Some nonlinear boundary value problems describing conflict control with continuous feedback and nonconvex target set Original Research Article

Nondegenerate differential game examples on plane with nonconvex target set are considered. These examples show that the evasion problem with continuous strategies in the phase vector can be either solvable or not solvable. Solvability depends on the properties of the conflict-control system. Here, we call a differential game nondegenerate if the evasion problem is solvable with the help of discontinuous feedback control, while the evasion problem is not solvable with the help of program controls. Unsolvability is established in the class of strategies (with deviation of argument) which satisfy the Carathéodory conditions. The unsolvability is shown with the help of rather simple mathematical techniques on the basis of the Schauder fixed point theorem. The unsolvability is reduced to the solvability of a boundary value problem with deviating argument.