| Mean field linear quadratic optimal control of multi-agent systems have attracted a lot of attention due to its wide application background in many fields,such as astrophysics,statistical analysis,finance,and biology.In addition,as is well known,team decision theory is an important optimization theory that plays a key role in the team decision optimality of multi-agent systems.It involves finding a control strategy for each agent in a multi-agent systems to minimize a common payoff,basecd on certain information it observed.If the multi-agent systems evolve dynamically in time and the decisions of the agents interact with the payoff as well as the information observed by the agents,then we have a team decision problem in which optimal control and information structure plays a decisive role.On the one hand,our work is on the basis of both above,by using the theory of dynamic team theory,we solve the linear quadratic average field teamoptimal control problem for multi-agent systems of arbitrary number(small or moderate population).To distinguish the case of large number,we use "Average field" to substitute with "Mean field".On the other hand,we deal with the infinite time horizon mean-field linear quadratic game for continuum-parameterized multi-agent systems driven by multiplicative noise.This paper mainly discusses the average field the linear quadratic team-optimal control problem and infinite time horizon mean-field linear quadratic game for continuum-parameterized multi-agent systems driven by multiplicative noise.With regard to the linear quadratic team-optimal control problem,the existing methods mainly use the method called social certainty equivalence principle.For detail,under certain conditions(large population multi-agent systems and dynamic parameters satisfy some statistical characteristics),use the mean field term to approximate the population state average term,and then decentralized control strategies are designed.Naturally,how to solve the team-optimal situations of the small populations or medium populations becomes an interesting problem to be solved.On the other hand,each agent can only obtain local information(says,to obtain local information is usually cheaper than obtaining global one)to implement control actions,so how to rely only on local information to design decentralized control strategies are particularly important.In addition,since the state process of stochastic dynamic system is usually not easily accessible,instead of an measure process related to the dynamic system.So,how to design decentralized control strategies using only the measure data of each agent observed is a urgent problem to be solved.Regarding the mean field linear quadratic games,the current results are mostly discussed in the situation of additive noise.The results on multiplicative noise are still rare and the related results mainly focus on the case of the same dynamic parameter.Furthermore,the admissible control set considered usually is taken as some square-integrable processes.Comparing with the situations above,for one hand,the dynamic parameters of multi-agent systems often show slight differences;for another hand,the admissible control set considered is not very broad in terms of engineering requirements.How to deal with problems proposed above simultaneously?In response to the problems above,this thesis mainly focus on the following aspects:1)Average field LQ team-optimal control for multi-agent systems with additive noise.In this chapter,we use the team decision theory to deal with the average field linear quadratic team-optimal control problem for multi-agent systems driven by additive noise.The classical variational method is used to derive the necessary and sufficient condition for this optimization problem.Wherein,in order to obtain the decentralized control strategies,by a simple algebraic calculation on the cost functional,we terminally get the result that the necessary conditions satisfied by each control process and the backward stochastic differential equations satisfied by the adjoint process of its corresponding state process are separated respectively for each agent.In the aspect of the decentralized control strategy,by means of the filtering theories of forward stochastic differential equations and backward stochastic differential equations,we design the decentralized control strategy of state feedback type.This part mainly contains two situations:partial observation and full observation.And we design the decentralized control strategies under these two situations.In the former case,we discuss the link between it and the mean field linear quadratic optimal control with cost functional of social type[53].Specifically,when the population of the multi-agent system is small or moderate,the off-line calculation part needs to solve a boundary value problem of an ordinary differential equations instead,comparing with the results of[53].When the dynamical parameters are the same,the boundary value problem of the ordinary differential equations can be degenerated into a boundary value problem of an ordinary differential equation.It is interesting that when the population size tends to infinity,we get the same result as article[10].In the later case,with the help of the Kalman filter theory,we convert the problem into a fully observed situation,and then derive the decentralized control strategies.In special,we show that the separation principle holds.2)Average field LQ team-optimal control for multi-agent systems with multi-plicative noise.In this chapter,we use the team decision optimality theory to deal with average field linear quadratic multi-agent systems driven by multiplicative noise.This part is the generalization of the previous chapter.Based on the classical variational method,we derive the necessary and sufficient condition for this optimization problem.The necessary conditions satisfied by each control process and the backward stochastic differential equations satisfied by the adjoint process of its corresponding state process are separated respective for each agent by using a similar method in the previous chapter.Then,the decentralized control strategy of state feedback type is designed by means of the filtering theories of forward stochastic differential equations and backward stochastic differential equations.Wherein,different from the case of additive noise,the method of integrating the adjoint equation in the previous chapter and then taking the conditional expectation is no longer applicable due to the diffusion term containing the control process.Be inspired from the integrand function in the cost functional,we assume that the adjoint process is an affine transformation of the state processes and then use the undetermined coefficient method to solve it.The Riccati equations and the backward ordinary differential equations obtained are slightly more complex than the ones in the previous chapter.In addition,it is easy to see when the diffusion term does not contain state and control processes,the result obtained is the same as the previous chapter.3)Dynamic team theory of stochastic systems under partial observation and its application.In this chapter,we derive the minimum principle of this dynamic team decision optimality problem under partial observations and apply the results to the average field linear quadratic multi-agent systems driven by multiplicative noise under the partial observation.The minimum principle of this dynamic team decision optimality problem is derived under both of reference measure and original measure.Different from the method used in[12],we avoid to augmenting the systems,and the adjoint processes obtained are simpler than theirs.This relieves the calculation burden in application.Finally,we apply the results obtained to the average field linear quadratic multi-agent systems driven by multiplicative noise under partial observation,and derive the necessary condition for each control process.By utilizing the filtering theory for forward stochastic differential equations and backwards stochastic differential equations,we derive the decentralized control strategies in the form of state feedback.4)Mean field stochastic LQ games for continuum-parameterized multi-agent systems.In this chapter,we discuss the infinite time horizon mean-field linear quadratic game for continuum-parameterized multi-agent systems driven by multiplicative noise.The cost functional considered has a form of time-average and the admissible control set is defined as a set of all progressively measurable random processes such that square integrals of the state process have the same order with the terminal moment.For the above model,we deduce the asymptotic equilibrium decentralized control strategy,including the decentralized control strategy design and the asymptotic equilibrium analysis.Regarding the decentralized control strategy design,a linear quadratic optimal tracking problem with a given signal is discussed in advance,and then Nash certain equivalent principle is used to get the decentralized control strategy.Regarding the asymptotic equilibrium analysis,with the help of the Dynkin formula and comparison theorem,the stability of the closed-loop system is proved,and then the asymptotic equilibrium analysis is shown by a perturbation method. |
