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 coauthors,^{[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 ${\displaystyle B}$, 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 ${\displaystyle B^{\lambda }}$.

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

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

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

In the 1960s Kalman and others^[8] showed that solving Problem ${\displaystyle B^{\lambda N}}$ 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 ${\displaystyle B^{\lambda }}$ (and hence Problem ${\displaystyle B}$) more easily by discretizing first (Problem ${\displaystyle B^{N}}$) and dualizing afterwards (Problem ${\displaystyle B^{N\lambda }}$). 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 ${\displaystyle B^{N\lambda }}$ to Problem ${\displaystyle B^{\lambda N}}$ thus completing the circuit.

See also

• Legendre pseudospectral method
• Ross–Fahroo pseudospectral methods
• Ross–Fahroo lemma