# Covector mapping principle

The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by Ross and co-workers,[1][2] [3][4][5][6] It provides conditions under which dualization can be commuted with discretization in the case of computational optimal control.

## 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.