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, 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
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
λ
.
Now suppose one discretizes Problem
B
λ
. This generates Problem
B
λ
N
where
N
represents the number of discrete pooints. For convergence, it is necessary to prove that as
N
→
∞
,
Problem
B
λ
N
→
Problem
B
λ
In the 1960s Kalman and others showed that solving Problem
B
λ
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
λ
(and hence Problem
B
) more easily by discretizing first (Problem
B
N
) and dualizing afterwards (Problem
B
N
λ
). 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
λ
to Problem
B
λ
N
thus completing the circuit.
