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# 9 - L Karp Notes for Dynamics IX Linear Control Problems 1...

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L. Karp Notes for Dynamics IX. Linear Control Problems 1) One-state variable linear control problem. 2) Necessary and sufficient condition for optimality. 3) Most Rapid Approach Path (MRAP), and singular arc. 4) Fishing example. 5) Two-state variable problem, a durable non-renewable resource with decay. K & S Section I.16 and II.13, Clark Ch 2.7 General 1 state variable problem (1) max t 1 t 0 G ( t , x ) H ( t , x ) ˙ xd t s.t. A ( t , x ) x ˙ B ( t , x ), given x t 0 , x t 1 Euler equation F x d ( F ˙x ) dt G x H x d dt H H t H x (2) G x H t Any x * that satisfies (2) is singular solution (arc) Thrm. Suppose whenever . G x > < H t x > < x ( t ) The optimal solution is (i) If x 0 > x * ( t 0 ), set x ˙ ( t )= A ( ) until x * hit If x 0 < x * ( t 0 ), set x ˙ ( t B ( ) until x * hit

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Define t a as time you hit x * using (i) (ii) Analogous description at RHS, with t b as time you leave singular path. over t 0 , t a , set x ˙ = B , over t b ,t 1 , set x ˙ = A . (MRAP) over t a , t b , solution follows singular path.
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9 - L Karp Notes for Dynamics IX Linear Control Problems 1...

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