Proportional navigation and the game of two cars: The case of a pursuer with variable speed
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.
nonlinear parabolic differential equation
scientific article; zbMATH DE number 4128684
scientific article