# Endogeneous Heterogeneity and Periodicity in Dynamic Optimization Problems

**FWF Project No. I476-N13,**** ** (2010 − 2013), Project leader: V. Veliov

Keywords: biological resources, optimization, control, distributed systems, heterogeneity, endogenous economic growth

Heterogeneity is an important feature of most of the existing population, economic or social systems. Parameters of heterogeneity of human populations, for example, may be age, preferences, health status, wealth, etc. In a purely economic context, heterogeneity can be brought by the variety of existing technologies or products; since this variety depends on investments in R&D, we encounter heterogeneous models where the domain of heterogeneity is controllable. Such models arise also in the recently developing space economics.

The general goal of this project was to enhance the optimization theory for heterogeneous dynamical systems in two directions: (i) for problems where the domain of heterogeneity depends on the decision variables (the control); (ii) for problems involving periodic control. This goal is motivated by numerous new applications in economics and population dynamics, where the existing theories do not provide efficient solution instruments.

The investigation carried out within the project, required some non-standard extensions of the optimal control theory and the numerical methods for first order partial differential equations, where the spatial domain (the domain of heterogeneity) is endogenous. In addition, an important result of the project is a new approach for obtaining a complete set of necessary optimality conditions for infinite-horizon problems; it provides an important contribution also to the classical theory of optimal control of ordinary differential systems.

Using these theoretical contributions we gained important insights in several economic problems at micro and macro level. In particular, we solved the problem of optimal allocation of investments to capital accumulation and R&D. Moreover, in a general equilibrium model with an endogenously growing product variety we showed that the obsolescence of consumption products is important for boundedness of the gap between savings and investments. We also investigated problems of optimal harvesting of biological resources, where the heterogeneity is either with respect to age/size or with respect to the position in the physical space. We establish that the heterogeneity may be responsible for appearance of cyclical optimal solutions involving recovery periods of the resource.

Several complementary investigations are devoted to problems of health investments or optimal education in life-cycle models, optimal emission abatement by heterogeneous agents, markets of emission permits in a vintage capital framework, and optimal immigration age-patterns.

**Peer-reviewed publications (already published or accepted)**

[1] T. Brechet, T. Tsachev, and V.M. Veliov. Markets for emission permits with free endowment: a vintage capital analysis. Optimal Control Applications and Methods, 33(2): 214-231, 2012.

[2] A. Belyakov, T. Tsachev, and V.M. Veliov. Optimal control of heterogeneous systems with endogenous domain of heterogeneity. Applied Mathematics and Optimization, 64: 287-311, 2011.

[3] S. Aseev and V.M. Veliov. Maximum principle for problems with dominating discount. Dynamics of Continuous, Discrete and Impulsive Systems, ser. B, 19(1-2): 43-63, 2012.

[4] M. Kuhn, S. Wrzaczek, A. Prskawetz, and G. Feichtinger. Externalities in a life-cycle model with endogenous survival. Journal of Mathematical Economics, 47: 627-641, 2011.

[5] A. Prskawetz, T. Tsachev, and V.M. Veliov. Optimal education in an age-structured model under changing labor demand and supply, Macroeconomic Dynamics, 16 (2): 1-15, 2012.

[6] T. Brechet, C. Camacho, and V.M. Veliov. Adaptive Model-Predictive Climate policies in a multi-country setting. (To appear in The Oxford Handbook of the Macroeconomics of Climate Change, 2014.). Research Report 2012-02, ORCOS, TU Wien, 2012

[7] C. Simon, A. Belyakov, and G. Feichtinger. Minimizing the Dependency Ratio in a Population with Below-Replacement Fertility through Immigration. Theoretical Population Biology, 82: 158-169, 2012.

[8] S. Wrzaczek, E. Shevkoplyas, and S. Kostyunin. Differential game of pollution control with overlapping generations. Contributions to Game Theory, 5, 310-320, 2012.

[9] S.M. Aseev and V.M. Veliov. Necessary optimality conditions for improper infinite-horizon control problems. In Operations Research Proceedings 2011, D. Klatte, H.-J. Lüthi, K. Schmedders (Eds.), pp. 21--26, Springer, 2012.

[10] S. Wrzaczek and P. Kort. Anticipation in Innovative Investment under oligopolistic competition. Automatica, 48: 2812-2823, 2012.

[11] S. Aseev and V.M. Veliov. Needle variations in infinite-horizon optimal control. To appear in Contemporary Mathematics (AMS), 2014. Research Report 2012-04, ORCOS, TU Wien, 2012

[12] C. Simon, B. Skritek, and V.M. Veliov. Optimal immigration age-patterns in populations of fixed size. J. Math. Anal. and Appl., 405(1): 71-89, 2013.

[13] M. Quincampoix and V. Veliov. Metric Regularity and Stability of Optimal Control Problems for Linear Systems. SIAM J. Contr. Optim., 51(5): 4118-4137, 2013.

[14] B. Skritek, T. Tsachev, and V.M. Veliov. Optimality conditions and the Hamiltonian for a distributed optimal control problem on controlled domain. To appear in Large-scale scientific computing, Lecture Notes in Computer Science, Springer, 2014.

[15] J. Haunschmied, A. Pietrus, and V. Veliov. The Euler Method for Linear Control Systems Revisited. To appear in Large-scale scientific computing, Lecture Notes in Computer Science, Springer, 2014. Research Report 2013-02, ORCOS, TU Wien, 2013

[16] B. Skritek, T. Tsachev, and V.M. Veliov. Optimality conditions and the Hamiltonian for a distributed optimal control problem on controlled domain. Applied Mathematics and Optimization, to appear in 2014, DOI 10.1007/s00245-014-9237-5.

[17] J. Haunschmied, V.M. Veliov, and S. Wrzaczek, Eds. Dynamic Games in Economics.

Springer series Dynamic Modeling and Econometrics in Economics and Finance, 310 pages,Springer, 2014. In print.

[18] E. Moser, W. Semmler, G. Tragler, and V.M. Veliov, Eds. Dynamic Optimization in Environmental Economics. Springer series Dynamic Modeling and Econometrics in Economics and Finance, 350 pages, Springer, 2014. In print.

[19] A. Belyakov and V.M. Veliov. Constant versus periodic fishing: age structured optimal control approach. To appear in Mathematical Modelling of Natural Phenomena, 2014. Research Report 2014-01, ORCOS, TU Wien, 2014

[21] A. Belyakov, A. Davydov, and V.M. Veliov. Optimal cyclic exploitation of renewable resources. Submitted to Journal of Dynamical and Control Systems.

[published in Journal of Dynamical and Control Systems (2015) 21:475-494]

[22] M. Kuhn, S. Wrzaczek, A. Prskawetz, and G. Feichtinger. Optimal choice of health and retirement in a life-cycle model. Journal of Economic Theory Volume 158, Part A, July 2015, Pages 186–212

[23] B. Skritek and V.M. Veliov. On the infinite-horizon optimal control of age-structured systems.

[published in Journal of Optimization Theory and Applications, 2015, 167:243-271, DOI 10.1007/s10957-014-0680-x]

**Planned Peer-reviewed publications **

[20] A. Belyakov, J. Haunschmied and V.M. Veliov. A general equilibrium model with horizontal innovations and heterogeneous products. (Submitted.) Research Report 2012-01, ORCOS, TU Wien, 2012: orcos.tuwien.ac.at/fileadmin/t/orcos/Research_Reports/2012-01_Bely-VV-HS.pdf

**Project-related participation in scientific conferences and lectures held **

1. V.M. Veliov. Distributed optimal control models in population dynamics and economics.

Invited seminar lecture. University of Bayreuth, 11.11.2010.

2. V.M. Veliov. Optimization of a Class of Distributed Systems on a Controlled Domain.

Talk given at the conference “Advances in Optimization and related Topics”, Barcelona, 29.11-03.12.2010.

3. M. Kuhn, S. Wrzaczek. On optimal fertility and mortality in economies with age-structured population. Talk given at “Workshop on Optimal Fertility in Ageing Societies”, Vienna, Austria, December 9-10, 2010.

4. V.M. Veliov. Distributed dynamic optimization on a controlled domain. Invited talk given at the “5th Annual Meeting of the Bulgarian Section of SIAM", Sofia, Dec. 20, 21, 2010.

5. V.M. Veliov. Distributed models for sustainable growth. Invited plenary talk given at the international conference “Sustainable growth, technological progress and the environment”, Louvain-la-Neuve, Belgium, May 19-20, 2011.

6. A.O. Belyakov. Consumption of heterogeneous quality goods with increasing quality domain. Talk given at the “Doctoral Workshop", Louvain-la-Neuve, 30.05-31.05.2011.

7. V.M. Veliov. Distributed Optimal Control in Problems of Endogenous Economic Growth.

Invited lecture at the seminar on Operations Research at ETH Zurich, 30.05.2011.

8. M. Kuhn, S. Wrzaczek, A. Prskawetz, G. Feichtinger. Optimal choice of health and retirement in a life cycle model. Talk given at the “LEPAS workshop”, Alicante Spain, June 16-17, 2011.

9. S. Wrzaczek, E. Shevkoplyas, S. Kostyunin. Differential Game of Pollution Control with overlapping generations. Talk given at the “Fifth International Conference on Game Theory and Management” (GTM2010), Sankt Petersburg, Russia, June 27-29, 2011.

10. S. Aseev and V.M. Veliov. Pontryagin's maximum principle under weakened assumptions for the convergence of the integral functional. Talk given at the “International Conference on Mathematical Control Theory and Mechanics”, Suzdal (Russia), 05-10.07.2011.

11. V.M. Veliov. Distributed optimal control in economics: endogenous technological growth. Invited talk given at the “International Conference on Mathematical Control Theory and Mechanics”, Suzdal, Russia, 05-10.07.2011.

12. S. Wrzaczek, E. Shevkoplyas, S. Kostyunin. Differential Game of Pollution Control with overlapping generations. Talk given at the “Eighth International ISDG Workshop", Padova, Italy, July 21-23, 2011.

13. V.M. Veliov. Necessary optimality conditions for improper infinite-horizon control problems. Talk given at the “International Conference on Operations Research" OR 2011, Zurich, 30.08-02.09.2011.

14. C. Simon. Minimizing the Dependency Ratio in a Population with Below-Replacement Fertility through Immigration. Talk given at the “International Conference on Operation Research" OR 2011, Zurich, 30.08-02.09.2011.

15. M. Kuhn, S. Wrzaczek. On optimal fertility and mortality in economies with age-structured population. Talk given at the Conference “OR 2011”, Zurich, 30.08-02.09.2011.

16. V.M. Veliov. Maximum principle for infinite-horizon optimal control problems with dominating discount. Talk given at the “25th IFIP TC7 Conference on System Modeling and Optimization", Berlin, 12-16.09.2011.

17. V.M. Veliov. Maximum principle for infinite-horizon optimal control problems with dominating discount. Invited talk given at the workshop “Variational and optimal control problems on unbounded domains" in memory of Arie Leizarowitz, Haifa, 09-12.01.2012.

18. V.M. Veliov. Necessary optimality conditions and approximations for infinite horizon optimal control. Invited session talk given at the 5th international conference “High performance scientific computing", Hanoi, 05-09.03.2012.

19. A.O. Belyakov. Continuous time OLG Model with horizontal innovations and heterogeneous consumption products. Talk given at the “Workshop Overlapping Generations Days", Marseille, May 14-16, 2012.

20. A.O. Belyakov. General equilibrium model with horizontal innovations and heterogeneous products. Talk given at the “12th Viennese Workshop on Optimal Control, Dynamic Games and Nonlinear Dynamics", Vienna, May 30th - June 2nd, 2012.

21. T.Y. Tsachev, Optimal control of heterogeneous systems on controlled domains. Talk given at the “12th Viennese Workshop on Optimal Control, Dynamic Games and Nonlinear Dynamics", Vienna, May 30th - June 2nd, 2012.

22. V.M. Veliov Infinite-horizon optimal control: some new developments. Seminar talk at Faculte des Sciences at Techiques, Université de Bretagne Occidentale, Brest, France, Sept. 25, 2012.

23. V.M. Veliov. Adaptive Model-Predictive Climate Policies in a multi-country setting. Invited talk at the workshop “Economic Growth Research within the IIASA Network: an Overview", Dec. 7, 8, 2012.

24. V.M. Veliov. Metric regularity and approximations of bang-bang type optimal control. Invited lecture given at Université des Antilles et de la Guyane, Point-a-Pitre, 21.02.2013.

25. V.M. Veliov. Metric regularity, stability and approximations in optimal control. Invited lecture at Institute of Mathematics, Faculty Math. and Natural Sciences, TU Berlin, 18.04.2013.

26. V.M. Veliov. Metric regularity, stability and approximations in optimal control. Invited lecture at the seminar on Operations Research at ETH Zurich, 13.05.2013.

27. V.M. Veliov. The Euler method for linear control systems revisited. Talk given at the conference Large-scale Scientific Computations, Sozopol, June 3--7, 2013.

28. V.M. Veliov. Optimal spatially distributed harvesting. Talk given at the conference XXVI EURO-INFORMS, Rome, July 1-4, 2013.

29. V.M. Veliov. Metric Regularity and Stability of Optimal Control Problems for Linear Systems. Invited talk given at the 26th IFIP TC7 Conference on System Modelling and Optimization, Klagenfurt, Sept. 8--13, 2013.

30. A.O. Belyakov. Constant versus periodic fishing. Talk given at the “International Workshop on Heterogeneous Dynamic Models of Economic and Population Systems", Vienna, 25-26 November 2013.

31. V.M. Veliov. Optimal spatially distributed harvesting. Invited talk given at the workshop “Renewable resources, Sustainability, and Search", Internationales Wissenschaftsforum (IWR), Heidelberg, Dec. 12-14, 2013.

32. A.O. Belyakov. Periodic versus constant fishing. Invited talk given at the workshop “Renewable resources, Sustainability, and Search", Internationales Wissenschaftsforum (IWR), Heidelberg, Dec. 12-14, 2013.

33. V.M. Veliov. Optimal cyclic utilization of renewable resources. Talk given at the “Annual Meeting on OR", Institute of Mathematics and Informatics, Bulgarian Academy of Science, Sofia, Dec. 20, 2013.