Costate proportionality conjecture for the Douglas–Rachford projector

Let $clambda^{DR}(t)$ be the costate variable emerging from the projector onto $\mathcal{A}$ computed in the numerical projection algorithm, and let $clambda(t)$ be the costate variable associated with Problem (P). A junction time $coverline{t}_j$ is a time at which the control $u_j(t)$ simultaneously falls into two cases of the control law, corresponding to a transition between active and inactive control constraints. Let $coverline{t}_j$ be a junction time for some $u_j$, $j=1,\dots,m$, such that $b_j(coverline{t}_j)^Tclambda^{DR}(coverline{t}_j)\neq 0$, and define

$$\alpha=\dfrac{-r_j(coverline{t}_j)u_j(coverline{t}_j)}{b_j(coverline{t}_j)^Tclambda^{DR}(coverline{t}_j)}.$$

Costate proportionality conjecture. The costate variables satisfy

$$\lambda(t)=\alpha\lambda^{DR}(t).$$

The conjecture was formulated and tested using extensive numerical experiments and proposes that the costate from Problem (P) is a scalar multiple of the Douglas–Rachford projector costate at every time. Its validity beyond the reported numerical evidence remains open.