Estimates for the number of extrema of the extremal control integral in the time-optimal control problem for a class of bilinear systems (Q1281213)

A time-optimal control problem of the quick-acting to the origin of coordinates for a class of bilinear systems of the form \[ \dot x (t)= u(t)A x(t) + b u( t) + c \] where \(x\) is a trajectory, \(u\) is a scalar control function, \(A\) is a real \(n \times n\)-matrix and \(x(\cdot), b,c \in \mathbb{R}^n; c\neq 0 \) is considered. It is supposed that the scalar control function \(u\) is bounded in absolute value. A solution is obtained by the Pontryagin-maximum principle. It is shown, that each extreme control function is a piecewise continuous function with values \(-1\) and \(+1\) and with a finite number of switchings. Various cases of representation of the matrix \(A\) are considered.