Singular Optimal Control Theory

A singular arc is a nonzero-duration interval on which the standard first-order optimality conditions fail to determine the optimal control. Singular arcs are important because an optimal control may lie strictly inside the admissible set while still being unobtainable from the usual stationarity equation.

This section develops:

strong and weak Hamiltonian optimality conditions;
the definition of singular intervals;
the switching function;
differentiation-based recovery of singular controls;
order of a singular arc;
generalized Legendre–Clebsch conditions;
junction conditions between bang and singular arcs; and
numerical difficulties associated with singular solutions.

Strong and Weak Optimality Conditions¶

Consider

\min_{\boldsymbol{u}(\cdot)} J = \Phi(\boldsymbol{x}(t_f),t_f) + \int_{t_0}^{t_f} L(\boldsymbol{x},\boldsymbol{u},t)\,\mathrm{d} t

(1)

subject to

\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t), \qquad \boldsymbol{u}(t)\in\mathcal{U}.

(2)

The Hamiltonian is

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).

(3)

If the optimal control lies in the interior of $\mathcal{U}$ and the Hamiltonian is differentiable in $\boldsymbol{u}$ , then the stationarity condition is

\boxed{ \frac{\partial H}{\partial \boldsymbol{u}} = \boldsymbol{0}. }

(4)

However, this implication is one-way:

\frac{\partial H}{\partial \boldsymbol{u}}=\boldsymbol{0} \quad\Rightarrow\quad \boldsymbol{u}^*\text{ may be interior},

(5)

but an interior control does not guarantee that stationarity determines it uniquely.

The general pointwise minimum condition is

\boxed{ H(\boldsymbol{x}^*,\boldsymbol{u}^*,\boldsymbol{\lambda}^*,t) \leq H(\boldsymbol{x}^*,\boldsymbol{u},\boldsymbol{\lambda}^*,t) \quad \forall \boldsymbol{u}\in\mathcal{U}. }

(6)

Definition of a Singular Arc¶

Let

\boldsymbol{\sigma}(t) = \frac{\partial H}{\partial \boldsymbol{u}}

(7)

be the switching function.

A singular interval is a finite interval

, \qquad t_2>t_1,

(8)

on which

\boxed{ \boldsymbol{\sigma}(t)\equiv\boldsymbol{0}. }

(9)

On this interval, neither the sign of the switching function nor the stationarity condition directly determines the control.

An isolated zero,

\boldsymbol{\sigma}(t_s)=\boldsymbol{0},

(10)

is generally a switching time, not a singular arc.

Autonomous Hamiltonians¶

Along an optimal solution,

\frac{\,\mathrm{d} H^*}{\,\mathrm{d} t} = \frac{\partial H}{\partial t}.

(11)

If $H$ is not an explicit function of time,

\frac{\partial H}{\partial t}=0,

(12)

and therefore

\boxed{ H^*(t)=\text{constant}. }

(13)

This property is useful because repeated time differentiation of a constant Hamiltonian produces zero:

\frac{\,\mathrm{d}^q H^*}{\,\mathrm{d} t^q}=0, \qquad q\geq1.

(14)

In practice, singular-control derivations are more often formulated by differentiating the switching function rather than the Hamiltonian itself.

Control-Affine Systems¶

Consider

\dot{\boldsymbol{x}} = \boldsymbol{f}_0(\boldsymbol{x},t) + \boldsymbol{B}(\boldsymbol{x},t)\boldsymbol{u}

(15)

with running cost independent of $\boldsymbol{u}$ .

Then

H = L(\boldsymbol{x},t) + \boldsymbol{\lambda}^{\mathsf{T}}\boldsymbol{f}_0 + \boldsymbol{\lambda}^{\mathsf{T}}\boldsymbol{B}\boldsymbol{u}.

(16)

The switching function is

\boxed{ \boldsymbol{\sigma} = \boldsymbol{B}^{\mathsf{T}}\boldsymbol{\lambda}. }

(17)

For a scalar bounded control,

u_{\min}\leq u\leq u_{\max},

(18)

Hamiltonian minimization gives

u^* = \begin{cases} u_{\min}, & \sigma>0,\\ u_{\max}, & \sigma<0,\\ \text{undetermined}, & \sigma=0. \end{cases}

(19)

The third case is the singular candidate.

Recovering the Singular Control¶

On a singular interval,

\sigma=0.

(20)

Differentiate until the control appears explicitly:

\begin{aligned} \sigma&=0,\\ \dot{\sigma}&=0,\\ \ddot{\sigma}&=0,\\ &\vdots\\ \sigma^{(q)}&=0. \end{aligned}

(21)

Suppose the first derivative containing $u$ is

\sigma^{(q)} = a(\boldsymbol{x},\boldsymbol{\lambda},t) + b(\boldsymbol{x},\boldsymbol{\lambda},t)u.

(22)

Then the candidate singular control is

\boxed{ u_s = -\frac{a}{b}, }

(23)

provided $b\neq0$ .

The control must satisfy

u_s\in[u_{\min},u_{\max}].

(24)

If it violates the bounds, the candidate singular arc is inadmissible.

Order of a Singular Arc¶

For many control-affine problems, the control first appears in an even derivative:

\frac{\,\mathrm{d}^{2r}\sigma}{\,\mathrm{d} t^{2r}}.

(25)

Then the singular arc is said to have order $r$ .

A common structure is

\frac{\,\mathrm{d}^{2r}\sigma}{\,\mathrm{d} t^{2r}} = A(\boldsymbol{x},\boldsymbol{\lambda},t) + B(\boldsymbol{x},\boldsymbol{\lambda},t)u.

(26)

The singular control becomes

u_s=-B^{-1}A.

(27)

Generalized Legendre–Clebsch Condition¶

A singular control satisfying the differentiated switching equations is only a candidate.

For a minimization problem, a necessary higher-order condition is

\boxed{ (-1)^r \frac{\partial}{\partial u} \left( \frac{\,\mathrm{d}^{2r}}{\,\mathrm{d} t^{2r}} \frac{\partial H}{\partial u} \right) \geq0. }

(28)

This is the generalized Legendre–Clebsch condition.

Failure of this condition rules out the candidate singular arc as a local minimizer.

Bang, Singular, and Boundary Arcs¶

A constrained optimal trajectory can contain:

a lower-bound bang arc, $u=u_{\min}$ ;
an upper-bound bang arc, $u=u_{\max}$ ;
a singular arc, $u=u_s$ ;
a state-constraint boundary arc; or
combinations of these.

A typical structure is

u_{\max} \longrightarrow u_s \longrightarrow u_{\min}.

(29)

The entry and exit times are unknown event times that must be solved as part of the optimal-control problem.

Junction Conditions¶

At a junction between bang and singular arcs, the state is normally continuous:

\boldsymbol{x}(t_s^-)=\boldsymbol{x}(t_s^+).

(30)

The costate is also continuous when no interior point constraint or impulse is present:

\boldsymbol{\lambda}(t_s^-)=\boldsymbol{\lambda}(t_s^+).

(31)

The switching conditions satisfy

\sigma(t_s)=0,

(32)

and, depending on the singular order,

\dot{\sigma}(t_s)=0, \quad \ldots, \quad \sigma^{(2r-1)}(t_s)=0.

(33)

These equations help determine the unknown switching times and interface values.

Why Singular Arcs Are Difficult Numerically¶

Singular arcs are difficult because:

the switching interval is unknown;
$\sigma$ may be numerically small without being identically zero;
differentiation magnifies numerical noise;
bang and singular arcs may connect through sensitive junctions;
a discretized solution may approximate a singular arc by rapid switching;
the singular control may violate its bounds; and
multiple singular arcs may occur.

A Practical Singular-Arc Analysis Procedure¶

Form the Hamiltonian.
Derive state and costate equations.
Derive the switching function.
Determine the bang controls from the sign of the switching function.
Test whether $\sigma\equiv0$ is theoretically possible.
Differentiate $\sigma$ until the control appears.
Solve for the singular control.
check admissibility and higher-order conditions.
derive junction conditions.
solve for switching times and trajectory parameters.
verify all endpoint and Hamiltonian conditions.

Summary¶

Singular arcs occur when first-order conditions do not determine the control.
An isolated zero of the switching function is not a singular interval.
The control is recovered by differentiating the switching function until $u$ appears.
The resulting singular control is only a candidate.
Admissibility and generalized Legendre–Clebsch conditions must be checked.
Bang–singular–bang trajectories require unknown junction times.
Numerical singular-arc detection is intrinsically difficult.

Connection. The Goddard rocket problem turns the abstract singular-arc conditions into a physical thrust profile with bang, singular, and coast phases.