Mathematics for Economists
Optimal control By the maximum principleu(t)is maximizes H(t,x,u,p(t))=u^2 x+(t (^2) )u,u (^2) [ 0 , (^1) ]. The optimization i ...
Optimal control The partially optimized Hamiltonian is Hb(t,x,p(t)) = max u 2 [ 0 , 1 ] u^2 x+(t 2 )u = = x if t 2 [ 0 , 1 ...
Optimal control Example Solve the problem Z 1 0 2 x(t)x^2 (t)dt!max x(t)=u(t), x( 0 )= 0 ,x( 1 )= 0 ,u(t) 2 [ 1 , 1 ]. The Hami ...
Optimal control The optimal uis maximizes the Hamiltonian 2xx^2 +puover U=[ 1 , 1 ]so u(t)= 1 if p(t)> 0 1 if p(t)< 0. ...
Optimal control Ifp( 1 )>0 then aspis decreasingp(t)>0 for every 0<t< 1 , which implies thatu1. In this casex=u= ...
Optimal control Aspis continuous there is a pointt 2 ( 0 , 1 )wherep(t)= 0. p(t)>0 on( 0 ,t)hence on this intervalu=1 so ...
Optimal control As dp dt = 2 (x 1 )= (^2) (t (^1) ) if t< 1 / 2 2 ( 1 t 1 ) if t 1 / 2 p(t) = Zt 0 2 (s 1 )ds=t^2 2 t+C, ...
Optimal control, Bolza problem ZT 0 f(t,x(t),u(t))dt+φ(x(T))!max x(t)=g(t,x(t),u(t)),x( 0 )=x 0 By the fundamental theorem of t ...
Optimal control, Bolza problem Hence H=f(t,x(t),u(t))+φ^0 (x(t))g(t,x(t),u(t))+ +p(t)g(t,x(t),u(t))= =f(t,x(t),u(t))+ p(t)+φ^0 ...
Optimal control, Bolza problem The adjoint equation with the transversality condition p = H^0 x= fx^0 +φ^00 g+φ^0 gx^0 +pgx^0 ...
Optimal control, Bolza problem d dt ep(t)= d dt p(t)+φ^00 (x(t))x(t). Hence d dtep(t)= fx^0 +epgx^0 ,p(T)= 0. IfHe=f+epgthen ...
Optimal control, Bolza problem Example Solve Z 1 0 x^2 xdt+x^2 ( 1 )!min,x( 0 )= 0 as an optimal control problem. Z 1 0 u^2 xdt ...
Optimal control, Bolza problem H = u^2 x+pu,p=Hx= 1 ,p=t+C p( 1 ) = 1 +C= 2 x( 1 ),C= 2 x( 1 ) 1 ,p(t)=t+ 2 x( 1 ) 1 0 = Hu^0 = ...
Optimal control, Bolza problem x(t) = x( 0 )+ Zt 0 x( 1 )+ 1 / 2 t/ 2 dt= = x( (^1) )t+ 1 / 2 tt^2 / 4. 2 x( 1 ) =^12 1 ^14 ...
Optimal control, Bolza problem Example Solve the problem Z 1 0 u^2 dt+x^2 ( 1 )!min x=x+u,x( 0 )= 1 ...
Optimal control, Bolza problem H=u^2 +p(x+u), p=H^0 x=p, p=Cexp(t) p( 1 )= 2 x( 1 )=Ce, C= 2 ex( 1 ),p= 2 x( 1 )exp(t+ 1 ), ...
Optimal control, Bolza problem 0 =Hu^0 = 2 u+p= 2 u+ 2 x( 1 )exp(t+ 1 ), u=x( 1 )exp(t+ 1 ) ...
Optimal control, Bolza problem x=xx( 1 )exp(t+ 1 ),x( 0 )= 1 xx=x( 1 )exp(t+ 1 ),x( 0 )= 1 , x e txet=x( 1 )e^2 t+^1 xet=x( 1 ...
Optimal control, Bolza problem x( 1 ) = x(^1 ) 2 +Ce, 1 =x( 0 )=x(^1 ) 2 e+C, x( 1 ) 2 = Ce,C= 1 x(^1 ) 2 e,C= 1 Ce^2 ,C=^1 1 +e ...
Optimal control, Bolza problem We can also write as Z 1 0 xx 2 + 2 xx dt = Z 1 0 x^2 +x^2 dt!min,x( 0 )= 1. The EulerñLag ...
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