Optimal Control with State Constraint and Non-concave Dynamics: A Model Arising in Economic Growth

• Published in: Applied mathematics & optimization Vol. 76; no. 2; pp. 323 - 373
• Main Authors: Acquistapace, Paolo, Bartaloni, Francesco
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2017; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Constraints; Continuity (mathematics); Control; Dynamic programming; Economic development; Economic growth; Economic models; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Optimal control; Optimization; Proving; Simulation; State variable; Systems Theory; Theoretical; Utility maximization; Hamilton Jacobi Bellman equation; Optimal control; Non-concave production function; Viscosity solutions
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-016-9353-5

We consider an optimal control problem arising in the context of economic theory of growth, on the lines of the works by Skiba and Askenazy–Le Van. The framework of the model is intertemporal infinite horizon utility maximization. The dynamics involves a state variable representing total endowment of the social planner or average capital of the representative dynasty. From the mathematical viewpoint, the main features of the model are the following: (i) the dynamics is an increasing, unbounded and not globally concave function of the state; (ii) the state variable is subject to a static constraint; (iii) the admissible controls are merely locally integrable in the right half-line. Such assumptions seem to be weaker than those appearing in most of the existing literature. We give a direct proof of the existence of an optimal control for any initial capital k 0 ≥ 0 and we carry on a qualitative study of the value function; moreover, using dynamic programming methods, we show that the value function is a continuous viscosity solution of the associated Hamilton–Jacobi–Bellman equation.