The quadratic phase approximation method for solving optimal control problems (Q1902806)

We construct and justify a new method of second-order, which uses quadratic approximation with respect to phase increment and the procedure of needle variation of control. The main operations of the method of each step of improvement are the following ones: integration of a vector-matrix conjugate system; maximum of the Pontryagin function with respect to the control for a perturbed conjugate vector-function and a free phase variable; variation of the control, as function of time and state on level sets of the switching function; integration of the phase system with varying control; parametric search in order to decrease the functional (as a parameter we use the measure of the set variation with respect to time, or a constant which characterizes the level set of the switching function). The proposed approximation is exact for linear-quadratic (with respect to the state) problems.