Unmanned Aerial Vehicles(UAVs)enabled Aerial Base Stations(UABSs)have been studied widely in future communications.However,there are a series of challenges such as interference management,trajectory design and resourc...Unmanned Aerial Vehicles(UAVs)enabled Aerial Base Stations(UABSs)have been studied widely in future communications.However,there are a series of challenges such as interference management,trajectory design and resource allocation in the scenarios of multi-UAV networks.Besides,different performances among UABSs increase complexity and bring many challenges.In this paper,the joint downlink transmission power control and trajectory design problem in multi-type UABSs communication network is investigated.In order to satisfy the signal to interference plus noise power ratio of users,each UABS needs to adjust its position and transmission power.Based on the interactions among multiple communication links,a non-cooperative Mean-Field-Type Game(MFTG)is proposed to model the joint optimization problem.Then,a Nash equilibrium solution is solved by two steps:first,the users in the given area are clustered to get the initial deployment of the UABSs;second,the Mean-Field Q(MFQ)-learning algorithm is proposed to solve the discrete MFTG problem.Finally,the effectiveness of the approach is verified through the simulations,which simplifies the solution process and effectively reduces the energy consumption of each UABS.展开更多
This paper presents a novel model-free method to solve linear quadratic(LQ)mean-field control problems with one-dimensional state space and multiplicative noise.The focus is on the infinite horizon LQ setting,where th...This paper presents a novel model-free method to solve linear quadratic(LQ)mean-field control problems with one-dimensional state space and multiplicative noise.The focus is on the infinite horizon LQ setting,where the conditions for solution either stabilization or optimization can be formulated as two algebraic Riccati equations(AREs).The proposed approach leverages the integral reinforcement learning technique to iteratively solve the drift-coefficient-dependent stochastic ARE(SARE)and other indefinite ARE,without requiring knowledge of the system dynamics.A numerical example is given to demonstrate the effectiveness of the proposed algorithm.展开更多
Presently,we develop a simplified corticothalamic(SCT)model and propose a single-pulse alternately resetting stimulation(SARS)with sequentially applying anodic(A,“+”)or cathodic(C,“−”)phase pulses to the thalamic ...Presently,we develop a simplified corticothalamic(SCT)model and propose a single-pulse alternately resetting stimulation(SARS)with sequentially applying anodic(A,“+”)or cathodic(C,“−”)phase pulses to the thalamic reticular(RE)nuclei,thalamus-cortex(TC)relay nuclei,and cortical excitatory(EX)neurons,respectively.Abatement effects of ACC-SARS of RE,TC,and EX for the 2 Hz-4 Hz spike and wave discharges(SWD)of absence seizures are then concerned.The m∶n on-off ACC-SARS protocol is shown to effectively reduce the SWD with the least current consumption.In particular,when its frequency is out of the 2 Hz-4 Hz SWD dominant rhythm,the desired seizure abatements can be obtained,which can be further improved by our proposed directional steering(DS)stimulation.The dynamical explanations for the SARS induced seizure abatements are lastly given by calculating the averaged mean firing rate(AMFR)of neurons and triggering averaged mean firing rates(TAMFRs)of 2 Hz-4 Hz SWD.展开更多
This paper investigates a distributed optimal energy consumption control strategy under mean-field game based speed consensus.Large scale vehicles in a traffic flow is targeted instead of individual vehicles,and it is...This paper investigates a distributed optimal energy consumption control strategy under mean-field game based speed consensus.Large scale vehicles in a traffic flow is targeted instead of individual vehicles,and it is assumed that the propulsion power of vehicles is hybrid electric powertrain.The control scheme is designed in the following two stages.In the first stage,in order to achieve speed consensus,the acceleration control law is designed by applying the MFG(mean-field game)theory.In the second stage,optimal powertrain control for minimizing energy consumption is obtained through coordinate the engine and the motor under the acceleration constraint.The simulation is conducted to demonstrate the effectiveness of the proposed control strategy.展开更多
We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward-backward stochastic differential equations with jump processes,in which the coefficients depend on the ...We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward-backward stochastic differential equations with jump processes,in which the coefficients depend on the marginal law of the state process through its expected value.The control variable is allowed to enter both diffusion and jump coefficients.Moreover,the cost functional is also of mean-field type.Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques.As an application,time-inconsistent mean-variance portfolio selectionmixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.展开更多
This paper solves a mean-field type hierarchical optimal control problem in electricity market.The authors consider n-1 prosumers and one producer.The ith prosumer,for 1<i<n,is a leader of the(i-1)th prosumer an...This paper solves a mean-field type hierarchical optimal control problem in electricity market.The authors consider n-1 prosumers and one producer.The ith prosumer,for 1<i<n,is a leader of the(i-1)th prosumer and is a follower of the(i+1)th prosumer.The first player(agent)is the follower at the bottom whereas the nth is the leader at the top.The problem is described by a linear jump-diffusion system of conditional mean-field type,where the conditioning is with respect to common noise,and a quadratic cost functional involving,the square of the conditional expectation of the controls of the agents.The authors provide a semi-explicit solution of the corresponding meanfield-type hierarchical control problem with common noise.Finally,the authors illustrate the obtained result via a numerical example with two different scenarios.展开更多
This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differ...This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.展开更多
This paper is devoted to path-dependent kinetics equations arising, in particular, from the analysis of the coupled backward-forward systems of equations of mean field games. We present local well-posedness, global ex...This paper is devoted to path-dependent kinetics equations arising, in particular, from the analysis of the coupled backward-forward systems of equations of mean field games. We present local well-posedness, global existence and some regularity results for these equations.展开更多
We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “cons...We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative” investors that: 1) behave in a similar way, 2) try to avoid abrupt changes in their trading strategies, 3) want to limit the risk due to the fact of having open positions on the asset shares, 4) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and an optimal control problem. These models are coupled by the asset share price dynamic equation. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. This strategy is used to formulate the optimal control problem that determines the behaviour of the big trader. The previous mathematical models are solved using the dynamic programming principle. In some special cases explicit solutions of the previous models are found. An extensive numerical study of the trading execution model proposed is presented. The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website:?http://www.econ.univpm.it/recchioni/finance.展开更多
Motivated by various mean-field type linear-quadratic(MF-LQ,for short)multilevel Stackelberg games,we propose a kind of multi-level self-similar randomized dominationmonotonicity structures.When the coefficients of a ...Motivated by various mean-field type linear-quadratic(MF-LQ,for short)multilevel Stackelberg games,we propose a kind of multi-level self-similar randomized dominationmonotonicity structures.When the coefficients of a class of mean-field type forwardbackward stochastic differential equations(MF-FBSDEs,for short)satisfy this kind of structures,we prove the existence,the uniqueness,an estimate and the continuous dependence on the coefficients of solutions.Further,the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games,respectively.展开更多
This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equati...This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels mar-tingales associated with Lévy processes.In either case,we obtain the optimality system for the optimal controls in open-loop form,and by means of a decoupling technique,we obtain the optimal controls in closed-loop form which can be represented by two Riccati differen-tial equations.Moreover,the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.展开更多
An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional.The coefficients and the weighting matrices in the cost functional are all assumed to be ...An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional.The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic.Closedloop strategies are introduced,which require to be independent of initial states;and such a nature makes it very useful and convenient in applications.In this paper,the existence of an optimal closed-loop strategy for the system(also called the closedloop solvability of the problem)is characterized by the existence of a regular solution to the coupled two(generalized)Riccati equations,together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.展开更多
In this paper,a reduced globus pallidus internal(GPI)-corticothalamic(GCT)model is developed,and a tri-phase delay stimulation(TPDS)with sequentially applying three pulses on the GPI representing the inputs from the s...In this paper,a reduced globus pallidus internal(GPI)-corticothalamic(GCT)model is developed,and a tri-phase delay stimulation(TPDS)with sequentially applying three pulses on the GPI representing the inputs from the striatal D_(1)neurons,subthalamic nucleus(STN),and globus pallidus external(GPE),respectively,is proposed.The GPI is evidenced to control absence seizures characterized by 2 Hz–4 Hz spike and wave discharge(SWD).Hence,based on the basal ganglia-thalamocortical(BGCT)model,we firstly explore the triple effects of D_(1)-GPI,GPE-GPI,and STN-GPI pathways on seizure patterns.Then,using the GCT model,we apply the TPDS on the GPI to potentially investigate the alternative and improved approach if these pathways to the GPI are blocked.The results show that the striatum D_(1),GPE,and STN can indeed jointly and significantly affect seizure patterns.In particular,the TPDS can effectively reproduce the seizure pattern if the D_(1)-GPI,GPE-GPI,and STN-GPI pathways are cut off.In addition,the seizure abatement can be obtained by well tuning the TPDS stimulation parameters.This implies that the TPDS can play the surrogate role similar to the modulation of basal ganglia,which hopefully can be helpful for the development of the brain-computer interface in the clinical application of epilepsy.展开更多
We consider an optimal control problem which serves as a mathematical model for several problems in economics and management.The problem is the minimization of a continuous constrained functional governed by a linear ...We consider an optimal control problem which serves as a mathematical model for several problems in economics and management.The problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection part.The additional constraint is non-negativity of a solution of state equation.We construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in time.All these approximations keep the positivity of the solutions to mesh state problem,either unconditionally or under some additional constraints to mesh steps.This allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its implementation.Based on theoretical estimates and numerical results,we draw conclusions about the quality of the proposed mesh schemes.展开更多
This is an overview paper on the relationship between risk-averse designs based on exponential loss functions with or without an additional unknown(adversarial)term and some classes of stochastic games.In particular,t...This is an overview paper on the relationship between risk-averse designs based on exponential loss functions with or without an additional unknown(adversarial)term and some classes of stochastic games.In particular,the paper discusses the equivalences between risk-averse controller and filter designs and saddle-point solutions of some corresponding risk-neutral stochastic differential games with different information structures for the players.One of the by-products of these analyses is that risk-averse controllers and filters(or estimators)for control and signal-measurement models are robust,through stochastic dissipation inequalities,to unmodeled perturbations in controlled system dynamics as well as signal and the measurement processes.The paper also discusses equivalences between risk-sensitive stochastic zero-sum differential games and some corresponding risk-neutral three-player stochastic zero-sum differential games,as well as robustness issues in stochastic nonzero-sum differential games with finite and infinite populations of players,with the latter belonging to the domain of mean-field games.展开更多
基金co-supported by the National Natural Science Foundation of China(Nos.62001387,61901379)the Natural Science Basic Research Plan in Shaanxi Province(No.2019JQ253)+4 种基金the Key R&D Plan of Shaanxi Province(No.2020GY034)the Aerospace Science and Technology Innovation Fund of China Aerospace Science and Technology Corporationthe Shanghai Aerospace Science and Technology Innovation Fund(No.SAST2018045)the China Fundamental Research Fund for the Central Universities(No.3102018QD096)the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University(No.CX2020152)。
文摘Unmanned Aerial Vehicles(UAVs)enabled Aerial Base Stations(UABSs)have been studied widely in future communications.However,there are a series of challenges such as interference management,trajectory design and resource allocation in the scenarios of multi-UAV networks.Besides,different performances among UABSs increase complexity and bring many challenges.In this paper,the joint downlink transmission power control and trajectory design problem in multi-type UABSs communication network is investigated.In order to satisfy the signal to interference plus noise power ratio of users,each UABS needs to adjust its position and transmission power.Based on the interactions among multiple communication links,a non-cooperative Mean-Field-Type Game(MFTG)is proposed to model the joint optimization problem.Then,a Nash equilibrium solution is solved by two steps:first,the users in the given area are clustered to get the initial deployment of the UABSs;second,the Mean-Field Q(MFQ)-learning algorithm is proposed to solve the discrete MFTG problem.Finally,the effectiveness of the approach is verified through the simulations,which simplifies the solution process and effectively reduces the energy consumption of each UABS.
文摘This paper presents a novel model-free method to solve linear quadratic(LQ)mean-field control problems with one-dimensional state space and multiplicative noise.The focus is on the infinite horizon LQ setting,where the conditions for solution either stabilization or optimization can be formulated as two algebraic Riccati equations(AREs).The proposed approach leverages the integral reinforcement learning technique to iteratively solve the drift-coefficient-dependent stochastic ARE(SARE)and other indefinite ARE,without requiring knowledge of the system dynamics.A numerical example is given to demonstrate the effectiveness of the proposed algorithm.
基金Project supported by the National Natural Science Foundation of China(Nos.11702018,11932003,and 11672074)。
文摘Presently,we develop a simplified corticothalamic(SCT)model and propose a single-pulse alternately resetting stimulation(SARS)with sequentially applying anodic(A,“+”)or cathodic(C,“−”)phase pulses to the thalamic reticular(RE)nuclei,thalamus-cortex(TC)relay nuclei,and cortical excitatory(EX)neurons,respectively.Abatement effects of ACC-SARS of RE,TC,and EX for the 2 Hz-4 Hz spike and wave discharges(SWD)of absence seizures are then concerned.The m∶n on-off ACC-SARS protocol is shown to effectively reduce the SWD with the least current consumption.In particular,when its frequency is out of the 2 Hz-4 Hz SWD dominant rhythm,the desired seizure abatements can be obtained,which can be further improved by our proposed directional steering(DS)stimulation.The dynamical explanations for the SARS induced seizure abatements are lastly given by calculating the averaged mean firing rate(AMFR)of neurons and triggering averaged mean firing rates(TAMFRs)of 2 Hz-4 Hz SWD.
文摘This paper investigates a distributed optimal energy consumption control strategy under mean-field game based speed consensus.Large scale vehicles in a traffic flow is targeted instead of individual vehicles,and it is assumed that the propulsion power of vehicles is hybrid electric powertrain.The control scheme is designed in the following two stages.In the first stage,in order to achieve speed consensus,the acceleration control law is designed by applying the MFG(mean-field game)theory.In the second stage,optimal powertrain control for minimizing energy consumption is obtained through coordinate the engine and the motor under the acceleration constraint.The simulation is conducted to demonstrate the effectiveness of the proposed control strategy.
基金The first author was partially supported by Algerian CNEPRU Project Grant B01420130137,2014-2016.
文摘We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward-backward stochastic differential equations with jump processes,in which the coefficients depend on the marginal law of the state process through its expected value.The control variable is allowed to enter both diffusion and jump coefficients.Moreover,the cost functional is also of mean-field type.Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques.As an application,time-inconsistent mean-variance portfolio selectionmixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.
基金support from Tamkeen under the NYU Abu Dhabi Research Institute grant CG002U.S.Air Force Office of Scientific Research under Grant No.FA955017-1-0259。
文摘This paper solves a mean-field type hierarchical optimal control problem in electricity market.The authors consider n-1 prosumers and one producer.The ith prosumer,for 1<i<n,is a leader of the(i-1)th prosumer and is a follower of the(i+1)th prosumer.The first player(agent)is the follower at the bottom whereas the nth is the leader at the top.The problem is described by a linear jump-diffusion system of conditional mean-field type,where the conditioning is with respect to common noise,and a quadratic cost functional involving,the square of the conditional expectation of the controls of the agents.The authors provide a semi-explicit solution of the corresponding meanfield-type hierarchical control problem with common noise.Finally,the authors illustrate the obtained result via a numerical example with two different scenarios.
基金supported by the National Natural Science Foundation of China(Nos.11871121,11471079,11301177)the Natural Science Foundation of Zhejiang Province for Distinguished Young Scholar(No.LR15A010001)
文摘This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.
文摘This paper is devoted to path-dependent kinetics equations arising, in particular, from the analysis of the coupled backward-forward systems of equations of mean field games. We present local well-posedness, global existence and some regularity results for these equations.
文摘We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative” investors that: 1) behave in a similar way, 2) try to avoid abrupt changes in their trading strategies, 3) want to limit the risk due to the fact of having open positions on the asset shares, 4) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and an optimal control problem. These models are coupled by the asset share price dynamic equation. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. This strategy is used to formulate the optimal control problem that determines the behaviour of the big trader. The previous mathematical models are solved using the dynamic programming principle. In some special cases explicit solutions of the previous models are found. An extensive numerical study of the trading execution model proposed is presented. The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website:?http://www.econ.univpm.it/recchioni/finance.
基金This work is supported in part by the National Key R&D Program of China(Grant No.2018YFA0703900)the National Natural Science Foundation of China(Grant No.11871310).
文摘Motivated by various mean-field type linear-quadratic(MF-LQ,for short)multilevel Stackelberg games,we propose a kind of multi-level self-similar randomized dominationmonotonicity structures.When the coefficients of a class of mean-field type forwardbackward stochastic differential equations(MF-FBSDEs,for short)satisfy this kind of structures,we prove the existence,the uniqueness,an estimate and the continuous dependence on the coefficients of solutions.Further,the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games,respectively.
基金supported by the Key Projects of Natural Science Foundation of Zhejiang Province of China(no.Z22A013952)the National Natural Science Foundation of China(no.11871121)supported by the Natural Science Foundation of Zhejiang Province of China(no.LY21A010001).
文摘This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels mar-tingales associated with Lévy processes.In either case,we obtain the optimality system for the optimal controls in open-loop form,and by means of a decoupling technique,we obtain the optimal controls in closed-loop form which can be represented by two Riccati differen-tial equations.Moreover,the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.
基金supported by Hong Kong RGC under grants 519913,15209614 and 15224215Jingrui Sun was partially supported by the National Natural Science Foundation of China(11401556)+1 种基金the Fundamental Research Funds for the Central Universities(WK 2040000012)Jiongmin Yong was partially supported by NSF DMS-1406776.
文摘An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional.The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic.Closedloop strategies are introduced,which require to be independent of initial states;and such a nature makes it very useful and convenient in applications.In this paper,the existence of an optimal closed-loop strategy for the system(also called the closedloop solvability of the problem)is characterized by the existence of a regular solution to the coupled two(generalized)Riccati equations,together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.
基金supported by the National Natural Science Foundation of China(Nos.11932003,12072021,and 11672074)。
文摘In this paper,a reduced globus pallidus internal(GPI)-corticothalamic(GCT)model is developed,and a tri-phase delay stimulation(TPDS)with sequentially applying three pulses on the GPI representing the inputs from the striatal D_(1)neurons,subthalamic nucleus(STN),and globus pallidus external(GPE),respectively,is proposed.The GPI is evidenced to control absence seizures characterized by 2 Hz–4 Hz spike and wave discharge(SWD).Hence,based on the basal ganglia-thalamocortical(BGCT)model,we firstly explore the triple effects of D_(1)-GPI,GPE-GPI,and STN-GPI pathways on seizure patterns.Then,using the GCT model,we apply the TPDS on the GPI to potentially investigate the alternative and improved approach if these pathways to the GPI are blocked.The results show that the striatum D_(1),GPE,and STN can indeed jointly and significantly affect seizure patterns.In particular,the TPDS can effectively reproduce the seizure pattern if the D_(1)-GPI,GPE-GPI,and STN-GPI pathways are cut off.In addition,the seizure abatement can be obtained by well tuning the TPDS stimulation parameters.This implies that the TPDS can play the surrogate role similar to the modulation of basal ganglia,which hopefully can be helpful for the development of the brain-computer interface in the clinical application of epilepsy.
基金Shuhua Zhang was supported by the National Basic Research Program(No.2012CB955804)the Major Research Plan of the National Natural Science Foundation of China(No.91430108)+2 种基金the Natural Science Foundation of China(No.11771322)the Major Program of Tianjin University of Finance and Economics(No.ZD1302)Alexander Lapin was supported by Russian Foundation of Basic Researches(No.16-01-00408)and by program”1000 Talents”of China.
文摘We consider an optimal control problem which serves as a mathematical model for several problems in economics and management.The problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection part.The additional constraint is non-negativity of a solution of state equation.We construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in time.All these approximations keep the positivity of the solutions to mesh state problem,either unconditionally or under some additional constraints to mesh steps.This allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its implementation.Based on theoretical estimates and numerical results,we draw conclusions about the quality of the proposed mesh schemes.
基金the Air Force Office of Scientific Research(AFOSR)under Grant No.FA9550-19-1-0353the Army Research Office MURI under Grant No.AG285。
文摘This is an overview paper on the relationship between risk-averse designs based on exponential loss functions with or without an additional unknown(adversarial)term and some classes of stochastic games.In particular,the paper discusses the equivalences between risk-averse controller and filter designs and saddle-point solutions of some corresponding risk-neutral stochastic differential games with different information structures for the players.One of the by-products of these analyses is that risk-averse controllers and filters(or estimators)for control and signal-measurement models are robust,through stochastic dissipation inequalities,to unmodeled perturbations in controlled system dynamics as well as signal and the measurement processes.The paper also discusses equivalences between risk-sensitive stochastic zero-sum differential games and some corresponding risk-neutral three-player stochastic zero-sum differential games,as well as robustness issues in stochastic nonzero-sum differential games with finite and infinite populations of players,with the latter belonging to the domain of mean-field games.
