Tate curve

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 by Ross and co-workers, 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

$B^{\lambda }$

.

File:CMP-OptimalControl.png

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

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

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

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

See also

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

References

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