  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.

Optimization and Control