On the pursuit process in differential games (Q1076634)

The paper deals with a pursuit evasion game where the equations of motion are described by a linear ordinary differential equation of finite dimension: \(\dot z=Az-Bu+Cv\), \(z(0)=z_ 0\). The control u must satisfy the integral constraint \[ \int^{\infty}_{0}\| u(t)\|^ 2dt\geq p^ 2, \] with \(p>0\) and the control v satisfies v(t)\(\in Q\) where Q is a compact and convex subset of \({\mathbb{R}}^ q\) (v has q components). The u-player tries to steer the solution z(t) such that \(z(t_ 1)\in M\), where M is a given target set and \(t_ 1\) the given final time. A novelty of the paper is the way u(t) is allowed to depend on \(v: u(t)=u(v(s):\) \(s\in \Omega (t))\), where \(\Omega\) (t) is a time interval with \(\max_{s}(s|\) \(s\in \Omega (t))\leq t\). The time interval may be empty. If \(\Omega\) (t) would be empty for all t, then u(t) must be a constant function according to its definition. Whether the author would prefer to allow an explicit time-dependence in this case is not clear. As an example later on in the paper the dependence \(u(t)=u(v(t))\) was considered. The main theorem of the paper gives a sufficiency condition such that the u-player can indeed terminate the game on M. The sufficiency condition consists of four rather technical hypotheses. In section 3 the main theorem is specialized to more explicit cases, i.e. more specific conditions are given in a number of cases such that the differential game is completed after time \(t_ 1\). Two applications are given. One with the information structure mentioned above and with differential equations of the form \(\dot x+x=u\), \(\dot y+y=v\), \(x\in {\mathbb{R}}^ n\), \(y\in {\mathbb{R}}^ n\), \(\alpha,\beta >0\). The target set is defied as \(\| x(t_ 1-y(t_ 1)\| \leq \epsilon\) where \(\epsilon\) is a given positive number. The other application deals with the differential equations of the form ẍ\(+a_ 1\dot x+a_ 2x=u\), ÿ\(+b_ 1y_ 1+b_ 2y=v\), the information structure is \(u(t)=u(v(t))\) and the target set is \(x(t_ 1)=y(t_ 1)\). In both cases conditions on the parameters are given such that the game terminates.