Over 1.3 million people die annually in traffic accidents,and this tragic fact highlights the urgent need to enhance the intelligence of traffic safety and control systems.In modern industrial and technological applic...Over 1.3 million people die annually in traffic accidents,and this tragic fact highlights the urgent need to enhance the intelligence of traffic safety and control systems.In modern industrial and technological applications and collaborative edge intelligence,control systems are crucial for ensuring efficiency and safety.However,deficiencies in these systems can lead to significant operational risks.This paper uses edge intelligence to address the challenges of achieving target speeds and improving efficiency in vehicle control,particularly the limitations of traditional Proportional-Integral-Derivative(PID)controllers inmanaging nonlinear and time-varying dynamics,such as varying road conditions and vehicle behavior,which often result in substantial discrepancies between desired and actual speeds,as well as inefficiencies due to manual parameter adjustments.The paper uses edge intelligence to propose a novel PID control algorithm that integrates Backpropagation(BP)neural networks to enhance robustness and adaptability.The BP neural network is first trained to capture the nonlinear dynamic characteristics of the vehicle.Thetrained network is then combined with the PID controller to forma hybrid control strategy.The output layer of the neural network directly adjusts the PIDparameters(k_(p),k_(i),k_(d)),optimizing performance for specific driving scenarios through self-learning and weight adjustments.Simulation experiments demonstrate that our BP neural network-based PID design significantly outperforms traditional methods,with the response time for acceleration from 0 to 1 m/s improved from 0.25 s to just 0.065 s.Furthermore,real-world tests on an intelligent vehicle show its ability to make timely adjustments in response to complex road conditions,ensuring consistent speed maintenance and enhancing overall system performance.展开更多
For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smalle...For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.展开更多
In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdin...In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.展开更多
This article is concerned with blow-up solutions of the Cauchy problem of critical nonlinear SchrSdinger equation with a Stark potential. By using the variational characterization of corresponding ground state, the li...This article is concerned with blow-up solutions of the Cauchy problem of critical nonlinear SchrSdinger equation with a Stark potential. By using the variational characterization of corresponding ground state, the limiting behavior of blow-up solutions with critical and small super-critical mass are obtained in the natural energy space ∑ = {u ∈ H^1; fRN ｜x｜^2｜u｜^2dx 〈 ＋∞）}. Moreover, an interesting concentration property of the blow-up solutions with critical mass is gotten, which reads that ｜u（t, x）｜^2→ ｜｜Q｜｜L^2 2 δx=x1 as t → T.展开更多
Underwater Wireless Sensor Networks(UWSNs)are becoming increasingly popular in marine applications due to advances in wireless and microelectronics technology.However,UWSNs present challenges in processing,energy,and ...Underwater Wireless Sensor Networks(UWSNs)are becoming increasingly popular in marine applications due to advances in wireless and microelectronics technology.However,UWSNs present challenges in processing,energy,and memory storage due to the use of acoustic waves for communication,which results in long delays,significant power consumption,limited bandwidth,and packet loss.This paper provides a comprehensive review of the latest advancements in UWSNs,including essential services,common platforms,critical elements,and components such as localization algorithms,communication,synchronization,security,mobility,and applications.Despite significant progress,reliable and flexible solutions are needed to meet the evolving requirements of UWSNs.The purpose of this paper is to provide a framework for future research in the field of UWSNs by examining recent advancements,establishing a standard platform and service criteria,using a taxonomy to determine critical elements,and emphasizing important unresolved issues.展开更多
Background With the aim of addressing the difficulty in identifying temperatures in virtual chemistry experiments,we propose a temperature-sensing simulation method of virtual chemistry experiments.Methods We construc...Background With the aim of addressing the difficulty in identifying temperatures in virtual chemistry experiments,we propose a temperature-sensing simulation method of virtual chemistry experiments.Methods We construct a virtual chemistry experiment temperature simulation platform,based on which a wearable temperature generation device is developed.The typical middle school virtual experiments of concentrated sulfuric acid dilution and ammonium nitrate dissolution are conducted to verify the actual effect of the device.Results The platform is capable to indicate near real-world experimental situations.The performance of the device not only meets the temperature sensing characteristics of human skin,but also matches the temperature change of virtual chemistry experiments in real-time.Conclusions It is demonstrated that this temperature-sensing simulation method can represent exothermic or endothermic chemistry experiments,which is beneficial for students to gain understanding of the principles of thermal energy transformation in chemical reactions,thus avoiding the danger that may be posed in the course of traditional teaching of chemistry experiments effectively.Although this method does not have a convenient enough operation for users,the immersion of virtual chemical experiments can be enhanced.展开更多
Complex field modulation(CFM)has found a plethora of applications in physics,biomedicine,and instrumentation.Among existing methods,superpixel-based CFM has been increasingly featured because of its advantages in high...Complex field modulation(CFM)has found a plethora of applications in physics,biomedicine,and instrumentation.Among existing methods,superpixel-based CFM has been increasingly featured because of its advantages in high modulation accuracy and its compatibility with high-speed spatial light modulators(SLMs).Nonetheless,the mainstream approach based on binary-amplitude modulation confronts limitations in optical efficiency and dynamic range.To surmount these challenges,we develop binary phase-engraved(BiPE)superpixel-based CFM and implement it using the phase light modulator(PLM)—a new micro-electromechanical system-based SLM undergoing development by Texas Instruments in recent years.Using BiPE superpixels,we demonstrate highaccuracy spatial amplitude and phase modulation at up to 1.44 kHz.To showcase its broad utility,we apply BiPEsuperpixel-based CFM to beam shaping,high-speed projection,and augmented-reality display.展开更多
Based on the number of customers and the server’s workload,this paper proposes a modified Min(N,D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy.Applying ...Based on the number of customers and the server’s workload,this paper proposes a modified Min(N,D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy.Applying the well-known stochastic decomposition property of the steady-state queue size,the probability generating function of the steady-state queue length distribution is obtained.Moreover,the explicit expressions of the expected queue length and the additional queue length distribution are derived by some algebraic manipulations.Finally,employing the renewal reward theorem,the explicit expression of the long-run expected cost per unit time is given.Furthermore,we analyze the optimal policy for economizing the expected cost and compare the optimal Min(N,D)-policy with the optimal N-policy and the optimal D-policy by using numerical examples.展开更多
Background Low back pain （LBP） is a common clinical problem.Many researchers have demonstrated that LBP disorders have difference in sensory strategies for postural control.Optokinetic stimulation （OKS） of optic f...Background Low back pain （LBP） is a common clinical problem.Many researchers have demonstrated that LBP disorders have difference in sensory strategies for postural control.Optokinetic stimulation （OKS） of optic flow has been widely applied to study its effect on vision,but has not been applied to LBP.Here we used OKS on different surfaces to investigate the characteristics of chronic nonspecific LBP （CNLBP） posture control,so as to provide new theoretical and experimental data for further recognizing CNLBP and enriching its treatment.Methods Fifteen individuals with CNLBP （age range 25-40 years） and 15 age and gender-matched control subjects were recruited.Each subject,while standing on a stable or soft surface,was exposed to random-dot patterns projected on a large screen,with the dots displaying expansion （＋） and contraction （-） and velocities including 80°,40°,and 20° per second.The visual stimulus used a ＂stimuli-interval＂ pattern.The peak velocity,different phases＇ standard deviation （SD） of the anterior-posterior centre of pressure （COP） displacements and the total length of the medial-lateral COP sway （LML） for stable surface and soft surface were recorded by force platform.Results The main effect of surface on all parameters was significant,while the main effect of group and OKS showed no significance with the exception of peak velocity （F（3,95）=3.6,P=0.01） and A2 （F（5,140）=9.34,P ＜0.01） for which the effect of OKS was significant.The interactions of group by OKS of A2 （F（5,140）=3.65,P ＜0.01） and group by surface by OKS （F（5,140）=2.83,P=-0.02）,and surface by OKS of A1 and A3 （P ＜0.05） were significant.It was reported that significantly more SD in amplitude in the T2 phase was seen in persons with CNLBP when confronting the ＋ 40 stimuli on the soft surface （P ＜0.05） compared to healthy individuals.Conclusions There was no significance between persons with CNLBP and healthy people when using the stable surface.Subjects with LBP showed decreased efficiency of postural adjustment when exposed to more complicated tasks and environments,especially OKS in the expansion direction,and displayed a visual-dependent phenomenon.This result suggested that the treatment of abnormal motor patterns in people with LBP should take the properties of task and environment into account.展开更多
The instability property of the standing wave uω（t, x） = eiωtφ（x） for the Klein–Gordon– Hartree equation is investigated. For the case N≥3 and w2 〈2/N＋4-γ,it is shown that the standing wave eiwtφ（x） is...The instability property of the standing wave uω（t, x） = eiωtφ（x） for the Klein–Gordon– Hartree equation is investigated. For the case N≥3 and w2 〈2/N＋4-γ,it is shown that the standing wave eiwtφ（x） is strongly unstable by blow-up in finite time.展开更多
This paper studies an k-out-of-n:G system with redundant dependency and repair equipment procurement lead time where the operating times and repair times of components follow exponential distributions and phase-type d...This paper studies an k-out-of-n:G system with redundant dependency and repair equipment procurement lead time where the operating times and repair times of components follow exponential distributions and phase-type distributions,respectively.When one component breaks down,it is repaired by a repair equipment.The repair equipment may fail during the repair period and the following repair is not‘as good as new’.After a number of repairs,it is replaced by a new one.The new spare repair equipment for replacement is only available by an order,and the procurement lead time for delivering follows a phase-type distribution.Moreover,in the multi-component system,the redundant dependency is taken into account.Applying the matrix-analytical method,the system availability,the rate of occurrence of failures of the system,the expected number of broken components,the availability and the rate of occurrence of failures of the repair equipment are derived.Finally,numerical examples are given to show these theoretical results.展开更多
In this paper,we introduce the notion of a product structure on a 3-Bihom-Lie algebra,which is a Nijenhuis operator with some conditions.We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is...In this paper,we introduce the notion of a product structure on a 3-Bihom-Lie algebra,which is a Nijenhuis operator with some conditions.We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras.Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition.Similarly,we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures.Finally,we establish the relation between a complex structure and a product structure.展开更多
This paper investigates a multi-component repairable system with double threshold control policy.The system is composed of n identical and independent components which operate simultaneously at the beginning,and it is...This paper investigates a multi-component repairable system with double threshold control policy.The system is composed of n identical and independent components which operate simultaneously at the beginning,and it is down when the number of operating components decreases to k−1(k≤n).When the number of failed components is less than the value L,the repairman repairs them with a low repair rate.The high repair rate is activated as soon as L failed components present,and continues until the number of failed components drops to the value N−1.Applying the matrix analytical method,the Laplace transform technique and the properties of the phase type distribution,various performance measures including the availability,the rate of occurrence of failures,and the reliability are derived in transient and stationary regimes.Further,numerical examples are reported to show the behaviour of the system.展开更多
基金supported by the National Key Research and Development Program of China(No.2023YFF0715103)-financial supportNational Natural Science Foundation of China(Grant Nos.62306237 and 62006191)-financial support+1 种基金Key Research and Development Program of Shaanxi(Nos.2024GX-YBXM-149 and 2021ZDLGY15-04)-financial support,NorthwestUniversity Graduate Innovation Project(No.CX2023194)-financial supportNatural Science Foundation of Shaanxi(No.2023-JC-QN-0750)-financial support.
文摘Over 1.3 million people die annually in traffic accidents,and this tragic fact highlights the urgent need to enhance the intelligence of traffic safety and control systems.In modern industrial and technological applications and collaborative edge intelligence,control systems are crucial for ensuring efficiency and safety.However,deficiencies in these systems can lead to significant operational risks.This paper uses edge intelligence to address the challenges of achieving target speeds and improving efficiency in vehicle control,particularly the limitations of traditional Proportional-Integral-Derivative(PID)controllers inmanaging nonlinear and time-varying dynamics,such as varying road conditions and vehicle behavior,which often result in substantial discrepancies between desired and actual speeds,as well as inefficiencies due to manual parameter adjustments.The paper uses edge intelligence to propose a novel PID control algorithm that integrates Backpropagation(BP)neural networks to enhance robustness and adaptability.The BP neural network is first trained to capture the nonlinear dynamic characteristics of the vehicle.Thetrained network is then combined with the PID controller to forma hybrid control strategy.The output layer of the neural network directly adjusts the PIDparameters(k_(p),k_(i),k_(d)),optimizing performance for specific driving scenarios through self-learning and weight adjustments.Simulation experiments demonstrate that our BP neural network-based PID design significantly outperforms traditional methods,with the response time for acceleration from 0 to 1 m/s improved from 0.25 s to just 0.065 s.Furthermore,real-world tests on an intelligent vehicle show its ability to make timely adjustments in response to complex road conditions,ensuring consistent speed maintenance and enhancing overall system performance.
基金supported by the National Natural Science Foundation of China(11201324)the Fok Ying Tuny Education Foundation(141114)the Sichuan Technology Program(2022ZYD0011,2022NFSC1852).
文摘For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.
基金Supported by National Natural Science Foundation of China(60675039)National High Technology Research and Development Program of China(863 Program)(2006AA04Z217)Hundred Talents Program of Chinese Academy of Sciences
基金supported by National Science Foundation of China (11071177)
文摘In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.
基金Supported by National Science Foundation of China (11071177)Excellent Youth Foundation of Sichuan Province (2012JQ0011)
文摘This article is concerned with blow-up solutions of the Cauchy problem of critical nonlinear SchrSdinger equation with a Stark potential. By using the variational characterization of corresponding ground state, the limiting behavior of blow-up solutions with critical and small super-critical mass are obtained in the natural energy space ∑ = {u ∈ H^1; fRN ｜x｜^2｜u｜^2dx 〈 ＋∞）}. Moreover, an interesting concentration property of the blow-up solutions with critical mass is gotten, which reads that ｜u（t, x）｜^2→ ｜｜Q｜｜L^2 2 δx=x1 as t → T.
文摘Underwater Wireless Sensor Networks(UWSNs)are becoming increasingly popular in marine applications due to advances in wireless and microelectronics technology.However,UWSNs present challenges in processing,energy,and memory storage due to the use of acoustic waves for communication,which results in long delays,significant power consumption,limited bandwidth,and packet loss.This paper provides a comprehensive review of the latest advancements in UWSNs,including essential services,common platforms,critical elements,and components such as localization algorithms,communication,synchronization,security,mobility,and applications.Despite significant progress,reliable and flexible solutions are needed to meet the evolving requirements of UWSNs.The purpose of this paper is to provide a framework for future research in the field of UWSNs by examining recent advancements,establishing a standard platform and service criteria,using a taxonomy to determine critical elements,and emphasizing important unresolved issues.
基金the National Key Research and Development Program of China(2018YFB1004901)Zhejiang Natural Science Fund Project of China(LY20F020019,LQ19F020012,LQ20F020001)+1 种基金Zhejiang Basic Public Welfare Research Project of China(LGF19E050005)and Major Scientific Research Project of Zhejiang Lab(2019MC0AD01).
文摘Background With the aim of addressing the difficulty in identifying temperatures in virtual chemistry experiments,we propose a temperature-sensing simulation method of virtual chemistry experiments.Methods We construct a virtual chemistry experiment temperature simulation platform,based on which a wearable temperature generation device is developed.The typical middle school virtual experiments of concentrated sulfuric acid dilution and ammonium nitrate dissolution are conducted to verify the actual effect of the device.Results The platform is capable to indicate near real-world experimental situations.The performance of the device not only meets the temperature sensing characteristics of human skin,but also matches the temperature change of virtual chemistry experiments in real-time.Conclusions It is demonstrated that this temperature-sensing simulation method can represent exothermic or endothermic chemistry experiments,which is beneficial for students to gain understanding of the principles of thermal energy transformation in chemical reactions,thus avoiding the danger that may be posed in the course of traditional teaching of chemistry experiments effectively.Although this method does not have a convenient enough operation for users,the immersion of virtual chemical experiments can be enhanced.
基金supported in part by the Natural Sciences and Engineering Research Council of Canada(Grant Nos.RGPIN-2024-05551,ALLRP 592389-23)the Canada Research Chairs Program(Grant No.CRC-2022-00119)the Fonds de Recherche du Québec–Nature et Technologies(Grant Nos.203345–Centre d’Optique,Photonique,et Lasers).
文摘Complex field modulation(CFM)has found a plethora of applications in physics,biomedicine,and instrumentation.Among existing methods,superpixel-based CFM has been increasingly featured because of its advantages in high modulation accuracy and its compatibility with high-speed spatial light modulators(SLMs).Nonetheless,the mainstream approach based on binary-amplitude modulation confronts limitations in optical efficiency and dynamic range.To surmount these challenges,we develop binary phase-engraved(BiPE)superpixel-based CFM and implement it using the phase light modulator(PLM)—a new micro-electromechanical system-based SLM undergoing development by Texas Instruments in recent years.Using BiPE superpixels,we demonstrate highaccuracy spatial amplitude and phase modulation at up to 1.44 kHz.To showcase its broad utility,we apply BiPEsuperpixel-based CFM to beam shaping,high-speed projection,and augmented-reality display.
基金supported by the National Natural Science Foundation of China(No.71571127)the National Natural Science Youth Foundation of China(No.72001181).
文摘Based on the number of customers and the server’s workload,this paper proposes a modified Min(N,D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy.Applying the well-known stochastic decomposition property of the steady-state queue size,the probability generating function of the steady-state queue length distribution is obtained.Moreover,the explicit expressions of the expected queue length and the additional queue length distribution are derived by some algebraic manipulations.Finally,employing the renewal reward theorem,the explicit expression of the long-run expected cost per unit time is given.Furthermore,we analyze the optimal policy for economizing the expected cost and compare the optimal Min(N,D)-policy with the optimal N-policy and the optimal D-policy by using numerical examples.
文摘Background Low back pain （LBP） is a common clinical problem.Many researchers have demonstrated that LBP disorders have difference in sensory strategies for postural control.Optokinetic stimulation （OKS） of optic flow has been widely applied to study its effect on vision,but has not been applied to LBP.Here we used OKS on different surfaces to investigate the characteristics of chronic nonspecific LBP （CNLBP） posture control,so as to provide new theoretical and experimental data for further recognizing CNLBP and enriching its treatment.Methods Fifteen individuals with CNLBP （age range 25-40 years） and 15 age and gender-matched control subjects were recruited.Each subject,while standing on a stable or soft surface,was exposed to random-dot patterns projected on a large screen,with the dots displaying expansion （＋） and contraction （-） and velocities including 80°,40°,and 20° per second.The visual stimulus used a ＂stimuli-interval＂ pattern.The peak velocity,different phases＇ standard deviation （SD） of the anterior-posterior centre of pressure （COP） displacements and the total length of the medial-lateral COP sway （LML） for stable surface and soft surface were recorded by force platform.Results The main effect of surface on all parameters was significant,while the main effect of group and OKS showed no significance with the exception of peak velocity （F（3,95）=3.6,P=0.01） and A2 （F（5,140）=9.34,P ＜0.01） for which the effect of OKS was significant.The interactions of group by OKS of A2 （F（5,140）=3.65,P ＜0.01） and group by surface by OKS （F（5,140）=2.83,P=-0.02）,and surface by OKS of A1 and A3 （P ＜0.05） were significant.It was reported that significantly more SD in amplitude in the T2 phase was seen in persons with CNLBP when confronting the ＋ 40 stimuli on the soft surface （P ＜0.05） compared to healthy individuals.Conclusions There was no significance between persons with CNLBP and healthy people when using the stable surface.Subjects with LBP showed decreased efficiency of postural adjustment when exposed to more complicated tasks and environments,especially OKS in the expansion direction,and displayed a visual-dependent phenomenon.This result suggested that the treatment of abnormal motor patterns in people with LBP should take the properties of task and environment into account.
基金The first author is supported by the Key Project of Chinese Ministry of Education(Grant No.211162)Sichuan Province Science Foundation for Youths(Grant No.2012JQ0011)the second author is supported byNational Natural Science Foundation of China(Grant No.11371267)
文摘The instability property of the standing wave uω（t, x） = eiωtφ（x） for the Klein–Gordon– Hartree equation is investigated. For the case N≥3 and w2 〈2/N＋4-γ,it is shown that the standing wave eiwtφ（x） is strongly unstable by blow-up in finite time.
基金This research was supported by the National Natural Science Foundation of China[grantnumber 72001181],[grant number 71571127]the funding of V.C.&V.R.Key Lab of Sichuan Province(SCVCVR2019.05VS).
文摘This paper studies an k-out-of-n:G system with redundant dependency and repair equipment procurement lead time where the operating times and repair times of components follow exponential distributions and phase-type distributions,respectively.When one component breaks down,it is repaired by a repair equipment.The repair equipment may fail during the repair period and the following repair is not‘as good as new’.After a number of repairs,it is replaced by a new one.The new spare repair equipment for replacement is only available by an order,and the procurement lead time for delivering follows a phase-type distribution.Moreover,in the multi-component system,the redundant dependency is taken into account.Applying the matrix-analytical method,the system availability,the rate of occurrence of failures of the system,the expected number of broken components,the availability and the rate of occurrence of failures of the repair equipment are derived.Finally,numerical examples are given to show these theoretical results.
基金Supported by NNSF of China(No.12271085 and No.12071405)supported by Sichuan Science and Technology Program(No.2023NSFSC1287).
文摘In this paper,we introduce the notion of a product structure on a 3-Bihom-Lie algebra,which is a Nijenhuis operator with some conditions.We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras.Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition.Similarly,we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures.Finally,we establish the relation between a complex structure and a product structure.
基金This research was supported by the National Natural Science Foundation of China(No.71571127)the funding of V.C.&V.R.Key Lab of Sichuan Province(SCVCVR2019.05VS)the Sichuan Science and Technology Program(Nos.2020YFS0318,2019YFS0155,2019YFS0146,2020YFG0430,2020YFS0307).
文摘This paper investigates a multi-component repairable system with double threshold control policy.The system is composed of n identical and independent components which operate simultaneously at the beginning,and it is down when the number of operating components decreases to k−1(k≤n).When the number of failed components is less than the value L,the repairman repairs them with a low repair rate.The high repair rate is activated as soon as L failed components present,and continues until the number of failed components drops to the value N−1.Applying the matrix analytical method,the Laplace transform technique and the properties of the phase type distribution,various performance measures including the availability,the rate of occurrence of failures,and the reliability are derived in transient and stationary regimes.Further,numerical examples are reported to show the behaviour of the system.
