In this chapter we shall prove existence theorems for ordinary and relaxed versions of Problem 2.3.2, which we restate for the reader’s convenience.
J(ϕ, u) = g(e(ϕ)) +
f0(t, ϕ(t), u(t)) dt (5.1.1)
subject to dϕ
dt = f(t, ϕ(t), u(t)) (5.1.2)
and (t0, ϕ(t0), t1, ϕ(t1)) ∈ B u(t) ∈ Ω(t, ϕ(t)). (5.1.3)
The constraint sets Ω(t, x) depend on t and x and are not assumed to be convex.