1D Integrator

1D Integrator

Here is a simple deterministic problem, without discrete modes for for illustrating the process law. We consider a classical 1D integrator.

$\displaystyle \left \lbrace \begin{array}{l} \min \frac{1}{2}\int_{0}^{T} u(t)^2 dt +\frac{1}{2}{X(T)}^2\\ \dot{x}=u(t)\\ u, x \in[-1, 1] \\ x(0)=0.1, T = 1. \end{array} \right .$

The following figure shows the value function at initial time. As expected, the value function is 0 when x=0, and increases symmetrically when getting farther form it.

We set the state probabilities at initial time according to a discrete uniform distribution

$\displaystyle\mathcal{P}_0(x_i)_{i=1, \ldots, N_{grid}} = \frac{1}{N_{grid}}$

The following animated figure shows the evolution of the state probabilities over the grid for each time step. We observe a concentration of the distribution towards x=0.

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