# Covector mapping principle

Introduced by Ross and his co-workers,[1][2] [3][4][5][6] the covector mapping principle is a fundamental result in computational optimal control. It address the Ross–Fahroo lemma by providing the favorable conditions under which dualization can be commuted with discretization.

## Description

An application of Pontryagin's minimum principle to Problem $B$, a given optimal control problem generates a boundary value problem. According to Ross, this boundary value problem is a Pontryagin lift and is represented as Problem $B^\lambda$.

Illustration of the Covector Mapping Principle (adapted from Ross and Fahroo .[7]

Now suppose one discretizes Problem $B^\lambda$. This generates Problem$B^{\lambda N}$ where $N$ represents the number of discrete pooints. For convergence, it is necessary to prove that as

$N \to \infty, \quad \text{Problem } B^{\lambda N} \to \text{Problem } B^\lambda$

In the 1960s Kalman and others [8] showed that solving Problem $B^{\lambda N}$ is extremely difficult. This difficulty, known as the curse of complexity,[9] is complementary to the curse of dimensionality.

In a series of papers starting in the late 1990s, Ross and Fahroo showed that one could arrive at a solution to Problem $B^{\lambda}$ (and hence Problem $B$) more easily by discretizing first (Problem $B^{N}$) and dualizing afterwards (Problem $B^{N \lambda}$). The sequence of operations must be done carefully to ensure consistency and convergence. The covector mapping principle asserts that a covector mapping theorem can be discovered to map the solutions of Problem $B^{N \lambda}$ to Problem $B^{\lambda N}$ thus completing the circuit.

## Flight implementations

The covector mapping principle was first implemented in flight by NASA to verify the optimality of the pseudospectral solution generated by DIDO. The first flight implementation[10] was on November 5, 2006, when NASA used the Legendre pseudospectral method to maneuver the International Space Station to perform the Zero Propellant Maneuver. The Zero Propellant Maneuver was discovered by Nazereth Bedrossian using DIDO.