Stability and controllability in proportional navigation (Q1180886)

Introducing Riemann surfaces into considering the relative motion of the center of mass of the pursuing body described in the polar coordinates \((a,\eta)\); \(\dot a=v_ s F(\eta)\), \(a\dot\eta=-v_ s f(\eta)\), \(F(\eta)=\cos\eta-p \cos b(\eta)\), \(f(\eta)=\sin\eta+p \sin b(\eta)\), \(b(\eta)=(b-1)(\eta-\varepsilon_ 0)\), \(p=v/v-s\), enables the authors to carry out the analysis of stability and controllability of the motion and to extend all the properties of the motion with integral \(b\) [see the authors, Theory of proportional navigation (in Russian). Leningrad: Izdat. Sudostroenie (1965; Zbl 0131.38502)] to the case of the arbitrary navigation constant \(b\neq 1\). Three aspects of stability: stable roots of \(f(\eta)\), stable trajectories and influence of small variations in the initial data on the final result of the motion are discussed together with the analysis of dependence of the motion on the parameters \(b\), \(p\), \(\varepsilon_ 0\). An important role in the latter plays the curve in the \((p,\varepsilon_ 0)\)-plane along which \(f(\eta)=0\), \(F(\eta)=0\), \(\Omega(\eta)=pb\{p- \cos[b\eta-b(\eta)]\}=0\) have multiple roots. Assertions on controllability (see the cited book) of stable roots and its dependence on the parameters are established on base of analysis of \(\Phi(\eta)=f'(\eta)+F(\eta)\) and the absolute trajectory of the motion. In the cases of \(b<1\), \(1<b<2\), \(b=2\), \(b>2\), general conclusions on the nature of the motion are presented. The dependence of stability and controllability on \(p\) is considered in the example with \(b=2/3\) and \(\varepsilon_ 0=0\).