Switching Curve Research Articles

Neoclassical growth model with two fixed delays

Abstract“Delay” has been considered as one of destabilizing factors in economic dynamics for a long time. Dynamic macroeconomics is concerned with explaining growth and fluctuations. This paper shows how various dynamics involving cyclic fluctuations can emerge in the standard neoclassical growth model when two distinct delays, a delay in production and a delay in depreciation, are explicitly taken into account. We first confirm that the production delay has a stabilizing effect and the depreciation delay has a destabilizing effect in a one‐delay model. We then determine the stability switching curve analytically in the two delay model. It is shown that cyclic fluctuations emerge via Hopf bifurcation when stability is lost. It is also found that stability loss and gain repeatedly occur. Numerical examples verify the theoretical results when the Cobb‐Douglas production function is adopted.

When to Triage in Service Systems with Hidden Customer Class Identities?

In service systems with heterogeneous customers, prioritization with respect to the relative importance of customers is known to improve certain performance measures. However, in many applications, information necessary to determine the importance level of a customer may not be available immediately but can be revealed only through some preliminary investigation, which is sometimes called triage. This triage process is typically error-prone and may take substantial amount of time, and hence, it is not always clear if and when it should be implemented for purposes of priority assignment. To provide insights into this question, we study a stylized queueing model with a single server and two types of customers with hidden type identities and different rates of waiting costs. By means of a Markov decision formulation, we first show that the optimal dynamic policy on triage is characterized by a switching curve: if the number of unclassified customers is sufficiently large, then the server triages an unclassified customer; otherwise, the server skips triage and directly serves an unclassified customer. We also identify a mathematical condition under which triage should never be considered, which depends on the classification power of triage, mean triage and service times, and the heterogeneity of customers. Based on all our numerical and analytical results, we conclude that triage for purposes of prioritization should not be considered if it takes a significantly large amount of time in comparison to the actual service but if it is sufficiently fast, then implementing a policy that triages some or all customers can significantly reduce long-run average costs of waiting. In these cases where there is benefit to using triage, a policy that triages all arrivals can be used if the triage time is substantially short and the probability that a customer is classified into a higher priority is large; otherwise, a policy that triages customers dynamically depending on the system state would perform significantly better than the triage-all policy, especially when the traffic is heavy.

Analytical Solution to the Boundary Layer Slip Flow and Heat Transfer over a Flat Plate using the Switching Differential Transform Method

In this paper, the two-dimensional steady boundary layer flow and heat transfer over a flat plate with slip velocity and temperature jump conditions at the walls were analyzed. Using similarity transforms, the governing equations were reduced to a system of ordinary differential equations. Semi-analytical solutions to the resulting boundary value problem were obtained using the differential transform method (DTM). In order to cover the asymptotic boundary conditions, a method of switching curves was proposed. In this switching approach, the traditional solution to the DTM, which is valid for finite horizons, was followed by another path that was also an analytical solution to the problem. The main preference of the resulting closed form solution with respect to numerical solution is the possibility of parametric studies. The method was verified using some available numerical data, and the results showed that our proposed method had reasonable efficiency and accuracy.

Optimal Bayesian Price Fine-Tuning

We consider the problem of optimally fine-tuning the price near a historically fixed price (incumbent price) to maximize the expected revenue over a finite horizon. The demand follows a linear Gaussian model in which the price sensitivity is unknown and can be learned in a Bayesian fashion as the sales data accumulate. We formulate the problem as a discrete-time Bayesian dynamic program and connect it to the Gauss-Weierstrass Transform through a reformulation. We prove that the optimal pricing policy has a simple and intuitively appealing structure: the optimal policy adjusts the myopic price away from the incumbent price, except when the latter two prices coincide and the precision of the posterior belief exceeds a threshold. Under some priors, a fixed price is optimal for the entire horizon. Further, we show that incomplete learning is absent from the optimal policy if the belief precision stays under a switching curve, but may emerge when the belief precision jumps above that curve, especially near the end of the horizon. We also develop an efficient algorithm using a technique called the fast Gauss transform, which enables us to obtain the optimal policy with only moderate computation.

In situ XRD analyses for asymmetric responses of poled PLZT ceramics during electric fatigue

Electric fatigue behaviors of the poled lanthanum-doped lead zirconate titanate (PLZT) ceramics were investigated using a home-made electric loading apparatus in conjunction with a conventional X-ray diffractometer. The XRD data were measured under actions of the applied direct current electric fields on the as-received poled PLZT specimens being experienced various cycles (10N) of alternating current (AC) electric fields. Experimental data showed that in addition to apparent degradation in remnant polarization, asymmetries in hysteresis loops and fraction of 90° domain switching curves were observed. It was found that an offset electrical field (\(\Delta {{\text{E}}_N}\)) was induced in the poled specimen by polarization with its direction against the direction of the polarization. Furthermore, the magnitude of \(\Delta {{\text{E}}_N}\) and the degree of asymmetry (δN) decreased as the number of AC electric fatigue cycles (10N) varied from 100 to 106. In situ XRD data suggested that the poled specimen exhibited different abilities to reorient a-domains into c-domains when the applied electric fields were opposite. The δN not only depended linearly on \(\Delta {{\text{E}}_N}\), but also on abilities of the poled specimen to reorient a-domains into c-domains.

Zeno points in optimal control models with endogenous regime switching

The present paper considers a type of multi-stage problem where a decision maker can optimally decide to switch between two alternating stages (regimes). In such a problem, either one of the two stages dominates in the long run or it is optimal to infinitely switch between stages. A point where such a phenomenon occurs with optimal switching time zero is called Zeno point.By a reformulation of the multi-stage problem using an additional binary control to account for the alternating stages, the necessary matching conditions for optimal switching are derived. A relaxation with a linear control is considered to analyze the implications of the occurrence of a singular solution. In case of the control becoming singular, the matching conditions are fulfilled, i.e. it is optimal to switch between the stages. If the time derivative of the matching conditions is zero, it is shown that trajectories approaching this point are tangential to the switching curve. Furthermore, if the trajectories move into opposing directions, this implies Zeno behavior.A two-stage version of a standard capital accumulation problem is analyzed and a Zeno point is found analytically. Numerical calculations are performed to visualize the Zeno phenomenon. The economic relevance of the phenomenon is discussed.

Switching Characteristics Optimization of Two-Phase Interleaved Bidirectional DC/DC for Electric Vehicles

In electric vehicles (EVs), bidirectional DC/DC(Bi-DC/DC) is installed between the battery pack and the DC bus to step up the voltage. In the process of mode switching under step signal, the Bi-DC/DC will be affected by a large current inrush which threatens the safety of the circuit. In this paper, a Bi-DC/DC mode switching method based on the optimized Bézier curve is proposed. The Boost and Buck modes can be switched based on the proposed method with fast and non-overshoot switching performance. The experimental results show that the mode switching can be finished in 4 ms without overshoot based on the optimal switching curve.

Design and Implementation of an Efficient Tracking Differentiator

Differential signals play significant roles in control practices, but they are prone to noise corruption. A noise-tolerant time-optimal system-based tracking differentiator (TD) was first proposed by Jingqing Han, which is constructed in the form of state feedback control for a discrete-time and double-integral system. However, its performances of signal-tracking filtering and differentiation extraction are sensitive to the sampling period. To relax the sensitivity on the sampling period, a time-criterion-based feedback control algorithm is proposed to construct the TD. The control algorithm is derived by comparing the time that the initial state is driven to the switching curve or the origin with any given sampling period. The impact of the algorithm parameter setting on filtering is analyzed in the frequency domain. Meanwhile, a compensation scheme is introduced to deal with the trade-off between the filtering ability and phase delay. The simulation results show that the proposed TD has smaller static errors in signal-tracking filtering, and better differentiation acquisition compared with others. The experiments conducted on STM32F405 reveal that the proposed TD uses the shortest execution time among others for processing the same input signals.

Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system.

In this paper, the dynamics of a modified Leslie-Gower predator-prey system with two delays and diffusion is considered. By calculating stability switching curves, the stability of positive equilibrium and the existence of Hopf bifurcation and double Hopf bifurcation are investigated on the parametric plane of two delays. Taking two time delays as bifurcation parameters, the normal form on the center manifold near the double Hopf bifurcation point is derived, and the unfoldings near the critical points are given. Finally, we obtain the complex dynamics near the double Hopf bifurcation point, including the existence of quasi-periodic solutions on a 2-torus, quasi-periodic solutions on a 3-torus, and strange attractors.

Soft-Switching Proximate Time Optimal Heading Control for Underactuated Autonomous Underwater Vehicle

In this paper, a soft-switching proximate time-optimal control (PTOC) is proposed based on the model-compensation extended state observer (ESO) and Takagi-Sugeno fuzzy switching, for the fast set-heading tracking of underactuated autonomous underwater vehicle (AUV). First, based on Pontryagin's maximum principle, a time-optimal control (TOC) law is derived for the first-order Nomoto model of the underactuated AUV. Then, a model-compensation active disturbance rejection control (ADRC) is developed; the outstanding characteristic is that the nonlinear heading dynamic model is compensated to a first-order Nomoto model rather than by a double integral system through implementing a model-compensation ESO (MC-ESO). The regular ESO is replaced with a reduced-order ESO (RESO) to reduce complexity, and the model-compensation RESO (MC-RESO) is designed by adding the known partial model to RESO. Based on the controller scaling method, a parameter self-tuning strategy is proposed for model-compensation ADRC (MC-ADRC) with changed plant parameters at different velocities. Finally, the soft-switching PTOC is developed for heading control, and the MC-RESO is adopted to estimate the unmeasured velocity and unknown total disturbances for feedback and compensation. The TOC with an unsaturated region (RTOC) is employed to enhance the robustness by using a switching region to replace the switching curve, and a soft-switching strategy between ADRC (near the origin) and RTOC (far from origin) is designed based on the Takagi-Sugeno fuzzy mode. Several simulations are carried out, the effectiveness of self-tuning MC-ADRC is verified, and the proposed soft-switching PTOC shows better performance compared with MC-ADRC.

A novel use of value iteration for deriving bounds for threshold and switching curve optimal policies

AbstractIn this article, we develop a novel role for the initial function v0 in the value iteration algorithm. In case the optimal policy of a countable state Markovian queueing control problem has a threshold or switching curve structure, we conjecture, that one can tune the choice of v0 to generate monotonic sequences of n‐stage threshold or switching curve optimal policies. We will show this for three queueing control models, the M/M/1 queue with admission and with service control, and the two‐competing queues model with quadratic holding cost. As a consequence, we obtain increasingly tighter upper and lower bounds. After a finite number of iterations, either the optimal threshold, or the optimal switching curve values in a finite number of states is available. This procedure can be used to increase numerical efficiency.

A heterogeneous agent model of asset price dynamics with two time delays

This study constructs a heterogeneous agents model of a financial market in a continuous-time framework. There are two types of agents, fundamentalists and chartists. The former follows the traditional efficiency market theory and has a linear demand function, whereas the latter experiences delays in the formation of price trends and possesses a S-shaped demand function. The main feature of this study is a theoretical investigation on the effects caused by two time delays in a price adjustment process. In particular, two main results are demonstrated: One is that the stability switching curves are analytically derived, and the other is that the stability losses and gains can repeatedly occur when the shape of the curves are meandering. Although it is well known that a time delay has a destabilizing effect, these results imply that multiple delays can stabilize and destabilize a market price generating persistent deviations from the stationary price.

Effect of electrode interfaces on peak-drift switching current of PZT thin films

In this study, we investigated the impact of electrodes on the structural, crystal orientation, and electrical characteristics of lead zirconate titanate [Pb(Zr0.52Ti0.48)O3, PZT] thin films, that were deposited on Pt, Au, and LaNiO3 (LNO) electrodes through sol-gel processes. The peak voltage, that is, the voltage corresponding to the maximum switching current point, was developed to depict the novel peak-drift electric characteristics in ferroelectric thin films. It increases in the positive ferroelectric peak voltage, and decreases in the negative peak voltage with an increase in the driving voltage amplitude. Based on space-charge-limited bulk conduction, we evaluated the peak-drift phenomenon, Schottky contact and their influences on the switching current. When polarization orientation is reversed, the Schottky contact disappears at one side of the PZT/electrode, and forms at the other side synchronously. Our results indicate that the formation of a Schottky contact affects the shape of the switching curves, and its disappearance influences the magnitude of the current characteristics.

Resistive switching characteristics of graphene/NiO/highly ordered pyrolytic graphite resistive random access memory capacitors

Polycrystalline NiO thin films were deposited on single crystalline HOPG substrates via radio frequency sputtering. On the HOPG substrate, the NiO thin film exhibited (111) and (222) X-ray diffraction peaks, thereby indicating that the film was polycrystalline. The graphene/NiO/HOPG RRAM capacitor showed canonical RRAM behavior, which was confirmed by evaluating the electrical properties of RRAM switching curves, writing endurance tests, retention tests, and SET and RESET voltages as a function of switching cycles. In particular, the graphene/NiO/HOPG RRAM capacitor exhibited lower SET and RESET voltages than the Pt/NiO/Pt RRAM capacitor, resulting from of the low work function of graphene and HOPG electrodes. In addition, it was possible to reduce the dispersion of the LRS and HRS states and the SET and RESET voltages for repeated switching cycles. It was suggested that this small dispersion was influenced by the simple configuration of the conducting filaments as well as the low interface roughness values of the graphene/NiO/HOPG capacitor.

Threshold Absolute Value of a Relay Control when Time-Optimally Bringing a Satellite to a Gravitationally Stable Position

The time-optimal problem is considered for the orientation of a satellite in the plane of a circular orbit and bringing it to the desirable angular position. The constraint on the control torque module is considered as the main parameter of the problem. It turned out that even in the case when the admissible control torque exceeds the gravitational moment in magnitude, additional switching curves corresponding to the relay control with two switchings are possible in the phase plane. As a result, a simple numerical algorithm is proposed, and with its help for each terminal angular position the threshold absolute control value is found at which the indicated switching curves are infinitesimal. Their coordinates in the phase plane are found in the most well-known case of the gravitationally stable terminal angular position. A suboptimal control law, which does not need a numerical calculation of the switching curve, is offered.

Optimal routeing in two-queue polling systems

AbstractWe consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customercper time unit to wait in the busy queue (where the server is) anddper time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

Selection of financing strategies with a risk-averse supplier in a capital-constrained supply chain

This paper investigates a supply chain where the retailer is capital-constrained and the supplier is risk-averse. The supplier's risk-averse behavior is gauged by Conditional Value-at-Risk method under two financing strategies: partial credit guarantee(PCG) and trade credit financing(TCF). We obtain the equilibrium solutions and characterize the preference of two financing strategies by the switching curves in two-dimensional space of credit guarantee coefficient and risk aversion degree. We find that there exists a region where TCF outperforms PCG for both players. Finally, we extend the model to the case in which both players are risk-averse and obtain similar results.

Optimal Control of Membrane-Filtration Systems

This paper studies an optimal control problem related to membrane-filtration processes. A generic mathematical model of membrane fouling is used to capture the dynamic behavior of the filtration process, which consists in the attachment of matter onto the membrane during the filtration period, and the detachment of matter during the cleaning period. We consider the maximization of the net water production of a membrane filtration system (i.e., the filtrate) over a finite time horizon, where the control variable is the sequence of filtration/backwashing cycles over the operation time of process. Based on the Pontryagin's maximum principle, we characterize the optimal control strategy and show that it exhibits a singular arc. Moreover, we prove the existence of an additional switching curve before reaching the terminal state, and also the possibility of having a dispersal curve as a locus where both the different strategies are optimal.

Optimal booking control in revenue management with two substitutable resources

This paper studies optimal booking policies for capacity control models in revenue management with two substitutable resources. Our model covers a broader class of problems than previous works including (i) flexible demand and opaque selling for (ii) both dynamic and static demand settings. We provide a unifying characterization of the structure of optimal booking control by exploiting concavity, submodularity, and subconcavity of the value function. Our characterization is based on the notion of optimal “booking paths” formalizing the idea that an optimal allocation of a demand batch decomposes into a sequence of optimal single-request allocations. In addition, we examine the relationship between our booking path-based and a switching curve-based policy, which has been known previously for the case with dynamic demand. We show that both these characterizations describe an optimal policy. Computationally, there is no advantage of implementing either switching curves or booking paths in the dynamic setting. In the static setting, however, one can resort to the simple criteria which we propose in order to construct the optimal booking paths, thereby accelerating the evaluation of the value function.

Hopping conduction distance of bipolar switching GdOx resistance random access memory thin films devices modified by different constant compliance current

Bipolar switching properties and electrical conduction mechanism of the gadolinium oxide thin film prepared by rf sputtering technology for applications in resistive random access memory devices was investigated in different constant compliance current. The electrical and physical properties of gadolinium oxide thin film were determined by current versus applied voltage (I-V) and XPS measurement. To discuss the electrical conduction mechanism for different constant compliance current, the typical electrical switching curves of gadolinium oxide thin films RRAM devices for different temperature was transferred to fitting. Finally, the activation barrier energy, hopping conduction distance and non-volatile characteristics between the electrons transmission switching behavior in initial metallic filament forming path of the gadolinium oxide thin film RRAM devices was described and explained in physical diagram model.