【作者】 王健；

【导师】 朱元国；

【作者基本信息】 南京理工大学 ， 数学， 2022， 博士

【摘要】 不确定微分方程作为模拟不确定动力系统的有效工具,是不确定理论及微分方程领域研究的一个重要课题。目前,不确定微分方程在理论领域和应用领域都取得了比较丰富的科研成果。然而,随着认知水平的提高,在考虑一些复杂不确定系统的时候,人们发现影响系统运动规律的并不只是系统的当前状态,有时还依赖于系统的过去状态;不确定时滞微分方程是刻画此类系统演化过程的一类有效工具。本文针对不确定时滞微分方程和不确定分数阶时滞微分方程,主要研究方程的解析解、解的逆分布、解的存在唯一性、稳定性等内容。论文的主要研究工作有如下方面:1、研究了两类不确定时滞微分方程的解。首先,基于不确定理论,利用构造验证的方法,给出了两类线性不确定时滞微分方程的解析解;其次,借助α-轨道的定义及拉普拉斯变换,进一步求出了解的逆不确定分布;随后,借助微分方程的格朗沃尔不等式,给出了解对初值的依赖关系式;最后,用所得结论求解了数值算例。2、研究了不确定时滞微分方程的稳定性。基于不确定时滞微分方程依测度稳定、依均值稳定、P-阶矩稳定、依分布稳定以及几乎必然稳定的定义,利用不等式的放缩技巧,依次提出并证明了不确定时滞微分方程稳定的新的充分条件。一些算例表明本文的判定条件与已有成果相比具有一定的优越性。3、研究了一类不确定分数阶时滞微分方程的解。本文针对一类特殊的线性不确定分数阶时滞微分方程,借助Mittag-Leffler函数,给出了其解的解析表达式;同时,进一步研究了其解的逆不确定分布及解对初值的依赖性。4、研究了不确定分数阶时滞微分方程解的存在唯一性及稳定性。首先,借助逐次逼近法,证明了方程系数在满足李普希茨和线性增长条件下,解的存在唯一性及样本连续性;接着,证明了在测度意义下解对初值的连续依赖性定理;随后,基于依测度稳定的定义,提出并证明了不确定分数阶时滞微分方程依测度稳定的充分条件;最后,通过典型算例对结论进行了验证。5、研究了一类不确定分数阶中立型时滞微分方程的解。首先,提出了不确定分数阶中立型时滞微分方程的一般形式;其次,针对一类特殊的线性不确定分数阶中立型时滞微分方程,推导出了其解析解;同时,还研究了其解的逆分布及解对初值的依赖性。更多还原

【Abstract】 As an effective tool to simulate uncertain dynamic systems,uncertain differential equations are an important topic in the fields of uncertainty theory and differential equations.At present,uncertain differential equations have achieved rich research results in both theoretical and application fields.However,with the improvement of cognitive level,when considering some complex uncertain systems,it is found that what affects the motion law of the system is not only the current state of the system,but also sometimes depends on the past state of the system.Uncertain delay differential equations are a kind of effective tools to describe the evolution process of such systems.For uncertain delay differential equations and uncertain fractional order delay differential equations,this paper mainly studies their analytical solutions,inverse distribution of a solution,existence and uniqueness of a solution,stability and so on.The main research work of this paper is as follows:1.The solutions of two kinds of uncertain delay differential equations are considered.Firstly,based on the uncertainty theory,the analytical solutions of two kinds of linear uncertain delay differential equations are obtained by using the method of construction verification;Secondly,with the help of the definition of α-path and Laplace transform,the inverse uncertainty distribution of the solution is further studied;Then,the dependence of the solution on the initial value is given by means of the Gronwall inequality of delay differential equation;Finally,some numerical examples are solved with the obtained results.2.The stability of uncertain delay differential equations is studied.Based on the definitions of stability in measure,stability in mean,stability in P-th moment,stability in distribution and almost sure stability for uncertain delay differential equations,and employing the technique of inequalities,new sufficient conditions for the stability of uncertain delay differential equations are proposed and proved.Some examples show that the judgment conditions in this paper are superior to the existing results.3.The solutions of a class of uncertain fractional delay differential equations are studied.Firstly,the analytical expression of the solution of a special class of linear uncertain fractional delay differential equations is obtained by means of Mittag-Leffler functions;After that,the inverse uncertainty distribution of the solution and the dependence of the solution on the initial value are further investigated.4.The existence and uniqueness and stability of solutions to uncertain fractional delay differential equations are investigated.Firstly,with the help of successive approximation method,the existence and uniqueness and sample continuity of the solution are proved when the coefficients satisfy Lipschitz and linear growth conditions;Subsequently,the continuous dependence theorem of the solution on the initial value in the sense of uncertainty measure is established;After that,based on the definition of stability in measure,the sufficient conditions for the stability of uncertain fractional delay differential equations are proposed and proved;Finally,the conclusion is verified by a typical example.5.The solutions of a class of uncertain fractional order neutral delay differential equations are discussed.Firstly,the general form of uncertain fractional order neutral delay differential equations is introduced;Secondly,for a special class of linear uncertain fractional order neutral delay differential equations,the analytical solutions are derived;Eventually,the inverse distribution of the solution and the dependence of the solution on the initial value are also obtained.更多还原