Flocking Control of Multi-Agent Systems with Permanent Obstacles in Strictly Confined Environments

Considering the application of flocking control on connected and automated vehicle (CAV) systems, the persistent interactions between CAVs (flocking agents) and road boundaries (permanent obstacles) are critical, due to flocking behaviors in a strictly confined environment. However, the existing flocking theories attempt to model and animate natural flocks by only considering temporary obstacles, which only describe interactions between agents and obstacles that will eventually disappear during flocking. This paper proposes a novel flocking control algorithm to extend existing flocking theories and guarantee the desired flocking coordination of multi-agent systems (e.g., CAV systems) with permanent obstacles (constraints). By analyzing comprehensive behaviors of flocks via Hamiltonian functions, a zero-sum obstacle condition is developed to ensure the satisfaction of permanent obstacle avoidance. Then, an additional control term representing the resultant forces of permanent obstacles is introduced to tackle interactions between agents and permanent obstacles. Demonstrated and compared through simulation results, a CAV system steered by the proposed flocking control protocol can successfully achieve the desired flocking behaviors with permanent obstacles avoidance in a three-lane traffic environment, which is failed by existing flocking control theories solely considering temporary obstacles.

Energy and Personality: A Bridge between Physics and Psychology

The objective of this paper is to present a mathematical formalism that states a bridge between physics and psychology, concretely between analytical dynamics and personality theory, in order to open new insights in this theory. In this formalism, energy plays a central role. First, the short-term personality dynamics can be measured by the General Factor of Personality (GFP) response to an arbitrary stimulus. This GFP dynamical response is modeled by a stimulus–response model: an integro-differential equation. The bridge between physics and psychology appears when the stimulus–response model can be formulated as a linear second order differential equation and, subsequently, reformulated as a Newtonian equation. This bridge is strengthened when the Newtonian equation is derived from a minimum action principle, obtaining the current Lagrangian and Hamiltonian functions. However, the Hamiltonian function is non-conserved energy. Then, some changes lead to a conserved Hamiltonian function: Ermakov–Lewis energy. This energy is presented, as well as the GFP dynamical response that can be derived from it. An application case is also presented: an experimental design in which 28 individuals consumed 26.51 g of alcohol. This experiment provides an ordinal scale for the Ermakov–Lewis energy that predicts the effect of a single dose of alcohol.

Energy and Personality: A Bridge between Physics and Psychology

The objective of this paper is to present a mathematical formalism that states a bridge between Physics and Psychology, concretely between analytical dynamics and personality theory in order to open new insights in this theory. In this formalism energy plays a central role. First, the short-term personality dynamics can be measured by the General Factor of Personality (GFP) response to an arbitrary stimulus. This GFP dynamical response is modelled by a stimulus-response model: an integro-differential equation. The bridge between Physics and Psychology is provided when the stimulus-response model can be formulated as a linear second order differential equation and, subsequently, reformulated as a Newtonian equation. This bridge is strengthened when the Newtonian equation is derived from a minimum action principle, obtaining the current Lagrangian and Hamiltonian functions. However, the Hamiltonian is a non-conserved energy. Then, some changes provide a conserved Hamiltonian function: the Ermakov-Lewis energy. This energy is presented, as well as the GFP dynamical response that can be derived from it. An application case is presented: an experimental design in which 28 individuals consumed 26.51 g of alcohol. This experiment provides an ordinal scale for the Ermakov-Lewis energies that predicts the effect of a single dose of alcohol.

Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

In the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

<p style='text-indent:20px;'>In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the <i>kinetic</i> and <i>potential</i> equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the <inline-formula><tex-math id="M1">\begin{document}$ C^{\infty} $\end{document}</tex-math></inline-formula> category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.</p>

Non-contractible orbits for Hamiltonian functions on Riemann surfaces

We consider two disjoint and homotopic non-contractible embedded loops on a Riemann surface and prove the existence of a non-contractible orbit for a Hamiltonian function on the surface whenever it is sufficiently large on one of the loops and sufficiently small on the other. This gives the first example of an estimate from above for a generalized form of the Biran–Polterovich–Salamon capacity for a closed symplectic manifold.

On the Topology of Isochronous Centers of Hamiltonian Differential Systems

In this paper, we study the topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions [Formula: see text] such that the isochronous center lies on the level curve [Formula: see text]. We prove that, in the one-dimensional homology group of the Riemann surface (removing the points at infinity) of level curve [Formula: see text], the vanishing cycle of an isochronous center cannot belong to a subgroup generated by those small loops such that each of them is centered at a removed point at infinity of having one of the two special types described in the paper, where [Formula: see text] is sufficiently close to [Formula: see text]. Besides, we present some topological properties of isochronous centers for a large class of Hamiltonian systems of degree [Formula: see text], whose homogeneous parts of degree [Formula: see text] contain factors with multiplicity of no more than [Formula: see text]. As applications, we study the nonisochronicity for some Hamiltonian systems with quite complicated forms which are usually very hard to handle by the classical tools.

A geometric Hamilton–Jacobi theory on a Nambu–Jacobi manifold

In this paper, we propose a geometric Hamilton–Jacobi (HJ) theory on a Nambu–Jacobi (NJ) manifold. The advantage of a geometric HJ theory is that if a Hamiltonian vector field [Formula: see text] can be projected into a configuration manifold by means of a one-form [Formula: see text], then the integral curves of the projected vector field [Formula: see text] can be transformed into integral curves of the vector field [Formula: see text] provided that [Formula: see text] is a solution of the HJ equation. This procedure allows us to reduce the dynamics to a lower-dimensional manifold in which we integrate the motion. On the other hand, the interest of a NJ structure resides in its role in the description of dynamics in terms of several Hamiltonian functions. It appears in fluid dynamics, for instance. Here, we derive an explicit expression for a geometric HJ equation on a NJ manifold and apply it to the third-order Riccati differential equation as an example.

On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

We summarize the known results on the integrability of the complex Hamiltonian systems with two degrees of freedom defined by the Hamiltonian functions of the form$$\begin{array}{} \displaystyle H=\frac{1}{2}\sum_{i=1}^{2}p_i^2+V(q_1,q_2), \end{array} $$where V(q1,q2) are homogeneous polynomial potentials of degree k.