Mathematics for Economists
Calculus of variations The general solution is x(t)=Aexp(t/C)+Bexp(+t/C) Writing back again to Euler-Lagrange 4 AB = C^2 x = αc ...
Calculus of variations 2 π ZT 0 x(t) r 1 + x 2 dt!min x( 0 )=a The solution of the EulerñLagrange equation is still of the ...
Calculus of variations Example Solve the problem Z 1 0 tx+ x 2 dt!min,x( 0 )= 1. The EulerñLagrange is 0 =Fx^0 = d dt Fx^0 ...
Calculus of variations The transversality condition is 0 = Fx^0 1 ,x( 1 ),x( 1 ) = t+ 2 x ( 1 )= = 1 + 2 t 2 +C ...
Calculus of variations Theorem If the kernel function F t,x,x is convex in x,x then every solution of the EulerñLagran ...
Calculus of variations Henceψis convex. on the real line. AsxsatisÖes the Euler-Lagrange equationψ^0 ( (^0) )=0 and asψis conve ...
Calculus of variations Example Study the problem with kernelF t,x,x = r 1 + x 2 . This is a convex function as d du p ...
Calculus of variations Example Study the problem with kernel F t,x,x =x r 1 + x 2 This a product of two convex functio ...
Calculus of variations λ^2 λx 1 + x 2 ^3 /^2 x 2 1 + x 2 x 1 + x 2 ^3 /^2 vu uu uu t x^2 1 + x ...
Homework Analyze the problems Z 1 0 x^2 + x 2 dt! maxmin , x( (^0) )=x( (^1) )= 0. Analyze the problems ZT 0 U cx e rt ...
Homework Solve the problem ZT 0 et/^4 ln 2 K K dt!max, K( (^0) )=K 0 ,K(T)=KT. Solve the problem ZT 0 et/^10 1 100 tK K ...
Homework Find the general solution of the EulerñLagrange equation of the functional Zb a x 2 t^3 dt wherea> 0. Find the ...
Homework Find the extremal solutions Zb a xy^0 + y^0 2 dx Find the extremal solutions Zπ 0 4 xcost+x 2 x^2 dt, x( (^0) )=x(π ...
Optimal control DeÖnition The problem Zt 1 t 0 f(t,x(t),u(t))dt!max x(t)=g(t,x(t),u(t)), x(t 0 )=x 0 u(t) 2 U, is called the op ...
Optimal control Example The problem Zt 1 t 0 f(t,x(t),u(t))dt!max x(t)=u(t), x(t 0 )=x 0 ,x(t 1 )=x 1 ,u(t)^2 [^0 ,^1 ] is an o ...
Optimal control DeÖnition The expression H(t,x,u,p)=f(t,x,u)+pg(t,x,u) is called the Hamiltonian. The variablepis called the ad ...
Optimal control Theorem (Maximum principle) If an(x(t),u(t))is an optimal solution of the problem then there is some adjoint f ...
Optimal control Theorem (Mangasarian) If U is a convex set and H(t,x,u,p(t))is concave in(x,u)then the solution of the Maximum P ...
Optimal control, regularity Example Solve the problem ZT 0 udt!max x=u^2 , x( 0 )=x(T)= 0. Asx0 andx( 0 )=x(T)=0 the only fea ...
Optimal control, regularity Assume thatx(T)is free. In this casep(T)=0 and ifp 0 =0 then d dtp(t) = H x^0 (t,x(t),u(t),p(t))= ...
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