Consider the following optimal control problem x = u, x(0) = 1 1 - = 45...

 

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Consider the following optimal control problem x = u, x(0) = 1 1 - = 45 (u - qx) dt where q is a constant. (a) Explain whether or not this is a linear quadratic regulator problem if q > 0. (b) Let q = 0 in this part. Pretend that you have not taken this course and learned the Minimum Principle. Determine the solution (u*(t) and x*(t)) to the problem by using just logic and observation. (c) Solve the problem with a q = 8 by finding the optimal pair {x*(t), u* (t)} Hint: the solution of differential equation * + kx = 0 with k > 0 is x(t) = c sin(at) + C cos(at) where = k Consider the following optimal control problem x = u, x(0) = 1 1 - = 45 (u - qx) dt where q is a constant. (a) Explain whether or not this is a linear quadratic regulator problem if q > 0. (b) Let q = 0 in this part. Pretend that you have not taken this course and learned the Minimum Principle. Determine the solution (u*(t) and x*(t)) to the problem by using just logic and observation. (c) Solve the problem with a q = 8 by finding the optimal pair {x*(t), u* (t)} Hint: the solution of differential equation * + kx = 0 with k > 0 is x(t) = c sin(at) + C cos(at) where = k