Linear Quadratic Regulator

Linear Quadratic Regulator
State dimension:	1
Differential states:	1
Discrete control functions:	1

The Linear Quadratic Regulator problem is a popular one-dimensional toy ODE model which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term.

The optimal control function exhibits a singular arc.

Mathematical formulation

$\begin{array}{lll}\underset{x,w}{\min }& & \underset{0}{\overset{10}{\int }}10\cdot (x(t)-3{)}^2+0.1\cdot u(t{)}^{2}dt\\ \text{subject to}\\ \dot{x}(t)& =& a\cdot x(t)+b\cdot u(t),\\ x(0)& =& 1\end{array}$

Parameters

We choose $a=-1$ and $b=1$.

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 problem description and further references can be found on Wikipedia [1].

References

[1] https://en.wikipedia.org/wiki/Linear%E2%80%93quadratic_regulator