On the optimal control of the $n$-fold integrator

The optimal control problem $n$-fold integrator with arbitrary boundary conditions and functionals of type norms in spaces of $L_q[t_0,t_f]$, $q=1, 2, \infty$ is considered. First, it is the problem of minimizing the total controling impulse, which boils down to $L_\infty$-problem of moments; secondly, the problem of minimizing the maximum values of the control parameter (represented as $L_1$-problem of moments), and, finally, it is the problem of minimizing "generalized work control" (as $L_2$-problem of moments). Solving problems is obtained by using the method of moments in the form of the maximum principle by N. N. Krasovsky. It is shown that optimal control in the first problem is approximated by a $\delta$-impulsive control. Conditions for the existence of regular and singular solutions to this problem depending on the boundary conditions are also specified. The general solution of the second problem, which is the conditions for existence of regular and singular solutions and not equivalence with the mutual problem of time-optimal control is obtained. Examples of solution for the considered control tasks are given. In case of a quadratic functional general relations required for constructing a program optimal control were obtained.