Appl Math Optim 51:123–162 (2005)
2005 Springer Science+Business Media, Inc.
Regular Solutions of First-Order Hamilton–Jacobi Equations
for Boundary Control Problems and Applications to Economics
Dipartimento di Matematica, Universit`a di Pisa,
Via F. Buonarroti 2, I-56127 Pisa, Italy
Abstract. This is the ﬁrst of two papers regarding a family of linear convex control
problems in Hilbert spaces and the related Hamilton–Jacobi–Bellman equations. The
framework is motivated by an application to boundary control of a PDE modeling
investments with vintage capital. Existence and uniqueness of a strong solution
(namely, the limit of classic solutions of approximating equations, introduced by
Barbu and Da Prato) is investigated. Moreover, such a solution is proved to be
the space variable.
Key Words. Linear convex control, Boundary control, Hamilton–Jacobi–Bellman
AMS Classiﬁcation. 49J15, 49J20, 35B37.
This is the ﬁrst of two papers regarding a family of linear convex boundary control
problems in Hilbert spaces and the related Hamilton–Jacobi–Bellman (brieﬂy, HJB)
In this ﬁrst part we study existence and uniqueness for a Hamilton–Jacobi equation
of the following type:
(t, x) + F(t,ϕ
(t, x)) − (A
x | ϕ
= g(t, x ),
ϕ(0, x) = ϕ
This work has been partially supported by the “Landesforschungsschwerpunkt Evolutionsgleichungen”
ﬁnanced by the State of Baden-W¨urttemberg.