148 problems

Let a hybrid dynamical system with inputs and disturbances be controlled by the proposed state-feedback law with pointwise minimum norm, and formulate the robust stabilization prob…

Continuous-time dissipativity conjecture. Continuous-time counterparts of these discrete-time results should be feasible.

Let $W$ be a Wiener process, and consider adaptive algorithms that construct a trajectory $h(t)$ using only the observed values $W(s)$ for $s\leq t$. The energy of a trajectory is…

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 $N$ be a positive integer, let $s_n$ be the switching parameters from the paper, and let $A(N+1;r,s,t)$ be the corresponding alternating factorization. For…

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…

Consider a two-dimensional slip system with convex polytope … where $\det(S^\alpha)=0$ and $0\in\operatorname{int}(\mathcal{U})$. Let $f_0,f_1,f_2$ be the numbers of vertices, edge…

Let $\mathcal{U}=\operatorname{conv}(X_1,\dots,X_m)\subseteq\mathop{\mathfrak{sl}}\nolimits(2)$, with $\det(X_j)=0$ for every $j$. Piecewise-constant control conjecture. Optimal co…

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

Localization conjecture. Analogous spatial localization phenomena should also occur beyond the setting covered by the existing theory, including for two-dimensional hyperbolic prob…

In a defined contribution pension plan, let the withdrawal control specify withdrawals between the minimum amount $q_{\min}$ and maximum amount $q_{\max}$ at discrete rebalancing a…

State-error convergence conjecture. The order of convergence in Lemma can be doubled, yielding an approximation of order

Weak-assumption conjecture. The results in the paper can be derived under assumptions on the coefficients that are similarly weak to those used in the cited prior work.

Consider the unconstrained optimal output sequence $Y^\infty$ arising in the recurrent neural network training problem, and suppose that $Y^\infty$ is non-unique. Let $\mathcal{T}$…

The paper considers recurrent neural network training with truncated backpropagation through time (TBPTT), together with optimal output sequences that may satisfy a geometric relat…

Let $a(t)$ be a $T$-periodic function and decompose it as … where $A>0$ is constant and … For a $T$-periodic fishing strategy $f(t)$, let $\mu_0$ denote the maximal parameter along…

Let $t$ be a time, let $h_t$ be the history at time $t$, and let $G_{t,\beta^v,\sigma^\nu}(t,h_t)$ and $G_{t,\sigma^v,\sigma^\nu}(t,h_t)$ denote the expected utilities when the foc…

In the discretionary-bonus contracting model, consider the sign-on bonus as the number of bonus payments increases. Vanishing sign-on bonus conjecture. As the number of payments be…

Let $u_1\leq u_2<w_0=\infty$ be the thresholds in the collaborating business-lines model, let $M_1<\infty$ be the upper bound on the reinsurance control, and let $\pi^*_1(x)$ denot…

Let $u_b$ denote the upper control bound, and consider the optimal controls for the bang-bang control problem studied in the paper. Bang-bang control conjecture. Optimal controls a…

Consider a control component with activity sets … where … and suppose that the cost function is strongly convex on each restriction $g_i|_{[a_i^k,b_i^k]}$. Tangent-compatibility co…

Let $(t_0,gamma_0)\in\mathcal{O}$ be an initial condition for the optimal control problem, and let … be an initial condition for the gradient descent flow … Assume that…

Minimum attention is defined by the control-change functional … Here $u$ is a control varying over state space $\Omega$ and time interval $[0,T]$, and the functional penalizes chan…

The paper considers higher-order necessary conditions for minima in optimal control problems, including Goh and Legendre–Clebsch conditions obtained using set-valued Lie brackets o…