hw3 - EE363 Prof S Boyd EE363 homework 3 1 Solution of a...

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EE363 Prof. S. Boyd EE363 homework 3 1. Solution of a two-point boundary value problem. We consider a linear dynamical system ˙ x = Ax , with x ( t ) R n . There is an n -dimensional subspace of solutions of this equation, so to single out one of the trajectories we can impose, roughly speaking, n equations. In the most common situation, we specify x (0) = x 0 , in which case the unique solution is x ( t ) = e tA x 0 . This is called an initial value problem since we specify the initial value of the state. In a Fnal value problem , we specify the Fnal state: x ( T ) = x f . In this case the unique solution is x ( t ) = e ( t - T ) A x f . In a two-point boundary value problem we impose conditions on the initial and Fnal states. (a) ±ind the solution to the two-point boundary value problem ˙ x = Ax, Fx (0) + Gx ( T ) = h. Here F, G R n × n , and h R n . Express your answer in terms of A , F , G , and h . Your answer can contain a matrix exponential. What condition must hold to ensure that there is a unique solution to this equa- tion? (b) Express the two-point boundary value problem that arises in the continuous time LQR problem ( i.e. , with the Hamiltonian system) in the form given above, and then Fnd the solution to this boundary value problem. (You may leave matrix
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hw3 - EE363 Prof S Boyd EE363 homework 3 1 Solution of a...

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