Optimal Control
Here we consider the Bolza control problem whose cost function depends on the observable values
\[\min \limits_{\{\mathbf{u}_n\}_{n = 0,1,\cdots,N-1}}
J[\{\mathbf{u}_n\}] = \Phi(\mathbf{g}(\mathbf{x}_N)) +
\sum\limits_{n = 1}^N L_n(\mathbf{g}(\mathbf{x}_n), \mathbf{u}_{n-1}).\]
\[ \mathrm{s.t.} \quad \mathbf{x}_{n+1} = \mathbf{f}(\mathbf{x}_n, \mathbf{u}_n). \]
where the initial condition, the terminal cost \(\Phi\) and the running cost \(L_n\) are given, but the dynamics \(\mathbf{f}\) is unknown. \(\mathbf{g}(\mathbf{x})\) is the observable function, and \(\{\mathbf{u}_n\}_{n = 0}^{N-1}\) the controls.
Optimal Control problems under PK-NNΒΆ
PK-NN transforms the above optimal control problem into
\[
\min_{\{\mathbf{u}_{n}\}_{n=0,1,\ldots,N-1}} J [\{\mathbf{u}_{n}\}]
= \Phi(B\hat{\Psi}_{N}) +
\sum_{n=1}^{N} L_{n} (B\hat{\Psi}_{n},\mathbf{u}_{n-1})
\]
\[
\mathrm{s.t.} \quad \hat{\Psi}_{n+1} =
K(\mathbf{u}_{n})\hat{\Psi}_{n}, \quad
\hat{\Psi}_{0}=\Psi(\mathbf{x}_{0}),
\]