Method:
The minimax method aims at finding a control \(v\) defined by:
\(v = arg\,min_{u\in U^K} \max_{x(\cdot)} {\cal J}_{z_0,\varepsilon}(x(\cdot),u(\cdot))\)
with:
\begin{multline}
{\cal J}_{z_0,\varepsilon}(x(\cdot),u(\cdot))\equiv \big\{\int_0^TC(x(t),u(t))dt \mid \exists d(\cdot)\in D:\\ \dot{x}(t)=f(x(t),u(t),d(t)) \mbox{ for } t\in[0,T]\ \wedge x(0)\in B(z_0,\varepsilon)\big\}.
\end{multline}
Since the minimax methods are excessively complex
(even in the sampled and finite-horizon setting),
simplified variants of the minimax problem have been developed.
We propose here such a simplified method composed of two steps.
Step 1:
In a first step, using an Euler-based symbolic computation method,
we will first obtain an upper-bound. So we get, for all \(u(\cdot)\in U^K\):
\(\max_{x(\cdot)} {\cal J}_{z_0,\varepsilon}(x(\cdot),u(\cdot))\leq {\cal K}_{z_0,\varepsilon}(u(\cdot))\),
with:
\({\cal K}_{z_0,\varepsilon}(u(\cdot))\)
\(\equiv \max_{x(\cdot)\in B(\tilde{x}_{z_0}^u(\cdot),\delta_{\varepsilon,D}^{u(\cdot)}(\cdot))}
\{\int_0^TC(x(t),u(t)) dt\}\),
where:
- \(\tilde{x}_{z_0}^u(\cdot)\) denotes Euler's approximate solution
of \(\dot{x}(t)=f(x(t),u(t),{\bf 0})\) for \(t\in[0,T]\) with
null perturbation (i.e. \(d(\cdot)=0\)) and initial condition \(z_0\in\mathbb{R}^n\),
- \(\delta_{\varepsilon,D}^{u(\cdot)}(\cdot)\) denotes the function defined in
the next section.
- \(x(\cdot)\in B(\tilde{x}_{z_0}(\cdot),\delta_{\varepsilon,D}^{u(\cdot)}(\cdot))\)
means, for all \(t\in[0,T]\):
\(x(t)\in B(\tilde{x}_{z_0}(t),\delta_{\varepsilon,D}^{u(\cdot)}(t))\). In particular \(x(0)\in B(z_0,\varepsilon)\)*.
* \(y\in B(z,a)\) with \(y,z\in\mathbb{R}^n\) and \(a\geq 0\) means \(\| y-z\| \leq a\)
where \(\|\cdot \|\) denotes the Euclidean norm.
Step 2:
In a second step, as the number of controls
\(u(\cdot)\in U^K\) is exponential in \(K\), and therefore explodes combinatorially, we will not consider the absolute minimum, but a probable near-minimum of
\({\cal K}_{z_0,\varepsilon}(u(\cdot))\).
The probably approximate near-minimum
of \({\cal K}_{z_0,\varepsilon}\)
is obtained by drawing randomly \(N\) control \(u_1,\cdots, u_N\) of \(U^K\), i.e. by generating \(N\) independent identically distributed (i.i.d.) samples \(u_1,\cdots,u_N\) of \(U^K\),
with a uniform probability (i.e. with probability
\(1/|U|^N\)) then by taking \({\cal K}_{z_0,\varepsilon}(u^*_N)\) with
\(u^*_N=arg\,min_{u_1,\cdots,u_N} {\cal K}_{z_0,\varepsilon}(u_i)\).