the ``game of two cars'' is considered. In this version the differential game is based on a noise-perturbed kinematic model. The speed \(V_ E\) of evader E is fixed and his maneuverability is determined by his yaw axis lateral acceleration: \(a_ E=\lambda_ EgV^ 1_ E\phi_ E\), where \(\phi_ E\in [-1,+1]\) is E's control function. Pursuer P has a variable speed \(V_ P\) and his maneuverability is determined by \(V_ P\) and his yaw lateral acceleration \(a_ P=\lambda_ PgV_ P^{-1}\phi_ P\) where \(\phi_ P\in [-1,+1]\) is P's control function. It is assumed that P can maneuver as long as \(v_ 1<V_ P<v_ 2\), has an ``effective operation zone'' \(D_ P\) defined by \(0<r<R_ 0\), \(v_ 1<V_ P<v_ 2\) and \(dr/dt<-v_ 0\), and a ``killing zone'' \(K_ P\) defined by \(0<r<R_ 0\), \(v_ 1<V_ P<v_ 2\) and \(dr/dt<-v_ 0\). On the other hand player E has a ``safe zone'' given by \(r\geq R_ 0\) or \(V_ P\leq v_ 1\) or \(V_ P\geq v_ 2\) or \(dr/dt\geq - v_ 0.\) Player E (resp. player P) looks for minimizing (resp. maximizing) \(P(E_ V)\) by choosing \(\phi_ E\) (resp. \(\phi_ P)\) where \(E_ V\) is the event ``before E enters his safe zone, player E enters \(K_ P\) at some t, \(0\leq t\leq T\) or player E stays at the domain \(D_ P-K_ P\) for all \(t\in [0,t_ 0]\) for some \(t_ 0>T''.\) Sufficient conditions for saddle-point feedback strategies are given which require the existence of a properly smooth solution to a nonlinear parabolic differential equation (PDE) with boundary conditions on a generalized torus in \({\mathbb{R}}^ 4\). It is then shown by numerically solving this PDE that a proportional navigation guidance law is a suboptimal feedback pursuit strategy.
Mathematics Subject Classification ID: 91A24
Mathematics Subject Classification ID: 91A99
Mathematics Subject Classification ID: 91A60
Mathematics Subject Classification ID: 91A23
Mathematics Subject Classification ID: 91A15
zbMATH DE Number: 4128684
zbMATH Keywords: stochastic version
zbMATH Keywords: game of two cars
zbMATH Keywords: noise-perturbed kinematic model
zbMATH Keywords: saddle-point feedback strategies
zbMATH Keywords: nonlinear parabolic differential equation
zbMATH Keywords: proportional navigation guidance law
zbMATH Keywords: suboptimal feedback pursuit strategy
reviewed by: J.-P. Lepeltier
MaRDI profile type: Publication
full work available at URL: 0898-1221(89)90125-9
OpenAlex ID: W1996071232
cites work: Q5532185
cites work: The game of two identical cars
cites work: The geometry of the barrier in the ‘game of two cars’
cites work: Two-target pursuit-evasion differential games in the plane
cites work: Capturability in a two-target 'game of two cars'
cites work: A two-target game analysis in line-of-sight coordinates
cites work: The game of two elliptical ships
cites work: A note on Three-dimensional pursuit-evasion game with bounded curvature
cites work: Optimal strategies in pursuit games with limited maneuverability
cites work: On the game of two cars
cites work: Game of two cars with containment probability as cost function
cites work: Proportional navigation and the game of two cars
cites work: The game of two cars with a containment probability as a cost function: The case of variable speed
cites work: Q3862204
cites work: Pursuit-evasion in medium-range air-combat scenarios
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 4128684