Linear-quadratic approximation of the main optimal control problem in a discrete variant

The problem of optimization of a nonlinear dynamical system on a set of discrete controls of a piecewise constant structure is considered. The linear-quadratic approximation of the functional is implemented within the framework of classical variations based on the Pontryagin function and the Gabasov matrix function. The formalization of the transformation procedure has been carried out. Explicit expressions for variations with respect to control parameters have been obtained. As a result, it is possible to use gradient procedures and Newton method for the numerical solution of the problem. Non-standard optimality condition based on the second variation of the functional is obtained. It combines elements of classical results for special controls. On the basis of the second variation of the functional, a non-standard optimality condition is obtained, which combines elements of classical results for singular controls.