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1. In an in finite horizon capital/consumption model, if kt and ct are the capital stock and consumption at time t, we have f(kt) = ct+kt+1 for t ≥ 0 where f is a given production function, and the total utility to be maximized is
where U is a given period utility function and β ? (0; 1) is a discount factor. Rephrase this as a standard (in finite horizon) control problem and write its Bellman equation.
2. Consider the discrete time control problem:
subject to x0 = x; xt+1 = g(t; xt; ut) for t = 0; : : : ; T - 1 (here f; g are C1, xt; ut ? R, x ? R given). Rewrite this as a Lagrangian optimization problem with 2T +2 variables (x0; : : : ; xT ; u0; : : : ; uT ) and T + 1 constraints. By applying the Lagrange condition to this problem, recover the maximum principle for the control problem (necessary conditions).
3. Consider the problem
subject to the initial and terminal conditions x0 = a; xT = b. One may think of it as a control problem by setting ut = xt+1-xt. Find the minimum and the optimal x *0 ; : : : ; x*T in two ways: directly (eg by Lagrangian method); and by writing the fundamental equation of dynamic programming for and computing Js(x) by backwards induction.
4. Consider the dynamic programming problem with \extended memory":
subject to xt+1 = g(t; xt; xt-1; ut) (x0; x-1 are given). Rephrase as a standard dynamic programming problem (with twice as many state variables).
A pole has to be erected at a point on the boundary of a circular park of diameter 13m in such a way that the differences of its distances from two diametrically opposite fixed gat
I need help solving this question...You have to design a rectangular flyer. The top and bottom must have 5" margins and the left and right sides must have 2" margins. If you must
An elliptical galaxy has gravitational boundaries defiend by 9x 2 +16y 2 +144z 2 =144. A black hole at the center of the galaxy is interacting with dark matter producing a radiatio
conclusion for the shares nd dividends
1/4 divided by (9/10 divided by 8/9)
i love math..but i am afraid to study it... i mean i ma afraid that it may leave me in clay...what can you suggest me?
y'-5y=0
I have difficuties in working out those 3D trigomentry problems within teh shortest possible time. Are there any tricks to get through such problems as soon as possible?
Advantages and disadvantages of operation researchs
[3+tan20+tan80]/tan20+tan80
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