Robbins

Robbins
State dimension:	1
Differential states:	3
Discrete control functions:	1

The Robbins problem is a classical benchmark in optimal control. This description is taken from [1].

Mathematical formulation

$\begin{array}{lll}\underset{u}{\min }& & \underset{0}{\overset{T}{\int }}(\alpha \cdot x_1(t)+\beta \cdot x_1(t{)}^2+\gamma \cdot u(t{)}^2)\\ \text{subject to}\\ \dot{x_1}(t)& =& x_2(t)),\\ \dot{x_2}(t)& =& x_3(t),\\ \dot{x_3}(t)& =& u(t),\\ x_1(t)& \ge & 0\mathrm{\forall }t\in [0,T]\\ x(0)& =& (1,-2,0{)}^T,\\ x(T)& =& (0,0,0{)}^T\\ \end{array}$

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$k_1$	1	Interaction between $x_1$ and $x_2$
$k_2$	10	Interaction between $x_1$ and $x_2$
$k_3$	1	Growth of $x_3$ under complementary control
$t_{\mathrm{f}}$	1	Horizon of the control problem

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

This formulation and a detailed description can be found in [1].

References

[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/jackson.md