The author considers a control system of the following form \[ \dot{x} = Ax + Bu \] on \([0,T]\) with the boundary conditions \[ x(0) = x_0,\quad x_j(T) = x_{T_j}, \] where \(j\) belongs to a given index set \(J \subset \{1,...,n\}.\) Here \(A\) is a constant \(n \times n\) matrix, \(B\) has the form \(B = diag[E_m,\mathbb{O}_{n-m}],\) \(m < n.\) It is supposed also that the discontinuity surface \(s(x,c) = 0_m\) is given, where the parameters \(c \in \mathbb{R}^l\) are unknown.
Under these conditions, it is assumed that the control functions have the form \[ u_i = - \alpha_i |x| sign (s_i(x,c)), \quad \alpha_i \in [\underline{a}_i,\overline{a}_i],\,\,i = 1,...,m. \] Then on the surfaces \(s_i(c,x) = 0,\) \(i = 1,...,m\) the system takes the form of the differential inclusion \[ \left\{ \begin{array}{l} \dot{x}_i \in A_ix + [- \overline{a}_i,\overline{a}_i]|x|,\quad i = 1,...,m; \\ \dot{x}_i \in A_ix, \quad i = m+1,...,n. \end{array} \right. \] It is required to find such a trajectory which moves along the discontinuity surface (the parameters \(c\) are to be determined as well) and satisfies the above differential inclusion and boundary conditions.
This task is reduced to a variational problem of finding a minimum of certain functionals defined on functional spaces. Some differentiability properties of these functionals are considered and necessary conditions of a minimum are presented.