Analytical method for solution of the game problem of soft landing for moving objects (Q5951342)

The dynamics of the pursuer and the evader is described by the differential equations \[ \ddot x=-\alpha\dot x+\rho u, \quad x(0)=x^0, \;\dot x(0)=\dot x^0, \] \[ \ddot y=-\beta\dot y+\sigma v, \quad y(0)=y^0, \;\dot y(0)=\dot y^0, \] where \(x,y,u,v\in \mathbb{R}^3\); \(\|u\|,\|v\|\leq 1\); \(\alpha,\rho,\beta,\sigma>0\). The phase constraints are \(x_3(t)\geq 0\), \(y_3(t)\equiv 0\) \((t\geq 0)\). A soft landing is treated as the fulfilment (at some instant \(T\geq 0\)) of the equalities \(x(T)=y(T)\), \(\dot x(T)=\dot y(T)\). The authors construct a measurable control of the pursuer \(u(.)\) which satisfies all given conditions and guarantees the soft landing for any control of the evador \(v(.)\), satisfying given requirements. The values of the system parameters and of the initial states of the pursuer are described for which the solution of this soft-landing game problem is possible for any admissible counteraction of the evader and for any of its initial states. Also, the two-stages pursuer's control is constructed in explicit form. At the first stage the pursuer and the evader velocities are equalized, and at the second stage the corresponding minimal time problem is solved.