ABSTRACT
The Fuller problem states: given a point a = (a 1, a 2) ∈ ℝ2, find a trajectory ( x ¯ 1 , x ¯ 2 , u ¯ ) ) : [ 0 , T ¯ ] → ℝ × ℝ × [ − 1 , + 1 ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745625/282a7636-99bc-47fe-9900-0c0a6434c0ca/content/eq3390.tif"/> of the system: d x 1 d t ( t ) = x 2 ( t ) , d x 2 d t ( t ) = u ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745625/282a7636-99bc-47fe-9900-0c0a6434c0ca/content/eq3391.tif"/> , starting at x ¯ ( 0 ) = a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745625/282a7636-99bc-47fe-9900-0c0a6434c0ca/content/eq3392.tif"/> and ending at x ¯ ( T ¯ ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745625/282a7636-99bc-47fe-9900-0c0a6434c0ca/content/eq3393.tif"/> , minimizing the cost 1 2 ∫ 0 T x 1 ( t ) 2 d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745625/282a7636-99bc-47fe-9900-0c0a6434c0ca/content/eq3394.tif"/> among all trajectories of the system starting at a and ending at 0.
