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.

An application of Pontryagin's minimum principle to Problem ${\displaystyle B}${\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 ${\displaystyle B^{\lambda }}${\displaystyle B^{\lambda }}.

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

${\displaystyle N\to \infty ,\quad {\text{Problem }}B^{\lambda N}\to {\text{Problem }}B^{\lambda }}${\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 ${\displaystyle B^{\lambda N}}${\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 ${\displaystyle B^{\lambda }}${\displaystyle B^{\lambda }} (and hence Problem ${\displaystyle B}${\displaystyle B}) more easily by discretizing first (Problem ${\displaystyle B^{N}}${\displaystyle B^{N}}) and dualizing afterwards (Problem ${\displaystyle B^{N\lambda }}${\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 ${\displaystyle B^{N\lambda }}${\displaystyle B^{N\lambda }} to Problem ${\displaystyle B^{\lambda N}}${\displaystyle B^{\lambda N}} thus completing the circuit.

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

• Optimal control

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