48 problems

Let $L>0$, $M\in\mathbb{R}\setminus\{0\}$, and $\varepsilon>0$. Consider the transport–diffusion equation $u_t+M u_x=\varepsilon u_{xx}$ on $(0,T)\times(0,L)$, with a Dirichlet bou…

Periodic-solution conjecture. If all solutions are uniform-bounded and uniform ultimate-bounded, then there exists a nonzero periodic solution with period $T$, which need not be un…

Let $G$ be an LTI system, let $k_N$ denote its Nyquist gain, and let $0<k<k_N$. Suppose there is no suitable LTI Zames-Falb multiplier for $G+1/k$. Zames-Falb multiplier equality c…

Let $G$ be a stable transfer function, let $k_N$ be its Nyquist value, and let $\phi$ be a memoryless nonlinearity in $S[0,k]$. Kalman conjecture. The feedback interconnection of…

Let $\gamma$ and $\delta$ be real analytic functions with $\gamma_4\neq0$, let $k$ and $\ell$ be integers, and let $\mathcal{E}^{\gamma,\delta}_{k,\ell}$ denote the partial differe…

Output-Lagrange stability and output asymptotic gain imply OL-IOS. Under these assumptions, the system is OL-IOS.

Let $\Sigma(A,[0,B_1])$ and $\Sigma(A,[0,B_2])$ denote the two linear systems associated with a bilinear system, and suppose that the bilinear system satisfies the hypothesis of Se…

Polynomial growth conjecture. There exist $N\in\mathbb{N}$ and $T,C>0$ such that

Let $m,p,n$ be positive integers with $n=mp$. Let $(\bar{A},\bar{B},\bar{C})$ be a minimal realization of the system represented through a coprime factorization $D^{-1}(s)N(s)$, wh…

Let $s_1,\ldots,s_n$ be distinct real numbers, and let $\gamma(s)$ be a parameterization of a rational normal curve in $\mathbf{R}^{m+p}$ of degree $m+p-1$. For integers $k_i$, def…

Let $H$ be the stabilizing positive semidefinite solution of the reduced discrete algebraic Riccati equation under Assumption, and let ${\mathcal V}_{\mathcal L}$ denote the struct…

Extension of the projected closed-loop ISS result. Under standard model-order-reduction assumptions, the closed-loop dynamics on the submanifold…

OZF multiplier conjecture. If such a multiplier exists, then $y_2\in\mathcal{L}_2\cap\mathcal{L}_{\infty}$.

Altshuller multiplier conjecture. If such a multiplier exists, then there exists a nonzero periodic solution with period $T$, which need not be unique or attracting.

Converse parametrization conjecture. If a Brunovsky transformation $(T,F,G)$ transforms the staircase linear system into Brunovsky form, then there exists an observation matrix $C$…

Consider the normalized delay differential equation referred to as, with solution $u(t;\beta,w_0)$ for parameter $\beta$ and initial velocity $w_0$. Let the comparison properties b…

Let $f(t,A,w)$ be the population distribution, let $f_\ell(z)$ be the leaders' opinion distribution, and define the population's average opinion and the leaders' average opinion by…

Let $\mathcal{C}_n$ be the space of full-order dynamic policies for $\mathcal{H}_\infty$ control. A policy $\mathsf{K}\in\mathcal{C}_n$ is Clarke stationary when…

Let $\mathcal{C}_n$ be the space of full-order dynamic policies, and let $\mathcal{C}_{\mathrm{nd}}$ be the subset of non-degenerate policies, namely those admitting a certificate…

The preceding theorem concerns a collection of generators for controlled Markovian quantum dynamics, with the relevant notation and generator class defined there. Converse characte…

Let $\Lambda_N>0$, $d\in\mathbb{N}$, $q\in[1,+\infty)$, and consider the linear difference delay system with delay parameters and matrices $A_N$ and $B$ described above. Let…

Detectability redundancy conjecture. The detectability condition in Theorem is redundant: if $(\Gamma,\Pi,\hat{\Sigma})$ solves the optimization in, then…

Let $(\mathcal{M},g)$ be a Riemannian manifold. The controlled wave maps equation is a map $\phi:\mathbb{R}\times\mathbb{S}^1\to\mathcal{M}$, with states in…

Two-delay minimal spectral abscissa conjecture. For arbitrary $a_1,a_2\in\mathbb{R}$, the minimal spectral abscissa is

Let a control system possess a control admissible symmetry group. EDFL-to-CFL conjecture. If the control system is EDFL, then it must be CFL. The conjecture concerns the relation b…