Consider the following optimal control problem x = -0.5x+u, 0 u 1 x(0) = 1 The...

 

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Consider the following optimal control problem x = -0.5x+u, 0 u 1 x(0) = 1 The final time tf is free, and the condition on the state is x(t) = 0 The performance index is 1 = u(t)dt (a) From the Minimum Principle, write down the necessary conditions (b) Show that the optimal control u* is either 0 or 1 for all t = [0,t,], or possibly a piecewise constant. Further argue that if u* is a piecewise constant, it can switch between 0 and 1 at most once. (c) It turns out that u* does not switch at all (remains a constant throughout [0, t,]). Solve this problem to find u", x*(t), and tf. Justify your choice of u*. Hint for part (c): For constant a (a + 0) and b, the solution to the 1st-order ODE is x = ax + b,x(0) = x0 b x(t) = (xo b/a)eat -- a Consider the following optimal control problem x = -0.5x+u, 0 u 1 x(0) = 1 The final time tf is free, and the condition on the state is x(t) = 0 The performance index is 1 = u(t)dt (a) From the Minimum Principle, write down the necessary conditions (b) Show that the optimal control u* is either 0 or 1 for all t = [0,t,], or possibly a piecewise constant. Further argue that if u* is a piecewise constant, it can switch between 0 and 1 at most once. (c) It turns out that u* does not switch at all (remains a constant throughout [0, t,]). Solve this problem to find u", x*(t), and tf. Justify your choice of u*. Hint for part (c): For constant a (a + 0) and b, the solution to the 1st-order ODE is x = ax + b,x(0) = x0 b x(t) = (xo b/a)eat -- a