Shooting Newton-Raphson method

Some times the stationarity condition = 0 of an optimal control problem can be solved to obtain an explicit expression for u in terms of the state y and the costate A. That expression, when substituted into state and costate equations, couples them. Thus, state equations become dependent on A and must be integrated simultaneously with the costate equations. The simultaneous integration constitutes a two point boundary value problem in which 

the state is specified at the first boundary, i.e., at the initial time  

the costate is specified at the second boundary, i. e., at the final time  

the state and costate equations depend on state as well as costate variables. 

The shooting Newton-Raphson method enables the solution of this problem. With a guessed initial costate, both state and costate equations are integrated forward or shot to the final time. The discrepancy between the final costate obtained in this way and that specified is improved using the Newton-Raphson 

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We explain the shooting Newton Raphson method with the help of an optimal control problem having one state, one control, and fixed final time. The objective of the problem is to find the control function u t) that minimizes the functional... 

Based on the above development, the computational algorithm for the shooting Newton-Raphson method is as follows. 

The optimal control solution is essentially the solution of the above differential equations with boimdary conditions at opposite end points of the time interval. In order to solve the equations using the shooting Newton-Raphson method, we need the derivative state equations to provide Aa, . ... 

The shooting Newton Raphson method will use the derivative A o at t[ to improve the guess A(0) = Aq, thereby zeroing out A(tf). We differentiate with respect to Aq the latest state and costate equations as well as the initial boundary conditions... 

A periodicity condition implies that the initial and final values of a state (or costate) variable are equal to a single value. Thus, in a optimal periodic control problem, the set of state as well as costate equations poses a two point boundary value problem. Either successive substitution or the shooting Newton-Raphson method may be used to integrate the periodic state and costate equations. 

In order to integrate the state and costate equations satisfying the periodicity conditions, we need the respective derivative differential equations for the shooting Newton-Raphson method. 

Following is the computational algorithm of the shooting Newton-Raphson method to solve the optimal periodic control problem. 

Eigenfunction expansions as used in Ref. 168 are not accurate near the critical point. Instead, we developed a shooting point method in order to make a direct numerical integration of Eq. (110) with the condition Eq. (112). Real energies (bound and virtual) were found by bisection methods, and for complex energies it was necessary to combine the Newton-Raphson and grid methods. 

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