A non-cooperative interpretation of the kernel

The purpose of the paper is to propose a bargaining game to interpret the kernel non-cooperatively. Based on the idea of the Davis-Maschler reduced game, a bilateral bargaining procedure is provided in our game model. We show that the set of all subgame perfect equilibrium outcomes of our non-cooperative game coincides with the kernel for transferable utility games.     

Scarcity,competition, and value

We suggest a value for finite coalitional games with transferable utility that are enriched by non-negative weights for the players. In contrast to other weighted values, players stand for types of agents and weights are intended to represent the population sizes of these types. Therefore, weights do not only affect individual payoffs but also the joint payoff. Two principles guide the behavior of this value. Scarcity: the generation of worth is restricted by the scarcest type. Competition: only scarce types are rewarded. We find that the types’ payoffs for this value coincide with the payoffs assigned by the Mertens value to their type populations in an associated infinite game.     

Dynamics of SIS Epidemic Model with Varying Total Population and Multivaccination Control Strategies

In this paper, two susceptible‐infected‐susceptible epidemic models with varying total population size, continuous vaccination, and state‐dependent pulse vaccination are formulated to describe the transmission of infectious diseases, such as diphtheria, measles, rubella, pertussis, and so on. The first model incorporates the proportion of infected individuals in population as monitoring threshold value; we analytically show the existence and orbital asymptotical stability of positive order‐1 periodic solution for this control model. The other model determines control strategy by monitoring the proportion of susceptible individuals in population; we also investigate the existence and global orbital asymptotical stability of the disease‐free periodic solution. Theoretical results imply that the disease dies out in the second case. Finally, using realistic parameter values, we carry out some numerical simulations to illustrate the main theoretical results and the feasibility of state‐dependent pulse control strategy.     

Fractional Order Systems Time-Optimal Control and Its Application

This paper deals with the time-optimal control problem for a class of fractional order systems. An analytic solution of the time-optimal problem is proposed, and the optimal transfer route is provided. Considering it is usually adopted in the discrete situation for actual control system, the sampling date may induce chattering phenomenon, an alternative sub-optimal solution is constructed. Additionally, the special and meaningful application of fractional order tracking differentiator is introduced to explain our main results. The effectiveness and advantages of the proposed method have been illustrated by numerical examples.     

Optimal Sliding Mode Robust Control for Fractional-Order Systems with Application to Permanent Magnet Synchronous Motor Tracking Control

This paper presents an optimal sliding mode output tracking control scheme for a class of fractional-order uncertain systems. Firstly, an augmented fractional-order system, composed of the original system and the external system, is constructed to transform the optimal output tracking issue into the design problem of linear quadratic regulator. The optimal tracking control problem for the nominal augmented fractional-order system is then studied. Secondly, the fractional-integral sliding mode controller is introduced to robustify the augmented fractional-order system, which satisfy the matching conditions. As a result, the original system output can track the external system output trajectory effectively even the uncertainties exist. Finally, the developed design techniques are applied to the tracking control of fractional-order permanent magnet synchronous motor. The simulation results demonstrate the validity of this approach.     

Homoclinic Orbits for a Class of Fractional Hamiltonian Systems via Variational Methods

This paper is devoted to the existence and multiplicity of homoclinic orbits for a class of fractional-order Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Here, we present a new approach via variational methods and critical point theory to obtain sufficient conditions under which the Hamiltonian system has at least one homoclinic orbit or multiple homoclinic orbits. Some results are new even for second-order Hamiltonian systems.     

Infinite Horizon Controlled Diffusions with Randomly Varying and State-Dependent Discount Cost Rates

This work focuses on optimal controls of diffusions in an infinite horizon. It has several distinct features in contrast to the existing literature. The discount factor is allowed to be randomly varying and state dependent. The existence and uniqueness of the viscosity solution to the associated Hamilton–Jacobi–Bellman equation are established. The verification theorem is also obtained. Because closed-form solutions are virtually impossible to obtain in most cases, we develop numerical methods. Using the Markov chain approximation methods, numerical schemes are constructed; viscosity solution methods are used to prove the convergence of the algorithm. In addition, examples are given for demonstration purpose.     

On weak* convergent sequences in duals of symmetric spaces of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

It is shown that the pre-dual of a σ-finite von Neumann algebra has property (k) in the sense of Figiel, Johnson and Pelczyński [12]. This resolves in the affirmative an open question raised in [12]. It is shown further that a weakly sequentially complete symmetric space E of τ-measurable operators affiliated with a semifinite σ-finite von Neumann algebra has property (k).     

Orthogonal and unitary tensor decomposition from an algebraic perspective

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors.More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras—associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting.