The explicit form of the switching surface in admissible synthesis problem

In this article we consider the problem related to positional synthesis and controllability function method more precisely admissible maximum principle. Unlike common approach principle gives discontinuous solutions problem. Let us canonical system of linear equations $\dot{x}_i=x_{i+1}, i=\overline{1,n-1}, \dot{x}_n=u$ with constraints $|u| \le d$. The for an arbitrary $\dot{x} = A x + b u$ can be simplified system. $\Theta(x)$ is given as a unique positive solution some equation $\Phi(x,\Theta) 0$. control chosen minimize derivative written $u(x) -d \text{ sign}(s(x,\Theta(x)))$. set points $s(x,\Theta(x)) 0$ called switching surface, it determines where changes its sign. Normally \mbox{contains} variable $\Theta$ which implicitly $\Phi(x, \Theta) Our aim in paper find representation surface that does not depend on $\Theta(x)$. We call explicit form. our case expressions \Theta)$ $s(x, are both polynomials respect $\Theta$, so finding conditions when two have root. Earlier 2-dimensional was known. But during exploration found out systems higher dimensions there exist certain difficulties. three dimensional presented researched. It shown sliding (according Filippov's definition). Also other ways constructing using interpolation approximation proposed used trajectories concrete points.