On the Dynamic Programming Approach for Optimal Control Problems of Pde's With Age Structure

Abstract

A survey and some new results are presented concerning the dynamic programming for a class of optimal control problems of partial differential equations with age-structure and of delay systems that include some applied examples from economic theory and from population dynamics. A general optimal control problem in Hilbert spaces applying to all examples is investigated, with particular stress on one family of applications: optimal investment models with vintage capital. Some new results are given for the case of constrained investments, including a study of the properties of the optimal trajectories.age-structured systems, optimal control, economic growth, vintage models,

Optimal investment models with vintage capital: Dynamic programming approach

“…to the book by Lasiecka and Triggiani [39], to the book by Bensoussan, Da Prato, Delfour and Mitter [14], and, for the case of nonautonomous systems, to the papers by Acquistapace, Flandoli and Terreni [1,2,3,4]. For the case of a linear system and a general convex cost, we mention the papers by Faggian [24,25,26,27,28], by Faggian and Gozzi [29]. On Pontryagin maximum principle for boundary control problems we mention again the book by Barbu and Precupanu (Chapter 4 in [10]).…”

“…[29,33,37,40]), general equilibrium with vintage capital (see e.g. [15,23] The problem has been already studied by Faggian and by Faggian and Gozzi in the papers [26,27,28,29] in the case of finite horizon, with and without constraints on the control and on the state, yielding a definition of generalized solutions of the associated evolutionary HJB equation. This paper studies instead the infinite horizon case.…”

“…We now describe the model of optimal investment with vintage capital, in the setting introduced by Barucci and Gozzi [11] [12], and later reprised and generalized by Feichtinger et al [30,31,32], and by Faggian [26,27] and Faggian and Gozzi [29].…”

“…That of optimal control of linear age structured equations is one of the possible approaches undertaken in the literature. Such framework has been introduced in in [11,12] and then studied in various papers, among which we mention [26,27,29,30,31,32]. There the optimal investment problem with vintage capital is treated in two main cases:…”

“…• In [11,12,29,31] the production function is linear and the representative investor is price taker (corresponding to an objective function which is linear in the capital stock). In this case the value function is linear and the optimal investment strategies (together with the corresponding capital stock trajectories) can be explicitly calculated.…”