The Goddard Rocket Problem and Singular Thrust

The Goddard rocket problem is a classical optimal-control problem in which a vertically ascending rocket uses a finite amount of fuel to maximize altitude. Despite its simple physical statement, the optimal thrust can contain a singular interval.

This section develops:

• the one-dimensional rocket model;

• the Mayer objective;

• the Hamiltonian and costate equations;

• the thrust switching function;

• bang and singular thrust conditions;

• differentiation of the singular condition;

• bang–singular–coast structure; and

• computational implications.

Problem Statement¶

Let:

• $h(t)$ be altitude;

• $v(t)$ be vertical velocity;

• $m(t)$ be mass;

• $T(t)$ be thrust;

• $D(h,v)$ be aerodynamic drag;

• $g$ be gravitational acceleration; and

• $c$ be effective exhaust velocity.

The objective is to maximize final altitude:

Using minimization notation,

The final time is free.

Dynamics¶

The one-dimensional ascent dynamics are

The thrust is bounded:

Typical initial conditions are

A fixed terminal mass is imposed:

The final altitude and final velocity are free.

Hamiltonian¶

Because the cost is in Mayer form and there is no running cost,

Therefore,

Group the thrust-dependent terms:

Hamiltonian Constancy¶

The Hamiltonian has no explicit time dependence:

Thus,

Since the final time is free and the terminal cost has no explicit dependence on $t_f$,

Therefore,

Costate Equations¶

The costate equations are

For altitude:

For velocity:

For mass:

Terminal Transversality Conditions¶

The terminal cost is

Since $h(t_f)$ and $v(t_f)$ are free,

The terminal mass is fixed, so $\lambda_m(t_f)$ is not prescribed directly.

Switching Function¶

The Hamiltonian is linear in thrust. Define

Pontryagin’s Minimum Principle gives

Thus:

• $\psi<0$ gives maximum thrust;

• $\psi>0$ gives zero thrust; and

• $\psi=0$ is the singular candidate.

Equivalent Singular Condition¶

Multiply the switching function by $cm$:

Define

On a singular arc,

Since $c>0$ and $m>0$, this condition is equivalent to $\psi=0$.

First Derivative of the Singular Condition¶

Differentiate

Then

Substitute

where

This gives

Rearranging:

On the singular arc,

Hence the thrust-dependent term vanishes and

The first derivative still does not determine $T$.

Second Derivative and Singular Thrust¶

Differentiate $\dot{I}=0$ once more:

After substituting the state and costate dynamics, the second derivative has the general form

Therefore, the candidate singular thrust is

provided $B\neq0$.

The explicit expressions depend on the selected drag law $D(h,v)$.

The candidate is admissible only if

A Common Drag Model¶

A frequently used model is

where

Then

These expressions simplify the costate and singular-thrust equations.

Expected Optimal Thrust Structure¶

The rocket must initially generate enough thrust to lift off, so the first arc is generally

A classical optimal structure is

This is a bang–singular–bang structure, where the final “bang” is the lower thrust bound.

Entry and Exit Conditions¶

The unknown singular entry time $t_1$ and exit time $t_2$ satisfy:

The state and costate must also be continuous:

These conditions, together with the endpoint conditions and integrated dynamics, determine the full multiphase solution.

Why Maximum Thrust Followed by Zero Is Not Always Optimal¶

A naive strategy is

This ignores the interaction among:

• fuel consumption;

• atmospheric drag;

• changing mass;

• velocity-dependent drag losses; and

• the value of thrust at different altitudes.

The singular arc balances these effects. It can reduce drag losses while still using the available fuel effectively.

Numerical Formulation¶

A practical numerical solution can be posed as a multiphase problem:

1. Phase 1: $T=T_{\max}$.

2. Phase 2: $T=T_s$ and $I=\dot{I}=0$.

3. Phase 3: $T=0$.

Unknowns include:

• initial costates;

• $t_1$;

• $t_2$;

• $t_f$;

• singular-arc state and costate values; and

• possibly parameters of the drag model.

Continuity and endpoint conditions form the nonlinear residual system.

Verification Conditions¶

A computed solution should satisfy:

1. state dynamics;

2. costate dynamics;

3. thrust bounds;

4. Hamiltonian minimization;

5. $H^*(t)=0$;

6. terminal transversality;

7. $I=\dot{I}=0$ on the singular arc;

8. continuity at $t_1$ and $t_2$; and

9. a generalized Legendre–Clebsch condition.

Common Errors¶

1. Using $\partial H/\partial T=0$ and assuming it directly gives thrust.

2. Ignoring the possibility $\psi=0$ over an interval.

3. Treating an isolated switching point as a singular arc.

4. Forgetting that final time is free.

5. Omitting the terminal Hamiltonian condition.

6. Using the wrong sign in the thrust switching law.

7. Assuming a candidate singular thrust is automatically admissible.

8. Failing to enforce continuity at the singular entry and exit points.

Summary¶

1. The Goddard rocket problem maximizes terminal altitude using bounded thrust.

2. The Hamiltonian is linear in thrust.

3. The switching function is

4. Negative $\psi$ gives maximum thrust and positive $\psi$ gives zero thrust.

5. $\psi\equiv0$ defines a singular candidate.

6. The first derivative of the singular condition does not contain thrust.

7. The second derivative introduces thrust and yields a candidate singular law.

8. The optimal trajectory may have a maximum-thrust, singular-thrust, zero-thrust structure.

9. Entry and exit times must be solved as unknown events.

10. The problem illustrates how simple dynamics can produce highly nontrivial optimal controls.

Connection. Even an exact set of state–costate equations may be numerically ill conditioned, so the coupled Hamiltonian dynamics and shooting sensitivity deserve separate study.