A method to solve nonlinear optimal control problems is proposed in this work. The method implements an approximating sequence of timevarying linear quadratic regulators that converge to the solution of the original, nonlinear problem. Each subproblem is solved by manipulating the state transition matrix of the statecostate dynamics. Hard, soft, and mixed boundary conditions are handled. The presented method is a modified version of an algorithm known as “approximating sequence of Riccati equations.” Sample problems in astrodynamics are treated to show the effectiveness of the method, whose limitations are also discussed.
Optimal control problems are solved with indirect or direct methods. Indirect methods stem from the calculus of variations [
This paper presents an approximate method to solve nonlinear optimal control problems. This is a modification of the method known as “approximating sequence of Riccati equations” (ASRE) [
The main feature of the presented method is that it does not require guessing any initial solution or Lagrange multiplier. In fact, iterations start by evaluating the state and controldependent functions using the initial condition and zero control, respectively. The way the dynamics and objective function are factorized recalls the statedependent Riccati equations (SDRE) method [
The optimal control problem requires that, given a set of
The problem consists in finding a solution that represents a stationary point of the augmented performance index
Let the controlled dynamics (
The initial step consists in solving
At a generic, subsequent iteration, problem
Iterations continue until a certain convergence criterion is satisfied. In the present implementation of the algorithm, the convergence is reached when
With the approach sketched in Section
Suppose that the following dynamics are given:
If both
In a hard constrained problem (HCP), the value of the final state is fully specified,
In a soft constrained problem (SCP), the final state is not specified, and thus
In a mixed constrained problem (MCP), some components of the final state are specified and some are not. Without any loss of generality, let the state be decomposed as
The MCP is solved by partitioning the state transition matrix in a suitable form such that, at final time, (
Two simple problems with nonlinear dynamics are considered to apply the developed algorithm. These correspond to the controlled relative spacecraft motion and to the controlled twobody dynamics for lowthrust transfers.
This problem is taken from the literature where a solution is available, for comparison’s sake [
The equations of motion are
The differential equations (
The two problems have been solved with the developed method. Table
Rendezvous solutions details.
Problem 

Iter  CPU time (s) 

HCP  0.9586  5  0.375 
SCP  0.5660  6  0.426 
Hard constrained rendezvous.
Soft constrained rendezvous.
In this problem, the controlled, planar Keplerian motion of a spacecraft in polar coordinates is studied. The dynamics are written in scaled coordinates, where the length unit corresponds to the radius of the initial orbit, the time unit is such that its period is
The two HCPs have been solved with the developed method. The solutions’ details are reported in Table
EarthMars transfer details.
Problem 

Iter  CPU time (s) 


0.5298  22  5.425 

4.8665  123  41.831 
Orbital transfer with
Transfer trajectory
Control profile
In this paper, an approximated method to solve nonlinear optimal control problems has been presented, with applications to sample cases in astrodynamics. With this method, the nonlinear dynamics and objective function are factorized in a pseudolinear and quadraticlike forms, which are similar to those used in the statedependent Riccati equation approach. Once in this form, a number of timevarying linear quadratic regulator problems are solved. A state transition matrix approach is used to deal with the timevarying linear quadratic regulators. The results show the effectiveness of the method, which can be used to either have suboptimal solutions or to provide initial solutions to more accurate optimizers.