Nonlocal improvement of controls in nonlinear discrete systems

A nonlinear optimal control problem for discrete system with both control function and control parameters (parameters are at the system's right side and at the initial condition) is considered. For the given optimization problem, the problem of control's improvement is studied. It's developed a known approach for non-local improvement of control based on construction of the exact (without residual terms w.r.t. state and control variables) formula for the cost functional's increment under some special conjugate system.
For the given optimization problem, it's considered the generalized Lagrangian following to the theory by V. F. Krotov. The function $\varphi(t,x)$ which plays an important role in the generalized Lagrangian is considered in this article in the linear w.r.t. $x$ form $\varphi(t,x) = \langle p(t), x \rangle$ where the function $p(t)$ is the solution of the mentioned conjugate system. Thus, first of all, the exact formula of the cost functional's increment is considered under the assumption on the solution $p(t)$ existence; and, secondly, the linear function $\varphi(t,x)$ is used here in connection with creation of the mentioned increment formula, and not for linear approximation of the generalized Lagrangian's increment. The corresponding condition of control's improvement is formulated in terms of the boundary value problem composed due to binding of the system given in the optimization problem together with the conjugate system. The obtained increment condition is similar to the increment conditions which were suggested before in the papers of the author for discrete problems without control parameters.
There is an example of control's improvement in some problem where the control to be improved gives the maximum of the Pontryagin's function for all values of $t$. The boundary value improvement problem is solved with help of the shooting method, and the calculations are made analytically.