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The paper concerns the study of equilibrium points, or steady states, of economic systems arising in modeling optimal investment with \textit{vintage capital}, namely, systems where all key variables (capitals, investments, prices) are indexed not only by time $\tau$ but also by age $s$. Capital accumulation is hence described as a partial differential equation (briefly, PDE), and equilibrium points are in fact equilibrium distributions in the variable $s$ of ages. Investments in frontier as well as non-frontier vintages are possible. Firstly a general method is developed to compute and study equilibrium points of a wide range of infinite dimensional, infinite horizon boundary control problems for linear PDEs with convex criterion, possibly applying to a wide variety of economic problems. Sufficient and necessary conditions for existence of equilibrium points are derived in this general context. In particular, for optimal investment with vintage capital, existence and uniqueness of a long run equilibrium distribution is proved for general concave revenues and convex investment costs, and analytic formulas are obtained for optimal controls and trajectories in the long run, definitely showing how effective the theoretical machinery of optimal control in infinite dimension is in computing explicitly equilibrium distributions, and suggesting that the same method can be applied in examples yielding the same abstract structure. To this extent, the results of this work constitutes a first crucial step towards a thorough understanding of the behavior of optimal controls and trajectories in the long run.

Art heritage cities are popular tourist destinations but for many of them overcrowding is becoming an issue. In this paper, we address the problem of modeling and analytically studying the flow of tourists along the narrow alleys of the historic center of a heritage city. We initially present a mean field game model, where both continuous and switching decisional variables are introduced to respectively describe the position of a tourist and the point of interest that he/she may visit. We prove the existence of a mean field equilibrium. A mean field equilibrium is Nash-type equilibrium in the case of infinitely many players. Then, we study an optimization problem for an external controller who aims to induce a suitable mean field equilibrium., Comment: 24 pages, published at https://link.springer.com/article/10.1007/s11067-019-09475-4

We develop the dynamic programming approach for a family of infinite horizon boundary control problems with linear state equation and convex cost. We prove that the value function of the problem is the unique regular solution of the associated stationary Hamilton--Jacobi--Bellman equation and use this to prove existence and uniqueness of feedback controls. The idea of studying this kind of problem comes from economic applications, in particular from models of optimal investment with vintage capital. Such family of problems has already been studied in the finite horizon case by Faggian. The infinite horizon case is more difficult to treat and it is more interesting from the point of view of economic applications, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is here performed through a nontrivial limiting procedure from the corresponding finite horizon problem., Comment: 31 pages

The paper concerns the study of equilibrium points, namely the stationary solutions to the closed loop equation, of an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. Sufficient conditions for existence of equilibrium points in the general case are given and later applied to the economic problem of optimal investment with vintage capital. Explicit computation of equilibria for the economic problem in some relevant examples is also provided. Indeed the challenging issue here is showing that a theoretical machinery, such as optimal control in infinite dimension, may be effectively used to compute solutions explicitly and easily, and that the same computation may be straightforwardly repeated in examples yielding the same abstract structure. No stability result is instead provided: the work here contained has to be considered as a first step in the direction of studying the behavior of optimal controls and trajectories in the long run.

The paper concerns the study of the Pontryagin Maximum Principle for an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. The optimal control model has already been studied both in finite and infinite horizon with Dynamic Programming methods in a series of papers by the same author, or by Faggian and Gozzi. Necessary and sufficient optimality conditions for open loop controls are established. Moreover the co-state variable is shown to coincide with the spatial gradient of the value function evaluated along the trajectory of the system, creating a parallel between Maximum Principle and Dynamic Programming. The abstract model applies, as recalled in one of the first sections, to optimal investment with vintage capital.

In this paper we consider a family of optimal control problems for economic models whose state variables are driven by Delay Differential Equations (DDE's). We consider two main examples: an AK model with vintage capital and an advertising model with delay effect. These problems are very difficult to treat for three main reasons: the presence of the DDE's, that makes them infinite dimensional; the presence of state constraints; the presence of delay in the control. Our main goal is to develop, at a first stage, the Dynamic Programming approach for this family of problems. The Dynamic Programming approach has been already used for similar problems in cases when it is possible to write explicitly the value function V. Here we deal with cases when the explicit form of V cannot be found, as most often occurs. We carefully describe the basic setting and give some first results on the solution of the Hamilton-Jacobi-Bellman (HJB) equation as a first step to find optimal strategies in closed loop form., Comment: 21 pages