The game of two cars with a containment probability as a cost function: The case of variable speed
type: Item
instance of: scholarly article
title: The game of two cars with a containment probability as a cost function: The case of variable speed
zbMATH Open document ID: 0688.90069
DOI: 10.1016/0898-1221(89)90124-7
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. The differential game is based on a noise-perturbed kinematic model. The speed \(V_ E\) of evader E is fixed, his maneuverability is determined by his yaw axis lateral acceleration \(a_ E=\lambda_ EgV_ E^{- 1}\phi_ E\) 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. P can maneuver as long as \(v_ 1<V_ P<v_ 2\) and has an effective operation zone defined by \(0<r<r_ 0\), \(v_ 1<V_ P<v_ 2\) and \(dr/dt\leq v-v_ 0.\) Denote by \(\lambda\) (t) the Lebesgue measure on the real line of the set \(\{\) \(s|\) \(0\leq s\leq t\) such that \(0<r(s)<r_ 0\), \(v_ 1<V_ P(s)<v_ 2\), \(dr(s)/ds\leq -v_ 0\}\) where r(t) is the range from P to E at the time t. We say that player P intercepts player E if for some \(t>0\), \(T_ 1\leq \lambda (t)\leq T_ 2\) and for all \(s\in [0,t]\) \(0<r(s)<v_ 0\) and \(v_ 1<V_ p(s)<v_ 2\) \((T_ 1,T_ 2\) given positive numbers). The cost functional is then: \(\Pr ob(\{T_ 1\leq \lambda (t)\leq T_ 2|\) for all \(s\in [0,t]:\) \(0<r(s)<r_ 0\) and \(v_ 1<V_ P(s)<v_ 2\})\). Then by numerically solving a nonlinear boundary problem on a generalized torus in \({\mathbb{R}}^ 5\), for \(r_ 0\leq 2000\) one can find values for \(v_ 0\), \(T_ 1\) and \(T_ 2\) such 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
zbMATH DE Number: 4128685
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: saddle-points
zbMATH Keywords: stochastic version
zbMATH Keywords: game of two cars
zbMATH Keywords: noise-perturbed kinematic model
zbMATH Keywords: nonlinear boundary problem
zbMATH Keywords: generalized torus
zbMATH Keywords: suboptimal feedback pursuit strategy
reviewed by: J.-P. Lepeltier
MaRDI profile type: Publication
full work available at URL: 0898-1221(89)90124-7
OpenAlex ID: W2041650686
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published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 4128685