We investigate the mean-field dynamics of stochastic McKean differential equations with heterogeneous particle interactions described by large network structures. To express a wide range of graphs, from dense to sparse structures, we incorporate the recently developed graph limit theory of graphops into the limiting McKean–Vlasov equations. Global stability of the splay steady state is proven via a generalised entropy method, leading to explicit graph structure-dependent decay rates. We highlight the robustness of the entropy approach by extending the results to the closely related Sakaguchi–Kuramoto model with intrinsic frequency distributions. We also present central examples of random graphs, such as power law graphs and the spherical graphop, and analyse the limitations of the applied methodology.

Conventional preytaxis systems assume that prey-tactic velocity is proportional to the prey density gradient. However, many experiments exploring the predator–prey interactions show that it is the predator’s acceleration instead of velocity that is proportional to the prey density gradient in the prey-tactic movement, which we call preytaxis with prey-induced acceleration. Mathematical models of preytaxis with prey-induced acceleration were proposed by Arditi et al. ((2001) Theor. Popul. Biol. 59(3), 207–221) and Sapoukhina et al. ((2003) Am. Nat. 162(1), 61–76) to interpret the spatial heterogeneity of predators and prey observed in experiments. This paper is devoted to exploring the qualitative behaviour of such preytaxis systems with prey-induced acceleration and establishing the global existence of classical solutions with uniform-in-time bounds in all spatial dimensions. Moreover, we prove the global stability of spatially homogeneous prey-only and coexistence steady states with decay rates under certain conditions on system parameters. For the parameters outside the stability regime, we perform linear stability analysis to find the possible patterning regimes and use numerical simulations to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise from the preytaxis system with prey-induced acceleration. Noticing that conventional preytaxis systems are unable to produce spatial patterns, our results imply that the preytaxis with prey-induced acceleration is indeed more appropriate than conventional preytaxis to interpret the spatial heterogeneity resulting from predator–prey interactions.

Selective compliance articulated robot arms (SCARA) robotic manipulators find wide use in industry. A nonlinear optimal control approach is proposed for the dynamic model of the 4-degrees of freedom (DOF) SCARA robotic manipulator. The dynamic model of the SCARA robot undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the system, a stabilizing optimal (H-infinity) feedback controller is designed. To compute the controller’s feedback gains, an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed control method is advantageous because: (i) unlike the popular computed torque method for robotic manipulators, it is characterized by optimality and is also applicable when the number of control inputs is not equal to the robot’s number of DOFs and (ii) it achieves fast and accurate tracking of reference setpoints under minimal energy consumption by the robot’s actuators. The nonlinear optimal controller for the 4-DOF SCARA robot is finally compared against a flatness-based controller implemented in successive loops.

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝn. We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

The article proposes a nonlinear optimal control method for the model of the wheeled inverted pendulum (WIP). This is a difficult control and robotics problem due to the system’s strong nonlinearities and due to its underactuation. First, the dynamic model of the WIP undergoes approximate linearization around a temporary operating point which is recomputed at each time step of the control method. The linearization procedure makes use of Taylor series expansion and of the computation of the associated Jacobian matrices. For the linearized model of the wheeled pendulum, an optimal (H-infinity) feedback controller is developed. The controller’s gain is computed through the repetitive solution of an algebraic Riccati equation at each iteration of the control algorithm. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, by using the H-infinity Kalman Filter as a robust state estimator, the implementation of a state estimation-based control scheme becomes also possible.

We study the non-autonomously forced Burgers equation

$$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$

This paper presents a new method to design global path-following controllers for underactuated ships under both deterministic and stochastic sea loads. The path-following errors are first interpreted in a moving frame attached to the path. These errors are then to be stabilized at the origin by a design of controllers based on backstepping and Lyapunov's direct methods. Weak and strong nonlinear Lyapunov functions are introduced to overcome difficulties caused by underactuation and Hessian terms induced by stochastic differentiation rule, and to guarantee boundedness of the sway velocity. Potential projection functions are introduced to design update laws that provide bounded estimates of the mean values and covariances of the disturbances. Simulations are included to illustrate the effectiveness of the proposed approach.

An underactuated mechanical system is generally a good test bed for advanced nonlinear controllers and can be applied to design a novel mechanical system with better energy efficiency and good controllability. It has been shown that the dynamics of many underactuated mechanical systems could be transformed into the chained canonical form. To improve the performance of the controllers presented in the literature, a novel controller design method is proposed in this paper. It is shown that the set-point stabilization problem of the second-order chained form systems can be changed into a trajectory-tracking problem based on the nonsmooth Hölder continuous feedback. By designing the tracked trajectory, the presented controller permits the achievement of exponential stability. Some numerical simulations demonstrate the stability of the proposed controller for an underactuated Hovercraft system.

This paper is devoted to the study of spreading speeds and traveling waves for a class ofreaction-diffusion equations with distributed delay. Such an equation describes growth anddiffusion in a population where the individuals enter a quiescent phase exponentially andstay quiescent for some arbitrary time that is given by a probability density function.The existence of the spreading speed and its coincidence with the minimum wave speed ofmonostable traveling waves are established via the finite-delay approximation approach. Wealso prove the existence of bistable traveling waves in the case where the associatedreaction system admits a bistable structure. Moreover, the global stability and uniquenessof the bistable waves are obtained in the case where the density function has zerotail

In order to better understand the dynamics of acute leukemia, and in particular to findtheoretical conditions for the efficient delivery of drugs in acute myeloblastic leukemia,we investigate stability of a system modeling its cell dynamics.

The overall system is a cascade connection of sub-systems consisting of distributeddelays and static nonlinear feedbacks. Earlier results on local asymptotic stability areimproved by the analysis of the linearized system around the positive equilibrium. For thenonlinear system, we derive stability conditions by using Popov, circle and nonlinearsmall gain criteria. The results are illustrated with numerical examples andsimulations.

In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDSepidemic model with an imperfect HIV vaccine, varying total population size anddistributed time delay to become infectious due to intracellular delay between initialinfection of a cell by HIV and the release of new virions. Here, we have established somesufficient conditions on the permanence and extinction of the disease by using inequalityanalytical technique. We have obtained the explicit formula of the eventual lower boundsof infected persons. We have introduced some new threshold valuesR 0 and R ∗ and further obtainedthat the disease will be permanent when R 0 > 1 and thedisease will be going to extinct when R ∗ < 1. ByLyapunov functional method, we have also obtained some sufficient conditions for globalasymptotic stability of this model. The aim of the analysis of this model is to trace theparameters of interest for further study, with a view to informing and assistingpolicy-maker in targeting prevention and treatment resources for maximumeffectiveness.

Hirofumi Uzawa is one of the giants of modern economic theory. Hiro is probably best known to the readers of Macroeconomic Dynamics (MD) for his seminal articles on two-sector economic growth. The two-sector technology is more general than the one-sector technology: it allows a production possibility frontier that is strictly concave to the origin as opposed to being necessarily flat. This generality allows richer and more complex dynamics. This makes it especially useful for the analysis of economic fluctuations. The two-sector model is perfect for dynamic international trade.

Hiro is also well known to macroeconomists for his seminal contribution to endogenous growth. In his article in the 1965 IER, productivity permanently increases as the result of permanent accumulation of human capital. Uzawa was thus a first mover in the new growth theory. The symbol H (for Human Capital, or for Hiro?) is today everywhere in models of economic dynamics.

On his own and through his many students and mentees, Hiro has been the major inspiration for the modern theory of optimal economic growth. He taught a generation of pure and applied economists how to apply Pontryagin's maximum principle in economic dynamics. It seems that Uzawa introduced—or at least pushed the use of—phase diagrams in economic dynamics. Where would we be without this essential tool?

Most readers of MD are likely to think first of Uzawa's contributions to macro, but Hiro is equally well known for his superb works on mathematical economics, general equilibrium, and demand theory. Hiro's mathematics is elegant and often very deep. Like the quality mathematician that he is, he does not apply technique for technique's sake.

Hiro has made fundamental contributions to nonlinear programming. For the convex (but not necessarily smooth) case, he employed Slater's condition to obtain Kuhn–Tucker multipliers that satisfy the saddlepoint property necessary for an optimum. For the smooth (but not necessarily convex) case, Arrow, Hurwicz, and Uzawa introduced the current version of the constraint qualification, which ensures that optimality implies the existence of Kuhn–Tucker multipliers satisfying the saddlepoint property.

Hiro's paper “Walras's Existence Theorem and Brouwer's Fixed Point Theorem” in the Economic Studies Quarterly (1962) is a hidden gem on general equilibrium. This paper can be seen as foreshadowing Sonnenschein's result on excess demand functions. Hiro clarified old, important questions about recovering preference maps from demand functions. Hiro was probably the first to convincingly show—in the context of tatonnement adjustment—the important distinction between local stability and global stability in economic dynamics.

We have given here only a glimpse into the very large body of beautiful, influential Uzawa papers. Hiro's splendid bibliography is given at the end of the interview. Some of the work that Hiro has pursued energetically has yet to be widely recognized. One thinks, for example, of the Penrose Effect, Hiro's modeling of the organizational costs incurred in adding capital or making other changes in the way a firm does business.

Hiro has had many successful students and mentees. Your MD interviewers are lucky to have been among those whom Hiro has influenced profoundly. A very incomplete list of the others would also include Dave Cass, Steve Goldman, Harl Ryder, Hajime Oniki, Bob Lucas, George Akerlof, Joe Stiglitz, Miguel Sidrauski, Morris Teubal, Assaf Razin, Guillermo Calvo, Bill Ethier, and Lenny Mirman.

Hiro is widely recognized and even revered in Japan. He was elected to the very selective Japan Academy in 1989 at a remarkably young age. He was named “A Person of Cultural Merit” in 1983 and elected to the Order of Culture in 1997. Hiro has received significant international recognition. He was President of the Econometric Society. He is a Fellow of the Econometric Society, Member of the American Academy of Arts and Sciences, Foreign Honorary Member of the American Economic Association, and Foreign Associate of the U.S. National Academy of Sciences.

This interview took place nearly 10 years ago. We apologize to the readers and to Professor Uzawa for the delay in getting the transcript to the editor. The interview was held at the Research Center on Global Warming of the Development Bank of Japan, at which Hiro plays an important role. Four of us—Uzawa, the two interviewers, and Yumiko Baba, who was then a post-doc in economics at the University of Tokyo, there to operate the tape recorder—were collected at the Meiji Gakuin University in central Tokyo and whisked away in a large black automobile to Hiro's home court at the Bank. Hiro is an imposing figure: tall and erect with a very long, pointed white beard. His eyes are very active. He strokes his beard in a soothing manner. It is not difficult to be in awe of him. The interview took an even more formal tack because there were two in the room with the nickname “Hiro.” It was hence efficient to use last names at times.

The interviewers had agreed to try to steer Uzawa toward a discussion of his well-known basic technical contributions and away from his less well-known and more political contributions. In the end, we failed to steer Hiro onto any course other than his own. This is mostly as it should be. In this interview, you will hear about some of the technical contributions for which Hiro is widely known. You will also hear about what motivated him to enter economics, his strong social concerns and strong political views, the turbulence of the war years and the postwar years, and his recent work and interests. A few of the paragraphs at the end of the interview were added to bring the record up to date. What comes through is a picture of Hirofumi Uzawa, a truly distinguished scholar and a person dedicated to human betterment.

Hiro talked in his usual warm, friendly voice. He peppered the interview with his strong opinions about other major economists, often with lively anecdotes. Of course, Hiro's opinions are his own, not those of the interviewers or the editors. We hope that the readers will get as much out of this conversation with Hiro as we did.

Almost all industrial robot applications in use today are controlled using a control law that is simple and computationally efficient, local joint error feedback. When two or more open chain manipulators cooperate to manipulate the same object - such as in mechanical grippers, walking machines, and cooperating manipulator systems - closed kinematic chain, redundantly actuated mechanisms are formed. Control approaches for this type of system focus on the more computationally intensive computed torque or inverse plant control laws, due to the concern over instability caused by the unspecified distribution of control forces in the redundant actuator space, and due to the constrained motion caused by the closed kinematic chains.

This paper presents mathematical models for tuberculosis and its dynamics under theimplementation of the direct observation therapy strategy (DOTS) in Nigeria. The models establishconditions for the eradication of tuberculosis in Nigeria based on the fraction of detectedinfectious individuals placed under DOTS for treatment. Both numerical and qualitative analysisof the models were carried out and the effect of the fraction of detected cases of active TB on thevarious epidemiological classes is investigated. The results showed that, provided that the fractionof detected infectious individuals exceeded a critical value, there exists a globally stable diseasefree equilibrium. However, if this critical detection level is not reached, the disease-free equilibriumwill be unstable even with the very high probability successful treatment under DOTS. Theresults showed that DOTS expansion in Nigeria must include significant increase in the detectionrate of infectious individuals; otherwise the effect in reducing the incidence in Nigeria will notbe achieved disregarding the tremendous efforts in any other direction, and the huge number ofundetected cases will make DOTS insignificant with respect to tuberculosis control.

We give a survey of results on global stability for deterministic compartmental epidemiologicalmodels. Using Lyapunov techniques we revisit a classical result, and give a simple proof.By the same methods we also give a new result on differential susceptibility and infectivity modelswith mass action and an arbitrary number of compartments. These models encompass the so-calleddifferential infectivity and staged progression models. In the two cases we prove that if the basicreproduction ratio $\mathcal{R}_0$ ≤ 1, then the disease free equilibrium is globally asymptotically stable. If $\mathcal{R}_0$ > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.

In this paper, the global stability problem for constrained robot motions in the presence of constraint uncertainties is investigated. We focus on the uncertainties in the constraint functions and their effects on the global stability. PD type controllers are used and conditions for global stability are developed using Lyapunov's direct approach. In the presence of the constraint uncertainties under investigation, the desired position and constraint force can be guaranteed with global asymptotic convergence. The developed conditions for feedback gain selections clearly show the effects of the constraint uncertainties. For the case when the velocity measurements are not available, conditions for global stability regulation are also established and the robot controller uses only the measurements of the position angles. Finally, we consider the case where the robot joints are flexible and global stability conditions are given.