20 problems

Let $v(x)$ be an antisymmetric velocity field with $v(x)>0$ for $x>0$, and suppose … Let $b(x)$ denote the diffusion-off boundary. Near-field scaling conjecture. Then … This predic…

Let $A_n(s)$ be the alternating factorization defined in the paper, and let $\widetilde{\operatorname{SL}}(2,\mathbb{R})$ be the universal covering group of…

Let $n\in\mathbb{N}$ with $n\geq3$. Let $A_n(s)$ and $A(n;r,s,t)$ denote the alternating factorizations used in the paper, let $s_n$ be the associated switching parameters, and let…

In the terminology of Constantin Carathéodory, two problems are Carathéodory-equivalent when their respective Lagrangians differ by a total derivative; the corresponding extremals…

Let $\bar{\eta}_1$ and $\bar{\eta}_2$ be cumulative-effort constraints satisfying $0<\bar{\eta}_1<\bar{\eta}_2$, and let $\mu_{\alpha_{opt},\bar{\eta}_i}$ be the measures associate…

Approximate multiplier convergence conjecture. Under the direct-adjoining and secant-function-augmentation formulations, $\tilde{\eta}_i$ approaches $\eta_i$ as $\rho_i\to 0$:

Costate proportionality conjecture. The costate variables satisfy

Divergence conjecture. Such collocations will likely diverge as the mesh size grows large. For Gauss/Gauss-Radau discretizations, the divergence is attributed to the divergence est…

Let $\Omega(t)$ be the evolving set with boundary parametrized by $x(t,\xi)$, and let the optimal control be the normal velocity control acting along this boundary. Denote the curv…

Let $A\in\mathbb{R}^{n \times n}$, $B\in\mathbb{R}^{n \times m}$, $T>0$, $\beta\in\\{1,2,\dots,m-1\\}$, and $\alpha_j\in(0,T]$ for $j=1,2,\dots,m$, as in Problem … exist. The exist…

Consider sets $Xt subseteqmathbb{R}^{ntimes T}$ for $0\leq tleq T$, a control set $Usubseteqmathbb{R}^m$, and cost functions … Let $\mathcal{F}$ denote the set of pairs $(f,x_0)$ su…

Cut-time equality conjecture. For any $\lambda \in C$

Exponential turnpike conjecture. There are constants $K>0$ and $\mu>0$, independent of $T$, such that

No-conjugate-points conjecture. For any $(x_0,\mu)\in M\times\mathbb{R}^{*}$, the conjugate locus is empty.

Consider the two-impulse space interception problem for a free-flight ballistic missile and the corresponding one-impulse problem, with the same numerical initial data and constrai…

Consider the population process with unbounded harvesting governed by … where $(Z_t)_{t\geq0}$ is a non-negative, increasing, right-continuous process adapted to…

Let $H(x,p,t)$ be a Hamiltonian satisfying $H(x,p,t)\in C^2$ and convexity with respect to $p$, and let (A5) denote the assumption referenced in the source. Let $\varphi$ be the vi…

Subdifferential condition. For almost every $t\in(0,\infty)$,

Let $(m,m\mathbf v)$ be a minimizer of the dual optimal-control problem, and let $\eta$ be the induced probability measure on continuous trajectories $\Gamma=C([0,T];\mathbb T^N)$,…

Gaussian decay conjecture. There exists a constant $\rho>0$ such that, for sufficiently small $h$,