arXiv subjects

Qinglong Zhou

Publications and source records attributed to Qinglong Zhou.

15 recordsLinked to original sources

arXiv2026

Square Roots of Symplectic Matrices

This paper characterizes the existence of real symplectic square roots for symplectic matrices. The decomposition of Wonenburger matrices with respect to their eigenvalues permits a partition of the problem into three primary cases. A symplectic matrix whose spectrum avoids the negative real axis is shown to always admit a real symplectic square root. For a negative hyperbolic matrix, such a root exists if and only if the half-dimension is even and the matrix itself is a $\diamond$-square. In the degenerate case of eigenvalue $-1$, a necessary and sufficient condition is established via a decomposition into specific standard blocks.

math.SG↗

arXiv2024

Optimal $H_{\infty}$ control based on stable manifold of discounted Hamilton-Jacobi-Isaacs equation

The optimal $H_{\infty}$ control problem over an infinite time horizon, which incorporates a performance function with a discount factor $e^{-\alpha t}$ ($\alpha > 0$), is important in various fields. Solving this optimal $H_{\infty}$ control problem is equivalent to addressing a discounted Hamilton-Jacobi-Isaacs (HJI) partial differential equation. In this paper, we first provide a precise estimate for the discount factor $\alpha$ that ensures the existence of a nonnegative stabilizing solution to the HJI equation. This stabilizing solution corresponds to the stable manifold of the characteristic system of the HJI equation, which is a contact Hamiltonian system due to the presence of the discount factor. Secondly, we demonstrate that approximating the optimal controller in a natural manner results in a closed-loop system with a finite $L_2$-gain that is nearly less than the gain of the original system. Thirdly, based on the theoretical results obtained, we propose a deep learning algorithm to approximate the optimal controller using the stable manifold of the contact Hamiltonian system associated with the HJI equation. Finally, we apply our method to the $H_{\infty}$ control of the Allen-Cahn equation to illustrate its effectiveness.

math.OC↗

arXiv2024

The positive fundamental group of ${\rm Sp}(2n)$

In this paper, we examine the homotopy classes of positive loops in ${\rm Sp}(2n)$. We demonstrate that two positive loops are homotopic if and only if they are homotopic through positive loops. As consequences, we can extend several results of McDuff \cite{McD} and Chance \cite{Cha} to higher dimensional symplectic manifolds without dimensional restrictions.

math.SG↗

arXiv2023

Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases

In this paper, we consider the elliptic relative equilibria of the restricted $N$-body problems, where the $N-1$ primaries form an Euler-Moulton collinear central configuration or a $(1+n)$-gon central configuration. We obtain the symplectic reduction to the general restricted $N$-body problem. For the first case, by analyzing the relationship between this restricted $N$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $N$-body problem by the $\omega$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters under $N=4$ and the symmetry of the central configuration. For the second case, there exist three positions $S_1,S_2$ and $S_3$ of the massless body (up to rotations of angle $\frac{2\pi}{n}$). For ${m_0\over m}$ sufficiently large, we show that the elliptic relative equilibria is linearly unstable if the eccentricity $0\le e<e_0$ and the massless body lies at $S_1$ or $S_2$; while the elliptic relative equilibria is linear stability if the massless body lies at $S_3$.

math.DS↗

arXiv2022

Linear stability of the elliptic relative equilibria for the restricted $4$-body problem: the Euler case

In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\mathbf{R}^2$. We obtain the symplectic reduction to the general restricted $N$-body problem. By analyzing the relationship between this restricted $4$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $4$-body problem by the $ω$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters for the symmetric cases.

math.DS↗

arXiv2021

The symplectic reduction of the linearized Hamiltonian systems at elliptic relative equilibria of four-body problem

In this paper, we consider the elliptic relative equilibria of four-body problem. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic relative equilibria of $4$-bodies splits into two independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and the other system is the essential part of the linearized Hamiltonian system, which is given implicitly. The reduction can be applied to the stability problem of such elliptic relative equilibria of four-body problem.

math-ph↗

arXiv2019

Linear stability of elliptic relative equilibria of four-body problem with two infinitesimal masses

In this paper, we consider the elliptic relative equilibria of four-body problem with two infinitesimal masses. The most interesting case is when the two small masses tend to the same Lagrangian point $L_4$ (or $L_5$). In \cite{Xia}, Z. Xia showed that there exist four central configurations: two of them are non-convex, and the other two are convex. We prove that the elliptic relative equilibria raised from the non-convex central configurations are always linearly unstable; while for the elliptic relative equilibria raised from the convex central configurations, the conditions of linear stability with respect to the parameters are given.

math-ph↗

arXiv2019

Linear Stability of Elliptic Relative Equilibria of Restricted Four-body Problem

In this paper, we consider the linear stability of the elliptic relative equilibria of the restricted 4-body problems where the three primaries form a Lagrangian triangle. By reduction, the linearized Poincaré map is decomposed to the essential part, the Keplerian part and the elliptic Lagrangian part where the last two parts have been studied in literature. The linear stability of the essential part depends on the masses parameters $α$, $β$ with $α\geq β>0$ and the eccentricity $e\in[0,1)$. Via $\om$-Maslov index theory and linear differential operator theory, we obtain the full bifurcation diagram of linearly stable and unstable regions with respect to $α$, $β$ and $e$. Especially, two linearly stable sub-regions are found.

math-ph↗

arXiv2019

Trace estimation of a family of periodic Sturm-Liouville operators with application to Robe's restricted three-body problem

In this paper, we consider a family of Sturm-Liouville operators on the $ω$-periodic domain. The bifurcation with respect to the parameter region is studied, and the elliptic regions are estimated by trace formula. At last, these results are used to study the linear stability of the elliptic equilibrium point along $z$-axis in Robe's restricted three-body problem.

math.SP↗

arXiv2016

Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem

In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the elliptic Euler solutions. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators we compute the ω-indices and obtain certain properties of linear stability of the Euler elliptic solutions of the classical three-body problem.

math.DS↗

arXiv2016

The reduction on the linear stability of elliptic Euler-Moulton solutions of the n-body problem to those of 3-body problems

In this paper, we consider the elliptic collinear solutions of the classical $n$-body problem, where the $n$ bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solution of $n$-bodies splits into $(n-1)$ independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and each of the other $(n-2)$ systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a $3$-body problem whose mass parameter is modified. Then the linear stability of such a solution in the $n$-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the $3$-body problems, which for example then can be further understood using numerical results of Martinéz, Samà and Simó in \cite{MSS1} and \cite{MSS2} on $3$-body Euler solutions in 2004-2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler-Moulton solution of the $4$-body problem with two small masses in the middle.

math.DS↗

arXiv2016

The analytical aspect to the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem

We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in \cite{Robe}, which is used to model small oscillations of the earth's inner core taking into account the moon attraction. For the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem, earlier results of such linear stability problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic relative equilibrium point in our problem coincides with the linearized system near the Euler elliptic relative equilibria in the classical three-body problem except for the rang of the mass parameter. We first establish some relations from the linear stability problem to symplectic paths and its corresponding linear operators. Then using the Maslov-type $ω$-index theory of symplectic paths and the theory of linear operators, we compute $ω$-indices and obtain certain properties of the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem.

math.DS↗

arXiv2016

Time-Inconsistent Stochastic Linear-quadratic Differential Game

We consider a general time-inconsistent stochastic linear-quadratic differential game. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derived a sufficient conditions for equilibrium strategies via a system of forward-backward stochastic differential equations. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.

q-fin.MF↗

arXiv2015

A New Class of Problems in the Calculus of Variations

This paper investigates an infinite-horizon problems in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak. It is shown that optimal solutions a class of problems raising from time inconsistency problems in the framework of the neoclassical one-sector model of economic growth, and contains new results in environment economics. Without exogenous commitment mechanism, a notion of the equilibrium strategies instead of the optimal strategies is introduced. We characterized the equilibrium strategies by an integro-differential equation system. For two special criteria, the bi-exponential criteria and the Chichilnisky criteria, we established the existence of the equilibrium strategies.

econ.GN↗

arXiv2015

Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Mart\'ınez, Samà and Simó ([15] in J. Diff. Equa.) of 2006 and results of Hu, Long and Sun ([6] in Arch. Ration. Mech.Anal.) of 2014 can be applied to these solutions of the charged three-body problem to get their linear stability.

math.DS↗