On Diferential Inclusions Arising from Some Discontinuous Systems (Q6593312)

The author considers a control system of the following form\N\[\N\dot{x} = Ax + Bu\N\]\Non \([0,T]\) with the boundary conditions\N\[\Nx(0) = x_0,\quad x_j(T) = x_{T_j},\N\]\Nwhere \(j\) belongs to a given index set \(J \subset \{1,\dots,n\}\). Here, \(A\) is a constant \(n \times n\) matrix, \(B\) has the form \(B = \mathrm{diag}[E_m,\mathbb{O}_{n-m}],\) \(m < n\). 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.\N\NUnder these conditions, it is assumed that the control functions have the form\N\[\Nu_i = - \alpha_i |x| \mathrm{sign} (s_i(x,c)), \quad \alpha_i \in [\underline{a}_i,\overline{a}_i],\,\,i = 1,\dots,m.\N\]\NThen on the surfaces \(s_i(c,x) = 0,\) \(i = 1,\dots,m\), the system takes the form of the differential inclusion \N\[\N\begin{cases} \dot{x}_i \in A_ix + [- \overline{a}_i,\overline{a}_i]|x|,\quad i = 1,\dots,m; \\\N\dot{x}_i \in A_ix, \quad i = m+1,\dots,n. \end{cases} \N\]\NIt 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.\N\NThis 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.