Delayed Stochastic Linear-Quadratic Control Problem and Related Applications

Abstract

We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations (FBSDEs) with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.

1. Introduction

The problem of optimal control for delayed stochastic system has received a lot of attention recently. One of the reasons is that there are many phenomena which have the nature of pastdependence that is, their behavior at time 𝑡 not only depends on the current situation, but also on their past history. Such kinds of mathematical models described by stochastic delay differential equations (SDDEs) are ubiquitous and have wide range of applications in physics, biology, engineering, economics, and finance (see Arriojas et al. [1], Mohammed [2, 3], and the references therein).

In control problem, a delay term may arise when there is a time lag between observation and regulation or the aftereffect of control; that is, there may be delay in state or control variables. Although many papers came out to discuss the delayed control problem, the analysis of systems with delay is fraught with many difficulties, not only for the infinite dimensional problem, but also for the absence of Itô's formula to deal with the delayed part of the trajectory. In order to surmount these difficulties, one can consider specific classes of systems with aftereffect, such as Øksendal and Sulem [4].

This paper is concerned with optimal control of a linear system with delay under a quadratic cost criteria, namely, the quadratic problem for stochastic linear control system with delay. It is well known that linear-quadratic (LQ) control is one of the most important classes of optimal control and the solution of this problem has many real-world applications. Deterministic LQ control problem with delay has been discussed in Alekal et al. [5], Basin et al. [6], and so forth. However, there is little results on stochastic LQ problem with delay since the difficulties in exploring the optimal feedback regulator, such as the Riccati equation, are very different from the case without delay.

In Peng and Yang [7], a new type of backward stochastic differential equations (BSDEs) was introduced, which is named anticipated BSDEs. The anticipated BSDE provides a new method to deal with optimal control problem with delay (see Chen and Wu [8]). In our paper, we study the delayed stochastic LQ problem by virtue of the anticipated BSDEs combined with the SDDEs.

In the next section, we introduce a kind of generalized FBSDE and give the existence and uniqueness result for its solution. With the help of the FBSDE, we find the optimal control of the stochastic LQ problem with delay in both state and control variables and the quadratic cost also involves delay terms.

It is very important to design an optimal feedback regulator for LQ problem in practice. Traditionally, a fundamental tool to obtain the state feedback is the Riccati equation. In Section 3, we introduced a new type of generalized Riccati equations and give the feedback regulator of delayed stochastic LQ problem. To the best of our knowledge, it is the first result on the optimal feedback control for the delayed stochastic LQ problem, where the state and control variables with delay are involved not only in the system, but also in the cost functional. In the last section, we apply our theoretical result to a population optimal control problem. Some technical proofs are put in Appendix.

2. Stochastic LQ Problem with Delay

We first introduce some notations. For any Euclidean space H, we denote by ⟨⋅,⋅⟩ (resp. |⋅|) the scalar product (resp. norm) of H. Let ℝ^𝑛×𝑚 be the Hilbert space consisting of all 𝑛×𝑚 matrices with the inner product: ⟨𝐴,𝐵⟩∶=tr𝐴𝐵^𝖳,∀𝐴,𝐵∈ℝ^𝑛×𝑚.(2.1) Here, the superscript 𝖳 denotes the transpose of vectors or matrices. Particularly, we denote by 𝒮^𝑛 the set of all 𝑛×𝑛 symmetric matrices.

Let 𝑊(⋅) be a standard 1-dimensional Brownian motion on a complete probability space (Ω,ℱ,𝑃). The information structure is given by a filtration 𝔽={ℱ_𝑡}_𝑡≥0, which is generated by 𝑊(⋅) and augmented by all the 𝑃-null sets. We assume the dimension of Brownian motion 𝑑=1 just for the simplicity of notations. In fact, all the conclusions in this paper still hold true for the case that the dimension of Brownian motion 𝑑>1. For any 𝐾, the Euclidean spaces or sets of matrices, the following notations will be used throughout the paper: 𝐿²Ω,ℱ_𝑡,𝑃;𝐾∶=𝜉isa𝐾-valuedℱ_𝑡-measurablerandomvariables.t.𝔼𝜉²;𝐿<+∞_2𝔽𝑡₁,𝑡₂;𝐾∶=𝜑(𝑡),0𝑡₁≤𝑡≤𝑡₂𝔼,isa𝐾-valued𝔽-adaptedstochasticprocesss.t._𝑡2𝑡1𝜑(𝑡)²;𝐿𝑑𝑡<+∞_∞𝔽𝑡₁,𝑡₂;𝐾∶=𝜑(𝑡),𝑡₁≤𝑡≤𝑡₂,isa𝐾-valued𝔽-adaptedboundedprocess.(2.2)

In this section, we consider the following linear controlled system involving delays in both state variable and control variable: 𝑥(𝑡)=𝑎+^𝑡₀𝐴₁(𝑠)𝑥(𝑠)+𝐴₂(𝑠)𝑥(𝑠−𝛿)+𝐵₁(𝑠)𝑣(𝑠)+𝐵₂(𝑠)𝑣(+𝑠−𝛿)𝑑𝑠_𝑡0𝐶₁(𝑠)𝑥(𝑠)+𝐶₂(𝑠)𝑥(𝑠−𝛿)+𝐷₁(𝑠)𝑣(𝑠)+𝐷₂(𝑠)𝑣((𝑠)[],𝑥(𝑡)𝑠−𝛿)𝑑𝑊,𝑡∈0,𝑇=𝑥₀(𝑡)[),𝑣(𝑡),𝑡∈−𝛿,0=𝑣₀(𝑡),𝑡∈[−𝛿,0),(2.3) where 𝛿>0 is a constant time delay, 𝑥₀(⋅)∈𝐿²_𝔽(−𝛿,0;ℝ^𝑛), 𝑣₀(⋅)∈𝐿²_𝔽(−𝛿,0;ℝ^𝑘) are deterministic functions, and 𝑎∈ℝ^𝑛. 𝐴₁(⋅),𝐴₂(⋅),𝐶₁(⋅),𝐶₂(⋅)∈𝐿^∞_𝔽(0,𝑇;ℝ^𝑛×𝑛), 𝐵₁(⋅),𝐵₂(⋅),𝐷₁(⋅),𝐷₂(⋅)∈𝐿^∞_𝔽(0,𝑇;ℝ^𝑛×𝑘), {𝑣(𝑡),𝑡∈[0,𝑇]} is an 𝔽-adapted square-integrable process taking values in ℝ^𝑘. Let 𝒰_𝑎𝑑 denote the set of stochastic processes 𝑣(⋅) of the form: 𝑣(𝑡)=𝑣₀(𝑡)𝑣(𝑡),−𝛿≤𝑡<0,∈𝐿²_𝔽0,𝑇;ℝ^𝑘,0≤𝑡≤𝑇.(2.4) An element of 𝒰_𝑎𝑑 is called an admissible control. By the classical results of SDDEs, we know that the system (2.3) has a unique solution for any admissible control 𝑣(⋅). The solution 𝑥^𝑣(⋅) of SDDE (2.3) is called the state trajectory corresponding to the control 𝑣(⋅)∈𝒰_𝑎𝑑 and (𝑥^𝑣(⋅),𝑣(⋅)) is called an admissible pair.

The cost functional is given by 𝐽(𝑣(⋅=1))2𝔼^𝑇₀𝑅₁(𝑡)𝑥(𝑡),𝑥(𝑡)+𝑅₂(𝑡)𝑥()(+𝑁𝑡−𝛿,𝑥𝑡−𝛿)₁(𝑡)𝑣(𝑡),𝑣(𝑡)+𝑁₂(𝑡)𝑣(𝑡−𝛿),𝑣(𝑡−𝛿)𝑑𝑡+⟨𝑄𝑥(𝑇),𝑥(𝑇)⟩.(2.5) Here, 𝑅₁(⋅),𝑅₂(⋅)∈𝐿^∞_𝔽(0,𝑇;𝒮^𝑛), 𝑁₁(⋅),𝑁₂(⋅)∈𝐿^∞_𝔽(0,𝑇;𝒮^𝑘), and 𝑄 is an ℱ_𝑇-measurable nonnegative symmetric bounded matrix. Moreover, we assume that, for any (𝜔,𝑡)∈Ω×[0,𝑇],  𝑅₁(𝑡)+𝑅₂(𝑡+𝛿) and 𝑄 are nonnegative definite, 𝑁₁(𝑡)+𝑁₂(𝑡+𝛿) is positive definite, and the inverse (𝑁₁(𝑡)+𝔼^ℱ𝑡[𝑁₂(𝑡+𝛿)])⁻¹ is also bounded.

The controller hopes to minimize the above cost functional 𝐽 by selecting an appropriate admissible control 𝑣(⋅); that is, the problem is to find 𝑢(⋅)∈𝒰_𝑎𝑑 such that 𝐽(𝑢(⋅))=inf_{𝑣(⋅)∈𝒰𝑎𝑑}𝐽(𝑣(⋅)).(2.6) We call the problem above the linear-quadratic optimal control problem of delayed system, and we denote it by Problem (LQD). An admissible pair (𝑥^𝑢(⋅),𝑢(⋅)) is called optimal for Problem (LQD) if 𝑢(⋅) achieves the infimum of 𝐽.

The Problem (LQD) we introduced above is a general type of LQ problem for stochastic system with delay. Not only the state variable and control variable are involved with delays, but also the cost functional contains delay terms.

By introducing a new type of BSDEs-anticipated BSDEs, we desire to solve Problem (LQD).

Theorem 2.1. The control 𝑢(𝑡)=−𝑁₁(𝑡)+𝔼^ℱ𝑡𝑁₂(𝑡+𝛿)⁻¹×𝐵^𝖳₁(𝑡)𝑦(𝑡)+𝐷^𝖳₁(𝑡)𝑧(𝑡)+𝔼^ℱ𝑡𝐵^𝖳₂(𝑡+𝛿)𝑦(𝑡+𝛿)+𝐷^𝖳₂(𝑡+𝛿)𝑧(𝑡+𝛿),𝑡∈[0,𝑇](2.7) is the unique optimal control of Problem (LQD). Here, (𝑥(⋅),𝑦(⋅),𝑧(⋅)) is the solution of the following generalized FBSDE: 𝑥(𝑡)=𝑎+^𝑡₀𝐴₁(𝑠)𝑥(𝑠)+𝐴₂(𝑠)𝑥(𝑠−𝛿)+𝐵₁(𝑠)𝑢(𝑠)+𝐵₂(𝑠)𝑢(+𝑠−𝛿)𝑑𝑠_𝑡0𝐶₁(𝑠)𝑥(𝑠)+𝐶₂(𝑠)𝑥(𝑠−𝛿)+𝐷₁(𝑠)𝑢(𝑠)+𝐷₂(𝑠)𝑢((𝑠)[],𝑦(𝑡)(𝑇)+𝑠−𝛿)𝑑𝑊,𝑡∈0,𝑇=𝑄𝑥^𝑇_𝑡𝐴^𝖳₁(𝑠)𝑦(𝑠)+𝐶^𝖳₁(𝑠)𝑧(𝑠)+𝑅₁(𝑠)𝑥(𝑠)+𝔼^ℱ𝑠𝐴^𝖳₂(𝑠+𝛿)𝑦(𝑠+𝛿)+𝐶^𝖳₂(𝑠+𝛿)𝑧(𝑠+𝛿)+𝑅₂()𝑥(𝑠−𝑠+𝛿)𝑑𝑠_𝑇𝑡𝑧(𝑠)(𝑠)[],𝑥(𝑡)𝑑𝑊,𝑡∈0,𝑇=𝑥₀(𝑡)[),𝑢(𝑡),𝑡∈−𝛿,0=𝑣₀(𝑡)[),𝑦(𝑡)(𝑡)(].,𝑡∈−𝛿,0=0,𝑧=0,𝑡∈𝑇,𝑇+𝛿(2.8)

Remark 2.2. If we set 𝑓𝑠,𝑥(𝑠),𝑥(𝑠−𝛿),𝑦(𝑠),𝑧(𝑠),𝑦(𝑠+𝛿),𝑧(𝑠+𝛿)=𝐴^𝖳₁(𝑠)𝑦(𝑠)+𝐶^𝖳₁(𝑠)𝑧(𝑠)+𝑅₁(𝑠)𝑥(𝑠)+𝔼^ℱ𝑠𝐴^𝖳₂(𝑠+𝛿)𝑦(𝑠+𝛿)+𝐶^𝖳₂(𝑠+𝛿)𝑧(𝑠+𝛿)+𝑅₂(𝑠+𝛿)𝑥(𝑠),(2.9) then the second equation in (2.8) can be rewritten as 𝑦(𝑡)=𝑄𝑥(𝑇)+^𝑇_𝑡𝑓(𝑠)()(𝑠)(𝑠)()()−𝑠,𝑥,𝑥𝑠−𝛿,𝑦,𝑧,𝑦𝑠+𝛿,𝑧𝑠+𝛿𝑑𝑠_𝑇𝑡𝑧(𝑠)𝑑𝑊(𝑠),𝑡∈[0,𝑇].(2.10)

Proof. Suppose (𝑥^𝑣(⋅),𝑣(⋅)) is an arbitrary admissible pair of system (2.3), then 𝐽(𝑣(⋅(𝑢(⋅=1))−𝐽))2𝔼^𝑇₀𝑅₁(𝑡)𝑥^𝑣(𝑡),𝑥^𝑣(𝑡)−𝑅₁(𝑡)𝑥(𝑡)(𝑡+𝑅,𝑥)₂(𝑡)𝑥^𝑣(𝑡−𝛿),𝑥^𝑣(𝑡−𝛿)−𝑅₂(𝑡)𝑥()(+𝑁𝑡−𝛿,𝑥𝑡−𝛿)₁(𝑡)𝑣(𝑡),𝑣(𝑡)−𝑁₁(𝑡)𝑢(𝑡)(𝑡+𝑁,𝑢)₂(𝑡)𝑣(𝑡−𝛿),𝑣(𝑡−𝛿)−𝑁₂(𝑡)𝑢()(+1𝑡−𝛿,𝑢𝑡−𝛿)𝑑𝑡2𝔼𝑄𝑥^𝑣(𝑇),𝑥^𝑣(𝑇−⟨(𝑇)(𝑇=1)𝑄𝑥,𝑥)⟩2𝔼^𝑇₀𝑅₁(𝑡)𝑥^𝑣(𝑡)−𝑥(𝑡),𝑥^𝑣(𝑡)(𝑡+𝑅−𝑥)₂(𝑡)𝑥^𝑣(𝑡−𝛿)−𝑥(𝑡−𝛿),𝑥^𝑣()(+𝑁𝑡−𝛿−𝑥𝑡−𝛿)₁(𝑡𝑣(𝑡)(𝑡(𝑡)(𝑡+𝑁)(−𝑢)),𝑣−𝑢)₂(𝑡𝑣()(()(𝑅)(𝑡−𝛿−𝑢𝑡−𝛿)),𝑣𝑡−𝛿−𝑢𝑡−𝛿)+2₁(𝑡)𝑥(𝑡),𝑥^𝑣(𝑡)−𝑥(𝑡)+2𝑅₂(𝑡)𝑥(𝑡−𝛿),𝑥^𝑣()(𝑁𝑡−𝛿−𝑥𝑡−𝛿)+2₁(𝑡)𝑢(𝑡),𝑣(𝑡)−𝑢(𝑡)+2𝑁₂(𝑡)𝑢()()(+1𝑡−𝛿,𝑣𝑡−𝛿−𝑢𝑡−𝛿)𝑑𝑡2𝔼𝑄𝑥^𝑣(𝑇)−𝑥(𝑇),𝑥^𝑣(𝑇)−𝑥(𝑇)+2𝑄𝑥(𝑇),𝑥^𝑣(𝑇)−𝑥(𝑇).(2.11) Since 𝑅₂(𝑡)≡0,𝑁₂(𝑡)≡0,𝑡∈(𝑇,𝑇+𝛿], by the time-shifting transformation and the initial conditions in (2.8), we derive that 𝐽(𝑣(⋅(𝑢(⋅=1))−𝐽))2𝔼^𝑇₀𝑅₁(𝑡)+𝔼^ℱ𝑡𝑅₂(𝑡+𝛿)𝑥^𝑣(𝑡)−𝑥(𝑡),𝑥^𝑣(𝑡)(𝑡+𝑁−𝑥)₁(𝑡)+𝔼^ℱ𝑡𝑁₂(𝑣(𝑡)(𝑡(𝑡)(𝑡𝑅𝑡+𝛿)(−𝑢)),𝑣−𝑢)+2₁(𝑡)+𝔼^ℱ𝑡𝑅₂(𝑡+𝛿)𝑥(𝑡),𝑥^𝑣(𝑡)(𝑡𝑁−𝑥)+2₁(𝑡)+𝔼^ℱ𝑡𝑁₂(𝑢(𝑡)(𝑡)(𝑡+1𝑡+𝛿),𝑣−𝑢)𝑑𝑡2𝔼𝑄𝑥^𝑣(𝑇)−𝑥(𝑇),𝑥^𝑣(𝑇)−𝑥(𝑇)+2𝑄𝑥(𝑇),𝑥^𝑣(𝑇)−𝑥(𝑇).(2.12) Applying Itô's formula to ⟨𝑥^𝑣(𝑡)−𝑥(𝑡),𝑦(𝑡)⟩, we have 𝔼𝑥^𝑣(𝑇)−𝑥(𝑇),𝑦(𝑇)=𝔼^𝑇₀−𝑅₁(𝑡)𝑥(𝑡),𝑥^𝑣(𝑡)−𝑥(𝑡)−𝔼^ℱ𝑡𝑅₂(𝑡+𝛿)𝑥(𝑡),𝑥^𝑣(𝑡)(𝑡+𝐵−𝑥)₁(𝑡)(𝑣(𝑡)−𝑢(𝑡)),𝑦(𝑡)+𝐷₁(𝑡𝑣(𝑡)(𝑡(𝑡+𝐵)(−𝑢)),𝑧)₂(𝑡)(𝑣(𝑡−𝛿)−𝑢(𝑡−𝛿)),𝑦(𝑡)+𝐷₂(𝑡)(𝑣(𝑡−𝛿)−𝑢(𝑡−𝛿)),𝑧(𝑡)𝑅₁𝑑𝑡.(2.13) Moreover, because 𝑣(𝑡)=𝑢(𝑡),𝑡∈[−𝛿,0) and 𝑦(𝑡)=𝑧(𝑡)=0,𝑡∈(𝑇,𝑇+𝛿], we can rewrite (2.13) as follows: 𝔼𝑥^𝑣(𝑇)−𝑥(𝑇),𝑦(𝑇)=𝔼^𝑇₀−𝑅₁(𝑡)+𝔼^ℱ𝑡𝑅₂(𝑡+𝛿)+𝑥(𝑡),𝑥^𝑣(𝑡)(𝑡+𝐵−𝑥)₁(𝑡)(𝑣(𝑡)−𝑢(𝑡)),𝑦(𝑡)+𝐷₁(𝑡)(𝑣(𝑡)−𝑢(𝑡)),𝑧(𝑡)+𝔼^ℱ𝑡𝐵₂(𝑡+𝛿)(𝑣(𝑡)−𝑢(𝑡)),𝑦(𝑡+𝛿)+𝐷₂(𝑡+𝛿)(𝑣(𝑡)−𝑢(𝑡)),𝑧(𝑡+𝛿)𝑑𝑡.(2.14) Consequently, by the fact that 𝑅₁(⋅)+𝑅₂(⋅+𝛿) and 𝑄 are nonnegative and 𝑁₁(⋅)+𝑁₂(⋅+𝛿) is positive, we have 𝐽(𝑣(⋅))−𝐽(𝑢(⋅))≥𝔼^𝑇₀𝑁₁(𝑡)+𝔼^ℱ𝑡𝑁₂(𝑢(𝑡)(𝑡)(𝑡+𝐵𝑡+𝛿),𝑣−𝑢)_𝖳1(𝑡)𝑦(𝑡)+𝐷^𝖳₁(𝑡)𝑧(𝑡)(𝑡)(𝑡+𝔼,𝑣−𝑢)_ℱ𝑡𝐵^𝖳₂(𝑡+𝛿)𝑦(𝑡+𝛿)+𝐷^𝖳₂(𝑡+𝛿)𝑧(𝑡+𝛿),𝑣(𝑡)−𝑢(𝑡)𝑑𝑡.(2.15) So letting 𝑢(𝑡)=−𝑁₁(𝑡)+𝔼^ℱ𝑡𝑁₂(𝑡+𝛿)⁻¹×𝐵^𝖳₁(𝑡)𝑦(𝑡)+𝐷^𝖳₁(𝑡)𝑧(𝑡)+𝔼^ℱ𝑡𝐵^𝖳₂(𝑡+𝛿)𝑦(𝑡+𝛿)+𝐷^𝖳₂(𝑡+𝛿)𝑧(𝑡+𝛿),𝑡∈[0,𝑇](2.16) we have 𝐽(𝑣(⋅))−𝐽(𝑢(⋅))≥0,∀𝑣(⋅)∈𝒰_𝑎𝑑,(2.17) that is, 𝑢(⋅) defined in (2.7) is the optimal control of Problem (LQD).
We will use the parallelogram rule to prove the uniqueness of the optimal control, and this method can also be seen in Wu [9]. We assume that 𝑢¹(⋅) and 𝑢²(⋅) are both optimal controls, and the corresponding trajectories are 𝑥¹(⋅) and 𝑥²(⋅). It is easy to know the trajectories corresponding to (𝑢¹(⋅)+𝑢²(⋅))/2 are (𝑥¹(⋅)+𝑥²(⋅))/2. Since 𝑁₁(⋅)+𝑁₂(⋅+𝛿) is positive, 𝑅₁(⋅)+𝑅₂(⋅+𝛿) and 𝑄 are nonnegative, we know that 𝐽(𝑢¹(⋅))=𝐽(𝑢²(⋅))=𝜆≥0, and 2𝜆=𝐽𝑢¹(⋅)+𝐽𝑢²(⋅𝑢)=2𝐽¹(⋅)+𝑢²(⋅)2+𝔼^𝑇₀𝑅₁(𝑡)+𝔼^ℱ𝑡𝑅₂(𝑥𝑡+𝛿)¹(𝑡)−𝑥²(𝑡)2,𝑥¹(𝑡)−𝑥²(𝑡)2+𝑁₁(𝑡)+𝔼^ℱ𝑡𝑁₂(𝑢𝑡+𝛿)¹(𝑡)−𝑢²(𝑡)2,𝑢¹(𝑡)−𝑢²(𝑡)2𝑄𝑥𝑑𝑡+𝔼¹(𝑇)−𝑥²(𝑇)2,𝑥¹(𝑇)−𝑥²(𝑇)2≥2𝜆+𝔼^𝑇₀𝑁₁(𝑡)+𝔼^ℱ𝑡𝑁₂(𝑢𝑡+𝛿)¹(𝑡)−𝑢²(𝑡)2,𝑢¹(𝑡)−𝑢²(𝑡)2𝑑𝑡.(2.18) Because of 𝑁₁(⋅)+𝑁₂(⋅+𝛿) being positive, we have 𝑢¹(⋅)=𝑢²(⋅). We complete our proof.

Remark 2.3. The existence of the optimal control is equivalent to the existence of solution for (2.8), which is a kind of complex generalized FBSDEs. The proof of the existence and uniqueness of the solution for this kind of FBSDE are put in Appendix.

3. Feedback Regulator of Delayed System

It is well known that the feedback representation of optimal control is very useful in applications. And in the classical case, the optimal feedback control can be represented via the Riccati equation. But, for stochastic systems with delay, it is not easy to find the feedback control because of the dependence of the history. What should then be an appropriate “Riccati equation” corresponding to our LQ problem with delay?

In this section, we will pay attention to the feedback regulator of the general delayed LQ problem discussed in Section 2 and try to get the appropriate Riccati equation associated with Problem (LQD). We remark that all given coefficients of the problem are assumed to be deterministic from now on.

Let us start with the following results about anticipated BSDE.

Assume that for all 𝑠∈[0,𝑡], 𝑓(𝑠,𝑤,𝑦,𝑧,𝜉,𝜂)∶[0,𝑇]×Ω×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²(Ω,ℱ_𝑟,𝑃;ℝ^𝑛)×𝐿²(Ω,ℱ_𝑟,𝑃;ℝ^𝑛×𝑑)→𝐿²(Ω,ℱ_𝑟,𝑃;ℝ^𝑛), where 𝑟∈[𝑠,𝑇+𝛿]. And 𝑓 satisfies the following conditions.(H3.1) There exists a constant 𝐶>0, such that for all 𝑠∈[0,𝑇], 𝑦,𝑦′∈ℝ^𝑛, 𝑧,𝑧′∈ℝ^𝑛×𝑑, 𝜉_⋅,𝜉^′_⋅∈𝐿²_𝔽(𝑠,𝑇+𝛿;ℝ^𝑛), 𝜂_⋅,𝜂^′_⋅∈𝐿²_𝔽(𝑠,𝑇+𝛿;ℝ^𝑛×𝑑), we have 𝑓𝑠,𝑦,𝑧,𝜉_𝑟,𝜂_𝑟−𝑓𝑠,𝑦^′,𝑧^′,𝜉^′_𝑟,𝜂^′_𝑟≤𝐶𝑦−𝑦^′+𝑧−𝑧^′+𝔼^ℱ𝑠𝜉_𝑟−𝜉^′_𝑟+𝜂_𝑟−𝜂^′_𝑟.(3.1)(H3.2)(𝔼[∫^𝑇₀|𝑓(𝑠,0,0,0,0)|²𝑑𝑠]<∞).

Let (𝑦_𝑖(⋅),𝑧_𝑖(⋅)), 𝑖=1,2, be, respectively, solutions of the following two anticipated BSDEs: 𝑦_𝑖(𝑡)=𝜉(𝑇)+^𝑇_𝑡𝑓_𝑖𝑠,𝑦_𝑖(𝑠),𝑧_𝑖(𝑠),𝑦_𝑖(𝑠+𝛿),𝑧_𝑖(𝑠+𝛿)𝑑𝑡−𝑧_𝑖(𝑠)(𝑠)[],𝑦𝑑𝑊,𝑡∈0,𝑇_𝑖(𝑡)=𝜉(𝑡),𝑧_𝑖(𝑡)=𝜂(𝑡),𝑡∈(𝑇,𝑇+𝛿].(3.2) We have the following lemma, which is a direct corollary of Theorem 2.2 in Yu [10].

Lemma 3.1. Assume that 𝑓₁,𝑓₂ satisfy (H3.1) and (H3.2), 𝜉(⋅)∈𝐿²_𝔽(𝑇,𝑇+𝛿;ℝ^𝑛), 𝜂(⋅)∈𝐿²_𝔽(𝑇,𝑇+𝛿;ℝ^𝑛×𝑘), and 𝔼^𝑇₀𝑓₁𝑡,𝑦₂(𝑡),𝑧₂(𝑡),𝑦₂(𝑡+𝛿),𝑧₂(𝑡+𝛿)−𝑓₂𝑡,𝑦₂(𝑡),𝑧₂(𝑡),𝑦₂(𝑡+𝛿),𝑧₂(𝑡+𝛿)²𝑑𝑡=0,(3.3) then 𝑦₁(𝑡)=𝑦₂(𝑡),𝑧₁(𝑡)=𝑧₂(𝑡),𝑎.𝑒.,𝑎.𝑠.(3.4)

In order to get the feedback regulator, we introduce the following generalized 𝑛×𝑛 matrix-valued Riccati equation system: −𝐾̇₁(𝑡)=𝐴^𝖳₁(𝑡)𝐾₁(𝑡)+𝐾₁(𝑡)𝐴₁(𝑡)−𝐾₁(𝑡)𝐵₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₁(𝑡)+𝐷^𝖳₁(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡)𝐵₂(𝑡)𝑁₁(t−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₁(𝑡)+𝐷^𝖳₂(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡+𝛿)𝐵₂(𝑡+𝛿)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₁(𝑡)+𝐷^𝖳₁(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡−𝛿)𝐵₁(𝑡−𝛿)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₁(𝑡)+𝐷^𝖳₂(𝑡)𝐻₁(𝑡)+𝐴^𝖳₂(𝑡)𝐾₁(𝑡)+𝐾₁(𝑡+𝛿)𝐴₂(𝑡+𝛿)−𝑒^−𝜆𝛿𝐾₂(𝑡+𝛿)+𝐾₂(𝑡)+𝐶^𝖳₁(𝑡)+𝐶^𝖳₂(𝑡)𝐻₁(𝑡)+𝑅₁(𝑡)+𝑅₂()[],−𝐾̇𝑡+𝛿,𝑡∈0,𝑇(3.5)₂(𝑡)=−𝜆𝐾₂(𝑡)−𝐾₁(𝑡)𝐵₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₂(𝑡)+𝐷^𝖳₁(𝑡)𝐻₂(𝑡)−𝐾₁(𝑡)𝐵₂(𝑡)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₁(𝑡)+𝐷^𝖳₂(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡−𝛿)𝐵₁(𝑡−𝛿)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₂(𝑡)+𝐷^𝖳₂(𝑡)𝐻₂(𝑡)−𝐾₁(𝑡+𝛿)𝐵₁(𝑡+𝛿)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₂(𝑡)+𝐷^𝖳₁(𝑡)𝐻₂(𝑡+𝐴_𝖳1)(𝑡)+𝐴^𝖳₂(𝑡)𝐾₂(𝑡)+𝐶^𝖳₁(𝑡)+𝐶^𝖳₂(𝑡)𝐻₂(𝑡)[],𝐻,𝑡∈𝛿,𝑇(3.6)₁(𝑡)=𝐾₁(𝑡)𝐶₁(𝑡)−𝐾₁(𝑡)𝐷₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₁(𝑡)+𝐷^𝖳₁(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡)𝐷₂(𝑡)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₁(𝑡)+𝐷^𝖳₂(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡−𝛿)𝐷₁(𝑡−𝛿)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₁(𝑡)+𝐷^𝖳₂(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡+𝛿)𝐷₂(𝑡+𝛿)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₁(𝑡)+𝐷^𝖳₁(𝑡)𝐻₁(𝑡)+𝐾₁(𝑡+𝛿)𝐶₂()[],𝐻𝑡+𝛿,𝑡∈0,𝑇(3.7)₂(𝑡)=−𝐾₁(𝑡)𝐷₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₂(𝑡)+𝐷^𝖳₁(𝑡)𝐻₂(𝑡)−𝐾₁(𝑡)𝐷₂(𝑡)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₂(𝑡)+𝐷^𝖳₂(𝑡)𝐻₂(𝑡)−𝐾₁(𝑡−𝛿)𝐷₁(𝑡−𝛿)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₂(𝑡)+𝐷^𝖳₂(𝑡)𝐻₂(𝑡)−𝐾₁(𝑡+𝛿)𝐷₂(𝑡+𝛿)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₂(𝑡)+𝐷^𝖳₁(𝑡)𝐻₂(𝑡),𝑡∈[0,𝑇].(3.8) With the boundary conditions 𝐾₁(𝑇)=𝑄,𝐾₁(𝑡)=0,𝑡∈(𝑇,𝑇+𝛿],𝐾₂(𝑡)[]∪(),𝐻=0,𝑡∈𝑇,𝑇+𝛿0,𝛿₁(𝑡)=𝐻₂(𝑡)=0,𝑡∈(𝑇,𝑇+𝛿].(3.9) Here, 𝜆 is some constant.

Then we have the following main result.

Theorem 3.2. Suppose that there exist matrix-valued deterministic processes (𝐾₁(⋅),𝐾₂(⋅),𝐻₁(⋅),𝐻₂(⋅)) satisfying the generalized Riccati equation system (3.5)–(3.8) with corresponding boundary conditions, and for system (2.3), 𝐴₂(𝑡)=𝐵₂(𝑡)=𝐶₂(𝑡)=𝐷₂(𝑡)=0, 𝑡∈[0,𝛿). Then the optimal feedback regulator for the delayed linear quadratic optimal Problem (LQD) is 𝑢(𝑡)=−𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝑦(𝑡)+𝐷^𝖳₁(𝑡)𝑧(𝑡)+𝔼^ℱ𝑡𝐵^𝖳₂(𝑡+𝛿)𝑦(𝑡+𝛿)+𝐷^𝖳₂(𝑡+𝛿)𝑧(𝑡+𝛿),𝑡∈[0,𝑇],(3.10) with 𝑦(𝑡)=𝐾₁(𝑡)𝑥(𝑡)+𝐾₂(𝑡)⁰_−𝛿𝑒^𝜆𝑠𝑥()𝑧(𝑡)𝑡+𝑠𝑑𝑠,(3.11)=𝐻₁(𝑡)𝑥(𝑡)+𝐻₂(𝑡)⁰_−𝛿𝑒^𝜆𝑠𝑥(𝑡+𝑠)𝑑𝑠.(3.12) And the optimal value function is 𝐽(𝑢(⋅=1))2𝐾₁(0)𝑎,𝑎.(3.13)

Proof. Let (𝐾₁(⋅),𝐾₂(⋅),𝐻₁(⋅),𝐻₂(⋅)) be the solution of Riccati equation system (3.5)–(3.8) and we set 𝑦₁(𝑡)=𝐾₁(𝑡)𝑥(𝑡)+𝐾₂(𝑡)⁰_−𝛿𝑒^𝜆𝑠𝑥()𝑧𝑡+𝑠𝑑𝑠,₁(𝑡)=𝐻₁(𝑡)𝑥(𝑡)+𝐻₂(𝑡)⁰_−𝛿𝑒^𝜆𝑠𝑥(𝑡+𝑠)𝑑𝑠.(3.14) Applying Itô's formula to 𝑦₁(⋅), we have 𝑦₁(𝑡)=𝑄𝑥(𝑇)+^𝑇_𝑡−𝐾̇₁(𝑠)𝑥(𝑠)+𝐾₁(𝑠)𝐴₁(𝑠)𝑥(𝑠)+𝐴₂(𝑠)𝑥(𝑠−𝛿)+𝐵₁(𝑠)𝑢(𝑠)+𝐵₂(𝑠)𝑢(+𝐾̇𝑠−𝛿)₂(𝑠)⁰_−𝛿𝑒^𝜆𝑟𝑥(𝑡+𝑟)𝑑𝑟+𝐾₂(𝑠)×𝑥(𝑠)−𝑒^−𝜆𝛿𝑥(𝑠−𝛿)−𝜆⁰_−𝛿𝑒^𝜆𝑟𝑥()−𝑡+𝑟𝑑𝑟𝑑𝑠_𝑇𝑡𝐾₁(𝑠)𝐶₁(𝑠)𝑥(𝑠)+𝐶₂(𝑠)𝑥(𝑠−𝛿)+𝐷₁(𝑠)𝑢(𝑠)+𝐷₂(𝑠)𝑢(𝑠−𝛿)𝑑𝑊(𝑠),𝑡∈[0,𝑇],(3.15) with 𝑢(𝑡)=−𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝑦₁(𝑡)+𝐷^𝖳₁(𝑡)𝑧₁(𝑡)+𝔼^ℱ𝑡𝐵^𝖳₂(𝑡+𝛿)𝑦₁(𝑡+𝛿)+𝐷^𝖳₂(𝑡+𝛿)𝑧₁(𝑡+𝛿),𝑡∈[0,𝑇].(3.16) Substituting (3.16) into (3.15), we have 𝑦₁(𝑡)=𝑄𝑥(𝑇)+^𝑇_𝑡𝑓₁𝑠,𝑥(𝑠),𝑥(𝑠−𝛿),𝑦₁(𝑠),𝑧₁(𝑠),𝑦₁(𝑠+𝛿),𝑧₁()−𝑠+𝛿𝑑𝑠_𝑇𝑡𝑧₂(𝑠)𝑑𝑊(𝑠),𝑡∈[0,𝑇],(3.17) where 𝑧₂(𝑠)=𝐾₁(𝑠)[𝐶₁(𝑠)𝑥(𝑠)+𝐶₂(𝑠)𝑥(𝑠−𝛿)+𝐷₁(𝑠)𝑢(𝑠)+𝐷₂(𝑠)𝑢(𝑠−𝛿)] and 𝑓₁𝑠,𝑥(𝑠),𝑥(𝑠−𝛿),𝑦₁(𝑠),𝑧₁(𝑠),𝑦₁(𝑠+𝛿),𝑧₁()𝐾̇𝑠+𝛿=−₁(𝑠)𝑥(𝑠)+𝐾₁(𝑠)𝐴₁(𝑠)𝑥(𝑠)+𝐴₂(𝑠)𝑥(𝑠−𝛿)+𝐵₁(𝑠)𝑢(𝑠)+𝐵₂(𝑠)𝑢(+𝐾̇𝑠−𝛿)₂(𝑠)⁰_−𝛿𝑒^𝜆𝑟𝑥(𝑡+𝑟)𝑑𝑟+𝐾₂(𝑠)𝑥(𝑠)−𝑒^−𝜆𝛿𝑥(𝑠−𝛿)−𝜆⁰_−𝛿𝑒^𝜆𝑟𝑥(𝑡+𝑟)𝑑𝑟.(3.18) The following analysis shows that 𝑧₁(⋅)=𝑧₂(⋅)a.e.,a.s. with the help of (3.7) and (3.8). In fact, taking (3.16) into the presentation of 𝑧₂(𝑡), we will have 𝑧₂(𝑡)=𝐾₁(𝑡)𝐶₁(𝑡)−𝐾₁(𝑡)𝐷₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₁(𝑡)+𝐷^𝖳₁(𝑡)𝐻₁(𝑡)−𝐾₁(𝑡)𝐷₂(𝑡)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₁(𝑡)+𝐷^𝖳₂(𝑡)𝐻₁(𝑡𝑥(𝑡)+𝐾₁)(𝑡)𝐶₂(𝑡)−𝐾₁(𝑡)𝐷₂(𝑡)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹×𝐵^𝖳₁(𝑡−𝛿)𝐾₁(𝑡−𝛿)+𝐷^𝖳₁(𝑡−𝛿)𝐻₁(𝑥()−𝐾𝑡−𝛿)𝑡−𝛿₁(𝑡)𝐷₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₂(𝑡+𝛿)𝐾₁(𝑡+𝛿)+𝐷^𝖳₂(𝑡+𝛿)𝐻₁(𝑡+𝛿)×𝔼^ℱ𝑡[𝑥(−𝐾𝑡+𝛿)]₁(𝑡)𝐷₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₂(𝑡)+𝐷^𝖳₁(𝑡)𝐻₂(𝑡)+𝐾₁(𝑡)𝐷₂(𝑡)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₂(𝑡)𝐾₂(𝑡)+𝐷^𝖳₂(𝑡)𝐻₂(𝑡)⁰_−𝛿𝑒^𝜆𝑠𝑥()−𝐾𝑡+𝑠𝑑𝑠₁(𝑡)𝐷₁(𝑡)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₂(𝑡+𝛿)𝐾₂(𝑡+𝛿)+𝐷^𝖳₂(𝑡+𝛿)𝐻₂(𝑡+𝛿)×⁰_−𝛿𝑒^𝜆𝑠𝑥()−𝐾𝑡+𝛿+𝑠𝑑𝑠₁(𝑡)𝐷₂(𝑡)𝑁₁(𝑡−𝛿)+𝑁₂(𝑡)⁻¹𝐵^𝖳₁(𝑡−𝛿)𝐾₂(𝑡−𝛿)+𝐷^𝖳₁(𝑡−𝛿)𝐻₂(𝑡−𝛿)×⁰_−𝛿𝑒^𝜆𝑠𝑥(𝑡−𝛿+𝑠)𝑑𝑠.(3.19) By the conditions 𝐾₁(𝑡)=𝐻₁(𝑡)=0, 𝑡∈(𝑇,𝑇+𝛿], 𝐾₂(𝑡)=𝐻₂(𝑡)=0, 𝑡∈[T,𝑇+𝛿] and 𝐵₂(𝑡)=𝐶₂(𝑡)=𝐷₂(𝑡)=0, 𝑡∈[0,𝛿), we derive 𝔼^𝑇₀𝑧₁(𝑠)−𝑧₂(𝑠)²𝑑𝑠=𝔼^𝑇₀𝐻₁(𝑠)−𝐾₁(𝑠)𝐶₁(𝑠)+𝐾₁(𝑠)𝐷₁(𝑠)×𝑁₁(𝑠)+𝑁₂(𝑠+𝛿)⁻¹𝐵^𝖳₁(𝑠)𝐾₁(𝑠)+𝐷^𝖳₁(𝑠)𝐻₁(𝑠)+𝐾₁(𝑠)𝐷₂(𝑠)𝑁₁(𝑠−𝛿)+𝑁₂(𝑠)⁻¹𝐵^𝖳₂(𝑠)𝐾₁(𝑠)+𝐷^𝖳₂(𝑠)𝐻₁(𝑠)−𝐾₁(𝑠+𝛿)𝐶₂(𝑠+𝛿)+𝐾₁(𝑠+𝛿)𝐷₂()×𝑁𝑠+𝛿₁(𝑠)+𝑁₂(𝑠+𝛿)⁻¹𝐵^𝖳₁(𝑠)𝐾₁(𝑠)+𝐷^𝖳₁(𝑠)𝐻₁(𝑠)+𝐾₁(𝑠−𝛿)𝐷₁(𝑠−𝛿)𝑁₁(𝑠−𝛿)+𝑁₂(𝑠)⁻¹𝐵^𝖳₂(𝑠)𝐾₁(𝑠)+𝐷^𝖳₂(𝑠)𝐻₁(𝑠𝑥(𝑡)+𝐻₂)(𝑠)+𝐾₁(𝑠)𝐷₁(𝑠)𝑁₁(𝑠)+𝑁₂(𝑠+𝛿)⁻¹𝐵^𝖳₁(𝑠)𝐾₂(𝑠)+𝐷^𝖳₁(𝑠)𝐻₂(𝑠)+𝐾₁(𝑠)𝐷₂(𝑠)𝑁₁(𝑠−𝛿)+𝑁₂(𝑠)⁻¹𝐵^𝖳₂(𝑠)𝐾₂(𝑠)+𝐷^𝖳₂(𝑠)𝐻₂(𝑠)+𝐾₁(𝑠−𝛿)𝐷₁(𝑠−𝛿)𝑁₁(𝑠−𝛿)+𝑁₂(𝑠)⁻¹𝐵^𝖳₂(𝑠)𝐾₂(𝑠)+𝐷^𝖳₂(𝑠)𝐻₂(𝑠)+𝐾₁(𝑡+𝛿)𝐷₂(𝑡+𝛿)𝑁₁(𝑡)+𝑁₂(𝑡+𝛿)⁻¹𝐵^𝖳₁(𝑡)𝐾₂(𝑡)+𝐷^𝖳₁(𝑡)𝐻₂(𝑡×_0−𝛿)𝑒^𝜆𝑟𝑥(𝑡+𝑟)𝑑𝑟²𝑑𝑠.(3.20) We can see that (3.7) and (3.8) ensure that 𝔼^𝑇₀𝑧₁(𝑠)−𝑧₂(𝑠)²𝑑𝑠=0.(3.21) In other words, 𝑧₁(⋅)=𝑧₂(⋅)a.e.,a.s.
So we use the following equation instead of (3.17): 𝑦₁(𝑡)=𝑄𝑥(𝑇)+^𝑇_𝑡𝑓₁𝑠,𝑥(𝑠),𝑥(𝑠−𝛿),𝑦₁(𝑠),𝑧₁(𝑠),𝑦₁(𝑠+𝛿),𝑧₁()−𝑠+𝛿𝑑𝑠_𝑇𝑡𝑧₁(𝑠)(𝑠)[],𝑦𝑑𝑊,𝑡∈0,𝑇₁(𝑡)=𝑧₁(𝑡)=0,𝑡∈(𝑇,𝑇+𝛿].(3.22) Namely, (𝑦₁(⋅),𝑧₁(⋅)) introduced in (3.14) satisfies the above anticipated BSDE. From now on, we pay attention to the anticipated BSDEs (2.8) and (3.22).
Applying the similar method we used above, we derive that (3.5) and (3.6) will lead to the following: 𝔼^𝑇₀𝑓𝑠,𝑥(𝑠),𝑥(𝑠−𝛿),𝑦₁(𝑠),𝑧₁(𝑠),𝑦₁(𝑠+𝛿),𝑧₁(𝑠+𝛿)−𝑓₁𝑠,𝑥(𝑠),𝑥(𝑠−𝛿),𝑦₁(𝑠),𝑧₁(𝑠),𝑦₁(𝑠+𝛿),𝑧₁(𝑠+𝛿)²𝑑𝑠=0.(3.23) Thus, we get 𝑦(𝑡)=𝑦₁(𝑡),𝑧(𝑡)=𝑧₁(𝑡) by Lemma 3.1 immediately, and feedback representation of optimal control in Theorem 3.2 is proved.

Applying Itô's formula to ⟨𝑥(𝑇),𝑦(𝑇)⟩ in 𝐽(𝑢(⋅)), it is easy to get (3.13).

Remark 3.3. (i) The condition 𝐴₂(𝑡)=𝐵₂(𝑡)=𝐶₂(𝑡)=𝐷₂(𝑡)=0,𝑡∈[0,𝛿) is reasonable in practice: the condition implies that there is no delay on the time interval [0,𝛿). That is to say there is no history of the system on [−𝛿,0), that is, 𝑥(𝑡)=0,𝑡∈[−𝛿,0).
(ii) The matrix-value differential equation (3.5)–(3.8) is a generalized type of Riccati equations which is different from the classical one. When there are no delays in the system, that is, 𝐴₂(𝑡)=𝐵₂(𝑡)=𝐶₂(𝑡)=𝐷₂(𝑡)=𝑁₂(𝑡)=𝑅₂(𝑡)=0, 𝐾₂(𝑡)=𝐻₂(𝑡)=0, our generalized anticipated Riccati equation will degenerate to the Riccati equation in Wu [9].

4. Application

The solvability of Riccati equation (3.5)–(3.8) is not easy to obtain in general. In this section, we derive the unique solvability of the problem for some special cases. As an application of our above results, we will consider a kind of stochastic LQ problem with delay arising from the population control model in this section. The optimal feedback control is given by the new type of Riccati equation and we can obtain its existence and uniqueness.

The following population growth model comes from Mohammed [3]: 𝑥(𝑡)=𝑎₀+^𝑡₀−𝑎₁𝑥(𝑠)+𝑎₂𝑥(𝑠−𝛿)𝑑𝑠+^𝑡₀(𝑠)[],𝑥(𝑡)𝜎𝑑𝑊,𝑡∈0,𝑇=𝑥₀(𝑡),𝑡∈[−𝛿,0),(4.1) where 𝑥(𝑡) is the population at time 𝑡, 𝑎₁ and 𝑎₂>0 are constant death rate and birth rate per capita, respectively. 𝛿>0 denotes the development period of each individual, and the diffusion part describes the migration of the population; that is, there is migration whose overall rate is distributed like 𝜎𝑊̇.

Based on the population growth model, we consider the following population control problem: 𝑥(𝑡)=𝑎₀+^𝑡₀−𝑎₁(𝑠)𝑥(𝑠)+𝑎₂(𝑠)𝑥(𝑠−𝛿)+𝑏₁(𝑠)𝑣(𝑠−𝛿)𝑑𝑠+^𝑡₀𝜎(𝑠)𝑥(𝑠)(𝑠)[],𝑥(𝑡)𝑑𝑊,𝑡∈0,𝑇=𝑥₀(𝑡)[),𝑣(𝑡),𝑡∈−𝛿,0=𝑣₀(𝑡),𝑡∈[−𝛿,0).(4.2) Here, we change the original constant death and birth rate to be time-varying 𝑎₁(⋅),𝑎₂(⋅),𝑏₁(⋅),𝜎(⋅) are deterministic bounded functions on [0,𝑇] and 𝑎₂(⋅)=𝑏₁(⋅)=0 on [0,𝛿). 𝑥₀(⋅),𝑣₀(⋅)∈𝐿²_𝔽(−𝛿,0;ℝ). And the control 𝑣(⋅) models the intensity of the spending on controlling the population. We take into account that there will be a time lag between control expenditure and the corresponding effect on the population level, so the system is involved with control delay. 𝑊(⋅) is a 1-dimensional B.M.

The objection is to minimize the following cost functional: 𝐽(𝑣(⋅=1))2𝔼^𝑇₀𝑟₁(𝑡)𝑥²(𝑡)+𝑛₁(𝑡)𝑣²(𝑡)𝑑𝑡+𝑞𝑥²(𝑇),(4.3) over the admissible control set 𝒰_𝑎𝑑={𝑣(𝑡)∣𝑣(𝑡)∈𝐿²_𝔽(0,𝑇;ℝ)}. 𝑟₁(⋅) and 𝑛₁(⋅) are nonnegative- and positive-valued bounded functions on [0,𝑇], respectively, and 𝑞≥0 is a given constant. And all the coefficients in our model are equal to zero outside of the interval [0,𝑇]. About the cost functional, one can consider that we are going to control some pests' quantity. We want to pay less expenditure and make the quantity of pests on lower level.

The above population control problem is a special case of the Problem (LQD) we discussed in Sections 2 and 3. From Theorems 2.1 and 3.2, we have the following results.

Proposition 4.1. The control 𝑢(𝑡)=−𝑛₁⁻¹(𝑡)𝑏₁(𝑡+𝛿)𝑦(𝑡+𝛿),𝑡∈[0,𝑇](4.4) is the unique optimal control. Here, (𝑥(⋅),𝑦(⋅),𝑧(⋅)) is the solution of the following generalized FBSDE: 𝑥(𝑡)=𝑎+^𝑡₀𝑎₁(𝑠)𝑥(𝑠)+𝑎₂(𝑠)𝑥(𝑠−𝛿)+𝑏₁(𝑠)𝑢(+𝑠−𝛿)𝑑𝑠_𝑡0𝜎(𝑠)𝑥(𝑠)(𝑠)[],𝑦(𝑡)(𝑇)+𝑑𝑊,𝑡∈0,𝑇=𝑞𝑥^𝑇_𝑡𝑎₁(𝑠)𝑦(𝑠)+𝜎(𝑠)𝑧(𝑠)+𝑟₁(𝑠)𝑥(𝑠)+𝔼^ℱ𝑠𝑎₂()𝑦(−𝑠+𝛿𝑠+𝛿)𝑑𝑠_𝑇𝑡𝑧(𝑠)(𝑠)[],𝑥(𝑡)[),𝑢(𝑡)[),𝑦(𝑡)(𝑡)(].𝑑𝑊,𝑡∈0,𝑇=0,𝑡∈−𝛿,0=0,𝑡∈−𝛿,0=0,𝑧=0,𝑡∈𝑇,𝑇+𝛿(4.5)

Proposition 4.2. The feedback control regulator of the population control problem is given by 𝑢(𝑡)=−𝑛₁⁻¹(𝑡)𝑏₁(𝑡+𝛿)𝑘₁(𝑡+𝛿)𝑥(𝑡+𝛿)+𝑘₂(𝑡+𝛿)⁰_−𝛿𝑒^𝜆𝑠𝑥(𝑡+𝑠)𝑑𝑠,𝑡∈[0,𝑇],(4.6) where (𝑘₁(⋅),𝑘₂(⋅)) satisfies −𝑘̇₁(𝑡)=2𝑎₁(𝑡)+𝑎₂(𝑡)𝑘₁(𝑡)+𝑎₂(𝑡+𝛿)𝑘₁(𝑡+𝛿)−𝑛₁⁻¹(𝑡−𝛿)𝑏²₂(𝑡)𝑘²₁(𝑡)+𝜎²(𝑡)𝑘₁(𝑡)+𝑟₁(𝑡)+𝑘₂(𝑡)−𝑒^−𝜆𝛿𝑘₂()[],𝑘̇𝑡+𝛿,𝑡∈0,𝑇₂(𝑡)=𝜆−𝑎₁(𝑡)−𝑎₂(𝑡)𝑘₂(𝑡)+𝑛₁⁻¹(𝑡−𝛿)𝑏²₂(𝑡)𝑘²₂(𝑡)[],𝑘,𝑡∈𝛿,𝑇₁(𝑇)=𝑞𝑥(𝑇),𝑘₁(𝑡)(],𝑘=0,𝑡∈𝑇,𝑇+𝛿₂(𝑡)=0,𝑡∈[0,𝛿)∪[𝑇,𝑇+𝛿].(4.7) Moreover, the above Riccati (4.7) admits a unique solution, so the presentation form of (4.6) exists and it is unique. Here, 𝜆 is a parameter satisfying 𝜆>𝑎₁(𝑡)+𝑎₂(𝑡), for all 𝑡∈[0,𝑇].

Proof. Obviously, the existence and uniqueness of presentation (4.6) are equivalent to Riccati (4.7) which has a unique solution, so we pay attention to (4.7).
Let us start with the second equation in (4.7). It is a classical Riccati-type differential equation with bounded coefficients, thence, by Theorem 7.7 of Chapter 6 in Yong and Zhou [11], we get that it admits a unique solution 𝑘₂(𝑡) on [0,𝑇] and also we have 0≤𝑘₂(𝑡)≤𝑀 with 𝑀 which is big enough positive constant.
Now, let us turn to the first equation in (4.7). It is anticipated. In order to derive its solvability, we discuss the equation backward step by step on the time interval [𝑇−𝛿,𝑇), [𝑇−2𝛿,𝑇−𝛿), ….
For example, when 𝑡∈[𝑇−𝛿,𝑇), the equation is equivalent to the following: −𝑘̇₁(𝑡)=2𝑎₁(𝑡)+12𝑎₂(𝑡)𝑘₁(𝑡)+𝜎²(𝑡)𝑘₁(𝑡)−𝑛₁⁻¹(𝑡)𝑏²₂(𝑡)𝑘²₁(𝑡)+𝑟₁(𝑡)+𝑘₂(𝑡)[),𝑘,𝑡∈𝑇−𝛿,𝑇₁(𝑇)=𝑞𝑥(𝑇),𝑘₁(𝑡)=0,𝑡∈(𝑇,𝑇+𝛿].(4.8)
Because 𝑘₂(𝑡) can be solved in advance, we get that there exists a unique solution of (4.8) and 𝑘₁(𝑡)≥0 over [𝑇−𝛿,𝑇) also by Theorem 7.7 of Chapter 6 in Yong and Zhou [11].
The discussions on the other time intervals are similar; we omit them. All in all, the Riccati equation (4.7) is solved backward in time from [𝑇,𝑇+𝛿] as an initial value problem and the solution is unique. This will lead to the existence and uniqueness of our feedback control immediately. The proof is completed.

Appendix

Let us pay attention to the generalized FBSDE (2.8). For the classical FBSDE, Hu, and Peng [12] and Peng, Wu [13] obtained the existence and uniqueness results under some monotonicity conditions. Yong [14] let the method in [12, 13] be systematic and called it the “continuation method." Then Wu [9, 15] discussed the applications of FBSDEs in LQ problem and maximum principle for optimal control problems of FBSDE systems. While if we substitute the optimal control (2.7) into (2.8), we notice that our FBSDE (2.8) is different from the classical one; it contains more delayed and anticipated part in the coefficients. Now, we try to analyze this generalized kind of FBSDEs.

For more general case, we consider the following FBSDE. (Just for simplicity, in this appendix we change the notations with writing the time variable 𝑡 as a subscript).

Consider the following: 𝑑𝑥_𝑡=𝑏𝑡,𝑥_𝑡,𝑥_𝑡−𝛿,𝐵_𝑡𝑦_𝑡,𝐷_𝑡𝑧_𝑡,𝐵_𝑡−𝛿𝑦_𝑡−𝛿,𝐷_𝑡−𝛿𝑧_𝑡−𝛿,𝐵¹_𝑡𝑦_𝑡,𝐷¹_𝑡𝑧_𝑡,𝐵¹_𝑡+𝛿𝑦_𝑡+𝛿,𝐷¹_𝑡+𝛿𝑧_𝑡+𝛿𝑑𝑡+𝜎𝑡,𝑥_𝑡,𝑥_𝑡−𝛿,𝐵_𝑡𝑦_𝑡,𝐷_𝑡𝑧_𝑡,𝐵_𝑡−𝛿𝑦_𝑡−𝛿,𝐷_𝑡−𝛿𝑧_𝑡−𝛿,𝐵¹_𝑡𝑦_𝑡,𝐷¹_𝑡𝑧_𝑡,𝐵¹_𝑡+𝛿𝑦_𝑡+𝛿,𝐷¹_𝑡+𝛿𝑧_𝑡+𝛿𝑑𝑊_𝑡,𝑡∈(0,𝑇],−𝑑𝑦_𝑡=𝑓𝑡,𝑥_𝑡,𝑦_𝑡,𝑧_𝑡,𝑦_𝑡+𝛿,𝑧_𝑡+𝛿𝑑𝑡−𝑧_𝑡𝑑𝑊_𝑡[],𝑥,𝑡∈0,𝑇_𝑡=𝜑_𝑡,𝑡∈[−𝛿,0],𝑦_𝑡=𝑧_𝑡[),𝑦=0,𝑡∈−𝛿,0_𝑇=Φ𝑥_𝑇,𝑦_𝑡=𝑧_𝑡=0,𝑡∈(𝑇,𝑇+𝛿].(A.1) Here 𝐵,𝐵¹,𝐷,𝐷¹ are 𝑘×𝑛 matrices, (𝑥,𝑦,𝑧)∈ℝ^{𝑛+𝑛+𝑛} and 𝑏,𝑓,𝜎 are 𝔽-adapted with appropriate dimensions.

We will use the notations: 𝑥𝑦𝑧𝜆=,𝜆_𝛿𝑥∶=𝑦^⋅−𝛿_⋅−𝛿𝑧_⋅−𝛿,𝑦_⋅+𝛿𝑧_⋅+𝛿=𝛽𝛾𝐴,(A.2)𝑡,𝜆,𝜆_𝛿,𝛽,𝛾=𝑏−𝑓𝑡,𝜆,𝛽,𝛾𝑡,𝜆,𝜆_𝛿𝜎,𝛽,𝛾𝑡,𝜆,𝜆_𝛿,𝛽,𝛾,(A.3) where 𝜎=(𝜎₁,…,𝜎_𝑑). We also impose the following monotonicity conditions: (i)𝐴thereexistsaconstant𝐶>0,suchthat𝑡,𝜆,𝜆_𝛿,𝛽,𝛾−𝐴𝑡,𝜆^′,𝜆^′_𝛿,𝛽^′,𝛾^′≤𝐶𝜆−𝜆^′+𝜆_𝛿−𝜆^′_𝛿+𝔼^ℱ𝑡𝛽−𝛽^′+𝛾−𝛾^′∀𝜆,𝜆′,𝜆_𝛿,𝜆^′_𝛿(),𝛽,𝛽′,𝛾,𝛾′;iiforeach𝜆,𝜆_𝛿∈ℝ^3𝑛,𝛽,𝛾∈𝐿²Ω,ℱ_⋅+𝛿,𝑃;ℝ^𝑛,𝐴⋅,𝜆,𝜆_𝛿,𝛽,𝛾isin𝐿²_𝔽();()(𝑥)0,𝑇iiiforany𝑥,Φisin𝐿²Ω,ℱ_𝑇(),𝑃anditisuniformlyLipschitzcontinuouswithrespectto𝑥;ivforany𝑥_𝑡,^𝑇₀𝑙𝑡,𝑥_𝑡,𝑥_𝑡−𝛿,𝑦_𝑡,𝑧_𝑡,𝑦_𝑡−𝛿,𝑧_𝑡−𝛿,𝑦_𝑡+𝛿,𝑧_𝑡+𝛿−𝑙𝑡,𝑥_𝑡,𝑥_𝑡−𝛿,𝑦^′_𝑡,𝑧^′_𝑡,𝑦^′_𝑡−𝛿,𝑧^′_𝑡−𝛿,𝑦^′_𝑡+𝛿,𝑧^′_𝑡+𝛿𝑑𝑡≤𝐾^𝑇₀𝐵_𝑡𝑦_𝑡−𝑦^′_𝑡+𝐷_𝑡𝑧_𝑡−𝑧^′_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑦_𝑡+𝛿−𝑦^′_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑧_𝑡+𝛿−𝑧^′_𝑡+𝛿()𝑑𝑡,where𝐾>0,𝑙=𝑏,𝜎,HA.1_𝑇0𝐴𝑡,𝜆_𝑡,𝜆_𝑡−𝛿,𝑦_𝑡+𝛿,𝑧_𝑡+𝛿−𝐴𝑡,𝜆_𝑡,𝜆_𝑡−𝛿,𝑦_𝑡+𝛿,𝑧_𝑡+𝛿,𝜆_𝑡−𝜆_𝑡𝑑𝑡≤^𝑇₀−𝜈₁𝑥_𝑡²−𝜈₂𝐵𝑦_𝑡+𝐷𝑧_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑦_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑧_𝑡+𝛿²⟨Φ(𝑥)(𝑑𝑡,−Φ𝑥),𝑥−𝑥⟩()≥0,HA.2 for all 𝜆=(𝑥,𝑦,𝑧),𝜆=(𝑥,𝑦,𝑧),𝑥=𝑥−𝑥,𝑦=𝑦−𝑦,𝑧=𝑧−𝑧, where 𝜈₁ and 𝜈₂ are given nonnegative constants with 𝜈₁≥0,𝜈₂>0.

Then we have the following result.

Theorem A.1. Let (HA.1) and (HA.2) hold. Then there exists a unique solution 𝜆=(𝑥,𝑦,𝑧)∈𝐿²_𝔽(0,𝑇;ℝ^𝑛)×𝐿²_𝔽(0,𝑇;ℝ^𝑛)×𝐿²_𝔽(0,𝑇;ℝ^𝑛) satisfying the generalized FBSDE (A.1).

Proof. Uniqueness
let 𝜆_𝑠=(𝑥_𝑠,𝑦_𝑠,𝑧_𝑠) and 𝜆^′_𝑠=(𝑥^′_𝑠,𝑦^′_𝑠,𝑧^′_𝑠) be two solutions of (A.1). We set 𝜆=(𝑥−𝑥′,𝑦−𝑦′,𝑧−𝑧′)=(𝑥,𝑦,𝑧). And we apply Itô's formula to ⟨𝑥_𝑠,𝑦_𝑠⟩ on [0,𝑇]: 𝑑𝑥_𝑠,𝑦_𝑠=𝑥_𝑠,𝑑𝑦_𝑠+𝑦_𝑠,𝑑𝑥_𝑠+𝑑𝑥_𝑠,𝑑𝑦_𝑠=𝑥_𝑠,−𝑓𝑠,𝜆_𝑠,𝑦_𝑠+𝛿,𝑧_𝑠+𝛿−𝑓𝑠,𝜆^′_𝑠,𝑦^′_𝑠+𝛿,𝑧^′_𝑠+𝛿𝑑𝑠+𝑥_𝑠,𝑧_𝑠𝑑𝑊_𝑠+𝑦_𝑠,𝑏𝑠,𝜆_𝑠,𝜆_𝑠−𝛿,𝑦_𝑠+𝛿,𝑧_𝑠+𝛿−𝑏𝑠,𝜆^′_𝑠,𝜆^′_𝑠−𝛿,𝑦^′_𝑠+𝛿,𝑧^′_𝑠+𝛿+𝑦𝑑𝑠_𝑠,𝜎𝑠,𝜆_𝑠,𝜆_𝑠−𝛿,𝑦_𝑠+𝛿,𝑧_𝑠+𝛿−𝜎𝑠,𝜆^′_𝑠,𝜆^′_𝑠−𝛿,𝑦^′_𝑠+𝛿,𝑧^′_𝑠+𝛿𝑑𝑊_𝑠+𝜎𝑠,𝜆_𝑠,𝜆_𝑠−𝛿,𝑦_𝑠+𝛿,𝑧_𝑠+𝛿−𝜎𝑠,𝜆^′_𝑠,𝜆^′_𝑠−𝛿,𝑦^′_𝑠+𝛿,𝑧^′_𝑠+𝛿,𝑧_𝑠𝑑𝑠.(A.4) Integrating and taking expectations on both sides of the above equation, we have 𝔼Φ𝑥_𝑇−Φ𝑥^′_𝑇,𝑥_𝑇=𝔼^𝑇₀𝐴𝑡,𝜆_𝑠,𝜆_𝑠−𝛿,𝑦_𝑠+𝛿,𝑧_𝑠+𝛿−𝐴𝑠,𝜆^′_𝑠,𝜆^′_𝑠−𝛿,𝑦_𝑠+𝛿,𝑧_𝑠+𝛿,𝜆_𝑠𝑑𝑠.(A.5) Combining (HA.2), we obtain 𝜈₂𝔼^𝑇₀𝐵_𝑠𝑦_𝑠+𝐷_𝑠𝑧_𝑠+𝔼^ℱ𝑠𝐵¹_𝑠+𝛿𝑦_𝑠+𝛿+𝐷¹_𝑠+𝛿𝑧_𝑠+𝛿²𝑑𝑠≤0.(A.6) This implies 𝐵_𝑠𝑦_𝑠+𝐷_𝑠𝑧_𝑠+𝐵¹_𝑠+𝛿𝑦_𝑠+𝛿+𝐷¹_𝑠+𝛿𝑧_𝑠+𝛿≡0. Following (HA.1)-(𝑖𝑣) and the uniqueness of solution for SDDE (see [2, 3]), we have 𝑥_𝑠≡𝑥^′_𝑠. Consequently, Φ(𝑥_𝑇)=Φ(𝑥^′_𝑇). By the uniqueness of solution for BSDE, we have 𝑦_𝑠≡𝑦^′_𝑠,𝑧_𝑠≡𝑧^′_𝑠.

Existence. In order to prove the existence of the solution we first consider the following family of equations parameterized by 𝛼∈[0,1]: 𝑑𝑥^𝛼_𝑡=(1−𝛼)−𝐵^𝖳_𝑡𝐵_𝑡𝑦^𝛼_𝑡+𝐷_𝑡𝑧^𝛼_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑦^𝛼_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑧^𝛼_𝑡+𝛿−𝐵¹_𝑡^𝖳𝐵_𝑡−𝛿𝑦^𝛼_𝑡−𝛿+𝐷_𝑡−𝛿𝑧^𝛼_𝑡−𝛿+𝐵¹_𝑡𝑦^𝛼_𝑡+𝐷¹_𝑡𝑧^𝛼_𝑡+𝛼𝑏𝑡,𝜆^𝛼_𝑡,𝜆^𝛼_𝑡−𝛿,𝑦^𝛼_𝑡+𝛿,𝑧^𝛼_𝑡+𝛿+𝜙_𝑡+𝑑𝑡(1−𝛼)−𝐷^𝖳_𝑡𝐵_𝑡𝑦^𝛼_𝑡+𝐷_𝑡𝑧^𝛼_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑦^𝛼_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑧^𝛼_𝑡+𝛿−𝐷¹_𝑡^𝖳𝐵_𝑡−𝛿𝑦^𝛼_𝑡−𝛿+𝐷_𝑡−𝛿𝑧^𝛼_𝑡−𝛿+𝐵¹_𝑡𝑦^𝛼_𝑡+𝐷¹_𝑡𝑧^𝛼_𝑡+𝛼𝜎𝑡,𝜆^𝛼_𝑡,𝜆^𝛼_𝑡−𝛿,𝑦^𝛼_𝑡+𝛿,𝑧^𝛼_𝑡+𝛿+𝜓_𝑡𝑑𝑊_𝑡,𝑡∈[0,𝑇],−𝑑𝑦^𝛼_𝑡=𝛼𝑓𝑡,𝜆^𝛼_𝑡,𝑦^𝛼_𝑡+𝛿,𝑧^𝛼_𝑡+𝛿+𝜍_𝑡𝑑𝑡−𝑧^𝛼_𝑡𝑑𝑊_𝑡[],𝑥,𝑡∈0,𝑇_𝛼𝑡=𝜑_𝑡,𝑡∈[−𝛿,0],𝑦^𝛼_𝑡=𝑧^𝛼_𝑡[),𝑦=0,𝑡∈−𝛿,0_𝛼𝑇=𝛼Φ𝑥_𝑇+𝜉,𝑦^𝛼_𝑡=𝑧^𝛼_𝑡=0,𝑡∈(𝑇,𝑇+𝛿],(A.7) where 𝜙,𝜓 and 𝜍 are given processes in 𝐿²_𝔽(0,𝑇;ℝ^𝑛), 𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑛). Obviously, the existence of solutions for (A.7) with 𝛼=1 implies our desired conclusion. When 𝛼=0, we notice that (A.7) is in a decupled form, and we obtain the existence and uniqueness result from the classical SDE and BSDE theory. Detailedly, in this case, the second equation and the terminal condition 𝑦⁰_𝑇=𝜉 construct a standard BSDE which admits a unique solution (𝑦⁰,𝑧⁰). Substituting (𝑦⁰,𝑧⁰) into the first equation, we get a standard SDE and it admits a unique solution 𝑥⁰. Thence, we need to consider the following case: 𝑑𝑋_𝑡=1−𝛼₀−𝐵^𝖳_𝑡𝐵_𝑡𝑌_𝑡+𝐷_𝑡𝑍_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑌_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑍_𝑡+𝛿−𝐵¹_𝑡^𝖳𝐵_𝑡−𝛿𝑌_𝑡−𝛿+𝐷_𝑡−𝛿𝑍_𝑡−𝛿+𝐵¹_𝑡𝑌_𝑡+𝐷¹_𝑡𝑍_𝑡+𝛼₀𝑏𝑡,Λ_𝑡,Λ_𝑡−𝛿,𝑌_𝑡+𝛿,𝑍_𝑡+𝛿+𝜌𝐵𝑑𝑡_𝖳𝑡𝐵_𝑡𝑦_𝑡+𝐷_𝑡𝑧_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑦_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑧_𝑡+𝛿+𝐵¹_𝑡^𝖳𝐵_𝑡−𝛿𝑦_𝑡−𝛿+𝐷_𝑡−𝛿𝑧_𝑡−𝛿+𝐵¹_𝑡𝑦_𝑡+𝐷¹_𝑡𝑧_𝑡+𝑏𝑡,𝜆_𝑡,𝜆_𝑡−𝛿,𝑦_𝑡+𝛿,𝑧_𝑡+𝛿+𝜙_𝑡+𝑑𝑡1−𝛼₀−𝐷^𝖳_𝑡𝐵_𝑡𝑌_𝑡+𝐷_𝑡𝑍_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑌_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑍_𝑡+𝛿−𝐷¹_𝑡^𝖳𝐵_𝑡−𝛿𝑌_𝑡−𝛿+𝐷_𝑡−𝛿𝑍_𝑡−𝛿+𝐵¹_𝑡𝑌_𝑡+𝐷¹_𝑡𝑍_𝑡+𝛼₀𝜎𝑡,Λ_𝑡,Λ_𝑡−𝛿,𝑌_𝑡+𝛿,𝑍_𝑡+𝛿𝑑𝑊_𝑡+𝜌𝐷^𝖳_𝑡𝐵_𝑡𝑦_𝑡+𝐷_𝑡𝑧_𝑡+𝔼^ℱ𝑡𝐵¹_𝑡+𝛿𝑦_𝑡+𝛿+𝐷¹_𝑡+𝛿𝑧_𝑡+𝛿+𝐷¹_𝑡^𝖳𝐵_𝑡−𝛿𝑦_𝑡−𝛿+𝐷_𝑡−𝛿𝑧_𝑡−𝛿+𝐵¹_𝑡𝑦_𝑡+𝐷¹_𝑡𝑧_𝑡+𝜎𝑡,𝜆_𝑡,𝜆_𝑡−𝛿,𝑦_𝑡+𝛿,𝑧_𝑡+𝛿+𝜓_𝑡𝑑𝑊_𝑡,𝑡∈[0,𝑇],−𝑑𝑌_𝑡=𝛼₀𝑓𝑡,Λ_𝑡,𝑌_𝑡+𝛿,𝑍_𝑡+𝛿+𝜌𝑓𝑡,𝜆_𝑡,𝑦_𝑡+𝛿,𝑧_𝑡+𝛿+𝜍_𝑡𝑑𝑡−𝑍_𝑡𝑑𝑊_𝑡[],𝑋,𝑡∈0,𝑇_𝑡=𝜑_𝑡,𝑡∈[−𝛿,0],𝑌_𝑡=𝑍_𝑡[),𝑌=0,𝑡∈−𝛿,0_𝑇=𝛼₀Φ𝑋_𝑇+𝜌Φ𝑥_𝑇+𝜉,𝑌_𝑡=𝑍_𝑡=0,𝑡∈(𝑇,𝑇+𝛿].(A.8) Here, Λ_𝑠=(𝑋_𝑠,𝑌_𝑠,𝑍_𝑠). We want to prove that the mapping defined by 𝐼_𝛼0+𝜌(𝜆)=Λ∶𝐿²_𝔽−𝛿,𝑇;ℝ^𝑛×𝐿²_𝔽−𝛿,𝑇+𝛿;ℝ^𝑛×𝐿²_𝔽−𝛿,𝑇+𝛿;ℝ^𝑛→𝐿²_𝔽−𝛿,𝑇;ℝ^𝑛×𝐿²_𝔽−𝛿,𝑇+𝛿;ℝ^𝑛×𝐿²_𝔽−𝛿,𝑇+𝛿;ℝ^𝑛(A.9) is a contraction.

For the difference (𝑋,𝑌)=(𝑋−𝑋′,𝑌−𝑌′), we have the following result: 𝜈₂𝛼₀+1−𝛼₀𝔼^𝑇₀𝐵_𝑠𝑌_𝑠+𝐷_𝑠𝑍_𝑠+𝔼^ℱ𝑠𝐵¹_𝑠+𝛿𝑌_𝑠+𝛿+𝐷¹_𝑠+𝛿𝑍_𝑠+𝛿²𝑑𝑠≤𝜌𝐶₁𝔼^𝑇₀𝜆_𝑠²𝑑𝑠+𝜌𝐶₁𝔼𝑥_𝑇²+𝜌𝐶₁𝔼^𝑇₀Λ_𝑠²𝑑𝑠+𝜌𝐶₁𝔼𝑋_𝑇².(A.10) Here 𝐶₁ depends on the Lipschitz constants of 𝑏,𝜎,𝑓,Φ and 𝐵,𝐵¹,𝐷,𝐷¹. If we select 𝜈₂𝛼₀+(1−𝛼₀)≥𝐿₁, where 𝐿₁=min(1,𝜈₂)>0, then 𝔼^𝑇₀𝐵_𝑠𝑌_𝑠+𝐷_𝑠𝑍_𝑠+𝔼^ℱ𝑠𝐵¹_𝑠+𝛿𝑌_𝑠+𝛿+𝐷¹_𝑠+𝛿𝑍_𝑠+𝛿²𝑑𝑠≤𝜌𝐶₂𝔼^𝑇₀𝜆_𝑠²𝑑𝑠+𝜌𝐶₂𝔼𝑥_𝑇²+𝜌𝐶₂𝔼^𝑇₀Λ_𝑠²𝑑𝑠+𝜌𝐶₂𝔼𝑋_𝑇²(A.11) with 𝐶₂=𝐶₁/𝐿₁. For 𝑋, using the estimate of SDDE, we derive sup_{0≤𝑠≤𝑇}𝔼𝑥_𝑇²≤𝐶₃𝜌𝔼^𝑇₀𝜆_𝑠²𝑑𝑠+𝐶₃𝔼^𝑇₀𝐵_𝑠𝑌_𝑠+𝐷_𝑠𝑍_𝑠+𝔼^ℱ𝑠𝐵¹_𝑠+𝛿𝑌_𝑠+𝛿+𝐷¹_𝑠+𝛿𝑍_𝑠+𝛿²𝑑𝑠,𝔼^𝑇₀𝑥_𝑇²𝑑𝑠≤𝐶₃𝑇𝜌𝔼^𝑇₀𝜆_𝑠²𝑑𝑠+𝐶₃𝑇𝔼^𝑇₀𝐵_𝑠𝑌_𝑠+𝐷_𝑠𝑍_𝑠+𝔼^ℱ𝑠𝐵¹_𝑠+𝛿𝑌_𝑠+𝛿+𝐷¹_𝑠+𝛿𝑍_𝑠+𝛿²𝑑𝑠.(A.12) Similarly, for the difference of solutions (𝑌,𝑍), we have 𝔼^𝑇₀𝑌_𝑠²+𝑍_𝑠²𝑑𝑠≤𝐶₃𝜌𝔼^𝑇₀𝜆_𝑠²𝑑𝑠+𝐶₃𝔼^𝑇₀𝑥_𝑠²𝑑𝑠+𝐶₃𝔼𝑥_𝑇²,(A.13) by the anticipated BSDE estimates (see [7]). Here, the constant 𝐶₃ depends on the Lipschitz constants, 𝐾,𝐵,𝐵¹,𝐷,𝐷¹ and 𝑇. Combining the estimates (A.11)–(A.13), we have 𝔼^𝑇₀Λ_𝑠²𝑑𝑠+𝔼𝑥_𝑇²≤𝐶𝜌𝔼^𝑇₀𝜆_𝑠²𝑑𝑠+𝔼𝑥_𝑇²,(A.14) where the constant 𝐶 depends on 𝐶₁,𝐶₂,𝐶₃ and 𝑇. So we choose 𝜌₀=1/2𝐶, then for each 𝜌∈[0,𝜌₀], the mapping 𝐼_𝛼0+𝜌 is a contraction; that is, for all 𝜌∈[0,𝜌₀] the (A.7) for 𝛼=𝜌 has a unique solution. We repeat this process 𝑁 times with 1≤𝑁𝜌₀≤1+𝜌₀. It then follows that (A.7) for 𝛼=1 has a unique solution.

Remark A.2. It is verified that the FBSDE (2.8) satisfies the assumptions (HA.1) and (HA.2); therefore, there exists a unique solution (𝑥,𝑦,𝑧) for FBSDE (2.8). Consequently, the optimal control of our Problem (LQD) exists and is unique.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11101242, 11126214, 10921101, and 61174092), the National Science Fund for Distinguished Young Scholars of China (11125102), the Natural Science Foundation of Shandong Province, China, (ZR2010AQ004), the Independent Innovation Foundation of Shandong University (IIFSDU, no. 2010TS060), and Doctoral Fund of Education Ministry of China.