20 problems

Homogeneous Lyapunov converse conjecture. For homogeneous vector fields, asymptotic stability implies existence of a polynomial Lyapunov function. Equivalently, existence of a homo…

The paper studies slowly time-varying systems for which uniform global asymptotic stability and input-to-state stability can be established using Lyapunov functions, including syst…

The paper considers methods for constructing closed-form strict Lyapunov functions for hybrid systems from suitable nonstrict Lyapunov functions, including systems with decay condi…

Let $\dot{x}=f(x)$ be a globally asymptotically stable homogeneous polynomial vector field, and let a homogeneous polynomial Lyapunov function mean a homogeneous polynomial certify…

Let ${\bf p}(t)$ be a solution of the mean-field ODE system - with initial datum ${\bf p}(0)\in\mathcal{A}$, and let $\mathcal{H}_{ab}[{\bf p}]$ be the candidate Lyapunov functiona…

Consider the globally rationally stable polynomial vector field … and the SOS tests for rational stability applied globally on $\Omega=\mathbb{R}^3$, with fixed gain bound $M=1.5$…

Lyapunov-function conjecture. The function

Let $\mathcal{V}=(\nu_1,\ldots,\nu_N)$ be the natural-frequency vector, let $\mathcal{D}(\mathcal{V})$ denote its frequency-diameter quantity, and let $\kappa_c(\mathcal{V})$ be th…

Let $\nu$ be a probability measure on $\mathbb{R}^3$, let $\bar{\nu}$ be its $(0,1)$-marginal, and let $\pi$ and $\bar{\pi}$ denote the invariant measure of the Markov local-field…

Let $d4a2,d4a2$ be path-complete Lyapunov functions in $d52f\mathrm{pc}(\Sigma)$ satisfying Assumption. Suppose that $d53d$ is a subset of linear invertible maps. The composition-l…

Let $G$ be the synchronisation diffeomorphism of the torus considered in the paper, and let $S$ and $R$ be the open basins of attraction of its corresponding asymptotically stable…

Weak-Lyapunov-functional conjecture. Either there is a constant $c\ge\frac12$ such that, whenever $\kappa>c\mathcal{D}(\mathcal{V})$, both models admit weak Lyapunov functionals, o…

Generalization conjecture. The results on the existence and properties of a discrete Lyapunov function for cyclic systems can be further generalized to these time-dependent-delay s…

Let $U\subseteq\mathbb{R}^d$ be an open domain, and consider the Fleming–Viot particle system with McKean–Vlasov dynamics whose particles interact through a drift depending on thei…

Let a mass-action system (MAS) admit a stable positive equilibrium. For a suitable set of boundary complexes, consider the Lyapunov function partial differential equations induced…

IQC improvement conjecture. By using off-by-one IQCs or other IQCs developed in the cited work within this Lyapunov framework, one can further improve the numerical bounds.

Consider the polynomial system … with $f(0)=0$, and let $S_f$ be a nonempty, bounded region of attraction of the equilibrium $0$. For $r\geq 0$, let $B_r(0)=\{x:u_r(x)\geq 0\}$, wh…

Let a polynomial dynamical system have a region of attraction, and suppose polynomial Lyapunov functions are sought on expanding bounded regions. Asymptotic infeasibility conjectur…

Divisibility conjecture. Every element of the matrix $\mathfrak Q$ is divisible by $\mathfrak Q_{11}$.

A strict GFN model consists of a family of trajectories satisfying conditions (a)–(e), including the concatenation property. Condition (e) conjecture. The condition described immed…