Time-optimal controlonasubspaceforthetwoand three-dimensional system

This article is devoted to the problem of the optimal synthesis on a subspace for the 2 and 3-dimensional case for the linear control system $\dot{x_1} = u, \dot{x}_i=x_{i-1}, i=\overline{2,n}$ with $|u|\le1$. This problem is related to the problem of optimal synthesis into the point, which solution was presented by V. I. Korobov and G. M. Sklyar and is based on the moment min-problem, but its difference with respect to the original problem is that the number of unknown functions is greater than the number of variables, which requires using the methods for the parametric optimization. As in the problem of optimal synthesis into the point, we search for the optimal solution in the form of piecewise function with $u=\pm 1$ and $n-1$ switching points, which is optimal according to Pontryagyn's maximum principle and the theorem on $n$ intervals. In this article we consider approaches and problems related to the finding of the general solution of the optimization problem and solve the cases of $n=2$ and $n=3$. The interest of the obtain results is the fact that unlike the solution for the single endpoint $x_T$ the general solution for the subspace may have less than $n-1$ switching points, or not have switching points at all, depending on the parameters of the subspace. In our work, we give the solution for the problem of synthesis of the two-dimensional system onto a subspace $G: {x_2 = k x_1}$ for all values of $k$ using the moment min-problem and the optimization methods. For the three-dimensional system we consider the problem of synthesis on a plane $x_3 = k_1 x_1 + k_2 x_2$ and obtain the number of the switching points depending on the values of $k_1$ and $k_2$, construct trajectories and present the equations the for optimal time $\Theta$ for different cases.