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OptControl_SL

# OptControl_SL - Optimal Control Prof Lutz Hendricks August...

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Optimal Control Prof. Lutz Hendricks August 6, 2009 L. Hendricks () Optimal Control August 6, 2009 1 / 39

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Topics Optimal control is a method for solving dynamic optimization problems in continuous time. L. Hendricks () Optimal Control August 6, 2009 2 / 39
Generic Optimal control problem Choose functions of time c ( t ) and k ( t ) so as to max Z T 0 v [ k ( t ) , c ( t ) , t ] dt (1) Constraints: 1 Law of motion of the state variable k ( t ) : ˙ k ( t ) = g [ k ( t ) , c ( t ) , t ] (2) 2 Feasible set for control variable c ( t ) : c ( t ) 2 Y ( t ) (3) 3 Boundary conditions: k ( 0 ) = k 0 ,given (4) k ( T ) ° k T (5) L. Hendricks () Optimal Control August 6, 2009 3 / 39

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Generic Optimal control problem c and k can be vectors. Y ( t ) is a compact, nonempty set. T could be in°nite. Then lim t ! k ( t ) ° k T . Important: the state cannot jump; the control can. L. Hendricks () Optimal Control August 6, 2009 4 / 39
Example A household chooses optimal consumption to max Z T 0 u [ c ( t )] dt (6) subject to ˙ k ( t ) = rk ( t ) ± c ( t ) (7) c ( t ) 2 [ 0 , ¯ c ] (8) k ( 0 ) = k 0 ,given (9) k ( T ) ° 0 (10) L. Hendricks () Optimal Control August 6, 2009 5 / 39

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A Recipe for Solving Optimal Control Problems L. Hendricks () Optimal Control August 6, 2009 6 / 39
A Recipe 1. Write down the Hamiltonian H ( t ) = v ( k , c , t ) + μ ( t ) g ( k , c , t ) (11) μ is essentially a Lagrange multiplier (called a ± co-state variable). L. Hendricks () Optimal Control August 6, 2009 7 / 39

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A Recipe 2. Derive the °rst order conditions which are necessary for an optimum: H / c = 0 (12) H / k = ± ˙ μ (13) L. Hendricks () Optimal Control August 6, 2009 8 / 39
A Recipe 3. Impose the transversality condition: for °nite horizon: μ ( T ) = 0 (14) for in°nite horizon: lim t ! H ( t ) = 0 (15) this depends on the terminal condition (see below). L. Hendricks () Optimal Control August 6, 2009 9 / 39

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A Recipe 4. A solution is the a set of functions [ c ( t ) , k ( t ) , μ ( t )] which satisfy the FOCs the law of motion for the state the boundary / transversality conditions L. Hendricks () Optimal Control August 6, 2009 10 / 39
Details First order conditions are necessary, not su¢ cient. They are necessary only if we assume that 1 a continuous, interior solution exists; 2 the objective function v and the constraint function g are continuously di/erentiable. Acemoglu 7 o/ers some insight into why the FOCs are necessary. L. Hendricks () Optimal Control August 6, 2009 11 / 39

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Details If there are multiple states and controls, simply write down one FOC for each separately: δ H / δ c i = 0 H / k j = ± ˙ μ j There is a large variety of cases depending on the length of the horizon (°nite or in°nite) and the kinds of boundary conditions.
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OptControl_SL - Optimal Control Prof Lutz Hendricks August...

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