Optimal synthesis in a two-dimensional problem with asymmetric constraints on the control

Abstract

A two-dimensional optimal control problem is considered on the assumption that the terminal time of the process is not fixed and the integral objective functional depends on a parameter. Asymmetric constraints are imposed on the control parameter. Two cases are considered: constraints of the same sign and constraints of different signs. In the case of constraints of different signs, if the parameters of the problem satisfy certain relations, one obtains chattering control, alternating with a control with two switchings and a first-order singular are when these relations are violated. In the case of sign-definite control the controllability domain is part of the plane bounded by two semiparabolas. Three types of control law are then possible, in two of which the system will hit the boundary of the controllability domain and move along it, while the third features a first-order singular are. As the parameter of the problem is varied, the phase portrait undergoes evolution and one of these three types is interchanged with another. The optimality of these control laws is rigorously established using a dynamic programming method.