Consider the system of two linear differential equations: y_ 1t y2t y_ 2t 2y1t

Consider the system of two linear differential equations: y_ 1ðtÞ ¼ y2ðtÞ y_ 2ðtÞ ¼ 2y1ðtÞ  y2ðtÞ þ uðtÞ which needs to be controlled to minimize: JðuÞ ¼ 1 2 ZT 0  y2 1ðtÞ þ 1 2 y2 2ðtÞ þ 1 4 u2 ðtÞ  dt:

Setting up the Hamiltonian Maximum Principle conditions, it turns out to have the following matrices: A ¼  0 1 2 1  ; B ¼  0 1  ; Q ¼  2 0 0 1  with R ¼ 1 2 : Solve the system of differential equations and at the same time minimize the functional, given the following initial conditions on the two state variables: yð0Þ ¼ ½4 4  and y(T) free. Set T ¼ 4.