# Near Optimality of Linear Delayed Doubly Stochastic Control Problem

Authors: Jie Xu; Ruiqiang Lin
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.


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## Body

## 1. Introduction

As known to all, stochastic differential equations and stochastic analysis develop rapidly. The theory of stochastic differential equations is widely used in economy, biology, physics, financial mathematics, and other fields. In order to give the probabilistic expression of stochastic partial differential equations, Pardoux and Peng [1] gave a class of double stochastic differential equations. Due to the wide applications of this kind of equation in many fields, more and more people pay attention to it. Han et al. [2] deduced the maximum principle for the backward doubly stochastic control system. Zhu and Shi [3] discussed the optimal control problem of the backward doubly stochastic system with partial information. And then they studied a type of forward-backward doubly stochastic differential equations with random jumps and applied their results to related games [4]. Many scholars have discussed the maximum principle of optimal control for different control systems [5].With the further exploration of stochastic problems, we find that many problems in the objective world are not only affected by the current state but also influenced by the past history. This kind of problem is called time delay problem. Time delay exists in many fields such as the latent period of infectious diseases, genetic problems, advertising effects, network transmission, and so on. The equation describing this kind of problem is called delay equation. Because of the importance of time delay, people try to study this kind of problem. Chen and Wu [6] considered the delayed backward stochastic system and obtained the maximum principle for this problem. Wu and Wang [7] studied the optimal control problem of the backward stochastic differential delay equation under partial information. Lv et al. [8] considered the maximum principle for optimal control of the anticipated forward-backward stochastic delayed system with regime switching. Wang and Wu [9] concerned with the optimal control problems of the forward-backward delay system involving impulse controls and established the stochastic maximum principle for this kind of system. Zhou [10] investigated the maximum principle for stochastic optimal control problems of the delay system with random coefficients involving both continuous and impulse controls. In previous work, we mainly studied the theory of doubly stochastic differential equations with time delay. We deduced the maximum principle for the double stochastic control system when all variables contain time delay variables [11]. And we concerned the expression of optimal control and value function by the solution of the Riccati equation for a special delayed doubly stochastic linear quadratic control system [12].When we study the control problems, we usually focus on finding optimal control. However, in practice, the optimal control may not exist or be difficult to obtain. Whether in theoretical analysis or numerical calculation, it is easier to obtain near optimal control than optimal control. Moreover, near optimal control has its unique advantages both in theory and practice. In order to solve the problem better, we need to pay attention to the research of near optimal control. Ekeland [13] discussed the necessary conditions for near optimality of the control system driven by ordinary differential equations. Zhou [14–16] discussed the dynamical system and gave the necessary and sufficient conditions for the existence of near optimal solutions for a kind of stochastic control problem. Bahlali et al. [17] considered a class of nonlinear forward-backward stochastic differential equations and gave the necessary conditions for near optimal control. Hafayed et al. [18] concerned with the stochastic maximum principle for near optimal control of nonlinear controlled mean-field forward-backward stochastic systems driven by Brownian motions and random Poisson martingale measure. Wang and Wu [19] and Zhang [20] discussed near optimal problem for the stochastic system with time delay, respectively. Li and Hu [21] concerned with a near optimal control problem for systems governed by mean-field forward-backward stochastic differential equations with mixed initial-terminal conditions.By consulting some literatures, we find that the near optimal control problem of the deterministic control system and stochastic system has relatively complete conclusions. However, the similar results about the doubly stochastic system are relatively few. Inspired by such problems, we try to study near optimal control problem of delayed doubly stochastic linear quadratic optimal control problem. We deduce the necessary condition of the near optimal control problem for the delayed system, which is similar to the maximum principle for the optimal control problem.The rest of our paper is organized as follows. In this section, we introduce the elementary introduction. In Section2, we give some common notations and necessary formulas, as well as the main conclusions to be used later. In Section 3, we give our main results of this paper. When we deal with the time delay problem, how to deal with the delay term reasonably is the key to our research. At the same time, different time delay variables will make our research more difficult. We define a function H˜ which is similar to the Hamiltonian function and discuss some estimates for the solution of the adjoint equations. Then, we deduce the conclusions according to Ekeland’s variational principle.


## 2. Preliminaries

Let us give some notations used in this paper. SetΩ,ℱ,P as a probability space and T>0 as fixed throughout our paper. Wt:0≤t≤T and Bt:0≤t≤T are two mutually independent standard Brownian motions which are defined on Ω,ℱ,P. The integral with respect to Wt is defined to be the forward Itô’s integral, and its value is in Rm. Note that the integral with respect to Bt is defined to be the backward Itô’s integral, and its value is in Rd. Let N denote the class of P-null sets of ℱ. For each t∈0,T, we define ℱt≐ℱtW∨ℱt,TB, where ℱtW=N∨σWr−W0:0≤r≤tandℱt,TB=N∨σBr−Bt:t≤r≤T. Note that the collection ℱt:t∈0,T is neither increasing nor decreasing, so it does not constitute a filtration.LetM20,T;Rn denote the set of all classes of (dt×dPa.e.equal) ℱt measurable stochastic process φt satisfying E∫0Tφt2dt<+∞. Similarly, S20,T;Rn denote the set of continuous n-dimensional ℱt measurable stochastic process φt satisfying Esupt∈0,Tφt2<+∞. ⋅,⋅ denotes the inner product. And ⊤ in the superscripts of the matrix means the transpose of the matrix. Moreover, Eℱt⋅=E⋅|ℱt denotes the conditional expectation under ℱt.For a convex subsetU⊂Rk, let U0,T=u:0,T×Ω⟶U|u is ℱt−measurable, E∫0Tut2dt<+∞.In general, the delayed doubly stochastic systems can be defined as follows:(1)dxt=ft,xt,xt−δ1,yt,yt−δ2,ut,ut−δ3dtquad+gt,xt,xt−δ1,yt,yt−δ2,ut,ut−δ3dWt⟶quad−ytdBt←,t∈0,T,xt=φt,t∈−δ1,0,yt=ψt,t∈−δ2,0,ut=ηt,t∈−δ3,0.Functionsf and g can be defined in different forms according to different problems. In this paper, we mainly investigate the delayed doubly stochastic linear quadratic control system, that is,(2)dxt=A1txt+B1txt−δ1+C1tyt+D1tyt−δ2+E1tut+F1tut−δ3dt+A2txt+B2txt−δ1+C2tyt+D2tyt−δ2+E2tut+F2tut−δ3dWt⟶−ytdBt←,t∈0,T,xt=φt,t∈−δ1,0,yt=ψt,t∈−δ2,0,ut=0,t∈−δ3,0,where the delayed variables δ1, δ2, and δ3 are not equal.Remark 1.
In this delayed doubly stochastic control system, the state variables and the control variables contain time delay at the same time, and the three delay variables are different. Time delay exists all the time in the system. However, we do nothing before the initial time. So, we give the assumption thatut=0 when the time t belongs to the interval before the intervention of the control variable.
The cost functional can be written as(3)Ju⋅=E∫0Tlt,xt,yt,utdt+ΦxT.
For better analysis and research, we give some definitions similar to these in reference [16].Definition 1.
The optimal control problem of the delayed doubly stochastic system can be described as minimizing the cost functional overU0,T to obtain the optimal control u∗⋅ satisfying(4)Ju∗⋅=V=infu⋅∈U0,TJu⋅,and the corresponding x∗⋅,y∗⋅,u∗⋅ is called an optimal triple.Definition 2.
For a givenε>0, an admissible triple xε⋅,yε⋅,uε⋅ or simply uε⋅ is called ε− optimal if Juε−V≤ε.Definition 3.
A family of admissible triplesxε⋅,yε⋅,uε⋅ or simply uε⋅ parameterized by ε>0 is called near optimal if Juε−V≤rε holds for sufficiently small ε, where rε⟶0 as ε⟶0. The estimate rε is called an error bound. If rε=cεγ for some γ>0 independent of the constant c, then uε is called near optimal with order εγ.We assume that the following conditions hold:(A1) Assume that the coefficient matricesAi, Bi, Ci, Di, Ei, and Fii=1,2 are bounded matrix processes with proper dimensions, i=1,2(A2) The functionΦ is continuously differentiable in x, and the partial derivative of Φ is bounded(A3) The functionl is continuously differentiable in x,y,u, and every partial derivative is boundedCorresponding to the delayed doubly stochastic control system, the adjoint equation can be written as(5)−dpt=A1⊤tpt+EℱtB1⊤t+δ1pt+δ1+A2⊤tqt−lxt+EℱtB2⊤t+δ1qt+δ1dt+lyt−C1⊤tpt−C2⊤tqt−EℱtD1⊤t+δ2pt+δ2−EℱtD2⊤t+δ2qt+δ2dBt←−qtdWt⟶,t∈0,T,pT=−ΦxxT,pt=0,t∈T,T+δ,qt=0,t∈T,T+δ,where the variable δ=maxδ1,δ2,δ3.Remark 2.
According to Theorem 3.1 in [11], the delayed doubly stochastic differential equation (2) admits a unique solution.Lemma 1.
Under the assumption (A1), the adjoint equation (5) admits a unique solution pt,qt for any u∈U0,T. And there exists a positive constant C>0 such that(6)Esup0≤t≤Tput2+∫0Tqut2dt≤C,∀u∈U0,T.Proof.
Adjoint equation (5) is a new kind of equation which is similar to the anticipated backward stochastic differential equation in [22]. We call it anticipated backward doubly stochastic differential equation. Theorem 2.2 in reference [23] introduces the conditions for the existence and uniqueness of solution of general anticipated backward doubly stochastic differential equation. In this paper, we only discuss the linear system, which is a special case in reference [23]. Characteristics of the linear system and boundedness of coefficient from assumption (A1) satisfy the condition of Theorem 2.3. We can directly deduce the existence and uniqueness of the solution from this theorem.
Under the premise of the existence of solutions, Theorem 2.5 in reference [24] gives the boundedness of solutions in general cases. When we discuss the linear system, the term ∫0Tft,0,0,0,02+gt,0,0,0,02dt=0. And the delayed terms pt=0,qt=0 when t∈T,T+δ. Then, we can deduce inequality (6) by Theorem 2.5 in reference [24] directly.Definition 4.
Let us define a metricd on U by du,v=E∫0Tut−vt2dt1/2.Obviously,U,d is a complete metric space. Next, we will discuss the relation by using the metric d.Lemma 2.
Assume (A1), then there exists a constantC>0 satisfying(7)Esup0≤t≤Txut−xvt2+∫0Tyut−yvt2dt≤Cdu,v2.Proof.
Applying Itô’s formula and Jensen inequality for the general delayed doubly stochastic system (1), we have(8)Exut−xvt2+∫0Tyut−yvt2dt≤E∫0T|ft,xut,xut−δ1,yut,yut−δ2,ut,ut−δ3−ft,xvt,xvt−δ1,yvt,yvt−δ2,vt,vt−δ3|2dt+E∫0T|gt,xut,xut−δ1,yut,yut−δ2,ut,ut−δ3−gt,xvt,xvt−δ1,yvt,yvt−δ2,vt,vt−δ3|2dt.
For the liner system (2), we can deal with the first term in (8) as the following:(9)E∫0T|ft,xut,xut−δ1,yut,yut−δ2,ut,ut−δ3−ft,xvt,xvt−δ1,yvt,yvt−δ2,vt,vt−δ3|2dt≤E∫0TA1txut−xvt2+B1txut−δ1−xvt−δ12+C1tyut−yvt2+D1tyut−δ2−yvt−δ22+E1tut−vt2+F1tut−δ3−vt−δ32dt.
Using variable substitution and paying attention to the initial conditions, we can get the following conclusions:(10)E∫0Txut−δ1−xvt−δ1dt=E∫−δ1T−δ1xut−xvtdt≤E∫0Txut−xvtdt.
Similarly,(11)E∫0Tyut−δ2−yvt−δ2dt=E∫−δ2T−δ2yut−yvtdt≤E∫0Tyut−yvtdt,(12)E∫0Tut−δ3−vt−δ3dt=E∫−δ3T−δ3ut−vtdt≤E∫0Tut−vtdt.
Then, substitute inequalities (10)–(12) into (9). Under the assumption (A1), there is a constant C>0 such that(13)E∫0T|ft,xut,xut−δ1,yut,yut−δ2,ut,ut−δ3−ft,xvt,xvt−δ1,yvt,yvt−δ2,vt,vt−δ3|2dt≤CE∫0Txut−xvt2dt+E∫0Tyut−yvt2dt+ut−vt2=CE∫0Txut−xvt2dt+E∫0Tyut−yvt2dt+du,v2.
Similarly, for the second term in (8), we have(14)E∫0T|gt,xut,xut−δ1,yut,yut−δ2,ut,ut−δ3−gt,xvt,xvt−δ1,yvt,yvt−δ2,vt,vt−δ3|2dt≤CE∫0Txut−xvt2dt+E∫0Tyut−yvt2dt+du,v2.
Using Gronwall’s inequality and Lemma 3.1 in [9], we can deduce conclusion (7) directly.Similarly, using the same method and Proposition 2.5 in reference [24], we can deduce the following conclusion directly.Lemma 3.
Assume (A1), then there exists a constantC>0 satisfying(15)Esup0≤t≤Tput−qvt2+∫0Tput−qvt2dt≤Cdu,v2.Lemma 4.
Assume (A1–A3), then there exists a constantC>0 satisfying(16)Ju−Jv≤Cdu,v,for all u,v∈U.Proof.
From (3) and the elementary inequality, we have(17)Ju−Jv≤E∫0Tluxut,yut,ut−lvxvt,yvt,vtdt+ΦxuT−ΦxvT.
From condition (A2), Lemma2, and Cauchy–Schwartz inequality, we find that(18)ΦxuT−ΦxvT=∫01ΦxxvT+λxuT−xvT,xuT−xvTdλ≤Cdu,v.
For the convenience of proof, we denote symbol(19)ςt=t,xvt+λxut−xvt,yvt+λyut−yvt,vt+λut−vt.
Then, we have(20)luxut,yut,ut−lvxvt,yvt,vt=∫01lxςt,xut−xvt+lyςt,yut−yvt+luςt,ut−vtdλ.
By using the same method, from the assumption (A3), Lemmas2 and 3, and the Definition 4, we can deduce that(21)E∫0Tluxut,yut,ut−lvxvt,yvt,vtdt≤Cdu,v.
Combining inequalities (18) and (21), we can prove the conclusion directly.Ekeland’s variational principle is an important tool for our study which can be seen in [25].Lemma 5.
(Ekeland’s variational principle). LetS,d be a complete metric space and ρ⋅:S⟶R1 be a lower-semicontinuous and bounded from below. For ε≥0, suppose uε∈S satisfies(22)ρuε≤infu∈Sρu+ε.Then, for anyλ>0, there exists uλ∈S such that(23)ρuλ≤ρuε,duλ,uε≤λ,ρuλ≤ρu+ελdu,uλ,for allu∈S.Assume thatuε∈U is a ε-optimal control; from Definition 2, we have Juε−V≤ε, that is, Juε≤V+ε. Then, from Definition 1, we have Juε≤infv∈UJv+ε. From assumption (A2), we know that J⋅ is a continuous bounded function and U,d is a complete metric space. From Lemma 5, we know that there is a uλ∈S, such that Juλ≤Juε, ∀λ>0. Take λ=ε and then uλ=u˜ε. We have Ju˜ε≤Juε and du˜ε,uε≤λ=ε.Then, we have(24)Ju˜ε≤Ju+εdu,u˜ε,∀u∈U.We discussu˜ε first and pay attention to uε. Let u∈M2−δ′,Tδ′=minδ1,δ2,δ3 satisfy u˜ε+u∈U. In the previous assumptions, we know that ut=0 for −δ3≤t≤0. Define uθ:=u˜ε+θu,θ∈0,1. Then, we have uθ=u˜ε+θu−u˜ε=1−θu˜ε+θu. From the convexity of U, we can deduce that uθ∈U for any θ∈0,1. Then, duθ,u˜ε=E∫0Tuθ−u˜ε2dt1/2=E∫0Tθu2dt1/2=θE∫0Tu2dt1/2. From the bounded of U, we know that there exist a constant β independent of ε and θ, such that duθ,u˜ε≤βθ.From inequality (24), we have(25)Ju˜ε≤Juθ+εduθ,u˜ε≤Juθ+βεθ.That is,(26)Juθ−Ju˜ε≥−βεθ.Let us introduce variational equations.(27)dx1t=A1tx1t+B1tx1t−δ1+C1ty1t+D1ty1t−δ2+E1tut+F1tut−δ3dt+A2tx1t+B2tx1t−δ1+C2ty1t+D2ty1t−δ2+E2tut+F2tut−δ3dWt⟶−y1tdBt←,t∈0,T,x1t=0,t∈−δ1,0,y1t=0,t∈−δ2,0.In order to simplify the symbols in proof, we denoteξθt:=t,xθt,yθt,uθt and ξε˜t:=t,xε˜t,yε˜t,uε˜t.


## 3. Main Results

Theorem 1.
Let (A1)–(A3) hold. Then, there exists a constantβ>0 independent of ε, such that(28)E∫0TEℱtF1⊤t+δ3p˜εt+δ3+F2⊤t+δ3q˜εt+δ3+E1⊤tp˜εt+E2⊤tq˜εt−luξ˜εt,v−u˜εtdt≤βε,∀v∈U.Proof.
From the definition of functionJ⋅ and inequality (26), we have(29)limθ⟶0Juθ−Ju˜εθ=limθ⟶0E∫0Tlξθt−lξ˜εtdtθ+limθ⟶0EΦxθT−Φx˜εTθ=E∫0Tlxξ˜εt,x1t+lyξ˜εt,y1t+luξ˜εt,utdt+Φxx˜εT,x1T≥−βε.
Next, we will deal with the termΦxx˜εT,x1T. We connect it with the solution of the adjoint equation. Using the Itô–Doeblin formula, we have(30)Ex1T,−Φxx˜εT=E∫0T−x1t,EℱtB1⊤t+δ1p˜εt+δ1+EℱtB2⊤t+δ1q˜εt+δ1−lx∗ξ˜εt+B1tx1t−δ1+D1ty1t−δ2+E1tut+F1tut−δ3,p˜εt+B2tx1t−δ1+D2ty1t−δ2+E2tut+F2tut−δ3,q˜εt+y1t,ly∗ξ˜εt−EℱtD1⊤t+δ2p˜εt+δ2−EℱtD2⊤t+δ2q˜εt+δ2dt.
Let us deal with the first term.(31)E∫0T−x1t,EℱtB1⊤t+δ1p˜εt+δ1dt=E∫δ1T+δ1−x1t−δ1,Eℱt−δ1B1⊤tp˜εtdt=E∫δ1T+δ1−x1t−δ1,B1⊤tp˜εtdt=E∫0T−x1t−δ1,B1⊤tp˜εtdt−∫0δ1−x1t−δ1,B1⊤tp˜εtdt+∫TT+δ1−x1t−δ1,B1⊤tp˜εtdt.
From the definition of adjoint equation (5) and variation equation (10), we have(32)E∫0T−x1t−δ1,B1⊤tp˜εtdt=0,(33)E∫TT+δ1−x1t−δ1,B1⊤tp˜εtdt=0.
Combining equalities (31)–(33), we deduce the following equality:(34)E∫0T−x1t,EℱtB1⊤t+δ1p˜εt+δ1+B1tx1t−δ1,p˜εtdt=0.
Similarly, we have(35)E∫0T−x1t,EℱtB2⊤t+δ1q˜εt+δ1+B2tx1t−δ1,q˜εtdt=0.
In the same way, we have(36)E∫0TD1ty1t−δ2,p˜εt+y1t,−EℱtD1⊤t+δ2p˜εt+δ2dt=0,(37)E∫0TD2ty1t−δ2,q˜εt+y1t,−EℱtD2⊤t+δ2q˜εt+δ2dt=0.
Substituting (34)–(37) into equality (30), we have(38)Ex1T,−ΦxxT=E∫0Tx1t,lxξ˜εt+E1tut+F1tut−δ3,p˜εt+E2tut+F2tut−δ3,q˜εt+y1t,lyξ˜εtdt.
Let us deal with delayed control variables.(39)∫0TF1tut−δ3,p˜εtdt=∫−δ3T−δ3F1t+δ3ut,p˜εt+δ3dt=∫−δ30F1⊤t+δ3p˜εt+δ3,utdt+∫0T−δ3F1⊤t+δ3p˜εt+δ3,utdt.
From the remark, we know thatut=0 when −δ3≤t≤0. From the adjoint equation (5), we have the terminal condition that pt=0 for T≤t≤T+δ, δ=maxδ1,δ2,δ3. Then, we have(40)E∫0TF1tut−δ3,p˜εtdt=E∫0TF1⊤t+δ3p˜εt+δ3,utdt−∫T−δ3TF1⊤t+δ3p˜εt+δ3,utdt=E∫0TEℱtF1⊤t+δ3p˜εt+δ3,utdt.
Equation (38) can be written as(41)Ex1T,−ΦxxT=E∫0Tx1t,lxξ˜εt+E1tut,p˜εt+E2tut,q˜εt+y1t,lyξ˜εt+EℱtF1⊤t+δ3p˜εt+δ3,ut+EℱtF2⊤t+δ3q˜εt+δ3,utdt.=E∫0Tx1t,lxξ˜εtE1⊤tp˜εt+E2⊤tq˜εt,ut+y1t,lyξ˜εt+EℱtF1⊤t+δ3p˜εt+δ3+F2⊤t+δ3q˜εt+δ3,utdt.
According to inequality (29) and equation (41), we can deduce that(42)E∫0Tluξ˜εt−E1⊤tp˜t−E2⊤tq˜t−EℱtF1⊤t+δ3p˜t+δ3−F2⊤t+δ3q˜t+δ3,utdt≥βε.
We know thatu is a variable such that uε+u∈U. Assume that uε+u=v∈U, then the desired conclusion (28) is deduced directly, that is,(43)E∫0TEℱtF1⊤t+δ3p˜εt+δ3+F2⊤t+δ3q˜εt+δ3+E1⊤tp˜εt+E2⊤tq˜εt−luξ˜εt,v−u˜εtdt≤βε,∀v∈U.
Theorem1 is proved.Next, we will show the necessary condition for the near optimal control of the delayed doubly stochastic control system.First, we give the definition of the Hamiltonian function of general delayed doubly stochastic system (1).(44)Ht,xt,xt−δ1,yt,yt−δ2,ut,ut−δ3=f⊤t,xt,xt−δ1,yt,yt−δ2,ut,ut−δ3pt+g⊤t,xt,xt−δ1,yt,yt−δ2,ut,ut−δ3qt.For linear system (2), we have(45)Hu=E1⊤tpt+E2⊤tqt,Huδ=F1⊤tpt+F2⊤tqt,Huδt+δ3=F1⊤t+δ3pt+δ3+F2⊤t+δ3qt+δ3.Assume that(46)H˜=Hu+EℱtHuδt+δ3.Then, we have the following conclusion.Theorem 2.
Assume (A1)–(A3). There exists a constantβ>0 such that for any ε>0, γ∈0,1/2, and the ε−optimal control triple xε,yε,uε of the delayed doubly stochastic control problems (2)–(4), we have(47)E∫0THuε˜,v−utεdt≤βεγ,∀v∈U.Proof.
From the definition of functionH˜, we have(48)H˜uε=E1⊤tpεt+E2⊤tqεt−luξεt+EℱtF1⊤t+δ3pεt+δ3+F2⊤t+δ3qεt+δ3.
Then, inequality (47) can be written as(49)E∫0TEℱtF1⊤t+δ3pεt+δ3+F2⊤t+δ3qεt+δ3+E1⊤tpεt+E2⊤tqεt−luξεt,v−uεtdt≤βε,∀v∈U.
We find that inequalities (28) and (49) are very similar. We need to focus on the differences between them.
We denote(50)Δ1=E∫0TE1⊤tpεt,v−uεt−E1⊤tp˜εt,v−u˜εtdt=E∫0TE1⊤tpεt−p˜εt,v−u˜εt+E1⊤tpεt,u˜εt−uεtdt=Δ11+Δ12,where(51)Δ11=E∫0TE1⊤tpεt−p˜εt,v−u˜εtdt,Δ12=E∫0TE1⊤tpεt,u˜εt−uεtdt.
Next, we will deal with these two terms. From the assumption (A1), Lemma3, and the bounded of the control domain, there exist a series of constants C′,C″,C1,C11,C12,C2,…, which are all independent of ε. We have(52)Δ11≤C′E∫0TE1⊤tpεt−p˜εtdt≤C11E∫0Tpεt−p˜εtdt≤C11duε,u˜ε≤C11ε.
And then from the assumption (A1), Lemmas1 and 3, and the Cauchy–Schwartz inequality, we can deduce that(53)Δ12≤C″E∫0Tpεtuεt−u˜εtdt≤C12ε.
Combining (52) and (53), we have(54)Δ1=Δ11+Δ12≤C1ε,whereC1=maxC11,C12.
We denoteΔ2 and prove it like Δ1. Then, we have(55)Δ2=E∫0TE2⊤tqεt,v−uεt−E2⊤tq˜εt,v−u˜εtdt≤C2ε.
Set(56)Δ3=E∫0TEℱtF1⊤t+δ3pεt+δ3,v−uεt−EℱtF1⊤t+δ3p˜εt+δ3,v−u˜εtdt.
Using variable substitution, we can deduce that(57)Δ3=E∫δ3T+δ3F1⊤tpεt,vt−δ3−uεt−δ3−F1⊤tp˜εt,vt−δ3−u˜εt−δ3dt=E∫δ3TF1⊤tpεt,vt−δ3−uεt−δ3−F1⊤tp˜εt,vt−δ3−u˜εt−δ3dt=E∫δ3TF1⊤tpεt−p˜εt,vt−δ3−u˜εt−δ3+F1⊤tpεt,u˜t−δ3−uεt−δ3dt.
Similar to the proof ofΔ1, the results can be obtained by using the boundedness of control domain and coefficients. We can deduce the result directly, that is,(58)Δ3≤C3ε.
Similarly, we have(59)Δ4=E∫0TEℱtF2⊤t+δ3pεt+δ3,v−uεt−EℱtF2⊤t+δ3p˜εt+δ3,v−u˜εtdt≤C4ε.
Then, we have(60)Δ5=E∫0T−luξεt,v−uεt−−luξ˜εt,v−u˜εtdt=E∫0Tluξ˜εt−luξεt,v−u˜εt−luξεt,u˜εt−uεtdt.
Set(61)Δ51=E∫0Tluξ˜εt−luξεt,v−u˜εtdt,Δ52=E∫0Tluξεt,uεt−u˜εtdt.
Then,Δ5=Δ51+Δ52.
We deal with the termΔ51 firstly. Similar to the previous proof, by the boundedness of U and inequality of (21), we have(62)Δ51≤C51duε,uε˜≤C51ε.
From assumption (A3), we know that the partial derivative of functionl is bounded, so we have(63)Δ52≤C52ε,(64)Δ5≤C5ε.
According to inequalities (54), (55), (58), (59), and (65), we can deduce that(65)Δ1+Δ2+Δ3+Δ4+Δ5≤Cε,whereC=maxC1,C2,C3,C4,C5.
Applying Theorem2, we can deduce the conclusion directly.Generally speaking, optimal control is limited by many conditions. Near optimal control is relatively easy to obtain and can be selected, analyzed, and applied to a wider range of fields. When the time delay variablesδ1=δ2=δ3, this is a special near optimal control problem with the same time delay variables. When the time delay variables δ1=δ2=δ3=0, we can deduce the conclusions directly for the common system which is described by doubly stochastic differential equations. In either case, we find that the results depending on the adjoint equation of the system. The adjoint equation is a new kind of equation which can be called anticipated double stochastic equations. Using the properties of this kind of equation, we deal with the delay terms reasonably. In the future, we should pay attention to the study of this kind of equation which can help us solve such problems relatively easy.



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*Source: 4487092-2021-08-19.xml*