Augmentation, Costates, and the Hamiltonian Formulation
A general optimal control problem contains a performance index, differential equations, endpoint constraints, and possibly path constraints. The objective cannot be minimized independently of these constraints because an unconstrained minimizer of the performance index may violate the dynamics or fail to satisfy the boundary conditions.
The central idea developed in this section is augmentation . Each constraint is appended to the performance index using an associated Lagrange multiplier. The resulting augmented functional contains the original constraints explicitly, allowing the entire problem to be treated using the calculus of variations.
This construction introduces two important multiplier variables:
the time-varying multiplier λ ( t ) \boldsymbol{\lambda}(t) λ ( t ) costate or adjoint ;
the static multiplier ν \boldsymbol{\nu} ν
The Hamiltonian
H ( x , u , λ , t ) = L ( x , u , t ) + λ ⊤ f ( x , u , t ) H(\boldsymbol{x},\boldsymbol{u},\boldsymbol{\lambda},t)
=
L(\boldsymbol{x},\boldsymbol{u},t)
+
\boldsymbol{\lambda}^{\top}\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t) H ( x , u , λ , t ) = L ( x , u , t ) + λ ⊤ f ( x , u , t ) then emerges naturally as a compact representation of the running cost and dynamic constraint.
General Optimal Control Problem ¶ Consider the Bolza-form problem
min u ( ⋅ ) , t 0 , t f J , \min_{\boldsymbol{u}(\cdot),\,t_0,\,t_f}
J, u ( ⋅ ) , t 0 , t f min J , where
J = Φ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) + ∫ t 0 t f L ( x ( t ) , u ( t ) , t ) d t . \begin{aligned}
J
&=
\Phi
\bigl(
\boldsymbol{x}(t_0),
t_0,
\boldsymbol{x}(t_f),
t_f
\bigr)
\nonumber\\
&\quad
+
\int_{t_0}^{t_f}
L
\bigl(
\boldsymbol{x}(t),
\boldsymbol{u}(t),
t
\bigr)\,\mathrm{d} t.
\end{aligned} J = Φ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) + ∫ t 0 t f L ( x ( t ) , u ( t ) , t ) d t . The dynamics are
x ˙ ( t ) = f ( x ( t ) , u ( t ) , t ) . \dot{\boldsymbol{x}}(t)
=
\boldsymbol{f}
\bigl(
\boldsymbol{x}(t),
\boldsymbol{u}(t),
t
\bigr). x ˙ ( t ) = f ( x ( t ) , u ( t ) , t ) . The endpoint constraints are
ϕ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) = 0 . \boldsymbol{\phi}
\bigl(
\boldsymbol{x}(t_0),
t_0,
\boldsymbol{x}(t_f),
t_f
\bigr)
=
\boldsymbol{0}. ϕ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) = 0 . Optional path constraints may also be present:
c ( x ( t ) , u ( t ) , t ) ≤ 0 . \boldsymbol{c}
\bigl(
\boldsymbol{x}(t),
\boldsymbol{u}(t),
t
\bigr)
\le
\boldsymbol{0}. c ( x ( t ) , u ( t ) , t ) ≤ 0 . For the derivation in this section, path constraints are omitted to avoid introducing complementary-slackness conditions and additional multipliers.
Why the Dynamics Are Constraints ¶ A differential equation is a constraint because it restricts the state derivative at every instant.
Equation (4)
f ( x , u , t ) − x ˙ = 0 . \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t)
-
\dot{\boldsymbol{x}}
=
\boldsymbol{0}. f ( x , u , t ) − x ˙ = 0 . This form makes the constraint character explicit. For every time t t t
x ˙ ( t ) \dot{\boldsymbol{x}}(t) x ˙ ( t ) cannot be chosen arbitrarily. It must equal the right-hand side of the system dynamics.
Augmentation with Lagrange Multipliers ¶ For each equality constraint, introduce a Lagrange multiplier.
The dynamic constraint receives a time-varying multiplier
λ ( t ) ∈ R n . \boldsymbol{\lambda}(t)\in\mathbb{R}^n. λ ( t ) ∈ R n . The endpoint constraint receives a static multiplier
ν ∈ R p . \boldsymbol{\nu}\in\mathbb{R}^p. ν ∈ R p . The augmented functional is defined by
J a = Φ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) − ν ⊤ ϕ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) + ∫ t 0 t f [ L ( x , u , t ) + λ ⊤ ( f ( x , u , t ) − x ˙ ) ] d t . \begin{aligned}
J_a
&=
\Phi
\bigl(
\boldsymbol{x}(t_0),
t_0,
\boldsymbol{x}(t_f),
t_f
\bigr)
-
\boldsymbol{\nu}^{\top}
\boldsymbol{\phi}
\bigl(
\boldsymbol{x}(t_0),
t_0,
\boldsymbol{x}(t_f),
t_f
\bigr)
\nonumber\\
&\quad
+
\int_{t_0}^{t_f}
\left[
L(\boldsymbol{x},\boldsymbol{u},t)
+
\boldsymbol{\lambda}^{\top}
\left(
\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t)
-
\dot{\boldsymbol{x}}
\right)
\right]\,\mathrm{d} t.
\end{aligned} J a = Φ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) − ν ⊤ ϕ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) + ∫ t 0 t f [ L ( x , u , t ) + λ ⊤ ( f ( x , u , t ) − x ˙ ) ] d t . On every feasible process,
f − x ˙ = 0 , \boldsymbol{f}-\dot{\boldsymbol{x}}=\boldsymbol{0}, f − x ˙ = 0 , and
ϕ = 0 . \boldsymbol{\phi}=\boldsymbol{0}. ϕ = 0 . Therefore,
for every feasible solution. In particular,
J a ∗ = J ∗ . J_a^*=J^*. J a ∗ = J ∗ . Why Augmentation Does Not Change the Optimal Cost ¶ The purpose of augmentation is not to modify the physical problem. It is to construct an unconstrained variational representation of the constrained problem.
At a feasible solution,
ν ⊤ ϕ = 0 , λ ⊤ ( f − x ˙ ) = 0. \begin{aligned}
\boldsymbol{\nu}^{\top}\boldsymbol{\phi}&=0,\\
\boldsymbol{\lambda}^{\top}
\left(
\boldsymbol{f}-\dot{\boldsymbol{x}}
\right)&=0.
\end{aligned} ν ⊤ ϕ λ ⊤ ( f − x ˙ ) = 0 , = 0. Hence, all augmented terms vanish on the feasible trajectory. The multipliers enlarge the set of unknowns, but they do not alter the original objective value at feasibility.
The Costate and Endpoint Multiplier ¶ The costate is time dependent because the dynamic constraint must hold separately at every time. Each instant has its own local multiplier associated with dynamic feasibility.
Unlike λ ( t ) \boldsymbol{\lambda}(t) λ ( t ) ν \boldsymbol{\nu} ν
Dimensional Bookkeeping ¶ Let
x ( t ) ∈ R n , u ( t ) ∈ R m , f ( x , u , t ) ∈ R n , ϕ ∈ R p . \begin{aligned}
\boldsymbol{x}(t)&\in\mathbb{R}^n,\\
\boldsymbol{u}(t)&\in\mathbb{R}^m,\\
\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t)&\in\mathbb{R}^n,\\
\boldsymbol{\phi}&\in\mathbb{R}^p.
\end{aligned} x ( t ) u ( t ) f ( x , u , t ) ϕ ∈ R n , ∈ R m , ∈ R n , ∈ R p . Since
λ ⊤ ( f − x ˙ ) \boldsymbol{\lambda}^{\top}
\left(
\boldsymbol{f}-\dot{\boldsymbol{x}}
\right) λ ⊤ ( f − x ˙ ) must be scalar,
λ ( t ) ∈ R n . \boldsymbol{\lambda}(t)\in\mathbb{R}^n. λ ( t ) ∈ R n . Similarly, because
ν ⊤ ϕ \boldsymbol{\nu}^{\top}\boldsymbol{\phi} ν ⊤ ϕ must be scalar,
ν ∈ R p . \boldsymbol{\nu}\in\mathbb{R}^p. ν ∈ R p . The endpoint mapping has the form
ϕ : R n × R × R n × R → R p . \boldsymbol{\phi}:
\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}
\rightarrow
\mathbb{R}^p. ϕ : R n × R × R n × R → R p . Functional Versus Function ¶ The quantity J a J_a J a
It depends on the entire functions
x ( ⋅ ) , u ( ⋅ ) , λ ( ⋅ ) , \boldsymbol{x}(\cdot),
\qquad
\boldsymbol{u}(\cdot),
\qquad
\boldsymbol{\lambda}(\cdot), x ( ⋅ ) , u ( ⋅ ) , λ ( ⋅ ) , and returns a scalar.
This distinction determines the correct stationarity operation:
δ J a = 0. \delta J_a=0. δ J a = 0. One does not take an ordinary derivative of J a J_a J a
Extremum Versus Extremal ¶ An extremum is a point that minimizes or maximizes an ordinary function.
An extremal is a function or trajectory satisfying first-order variational stationarity conditions.
Thus, the augmented optimal control derivation seeks an extremal process:
( x ∗ ( ⋅ ) , u ∗ ( ⋅ ) , λ ∗ ( ⋅ ) ) . \left(
\boldsymbol{x}^*(\cdot),
\boldsymbol{u}^*(\cdot),
\boldsymbol{\lambda}^*(\cdot)
\right). ( x ∗ ( ⋅ ) , u ∗ ( ⋅ ) , λ ∗ ( ⋅ ) ) . The first-order conditions identify stationary candidates but do not by themselves establish whether the candidate is a minimum or maximum.
Definition of the Hamiltonian ¶ Define the Hamiltonian
H ( x , u , λ , t ) = L ( x , u , t ) + λ ⊤ f ( x , u , t ) . \boxed{
H(\boldsymbol{x},\boldsymbol{u},\boldsymbol{\lambda},t)
=
L(\boldsymbol{x},\boldsymbol{u},t)
+
\boldsymbol{\lambda}^{\top}
\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t)
}. H ( x , u , λ , t ) = L ( x , u , t ) + λ ⊤ f ( x , u , t ) . Then the augmented functional becomes
J a = Φ − ν ⊤ ϕ + ∫ t 0 t f [ H − λ ⊤ x ˙ ] d t . \begin{aligned}
J_a
&=
\Phi
-
\boldsymbol{\nu}^{\top}\boldsymbol{\phi}
\nonumber\\
&\quad
+
\int_{t_0}^{t_f}
\left[
H
-
\boldsymbol{\lambda}^{\top}\dot{\boldsymbol{x}}
\right]\,\mathrm{d} t.
\end{aligned} J a = Φ − ν ⊤ ϕ + ∫ t 0 t f [ H − λ ⊤ x ˙ ] d t . This is only a change of notation. No new assumption has been introduced.
Quantities That May Vary Independently ¶ Before taking the first variation, identify all independently variable quantities.
For the unrestricted endpoint problem, these are
x ( ⋅ ) , u ( ⋅ ) , λ ( ⋅ ) , ν , t 0 , t f , x ( t 0 ) , x ( t f ) . \boldsymbol{x}(\cdot),
\quad
\boldsymbol{u}(\cdot),
\quad
\boldsymbol{\lambda}(\cdot),
\quad
\boldsymbol{\nu},
\quad
t_0,
\quad
t_f,
\quad
\boldsymbol{x}(t_0),
\quad
\boldsymbol{x}(t_f). x ( ⋅ ) , u ( ⋅ ) , λ ( ⋅ ) , ν , t 0 , t f , x ( t 0 ) , x ( t f ) . More precisely, the endpoint values and endpoint times are linked by the endpoint variation identities
δ x 0 = δ x ( t 0 ) + x ˙ ( t 0 ) δ t 0 , \begin{aligned}
\delta\boldsymbol{x}_0
&=
\delta\boldsymbol{x}(t_0)
+
\dot{\boldsymbol{x}}(t_0)\delta t_0,
\end{aligned} δ x 0 = δ x ( t 0 ) + x ˙ ( t 0 ) δ t 0 , δ x f = δ x ( t f ) + x ˙ ( t f ) δ t f . \begin{aligned}
\delta\boldsymbol{x}_f
&=
\delta\boldsymbol{x}(t_f)
+
\dot{\boldsymbol{x}}(t_f)\delta t_f.
\end{aligned} δ x f = δ x ( t f ) + x ˙ ( t f ) δ t f . Thus, the derivation must carefully distinguish:
Variation of the Endpoint Part ¶ Define the endpoint part of the augmented functional as
Ψ = Φ − ν ⊤ ϕ . \Psi
=
\Phi
-
\boldsymbol{\nu}^{\top}\boldsymbol{\phi}. Ψ = Φ − ν ⊤ ϕ . Then
δ Ψ = δ Φ − δ ( ν ⊤ ϕ ) . \delta\Psi
=
\delta\Phi
-
\delta
\left(
\boldsymbol{\nu}^{\top}\boldsymbol{\phi}
\right). δ Ψ = δ Φ − δ ( ν ⊤ ϕ ) . We derive these two pieces separately.
Variation of the Mayer Cost ¶ The Mayer cost is
Φ = Φ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) . \Phi
=
\Phi
\bigl(
\boldsymbol{x}(t_0),
t_0,
\boldsymbol{x}(t_f),
t_f
\bigr). Φ = Φ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) . Its first variation is
δ Φ = ∂ Φ ∂ x ( t 0 ) δ x ( t 0 ) + ∂ Φ ∂ t 0 δ t 0 + ∂ Φ ∂ x ( t f ) δ x ( t f ) + ∂ Φ ∂ t f δ t f . \begin{aligned}
\delta\Phi
&=
\frac{\partial\Phi}
{\partial\boldsymbol{x}(t_0)}
\delta\boldsymbol{x}(t_0)
+
\frac{\partial\Phi}{\partial t_0}
\delta t_0
\nonumber\\
&\quad
+
\frac{\partial\Phi}
{\partial\boldsymbol{x}(t_f)}
\delta\boldsymbol{x}(t_f)
+
\frac{\partial\Phi}{\partial t_f}
\delta t_f.
\end{aligned} δ Φ = ∂ x ( t 0 ) ∂ Φ δ x ( t 0 ) + ∂ t 0 ∂ Φ δ t 0 + ∂ x ( t f ) ∂ Φ δ x ( t f ) + ∂ t f ∂ Φ δ t f . Under the row-gradient convention,
∂ Φ ∂ x ( t 0 ) ∈ R 1 × n , \frac{\partial\Phi}{\partial\boldsymbol{x}(t_0)}
\in\mathbb{R}^{1\times n}, ∂ x ( t 0 ) ∂ Φ ∈ R 1 × n , while
δ x ( t 0 ) ∈ R n × 1 , \delta\boldsymbol{x}(t_0)\in\mathbb{R}^{n\times1}, δ x ( t 0 ) ∈ R n × 1 , so their product is scalar.
Variation of the Endpoint Constraint Term ¶ Using the product rule,
δ ( ν ⊤ ϕ ) = δ ν ⊤ ϕ + ν ⊤ δ ϕ . \delta
\left(
\boldsymbol{\nu}^{\top}\boldsymbol{\phi}
\right)
=
\delta\boldsymbol{\nu}^{\top}\boldsymbol{\phi}
+
\boldsymbol{\nu}^{\top}\delta\boldsymbol{\phi}. δ ( ν ⊤ ϕ ) = δ ν ⊤ ϕ + ν ⊤ δ ϕ . The first variation of ϕ \boldsymbol{\phi} ϕ
δ ϕ = ∂ ϕ ∂ x ( t 0 ) δ x ( t 0 ) + ∂ ϕ ∂ t 0 δ t 0 + ∂ ϕ ∂ x ( t f ) δ x ( t f ) + ∂ ϕ ∂ t f δ t f . \begin{aligned}
\delta\boldsymbol{\phi}
&=
\frac{\partial\boldsymbol{\phi}}
{\partial\boldsymbol{x}(t_0)}
\delta\boldsymbol{x}(t_0)
+
\frac{\partial\boldsymbol{\phi}}{\partial t_0}
\delta t_0
\nonumber\\
&\quad
+
\frac{\partial\boldsymbol{\phi}}
{\partial\boldsymbol{x}(t_f)}
\delta\boldsymbol{x}(t_f)
+
\frac{\partial\boldsymbol{\phi}}{\partial t_f}
\delta t_f.
\end{aligned} δ ϕ = ∂ x ( t 0 ) ∂ ϕ δ x ( t 0 ) + ∂ t 0 ∂ ϕ δ t 0 + ∂ x ( t f ) ∂ ϕ δ x ( t f ) + ∂ t f ∂ ϕ δ t f . The Jacobian dimensions are
∂ ϕ ∂ x ( t 0 ) ∈ R p × n , ∂ ϕ ∂ x ( t f ) ∈ R p × n . \begin{aligned}
\frac{\partial\boldsymbol{\phi}}
{\partial\boldsymbol{x}(t_0)}
&\in\mathbb{R}^{p\times n},\\
\frac{\partial\boldsymbol{\phi}}
{\partial\boldsymbol{x}(t_f)}
&\in\mathbb{R}^{p\times n}.
\end{aligned} ∂ x ( t 0 ) ∂ ϕ ∂ x ( t f ) ∂ ϕ ∈ R p × n , ∈ R p × n . Therefore,
ν ⊤ ∂ ϕ ∂ x ( t 0 ) δ x ( t 0 ) \boldsymbol{\nu}^{\top}
\frac{\partial\boldsymbol{\phi}}
{\partial\boldsymbol{x}(t_0)}
\delta\boldsymbol{x}(t_0) ν ⊤ ∂ x ( t 0 ) ∂ ϕ δ x ( t 0 ) has dimensions
( 1 × p ) ( p × n ) ( n × 1 ) , (1\times p)(p\times n)(n\times1), ( 1 × p ) ( p × n ) ( n × 1 ) , and is scalar.
Combined Variation of the Endpoint Terms ¶ Substituting (34) (37) (38) (32)
δ Ψ = [ ∂ Φ ∂ x ( t 0 ) − ν ⊤ ∂ ϕ ∂ x ( t 0 ) ] δ x ( t 0 ) + [ ∂ Φ ∂ t 0 − ν ⊤ ∂ ϕ ∂ t 0 ] δ t 0 + [ ∂ Φ ∂ x ( t f ) − ν ⊤ ∂ ϕ ∂ x ( t f ) ] δ x ( t f ) + [ ∂ Φ ∂ t f − ν ⊤ ∂ ϕ ∂ t f ] δ t f − δ ν ⊤ ϕ . \begin{aligned}
\delta\Psi
&=
\left[
\frac{\partial\Phi}
{\partial\boldsymbol{x}(t_0)}
-
\boldsymbol{\nu}^{\top}
\frac{\partial\boldsymbol{\phi}}
{\partial\boldsymbol{x}(t_0)}
\right]
\delta\boldsymbol{x}(t_0)
\nonumber\\
&\quad
+
\left[
\frac{\partial\Phi}{\partial t_0}
-
\boldsymbol{\nu}^{\top}
\frac{\partial\boldsymbol{\phi}}{\partial t_0}
\right]
\delta t_0
\nonumber\\
&\quad
+
\left[
\frac{\partial\Phi}
{\partial\boldsymbol{x}(t_f)}
-
\boldsymbol{\nu}^{\top}
\frac{\partial\boldsymbol{\phi}}
{\partial\boldsymbol{x}(t_f)}
\right]
\delta\boldsymbol{x}(t_f)
\nonumber\\
&\quad
+
\left[
\frac{\partial\Phi}{\partial t_f}
-
\boldsymbol{\nu}^{\top}
\frac{\partial\boldsymbol{\phi}}{\partial t_f}
\right]
\delta t_f
\nonumber\\
&\quad
-
\delta\boldsymbol{\nu}^{\top}\boldsymbol{\phi}.
\end{aligned} δ Ψ = [ ∂ x ( t 0 ) ∂ Φ − ν ⊤ ∂ x ( t 0 ) ∂ ϕ ] δ x ( t 0 ) + [ ∂ t 0 ∂ Φ − ν ⊤ ∂ t 0 ∂ ϕ ] δ t 0 + [ ∂ x ( t f ) ∂ Φ − ν ⊤ ∂ x ( t f ) ∂ ϕ ] δ x ( t f ) + [ ∂ t f ∂ Φ − ν ⊤ ∂ t f ∂ ϕ ] δ t f − δ ν ⊤ ϕ . This completes the variation of all terms outside the integral.
Why Endpoint Times Have Two Effects ¶ The endpoint time t 0 t_0 t 0
directly as an argument of Φ \Phi Φ ϕ \boldsymbol{\phi} ϕ
indirectly through the evaluated state x ( t 0 ) \boldsymbol{x}(t_0) x ( t 0 )
Therefore, perturbing t 0 t_0 t 0
The endpoint relation
δ x 0 = δ x ( t 0 ) + x ˙ ( t 0 ) δ t 0 \delta\boldsymbol{x}_0
=
\delta\boldsymbol{x}(t_0)
+
\dot{\boldsymbol{x}}(t_0)\delta t_0 δ x 0 = δ x ( t 0 ) + x ˙ ( t 0 ) δ t 0 is the mechanism that combines these effects later in the derivation.
The same reasoning applies at t f t_f t f
Perturbation-Parameter Interpretation ¶ A useful way to understand variations is to introduce a scalar perturbation parameter ϵ \epsilon ϵ
For example,
t 0 ( ϵ ) = t 0 + ϵ η t 0 . t_0(\epsilon)
=
t_0+\epsilon\eta_{t_0}. t 0 ( ϵ ) = t 0 + ϵ η t 0 . Then
δ t 0 = d t 0 ( ϵ ) d ϵ ∣ ϵ = 0 = η t 0 . \delta t_0
=
\left.
\frac{\,\mathrm{d} t_0(\epsilon)}
{\,\mathrm{d}\epsilon}
\right|_{\epsilon=0}
=
\eta_{t_0}. δ t 0 = d ϵ d t 0 ( ϵ ) ∣ ∣ ϵ = 0 = η t 0 . Similarly,
x ϵ ( t ) = x ( t ) + ϵ η x ( t ) , \boldsymbol{x}_{\epsilon}(t)
=
\boldsymbol{x}(t)
+
\epsilon\boldsymbol{\eta}_x(t), x ϵ ( t ) = x ( t ) + ϵ η x ( t ) , so
δ x ( t ) = η x ( t ) . \delta\boldsymbol{x}(t)
=
\boldsymbol{\eta}_x(t). δ x ( t ) = η x ( t ) . This construction clarifies that every independently variable quantity must receive its own admissible perturbation.
What Remains for the Full Derivation ¶ The complete first variation of J a J_a J a
∫ t 0 t f [ H − λ ⊤ x ˙ ] d t . \int_{t_0}^{t_f}
\left[
H
-
\boldsymbol{\lambda}^{\top}\dot{\boldsymbol{x}}
\right]\,\mathrm{d} t. ∫ t 0 t f [ H − λ ⊤ x ˙ ] d t . That derivation must account for:
moving lower and upper integration limits;
variations in x ( t ) \boldsymbol{x}(t) x ( t )
variations in u ( t ) \boldsymbol{u}(t) u ( t )
variations in λ ( t ) \boldsymbol{\lambda}(t) λ ( t )
variation of x ˙ ( t ) \dot{\boldsymbol{x}}(t) x ˙ ( t )
integration by parts;
endpoint state and time identities.
Once the complete first variation is separated into independent variations, the coefficients yield:
Chapter Summary ¶ The main results are:
The state equation is an equality constraint imposed at every time:
f ( x , u , t ) − x ˙ = 0 . \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t)
-
\dot{\boldsymbol{x}}
=
\boldsymbol{0}. f ( x , u , t ) − x ˙ = 0 . The constrained optimal control problem can be converted into an augmented variational problem using Lagrange multipliers.
The costate λ ( t ) \boldsymbol{\lambda}(t) λ ( t )
The static vector ν \boldsymbol{\nu} ν
The augmented functional is
J a = Φ − ν ⊤ ϕ + ∫ t 0 t f [ L + λ ⊤ ( f − x ˙ ) ] d t . \begin{aligned}
J_a
&=
\Phi-\boldsymbol{\nu}^{\top}\boldsymbol{\phi}
+
\int_{t_0}^{t_f}
\left[
L+
\boldsymbol{\lambda}^{\top}
(\boldsymbol{f}-\dot{\boldsymbol{x}})
\right]\,\mathrm{d} t.
\end{aligned} J a = Φ − ν ⊤ ϕ + ∫ t 0 t f [ L + λ ⊤ ( f − x ˙ ) ] d t . On feasible processes,
The Hamiltonian is
H = L + λ ⊤ f . H=L+\boldsymbol{\lambda}^{\top}\boldsymbol{f}. H = L + λ ⊤ f . The augmented functional becomes
J a = Φ − ν ⊤ ϕ + ∫ t 0 t f [ H − λ ⊤ x ˙ ] d t . J_a
=
\Phi-\boldsymbol{\nu}^{\top}\boldsymbol{\phi}
+
\int_{t_0}^{t_f}
\left[
H-\boldsymbol{\lambda}^{\top}\dot{\boldsymbol{x}}
\right]\,\mathrm{d} t. J a = Φ − ν ⊤ ϕ + ∫ t 0 t f [ H − λ ⊤ x ˙ ] d t . The correct stationarity operation is
δ J a = 0. \delta J_a=0. δ J a = 0. The endpoint variation must include the variations of endpoint states, endpoint times, and the endpoint multiplier.
The complete variation of the endpoint terms is given by (42)
Connection. After defining the augmented functional and its multipliers, we can begin the central derivation by varying its endpoint contribution.