Approximate Solutions to Nonlinear Optimal Control Problems in Astrodynamics

A method to solve nonlinear optimal control problems is proposed in this work. The method implements an approximating sequence of time-varying 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 state-costate 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. 1. Introduction Optimal control problems are solved with indirect or direct methods. Indirect methods stem from the calculus of variations [1, 2]; direct methods use a nonlinear programming optimization [3, 4]. Both methods require the solution of a complex set of equations (Euler-Lagrange differential equations or Karush-Kuhn-Tucker algebraic equations) for which iterative numerical methods are used. These iterative procedures implement some form of Newton’s method to find the zeros of a nonlinear function. They are initiated by providing an initial guess solution. Guessing an appropriate initial solution is not trivial and requires a deep knowledge of the problem at hand. In indirect methods, the initial value of the Lagrange multiplier has to be provided, whose lack of physical meaning makes it difficult to formulate a good guess. In direct methods, the initial trajectory and control have to be guessed at discrete points over the whole time interval. 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) [5, 6]. It transforms the nonlinear dynamics and objective function into a pseudolinear and quadratic-like structure, respectively, by using state- and control-dependent functions. At each iteration, these functions are evaluated by using the solutions at the previous iteration, and therefore, a series of time-varying linear quadratic regulators is treated. This sequence is solved with a state transition matrix approach, where three different final conditions are handled: final state fully specified, final state not specified, and final state not completely specified. These define hard, soft, and mixed constrained problems, respectively. 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