Equality Path Constraints in Optimal Control
This section introduces path constraints in optimal control and develops the necessary conditions for equality path constraints. It begins by interpreting the system dynamics themselves as path constraints, then distinguishes differential, algebraic, control, and state constraints. The augmented cost functional and augmented Hamiltonian are introduced, followed by the necessary conditions for control-only, state-only, and mixed state–control equality path constraints.
What Is a Path Constraint? ¶ A path constraint is a condition that must hold pointwise along the trajectory. The familiar dynamics
x ˙ ( t ) = f ( x ( t ) , u ( t ) , t ) \dot{\boldsymbol{x}}(t)=\boldsymbol{f}(\boldsymbol{x}(t),\boldsymbol{u}(t),t) x ˙ ( t ) = f ( x ( t ) , u ( t ) , t ) can be rewritten as
f ( x , u , t ) − x ˙ = 0 . \boxed{
\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t)-\dot{\boldsymbol{x}}=\boldsymbol{0}.
} f ( x , u , t ) − x ˙ = 0 . Thus, the dynamics themselves are equality path constraints because they must hold at every time.
A more general differential constraint may be written as
ψ ( x , x ˙ , u , t ) = 0 . \boxed{
\boldsymbol{\psi}(\boldsymbol{x},\dot{\boldsymbol{x}},\boldsymbol{u},t)=\boldsymbol{0}.
} ψ ( x , x ˙ , u , t ) = 0 . This is called a differential form because the derivative need not be explicitly isolated.
The inequality generalization is
ψ min ≤ ψ ( x , x ˙ , u , t ) ≤ ψ max , \boxed{
\boldsymbol{\psi}_{\min}
\leq
\boldsymbol{\psi}(\boldsymbol{x},\dot{\boldsymbol{x}},\boldsymbol{u},t)
\leq
\boldsymbol{\psi}_{\max},
} ψ m i n ≤ ψ ( x , x ˙ , u , t ) ≤ ψ m a x , which is a differential inclusion.
In classical optimal-control theory, the state rates are usually handled through explicit first-order dynamics, while additional constraints are treated as algebraic path constraints.
Algebraic Path Constraints ¶ An algebraic path constraint does not explicitly involve x ˙ \dot{\boldsymbol{x}} x ˙
c ( u , t ) = 0 , c ( x , t ) = 0 , c ( x , u , t ) = 0 , \begin{aligned}
\boldsymbol{c}(\boldsymbol{u},t)&=\boldsymbol{0},\\
\boldsymbol{c}(\boldsymbol{x},t)&=\boldsymbol{0},\\
\boldsymbol{c}(\boldsymbol{x},\boldsymbol{u},t)&=\boldsymbol{0},
\end{aligned} c ( u , t ) c ( x , t ) c ( x , u , t ) = 0 , = 0 , = 0 , for equality constraints, or corresponding inequalities.
Original Augmented Cost Functional ¶ Consider a Bolza problem with endpoint constraints. The standard augmented functional is
J A = Φ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) − ν T ϕ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) + ∫ t 0 t f [ L ( x , u , t ) + λ T ( f ( x , u , t ) − x ˙ ) ] d t . \begin{aligned}
J_A
={}&
\Phi(\boldsymbol{x}(t_0),t_0,\boldsymbol{x}(t_f),t_f)
-
\boldsymbol{\nu}^{\mathsf{T}}
\boldsymbol{\phi}(\boldsymbol{x}(t_0),t_0,\boldsymbol{x}(t_f),t_f)
\nonumber\\
&+
\int_{t_0}^{t_f}
\left[
L(\boldsymbol{x},\boldsymbol{u},t)
+
\boldsymbol{\lambda}^{\mathsf{T}}
\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 T ϕ ( x ( t 0 ) , t 0 , x ( t f ) , t f ) + ∫ t 0 t f [ L ( x , u , t ) + λ T ( f ( x , u , t ) − x ˙ ) ] d t . At any feasible solution, the added constraint terms vanish. Therefore, the original and augmented costs agree.
Control Equality Path Constraints ¶ Consider
c ( u , t ) = 0 , \boxed{
\boldsymbol{c}(\boldsymbol{u},t)=\boldsymbol{0},
} c ( u , t ) = 0 , with
c : R n u × R → R p . \boldsymbol{c}:\mathbb{R}^{n_u}\times\mathbb{R}\rightarrow\mathbb{R}^{p}. c : R n u × R → R p . Introduce a time-varying multiplier
μ ( t ) ∈ R p . \boldsymbol{\mu}(t)\in\mathbb{R}^p. μ ( t ) ∈ R p . The augmented functional becomes
J A = Φ − ν T ϕ + ∫ t 0 t f [ L + λ T ( f − x ˙ ) − μ T c ] d t . \begin{aligned}
J_A
={}&
\Phi
-
\boldsymbol{\nu}^{\mathsf{T}}\boldsymbol{\phi}
\nonumber\\
&+
\int_{t_0}^{t_f}
\left[
L
+
\boldsymbol{\lambda}^{\mathsf{T}}(\boldsymbol{f}-\dot{\boldsymbol{x}})
-
\boldsymbol{\mu}^{\mathsf{T}}\boldsymbol{c}
\right]\,\mathrm{d} t.
\end{aligned} J A = Φ − ν T ϕ + ∫ t 0 t f [ L + λ T ( f − x ˙ ) − μ T c ] d t . Augmented Hamiltonian ¶ Define the ordinary Hamiltonian
H ( x , u , λ , t ) = L ( x , u , t ) + λ T f ( x , u , t ) . \mathcal{H}(\boldsymbol{x},\boldsymbol{u},\boldsymbol{\lambda},t)
=
L(\boldsymbol{x},\boldsymbol{u},t)
+
\boldsymbol{\lambda}^{\mathsf{T}}\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t). H ( x , u , λ , t ) = L ( x , u , t ) + λ T f ( x , u , t ) . The augmented Hamiltonian is
H A = H − μ T c . \boxed{
\mathcal{H}_A
=
\mathcal{H}
-
\boldsymbol{\mu}^{\mathsf{T}}\boldsymbol{c}.
} H A = H − μ T c . The necessary conditions have the same structure as before, but with H A \mathcal{H}_A H A H \mathcal{H} H
Necessary Conditions for Control Equality Constraints ¶ The state equation remains
x ˙ = ∂ H A ∂ λ = f ( x , u , t ) . \dot{\boldsymbol{x}}
=
\frac{\partial \mathcal{H}_A}{\partial \boldsymbol{\lambda}}
=
\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t). x ˙ = ∂ λ ∂ H A = f ( x , u , t ) . Because c \boldsymbol{c} c x \boldsymbol{x} x
∂ H A ∂ x = ∂ H ∂ x . \frac{\partial \mathcal{H}_A}{\partial \boldsymbol{x}}
=
\frac{\partial \mathcal{H}}{\partial \boldsymbol{x}}. ∂ x ∂ H A = ∂ x ∂ H . Therefore,
λ ˙ = − ∂ H ∂ x . \boxed{
\dot{\boldsymbol{\lambda}}
=
-
\frac{\partial \mathcal{H}}{\partial \boldsymbol{x}}.
} λ ˙ = − ∂ x ∂ H . The stationarity condition changes:
∂ H A ∂ u = ∂ H ∂ u − ( ∂ c ∂ u ) T μ = 0 . \boxed{
\frac{\partial \mathcal{H}_A}{\partial \boldsymbol{u}}
=
\frac{\partial \mathcal{H}}{\partial \boldsymbol{u}}
-
\left(
\frac{\partial \boldsymbol{c}}{\partial \boldsymbol{u}}
\right)^{\mathsf{T}}\boldsymbol{\mu}
=
\boldsymbol{0}.
} ∂ u ∂ H A = ∂ u ∂ H − ( ∂ u ∂ c ) T μ = 0 . The path constraint itself must also hold:
c ( u , t ) = 0 . \boxed{
\boldsymbol{c}(\boldsymbol{u},t)=\boldsymbol{0}.
} c ( u , t ) = 0 . Thus, the unknown multiplier μ ( t ) \boldsymbol{\mu}(t) μ ( t )
Differential-Algebraic Structure ¶ For a control equality constraint, the optimality system includes:
state differential equations;
costate differential equations;
stationarity equations;
algebraic path constraints;
endpoint constraints and transversality conditions.
The result is generally a differential-algebraic boundary-value problem.
State Equality Path Constraints ¶ Now consider
c ( x , t ) = 0 . \boxed{
\boldsymbol{c}(\boldsymbol{x},t)=\boldsymbol{0}.
} c ( x , t ) = 0 . The augmented Hamiltonian remains
H A = H − μ T c . \mathcal{H}_A
=
\mathcal{H}
-
\boldsymbol{\mu}^{\mathsf{T}}\boldsymbol{c}. H A = H − μ T c . The costate equation becomes
λ ˙ = − ∂ H A ∂ x = − ∂ H ∂ x + ( ∂ c ∂ x ) T μ . \boxed{
\dot{\boldsymbol{\lambda}}
=
-
\frac{\partial \mathcal{H}_A}{\partial \boldsymbol{x}}
=
-
\frac{\partial \mathcal{H}}{\partial \boldsymbol{x}}
+
\left(
\frac{\partial \boldsymbol{c}}{\partial \boldsymbol{x}}
\right)^{\mathsf{T}}\boldsymbol{\mu}.
} λ ˙ = − ∂ x ∂ H A = − ∂ x ∂ H + ( ∂ x ∂ c ) T μ . However, because c \boldsymbol{c} c u \boldsymbol{u} u
∂ H A ∂ u = ∂ H ∂ u . \frac{\partial \mathcal{H}_A}{\partial \boldsymbol{u}}
=
\frac{\partial \mathcal{H}}{\partial \boldsymbol{u}}. ∂ u ∂ H A = ∂ u ∂ H . Therefore, the original stationarity condition contains no explicit information about the effect of the state constraint on the control.
Constraint Order ¶ To expose the control, differentiate the state constraint with respect to time.
Starting from
c ( x , t ) = 0 , \boldsymbol{c}(\boldsymbol{x},t)=\boldsymbol{0}, c ( x , t ) = 0 , differentiate:
d d t c ( x , t ) = ∂ c ∂ x x ˙ + ∂ c ∂ t = 0 . \frac{\,\mathrm{d}}{\,\mathrm{d} t}\boldsymbol{c}(\boldsymbol{x},t)
=
\frac{\partial \boldsymbol{c}}{\partial \boldsymbol{x}}\dot{\boldsymbol{x}}
+
\frac{\partial \boldsymbol{c}}{\partial t}
=
\boldsymbol{0}. d t d c ( x , t ) = ∂ x ∂ c x ˙ + ∂ t ∂ c = 0 . Substitute the dynamics:
c ( 1 ) = ∂ c ∂ x f ( x , u , t ) + ∂ c ∂ t = 0 . \boldsymbol{c}^{(1)}
=
\frac{\partial \boldsymbol{c}}{\partial \boldsymbol{x}}
\boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t)
+
\frac{\partial \boldsymbol{c}}{\partial t}
=
\boldsymbol{0}. c ( 1 ) = ∂ x ∂ c f ( x , u , t ) + ∂ t ∂ c = 0 . If the control still does not appear explicitly, differentiate again. Continue until the control appears.
Symbolically,
c ( 0 ) ( x , t ) = 0 , c ( 1 ) ( x , t ) = 0 , ⋮ c ( q − 1 ) ( x , t ) = 0 , c ( q ) ( x , u , t ) = 0 . \begin{aligned}
\boldsymbol{c}^{(0)}(\boldsymbol{x},t)&=\boldsymbol{0},\\
\boldsymbol{c}^{(1)}(\boldsymbol{x},t)&=\boldsymbol{0},\\
&\vdots\\
\boldsymbol{c}^{(q-1)}(\boldsymbol{x},t)&=\boldsymbol{0},\\
\boldsymbol{c}^{(q)}(\boldsymbol{x},\boldsymbol{u},t)&=\boldsymbol{0}.
\end{aligned} c ( 0 ) ( x , t ) c ( 1 ) ( x , t ) c ( q − 1 ) ( x , t ) c ( q ) ( x , u , t ) = 0 , = 0 , ⋮ = 0 , = 0 . The final relation is control dependent and can be used in the stationarity system.
Why All Lower-Order Conditions Matter ¶ It is not sufficient to enforce only the highest derivative condition. If
c ( q ) = 0 , \boldsymbol{c}^{(q)}=\boldsymbol{0}, c ( q ) = 0 , then integration introduces constants. Therefore, the original condition and all lower derivatives must also be satisfied:
c ( j ) = 0 , j = 0 , … , q . \boxed{
\boldsymbol{c}^{(j)}=\boldsymbol{0},
\qquad
j=0,\ldots,q.
} c ( j ) = 0 , j = 0 , … , q . Mixed State–Control Equality Constraints ¶ For
c ( x , u , t ) = 0 , \boxed{
\boldsymbol{c}(\boldsymbol{x},\boldsymbol{u},t)=\boldsymbol{0},
} c ( x , u , t ) = 0 , the augmented Hamiltonian is again
H A = H − μ T c . \mathcal{H}_A
=
\mathcal{H}-\boldsymbol{\mu}^{\mathsf{T}}\boldsymbol{c}. H A = H − μ T c . The costate equation is
λ ˙ = − ∂ H ∂ x + ( ∂ c ∂ x ) T μ . \boxed{
\dot{\boldsymbol{\lambda}}
=
-
\frac{\partial \mathcal{H}}{\partial \boldsymbol{x}}
+
\left(
\frac{\partial \boldsymbol{c}}{\partial \boldsymbol{x}}
\right)^{\mathsf{T}}\boldsymbol{\mu}.
} λ ˙ = − ∂ x ∂ H + ( ∂ x ∂ c ) T μ . The stationarity condition is
∂ H ∂ u − ( ∂ c ∂ u ) T μ = 0 . \boxed{
\frac{\partial \mathcal{H}}{\partial \boldsymbol{u}}
-
\left(
\frac{\partial \boldsymbol{c}}{\partial \boldsymbol{u}}
\right)^{\mathsf{T}}\boldsymbol{\mu}
=
\boldsymbol{0}.
} ∂ u ∂ H − ( ∂ u ∂ c ) T μ = 0 . The algebraic constraint must also hold:
c ( x , u , t ) = 0 . \boxed{
\boldsymbol{c}(\boldsymbol{x},\boldsymbol{u},t)=\boldsymbol{0}.
} c ( x , u , t ) = 0 . Summary ¶ The dynamics are equality path constraints.
Differential forms generalize explicit state equations.
Differential inclusions generalize differential equalities to inequalities.
Algebraic path constraints are imposed pointwise along the trajectory.
Equality path constraints introduce time-varying Lagrange multipliers.
The augmented Hamiltonian is
H A = H − μ T c . \mathcal{H}_A=\mathcal{H}-\boldsymbol{\mu}^{\mathsf{T}}\boldsymbol{c}. H A = H − μ T c . A control equality constraint modifies stationarity but not the costate equation.
A state equality constraint modifies the costate equation but initially provides no direct control relation.
State constraints must be differentiated until the control appears.
Equality path constraints lead naturally to differential-algebraic optimality systems.
Connection. Equality constraints establish the multiplier framework; inequalities add activity decisions, complementary slackness, and junction behavior.