Appl Math Optim 45:75–98 (2002)
2002 Springer-Verlag New York Inc.
A Class of Hamilton–Jacobi Equations
with Unbounded Coefﬁcients in Hilbert Spaces
Department of Mathematics, Tokyo Metropolitan University,
Hachioji, Tokyo 192-0397, Japan
Communicated by M. Nisio
Abstract. We establish existence and comparison theorems for a class of
Hamilton–Jacobi equations. The class of Hamilton–Jacobi equations includes and
is broader than those studied in . We apply the existence and uniqueness results
to characterizing the value functions associated with the optimal control of systems
governed by partial differential equations of parabolic type.
Key Words. Hamilton–Jacobi equations, Value functions, Perron’s method,
Optimal control problems, Viscosity solutions, Invariant sets.
AMS Classiﬁcation. 35K55, 35R15, 47N20, 49L20, 49L25.
Let H be a real Hilbert space with inner product ·,· and norm |·|. We are concerned
with the Hamilton–Jacobi equations
λu +∂ϕ(x )+ B(x), Du+F (x , Du) = 0inS (1.1)
+∂ϕ(x) + B(x, t), Du+F(x, t, Du) = 0inS × (0, T ). (1.2)
Here ϕ: H → [0,∞] is a proper lower semicontinuous convex function, ∂ϕ denotes
the subdifferential of ϕ, λ>0 is a given number, S is a closed subset of H which
satisﬁes S ⊂
D(∂ϕ), and, moreover, S ∩ D(∂ϕ) is dense in S. u(x) or u(x, t ) represents
a real unknown function on S or S × [0, T ], and Du denotes the Fr´echet derivative in