Time-Varying Optimal Solutions#

A trajectory \(\{\beta^*_k\}\) is a critical path [1] of a time-varying inclusion problem if

\[\begin{align} 0 \in \sum_{i=1}^{N_s} F_{ik}(\beta^*_k) & & \forall k \in \N. \end{align}\]

Time-variation in the operators \(F_k\) may arise from tracking a moving target.

The time-varying inclusion problem may be expressed as the existence of a pair \((\beta^*, w^*)\) with

\[\begin{align} 0 \in \sum_{i=1}^{N_s} w^i_k = 0, \qquad w^i \in F_{ik}(\beta^*_k) & & \forall k \in \N. \end{align}\]

opt-syn supports time-variation in \(\beta^*\) according to linear dynamics. Time-variation in \(w^*\) is not yet supported. This restriction is equivalent to the existence of time-independent operators \(F_\bullet\) such that

\[\begin{align} F_k(\beta) = F_{\bullet i}(\beta - \beta^*_k) & & \forall k \in \N. \end{align}\]

The linear system (signal generator) governing the path \(\beta^*\) is described by

\[\begin{split}\begin{align} \mat{c}{\eta_{k+1} \\ \beta^*_k} = \mat{c}{S_\beta \\ R_\beta} \eta_{k}. \end{align}\end{split}\]

Tracking of an the optimal solution is accomplished by setting the tracking field in opt_system to a struct with fields (Sbeta, Rbeta).

Time-variation of w^* is not yet supported.