bryson_denham.md

The Bryson–Denham problem is a classic optimal control benchmark involving a state inequality constraint.
It models the movement of a unit mass on a frictionless plane, where the goal is to relocate the mass over a fixed time interval with minimum control effort, while ensuring the position never exceeds a certain limit.
Originally introduced in Bryson et al. 1963, it is a standard test case for evaluating the ability of numerical solvers to handle state-constrained trajectories.

Mathematical formulation

The problem can be stated as

$$ \begin{aligned} \min_{x,u} \quad & J(x,u) = \int_0^1 \frac{1}{2} u^2(t) \,\mathrm{d}t \\\ \text{s.t.} \quad & \dot{x}_1(t) = x_2(t), \quad \dot{x}_2(t) = u(t), \\\ & x_1(0) = 0, \; x_1(1) = 0, \; x_2(0) = 1, \; x_2(1) = -1, \\\ & x_1(t) \le a. \end{aligned} $$

where $x_1$ represents position, $x_2$ velocity, $u$ acceleration (control), and $a$ is the maximum allowable displacement (e.g., $a=1/9$).

Qualitative behaviour

The problem features a second-order state constraint because the control $u$ only appears in the second derivative of the constrained state $x_1$:

$$ \ddot{x}_1(t) = u(t). $$

When the trajectory is on a boundary arc ($x_1(t) = a$), the derivatives $\dot{x}_1$ and $\ddot{x}_1$ must be zero. This implies that while the constraint is active, the velocity $x_2$ is zero and the control $u$ is zero:

$$ u(t) = 0. $$

The solution structure changes based on the value of the constraint parameter $a$:

• Unconstrained Case: If $a$ is sufficiently large ($a \ge 1/4$), the state constraint is never active.
• Touch Point: For a critical value of $a$, the trajectory just touches the boundary $x_1 = a$ at a single point $t=0.5$.
• Boundary Arc: For smaller values (e.g., $a=1/9$), the trajectory remains on the boundary for a finite time interval. During this interval, the control vanishes.

Characteristics

• Linear–quadratic dynamics with a second-order state inequality constraint.
• Demonstrates a "bang-off-bang" control structure where the control is zero on the constraint boundary.
• Used to test the accuracy of state constraint handling and the detection of switching times.

References

• Bryson, A.E., Denham, W.F., & Dreyfus, S.E. (1963). Optimal programming problems with inequality constraints I: necessary conditions for extremal solutions.
  AIAA Journal. doi.org/10.2514/3.2107
• Dymos Documentation: Bryson-Denham Problem. dymos/examples/bryson_denham