Gaussian Quadrature Weights and LGL Pseudospectral Methods

This section develops Gaussian quadrature weights and introduces the Legendre–Gauss–Lobatto (LGL) pseudospectral transcription for optimal control. It concludes by explaining why the differentiation matrix of the LGL method is singular and therefore does not constitute a true quadrature (integration) method.

Gaussian Quadrature Weights¶

For $N$ Legendre–Gauss (LG) points,

w_i=\frac{2}{(1-\tau_i^2)\left(P_N'(\tau_i)\right)^2}, \qquad i=1,\ldots,N.

(1)

For Legendre–Gauss–Radau (LGR),

w_1=\frac{2}{N^2},

(2)

and

w_i=\frac{1}{(1-\tau_i)\left(P'_{N-1}(\tau_i)\right)^2}, \qquad i=2,\ldots,N.

(3)

For Legendre–Gauss–Lobatto (LGL),

w_1=w_N=\frac{2}{N(N-1)},

(4)

and

w_i=\frac{2}{N(N-1)\left(P_{N-1}(\tau_i)\right)^2}, \qquad i=2,\ldots,N-1.

(5)

Model Optimal Control Problem¶

Consider

\min \Phi=x(1)

(6)

subject to

\dot x=f(x,u,t),

(7)

with

x(-1)=x_0.

(8)

The interval has already been normalized to

\tau\in[-1,1].

(9)

LGL State Approximation¶

Using all LGL nodes,

x(\tau)\approx X(\tau)= \sum_{i=1}^{N}X_iL_i(\tau),

(10)

where $L_i(\tau)$ are Lagrange basis polynomials.

Because

L_i(\tau_j)=\delta_{ij},

(11)

the coefficients satisfy

X(\tau_j)=X_j.

(12)

Differentiation Matrix¶

Differentiating,

\dot X(\tau) = \sum_{i=1}^{N} X_iL_i'(\tau).

(13)

Evaluating at each collocation point,

\sum_{i=1}^{N} D_{ji}X_i = f(X_j,U_j),

(14)

where

D_{ji}=L_i'(\tau_j).

(15)

Stacking every node gives

DX=F.

(16)

Boundary Conditions¶

The discretized initial condition becomes

X_1=x_0,

(17)

and the Mayer objective becomes

\Phi=X_N.

(18)

The resulting NLP is

\min X_N

(19)

subject to

DX-F=0,

(20)

and

X_1=x_0.

(21)

Integral Form¶

A true integration method would satisfy

X_j = X_1 + \sum_{i=1}^{N} A_{ji}F_i,

(22)

which discretizes

x(\tau_j) = x(-1) + \int_{-1}^{\tau_j}f(\tau)\,d\tau.

(23)

The matrix $A$ is obtained from Gaussian quadrature weights.

Why LGL Fails as a True Quadrature Method¶

The interpolation polynomial has degree

N-1.

(24)

Its derivative therefore has degree

N-2.

(25)

However, collocation is enforced at all $N$ LGL points.

Thus the differentiation system contains one more equation than is required to uniquely determine the derivative polynomial.

Consequently,

D

(26)

is singular.

Therefore,

D^{-1}

(27)

does not exist.

The differential system

DX=F

(28)

cannot be transformed into

X=X_1\mathbf 1+AF.

(29)

Hence the LGL differentiation equations are not equivalent to a Gaussian quadrature rule.

Interpretation¶

Differentiation destroys constants since

\frac{d}{dt}(c)=0.

(30)

Therefore constant functions lie in the null space of the differentiation operator.

The singularity of the differentiation matrix is the finite-dimensional manifestation of this null space.

Comparison with Euler¶

Euler forward,

\frac{x_{k+1}-x_k}{h}=f_k,

(31)

may be rewritten as

x_{k+1}=x_k+h f_k,

(32)

which is an explicit integration rule.

Therefore Euler possesses an equivalent integral form.

The LGL differentiation equations do not.

Consequences¶

The lecture emphasizes that although LGL nodes appear attractive because they include both endpoints, their differentiation matrix is singular. This motivates the development of Legendre–Gauss and Legendre–Gauss–Radau pseudospectral methods, which preserve Gaussian quadrature while still enforcing boundary conditions correctly.

Summary¶

Closed-form Gaussian quadrature weights were introduced.
The LGL pseudospectral transcription was derived.
The differentiation matrix was constructed.
The LGL differentiation matrix is singular.
Constants form the null space of differentiation.
Unlike Euler integration, the LGL differentiation equations cannot be converted into an equivalent quadrature rule.
This motivates LG and LGR pseudospectral methods.

Connection. Removing both endpoints from the collocation grid produces the LG method and reveals the relationship between rectangular differentiation and exact integration matrices.