# hw3 - 18 1 Dierentiability and the calculus of variations y...

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7. Consider the calculus of variations problem Φ ( z ) = t f t 0 φ ( z ( t ) , ˙ z ( t ) , t ) dt where t 0 , t f are fixed, z ( t 0 ) and z ( t f ) are free except for the following mixed end-point constraint z ( t 0 ) + 2 z ( t f ) = 1 . Obtain the correct transversality conditions on any stationary curve ¯ z for this objective.

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2.3 Exercises 29 1. In this problem you will derive the Algebraic Ricatti Equation (ARE) of the LQR problem using constrained finite dimensional optimization. a) Given a stable LTI system without input ˙ x ( t ) = Ax ( t ) , x (0) = x o , it is desired to evaluate the performance index J := 0 x * ( t ) Mx ( t ) dt. Show that J is given by J = x * o Px o , where P solves A * P + PA = - M. b) Now given an LTI system with input ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x o , it is desired to find the optimal static state feedback law, i.e. u ( t ) = Kx ( t ) (where K is a constant matrix), such that the following per- formance index is minimized J := 0 x * ( t ) Qx ( t ) + u * ( t ) Ru ( t ) dt, x (0) = x o . Using the answer to the first part above, recast this problem as a
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• Winter '09
• Bamieh
• Probability distribution, probability density function, Cumulative distribution function, performance index

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