ANNOUNCEMENTS:

- Retake ECRMMAT scheduled on Wednesday 1 February, 14:00-17:00. Location: room 610, WG (Mathematics Building)
- Complete solutions of quiz 1 can be found here (WISB372) and here (ECRMMAT).
- Complete solutions of quiz 2 can be found here (WISB372) and here (ECRMMAT).
- Complete solutions to the final of 9-11 can be found here for ECRMMAT and here for WISB372.
- Possibility to review your results in the final: please make an email appointment for 6 December or later.
- Literature for weeks 36-40: Chapters 20 (Calculus of Variations) and 21 (Optimal Control) from Introduction to Mathematical Economics (Schaum's Outline Series, 3rd edition, 2008) by E.T. Dowling, ISBN 0-07-135896.
- Literature for weeks 40-44: Chapters 1 (Dynamic Programming),3 (Deterministic Continuous-Time Optimal Control) and 4 (Problems with Perfect State Information) from: Dynamic Programming and Optimal Control, Volume I, Athena Scientific, 2005, ISBN 1-886529-26-4. For Mathematics students there will be additional theoretical material to cover (details TBA) and for Economics students there will be additional applied material (details TBA). Note that ONLY volume I is needed. It can be ordered, for instance, from the above-mentioned webstores. Used copies cost around 80 E. from amazon.co.uk, but new copies cost less (!) when bought directly from amazon.com in the USA. The Mathematics library has a copy of this book on its shelves (NOT on loan). Some of the material in weeks 40-44 can be found in the first 75 pages of the author's MIT lecture slides

Course Progress:

- Week 36: To show difference between CoV (Calculus of Variations) and OCT (Optimal Control Theory), an economic example (in OCT form) was shown to be reducible to CoV form; conditions for this to work were discussed. Notational pitfalls were emphasized. Euler's equation was presented in general and specialized to (but not yet solved for) the aforementioned example in its CoV form.
- Week 37: Gave examples for simplest problem in CoV. Gave "proof" of Euler's equation for this problem and used it to develop necessary conditions for two modifications (not in book): mod. 1: free position at end time, mod. 2: no fixed positions at initial and end times, but cost evaluation of initial and end positions. Gave example. To prepare next week's treatment of section 20.5 convexity/concavity of multivariate functions was discussed. Some exercises for week 37 can be found here (corrected version)
- Week 38: Completed section 20.5 (including a full proof of the sufficiency theorem -- not in book), pointing out shortcomings in Dowling. Applied it to "old" optimal consumption problem. Gave useful trick for autonomous problems (not in book) to sidestep Euler's equation; applied it to problem of Example 20.18. Discussed section 20.6 and gave application to hanging chain problem (not in book). Section 20.7 was skipped. Presented one example from 20.8 (pp. 468-469). All other examples/problems of 20.8 (except for problems 20.34 and 20.35) are now part of the course material as well. Note that solutions in Dowling should be improved for shortcomings in section 20.5! Gave a third modification of CoV's simplest problem. Started with Chapter 21 by introducing OCT. Discussed the "(too) heroic assumption" of internality by Dowling to contrast later with Bertsekas' treatment of OCT. More exercises for week 38 can be found here (corrected version)
- Week 39: Discussed the entire chapter 21, including a proof (not in book) of the Sufficiency Theorem. Introduced modifications "Mod 2a", "Mod 2b" (not in book), next to "Mod 1" (section 21.4 in book) and "Mod 4" (section 21.6). Apart from the examples and worked out problems in Dowling, more exercises for week 39 can be found here and the final example on 28-9 has been worked out here. Recall here also the agreement made in class on 28-10 about the "default dynamical system" and the "default space Omega".
- Week 40: Discussed how systems of 2 DE's can be solved by means of a 2nd order DE. Worked out several OC-examples ("mod 2b", alternative treatment of Problem 21.12, "mod 4", integration trick, ...), emphasizing solution methods for DE's. Started to discuss Bertsekas OC-formulation, including the new minimum principle, which does away with Dowling's "heroic interiority assumption". No new homework is given, because of the approaching quiz 1.
- Week 41: Discussed the minimum principle (p. 119) and its variants for the different "mod's". Treated Example 3.3.2 and sections 3.4.3 (including a concrete example of time-optimal control) and 3.4.4. Presented discrete-time optimal control. The homework can be found here, including a solution of exercise 4.
- Week 42: Presented the standard DP-model and the DP algorithm (see pp. 3-25 of Bertsekas). Examples: cheapest route model, discrete time optimal control (i.e., Ex 1.3.1, done both by today's DPA and by MP of week 41) and asset selling (p. 177). The homework can be found here.
- Week 43: Continued with selling an asset model. Included model with retention of offers (p. 185). Is special case of one step look ahead stopping sets on p. 183 ff. (note the importance of condition (4.50)!), which must read by the WISB372 students. Note: what happens for large N on p. 178 was sketched but is not part of exam material. Modeled for and solved by DPA : 1) optimal stopping exercise 6, homework week 42, 2) chess tournament problem (starts in Bertsekas on p.11). Modeled for DPA: 1) desperate in Las Vegas example, 2) knapsack problem, 3) stochastic knapsack problem (partly). Discussed inventory control, p. 162 ff.
- Week 44: roughly sketched route from DPA to MP via HJB-equation (sections 3.2 and 3.3.1), including interpretation of costate variable p(t). Followed up by machine replacement OC-model with economic interpretation of costate variable. Discussed solutions of quiz 2. Solved a problem with two functional constraints.

Intended audience: research master students USE and bachelor students Mathematics.

This course is given on Wednesdays, 13:15-17 h. in Room 013, U-Building, University College Campus.

Credit and presence: for Economics students passing this course yields 5 ECTS; their presence at lectures is mandatory. For mathematics students passing this course yields 7.5 ECTS; their presence is not mandatory.

Homework: each week some homework will be assigned. Even though it is extremely important to do well in these assignments, your work will not be graded. Naturally, to obtain feedback about your homework achievements, you should contact the teacher. Saving up questions until the end of the course is not a good strategy for several reasons (one being that the amount of teacher time available for each student is limited).

Office Hours: Monday afternoons from 14:00-16:00 in Room 612, Mathematics Building (= Wiskundegebouw), Budapestlaan 6, Uithof, across from the Botanical Garden.

Literature: as announced above. You should keep an eye on this webpage for the above Course Program, but also for comments, corrections, etc. and the above announcements in red.

Prequisites: Chapters 1,2,3,4,7,10,11,14 and 15 from Dowling plus sections 5.1, 5.2 and 5.3 are prerequisites for this course. Starting with week 37, chapters 16 and 18 are added to this list (see announcement above) and as of week 38 it will also include chapters 17 and 18.2.The material about solving systems of differential equations should also be studied now, although ECRMMAT students are not required to know it for quiz 1 (but see the preface to the set of exercises for week 39).

Learning Goals: Competence in problem solving in the Calculus of Variations, Optimal Control Theory and Dynamic Programming. Additionally, for Mathematics students: some competence in providing proofs plus extra emphasis. Additionally, for Economics students: competence in problem solving of anterior subjects, such as solving differential equations per se, plus extra emphasis on some economic models.