On the number of extremal control switchings for a class of bilinear systems with a nonsingular matrix (Q5938981)

For a class of functions defined by a set of equations of the form \[ dx/dt=Ax+\mu u(t)x(t)+ bu(t),\;|u(t) |< 1,\;t\in[0, T],\;T > 0,\;\mu\neq 0,\;x(.), b\in {\mathbb R}^n, \tag{1} \] a number of switchings of extremal controls is estimated. It is supposed that \(A\) is a regular \(n \times n\) matrix with real eigenvalues and linearly independent vectors \(b, Ab,\dots, A^{n-1}b\). The exact estimations of the number of switchings of extremal controls are obtained in two partial cases.