## Tesi etd-10192019-124556 |

Thesis type

Tesi di dottorato di ricerca

Author

BARTALONI, FRANCESCO

URN

etd-10192019-124556

Title

Infinite horizon optimal control problems with non-compact control space. Existence results and dynamic programming

Settore scientifico disciplinare

MAT/05

Corso di studi

MATEMATICA

Supervisors

**tutor**Prof. Acquistapace, Paolo

**correlatore**Prof. Gozzi, Fausto

Parole chiave

- convex-concave dynamics
- infinite horizon
- optimal control
- non-concavity
- non-compactness
- existence theorem
- uniform localization
- HJB equation
- viscosity solution

Data inizio appello

25/10/2019;

Consultabilità

Completa

Riassunto analitico

In optimal control theory, infinite horizon problems may be difficult to treat especially if associated with large classes of admissible controls or with state constraints. Such problems arise naturally in economic applications, and - in some cases - crucial questions such as the existence of solutions to the problem (namely of optimal controls) are left aside due to the technical issues that their handling involves, despite a large literature testifying to the interest of the scientific community about the subject.

In this work we develop a new technique for proving existence results and we apply the method to three different problems of increasing difficulty: the Ramsey-Skiba utility maximization problem, the monotone and the non-monotone Shallow Lake problem.

These are classical problems in economic theory: the Ramsey-Skiba model dates back to 1978 while the Shallow Lake model first appeared in 2003. These models have generated a consistent stream of literature, but, surprisingly, a complete mathematical analysis is still not available; in particular, an appropriate existence result seems to be missing in each case.

Our approach to the existence problem has three phases that are described in detail in the introduction.

For the Ramsey-Skiba model, which is a milestone in modern growth theory, we also provide a deep analysis of the value function which includes the classical necessary conditions expressed in terms of viscosity solutions to the HJB equation.

Although the problems addressed feature a convex-concave dynamics, our approach uses weaker assumptions that allow to cover also the purely concave case of the original Ramsey model.

Existence and uniqueness of the optimum among regular functions has been proven by Ekeland in 2010, with a technique strongly relying on the concavity of the dynamics. Thus, our method provides, in particular, a proof of the existence result alternative to the prior one, in the broader class of locally integrable functions.

In this work we develop a new technique for proving existence results and we apply the method to three different problems of increasing difficulty: the Ramsey-Skiba utility maximization problem, the monotone and the non-monotone Shallow Lake problem.

These are classical problems in economic theory: the Ramsey-Skiba model dates back to 1978 while the Shallow Lake model first appeared in 2003. These models have generated a consistent stream of literature, but, surprisingly, a complete mathematical analysis is still not available; in particular, an appropriate existence result seems to be missing in each case.

Our approach to the existence problem has three phases that are described in detail in the introduction.

For the Ramsey-Skiba model, which is a milestone in modern growth theory, we also provide a deep analysis of the value function which includes the classical necessary conditions expressed in terms of viscosity solutions to the HJB equation.

Although the problems addressed feature a convex-concave dynamics, our approach uses weaker assumptions that allow to cover also the purely concave case of the original Ramsey model.

Existence and uniqueness of the optimum among regular functions has been proven by Ekeland in 2010, with a technique strongly relying on the concavity of the dynamics. Thus, our method provides, in particular, a proof of the existence result alternative to the prior one, in the broader class of locally integrable functions.

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