5 problems

Let $d>3$ and let $f$ be the function appearing in the discrete-time $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra system. Let $C$ denote its Casimir function. An invariant is called t…

All-pass parametrization conjecture. The set of all minimal right-invertible spectral factors of $\Phi(z)$ is

Let $\Phi(z)\in\mathbb{R}(z)^{n\times n}$ be a spectrum of normal rank $\mathrm{rk}(\Phi)=r\neq 0$. Let $\mathscr{A}_p$ and $\mathscr{A}_z$ be two unmixed-symplectic sets. Let…

Consider a feedback system with $G\in\mathbf{RH}_\infty$, and let $c6$ be slope-restricted in $S[0,k]$ and monotonically increasing. The Nyquist plot of the linear part is the plot…

Lyapunov representation conjecture. There exists a symmetric matrix $P=P^{\top}$ satisfying