Regularization of optimal control problems on stratified domains using additional controls

• Academic Example: We provide an academic example illustrating numerically the convergence of approximated optimal solutions (to the penalized problem) to an optimal solution of the academic example.

We consider a dynamics in the plane defined over three strata:

$$ \dot{x} = \begin{cases} g_1(x,u) & \text{if } x_2 > 0, \\ g_2(x,u) & \text{if } x_1 > 0,; x_2 < 0, \\ g_3(x,u) & \text{if } x_1 < 0,; x_2 < 0, \end{cases} $$

where

$$ g_1(x,u) = \begin{pmatrix} 0 \ -2 - u \end{pmatrix}, \quad g_2(x,u) = \begin{pmatrix} -2 + u \ x_1 - 1 \end{pmatrix}, \quad g_3(x,u) = \begin{pmatrix} -\Bigl(x_1 + \frac{1}{2}\Bigr)^2 \ -2 + u \end{pmatrix}, $$

and $u$ is a control that takes values in $U = [-1, 1]$.

In this example, the three strata are defined by the signs of the functions $\varphi_1(x) = x_1$ and $\varphi_2(x) = x_2$. Following Section #sec-partition and its notations, four sets are associated with these two functions:

$$ X_1 = { x \in \mathbb{R}^2 \mid x_1 < 0,; x_2 < 0 } $$ $$ X_2 = { x \in \mathbb{R}^2 \mid x_1 < 0,; x_2 > 0 } $$ $$ X_3 = { x \in \mathbb{R}^2 \mid x_1 > 0,; x_2 < 0 } $$ $$ X_4 = { x \in \mathbb{R}^2 \mid x_1 > 0,; x_2 > 0 } $$

which define a stratification of the plane with ( N = 2^2 = 4 ) regions:

$$ \mathbb{R}^2 = \overline{X}_1 \cup \overline{X}_2 \cup \overline{X}_3 \cup \overline{X}_4. $$

We write the dynamics as:

$$ \dot{x} = f(x,u) = \begin{cases} f_1(x,u) = g_3(x,u), & \text{if } x \in X_1, \\ f_2(x,u) = g_1(x,u), & \text{if } x \in X_2, \\ f_3(x,u) = g_2(x,u), & \text{if } x \in X_3, \\ f_4(x,u) = g_1(x,u), & \text{if } x \in X_4. \end{cases} $$

The problem of interest:

Minimize:

$$ \int_0^T x_2^2(t) , \mathrm{d}t, $$

Subject to:

$$ \begin{cases} \dot{x}(t) = f(x(t), u(t)) \quad \text{for a.e. } t \in [0, T],\\ x(0) = (1,1). \end{cases} $$

where $T=\frac{7}{2}$ and $u$ belongs to the set of time-measurable functions from $[0, \frac{7}{2}]$ to $[-1,1]$.