Bryson Denham

Bryson Denham
State dimension:	1
Differential states:	2
Discrete control functions:	1

The Bryson-Denham problem is a two-dimensional toy ODE model. It aims to minimize a quadratic Lagrange term.

The optimal integer control functions exhibits a singular arc.

Mathematical formulation

$\begin{array}{lll}\underset{u}{\min }& & \underset{0}{\overset{1}{\int }}\frac{1}{2}\cdot w(t{)}^{2}dt\\ \text{subject to}\\ \dot{x}(t)& =& v(t),\\ \dot{v}(t)& =& u(t),\\ x(0)& =& 0,\\ v(0)& =& 1,\\ x(1)& =& 0,\\ v(1)& =& -1,\\ x(t)& \le & \frac{1}{9}\end{array}$

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and discretized control for a local optimum.

Miscellaneous and Further Reading

The Bryson-Denham problem is a variation of the double integrator problem [1]. This formulation detailed description can be found in [2].

References

[1] Arthur E Bryson and Yu-Chi Ho. Applied Optimal Control: Optimization, Estimation and Control. CRC Press, 1975.
[2] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/bryson_denham/bryson_denham.html