Seminar 7

Milestone reached: we have covered one full exam set

Exam 2009 Links to an external site. had nothing about discrete time nor nothing about after Easter (but you would be lost if you couldn't do the last few weeks).

Do that, but you should do some optimal control first.

New material: induction.

• Use induction to prove these classics for all natural n:
  • 1+…+n=(n+1)n/2
  • This matrix version of the geometric series formula - do it without multiplying both sides by I-M:
    I+M+…+Mⁿ=(I−M)⁻¹(I−Mⁿ⁺¹) when 1 is not an eigenvalue of the (square!) matrix M.
• Show that (9ⁿ-1)/8 is an integer. 
  Hint: Write 9ⁿ⁺¹=(8+1)9ⁿ

The optimal control problems posted earlier

Updates compared to posted: with a full exam problem set, the "Eric's lecture" problem might be a bit of workload and might be relegated. So I retract what I said that 9-17 will have lower priority than the Eric's problem. It will then reappear after Easter.
Also there is an update to 9-17.

• Exam 2016 Links to an external site. problem 4
• 9-12
• 9-17. 
• UPDATE: In 9-17, if x(0) were changed from 0 to z (with z≈0), what can you say about the change in optimal value? 
• This problem from Eric's lecture:
  $\max_{h(t)\geq 0}\int_0^T(h-\tfrac12h^2)e^{-rt}\,dt$  where $\dot x=x(1-x)-h,\qquad x(0)=x_0\in(0,1),\quad x(T)\geq 0$.
  Here the control is h (for "harvesting"). r is a constant, 0<r<1 (and T>0 is a constant too).
  Exercises for you:

• 1. If you increase the constant r, how would the phase diagram change?
     (In the rest of the problem, keep the r as Eric's diagram.)  
  2. Change the terminal condition to impose x(T) = x₀. That is, if this were a capital accumulation model: You are left to manage the economy but with the constraint that you hand it back with the same capital stock as when you started.
     
     • Take a stand on whether there even exists an admissible path for all T>0.  (You might not want to do this part first.)
     • Sketch a possible solution path for a small T>0.  
     • ... and one where T is increased and is very large (keeping the same x₀ as for the previous path).
     • When T -> infinity, what happens?
  3. Now instead, assume that you start with a fairly low x₀ but you are required to end put at x(T) greater than or equal to [somewhere to the right of the vertical nullcline, but not as large as 1].  
     Precisely the same four questions:
     
     • Take a stand on whether there even exists an admissible path for all T>0.  (You might not want to do this part first.)
     • Sketch a possible solution path
     • ... and one where T is increased and is very large (keeping the same x₀ as for the previous path).  
     • When T -> infinity, what happens?