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AOE 5244
HW Set 6 - Solution
- The question deals with the extension required to allow for explicit de-
pendence on time (the independent variable) in the statement of the optimal
control problem. Specifically, we have dynamics given by
x˙ = f (x, t, u)
u ∈ Ω ⊂ IR
m , with given initial condition (x(t 0 ) = x 0 ), the end-condition
θ ı
(x(t f
), t f
) = 0, ı = 1, 2 ,... , q,
and the Mayer cost-functional
g(x(t f
), t f
To treat this problem we introduce an augmented state vector
xˆ =
[
x
x n+
]
and transform to a standard problem, where the t variable does not explicitly
appear. To this end we define
xˆ =
f (ˆx, u) =
[
f (x, x n+
, u)
]
and
θ ı
(x(t f
), t f
) = θ ı
(x(t f
), x n+
alongwith a modified Mayer cost-functional
gˆ(x(tf ), tf ).
The variational Hamiltonian for the new problem is written in terms of
the extended costate
λ = (λ,
λn+1)
H(
λ, ˆx, u) = λ
T f (x, x n+
, u) +
λ n+
where the first term on the right is the variational Hamiltonian for the
original problem. The adjoint equations are
λ = −
(
H
∂ x
) T
(
∂H
∂ x
) T
λ n+
H
∂ x n+
∂H
∂ t
Since the modified problem does not explicitly include the independent vari-
abel (t), we have
H(t) = 0 constant.
Since
H is constant alongan extremal and since
H = H +
λ n+
we find
d
H
d t
d H
d t
λ n+
or
d H
d t
λn+1 =
∂ H
∂ t
The transversality conditions for the first n components of
λ are as be-
fore, but the last of these is
λ n+
(t f
) = λ 0
∂ ˆg
∂ x n+
q ∑
ı=
ν ı
θ ı
∂ x n+
Which can be written in terms of the original problem as
−H(t f
) = λ 0
∂ g
∂ t
q ∑
ı=
νı
∂ θ ı
∂ t
