The aim of this course is to present the main results of modern optimal control theory for the class of infinite horizon problems arising in economics. The focus will be on the Pontryagin maximum principle for these problems. The economic interpretation of the maximum principle will be discussed. Theorems on the existence of strongly optimal controls and sufficient conditions for weakly overtaking optimality will be proven. A number of illustrative examples are expected to be considered.

The presentation of the material is mostly self-contained. Participants are expected to have a basic knowledge of measure theory and Lebesgue integration, as well as ordinary differential equations. Familiarity with the Pontryagin maximum principle is desirable.

Program

1. Formulation of optimal control problems over finite and infinite time horizons. Reduction of the problem with random stopping time to a problem over an infinite time interval. Examples: Ramsey model, optimal investment model in fixed capital of an enterprise, model of optimal exploitation of a non-renewable resource.
2. Admissible processes. Regularity conditions for processes in optimal control problems. Strong optimality, finite optimality, and weakly overtaking optimality in infinite horizon problems.
3. Autonomous problem with exponential discounting. Time discounting and tine consistency. General version of the Pontryagin maximum principle for infinite horizon problems. Core relations of the maximum principle. Transversality conditions at infinity.
4. Sufficient conditions for weakly overtaking optimality for problems with an infinite horizon. Existence of strongly optimal control in an autonomous problem with exponential discounting. Finite-horizon approximations of autonomous problems with exponential discounting.
5. Dominating discount condition. Complete version of the Pontryagin maximum principle for an autonomous problem with exponential discounting under dominating discount condition.
6. Dynamic programming method and the Pontryagin maximum principle. Economic interpretation of the maximum principle. Growth condition. Conditional value function and its differentiability.
7. Complete version of the Pontryagin maximum principle for a general nonlinear problem with an infinite horizon under the growth condition.

Bibliography
1. Aseev S.M., Conditional cost function and necessary optimality conditions for infinite horizon optimal control problems, Dokl. Math., 2023, Vol. 108, № 3,  pp. 425–430.
2. Aseev S.M., Kryazhimskii A.V., The Pontryagin maximum principle and optimal economic growth problems, Proc. Steklov Inst. Math., 2007, Vol. 257, pp. 1–255.
3. Aseev S.M., Besov K.O., Kryazhimskiy A.V., Infinite-horizon optimal control problems in economics, Russian Math. Surveys, 2012, Vol. 67:2, pp. 195–253.
4. Aseev S.M., Veliov V.M., Another view of the maximum principle for infinite-horizon optimal control problems in economics, Russian Math. Surveys, 2019, Vol. 74:6, pp. 963–1011.
5. Barro R.J., Sala-i-Martin X., Economic growth, McGraw Hill: New York, 1995.
6. Pontryagin L.S., Boltyanskij V.G., Gamkrelidze R.V., Mishchenko E.F., The mathematical theory of optimal processes, Pergamon, Oxford, 1964.
7. Aseev S.M., Veliov V.M., Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions, Proc. Steklov Inst. Math. (Suppl.), 2015, Vol. 291, suppl. 1, pp. 22–39.
8. Caputo M.R., Foundations of dynamic economic analysis. Optimal control theory and applications, Cambridge: Cambridge University Press, 2005.
9. Carlson D.A., Haurie A.B., and Leizarowitz A., Infinite horizon optimal control. Deterministic and stochastic systems, Berlin: Springer, 1991.
10. Dorfman R., An economic interpretation of optimal control theory, American Economic Revew, 1969. Vol. 59, pp. 817–831.
11. Ramsey F.P., A mathematical theory of saving, Econ. J., 1928, Vol. 38, pp. 543–559.
12. Seierstad A., Sydsæter K., Optimal control theory with economic applications, North Holland, 1987.

Lecturer
Aseev Sergey Mironovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center