August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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Chapter 5
Existence Theorems; Non-Compact
Constraints
5.1 Introduction
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.
Minimize
J(ϕ, u) = g(e(ϕ)) +
Z
t
1
t
0
f
0
(t, ϕ(t), u(t)) dt (5.1.1)
subject to
dϕ
dt
= f(t, ϕ(t), u(t)) (5.1.2)
and
(t
0
, ϕ(t
0
), t
1
, ϕ(t
1
)) ∈ 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.
In Chapter 4, when studying this problem, the constraint sets Ω(t) were
assumed to depend on t alone and to be compact. The weak compactness of
relaxed controls in this case enabled us to pattern the pro ...
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