9 problems

Time-dependent Riccati feedback conjecture. The optimal control retains the same feedback form as in the time-independent case, with $\bm{\chi}^{\mathbb{X}}$ solving the Riccati eq…

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…

Let $\mathbf{P}_{\lambda}(\cdot)\equiv\left[\mathcal{P}_{\lambda}(\cdot,1),\cdots,\mathcal{P}_{\lambda}(\cdot,D)\right]$ denote the collection of solutions associated with the coup…

In the Riccati formulation, let $W = \zeta\gamma^{-1}$ be the backward Riccati matrix and let $Q = \gamma\zeta^{-1}$ be the forward Riccati matrix. Riccati-divergence conjecture. D…

Let $K_{[t_0,T]}$ be the reproducing kernel of the LQ state space, let $K_{[t_0,T]}(t_0,t_0)^{\ominus}$ denote its pseudoinverse for the associated seminorm, and let…

Let $\Phi^N(t)$ be the finite-dimensional matrix-valued function defined by the Riccati system in the source, with entries $\Phi^{N,i,j}(t)$. Consider the associated functions…

Let $(X(s),Y(s))$ be the state and adjoint components of the mean-field forward-backward stochastic differential equation associated with the optimal control problem. Let…

Solvability characterization conjecture. In the stochastic case, the equation subject to the constraint $\varDelta(P)\gg 0$ should be solvable if and only if it has a subsolution s…

Converse solvability conjecture. If the singular LQ problem admits a regular solution for every initial state $x_0\in\mathbb{R}^n$, then the generalized continuous algebraic Riccat…