Backward Stochastic 𝐻₂/𝐻_∞ Control: Infinite Horizon Case

Abstract

The mixed 𝐻₂/𝐻_∞ control problem is studied for systems governed by infinite horizon backward stochastic differential equations (BSDEs) with exogenous disturbance signal. A necessary and sufficient condition for the existence of a unique solution to the 𝐻₂/𝐻_∞ control problem is derived. The equivalent feedback solution is also discussed. Contrary to deterministic or stochastic forward case, the feedback solution is no longer feedback of the current state; rather, it is feedback of the entire history of the state.

1. Introduction

𝐻_∞ control is one of the most important robust control approaches in which control law is sought to efficiently eliminate the effect of the exogenous disturbance in the practical system. We refer the reader to [1–3] and the references therein. If the purpose is to select control not only to restrain the exogenous disturbance, but also to minimize a cost function when the worst case disturbance 𝑑^∗ is implemented, this is the so-called mixed 𝐻₂/𝐻_∞ control problem. Mixed 𝐻₂/𝐻_∞ control problem has attracted much attention and has been widely applied to various fields. Please refer to [4, 5] for more information.

It should be pointed out that the above-mentioned works are concerned only with the forward stochastic systems. The case of systems governed by backward stochastic differential equations with exogenous disturbance signal, to our best knowledge, seems to be open. The objective of this paper is to develop an 𝐻₂/𝐻_∞ control theory for infinite horizon backward stochastic systems.

A BSDE is an Itô stochastic differential equation (SDE) for which a random terminal condition on the state has been specified. Since BSDEs are well-defined dynamic systems, it is very natural and appealing to study the control problems involving BSDEs as well as their applications in lots of different fields, especially in finance, economics, insurance, and so forth. Please refer to [6–12] for more details. This paper is concerned with mixed 𝐻₂/𝐻_∞ control of backward systems governed by infinite horizon linear BSDEs, namely, an infinite horizon backward stochastic 𝐻₂/𝐻_∞ control problem. This means that our purpose is to study mixed 𝐻₂/𝐻_∞ backward stochastic control problem in infinite horizon which presents more robust and stable sense in practise. For that, as preliminaries, we first need to review some results on infinite horizon BSDEs in Section 2. Chen and Wang [13] gave an existence and uniqueness result under a kind of Lipschitz condition suitable for one-dimensional infinite horizon BSDEs. Wu [14] generalized the result of [13] into the poisson jump process case in unbounded stopping time duration and obtained the corresponding comparison theorem. In this section, under this frame, we get the existence and uniqueness result for the infinite horizon matrix-valued BSDEs.

In Section 3, similar to the deterministic or stochastic forward case, we formulate the infinite horizon backward stochastic 𝐻₂/𝐻_∞ control problem. In Section 4, a necessary and sufficient condition for the existence of a unique solution to the 𝐻₂/𝐻_∞ control problem is derived. It is shown that the existence of a unique solution to the control problem is equivalent to the corresponding uncontrolled perturbed system to have a 𝕃₂-gain less than or equal to 𝛾 and the resulting solution is characterized by the solution of an uncontrolled forward backward stochastic differential equation (FBSDE). Under some monotone assumptions, Hu and Peng [15] and Peng and Wu [16] obtained the existence and uniqueness results in an arbitrarily prescribed time duration. Wu and Xu [17] gave some comparison theorems for FBSDEs. Riccati equation plays an important role to get the feedback form of the optimal control; please refer to Yong and Zhou [18] for the details. Section 5 gives the equivalent linear feedback solution by virtue of the solution of a Riccati-type equation. As it turns out, the infinite horizon backward stochastic 𝐻₂/𝐻_∞ control can no longer be expressed as a linear feedback of the current state like that in deterministic or stochastic forward case. Rather, it depends, in general, on the entire past history of the state pair (𝑥(⋅),𝑧(⋅)).

2. Notations and Preliminary Results of Infinite Horizon BSDEs

To treat the infinite horizon backward stochastic 𝐻₂/𝐻_∞ control problem, we need the following preliminary results of infinite horizon BSDEs.

Let (Ω,ℱ,ℱ_𝑡,𝑃) be a completed filtering probability space; let (𝑊_𝑡)_𝑡≥0 be a standard one-dimensional Wiener process (our assumption that 𝑊(⋅) is scalar-valued is for the sake of simplicity; no essential difficulties are encountered when extending our analysis to the case of vector-valued Wiener process). {ℱ_𝑡}_𝑡≥0 is the natural filtration generated by this Wiener process 𝑊(⋅) up to time 𝑡, where ℱ₀ contains all 𝑃-null sets of ℱ and ℱ_∞=∨_𝑡≥0ℱ_𝑡.

Throughout this paper, we adopt the following conventional notations. 𝑆^𝑛: the set of symmetric 𝑛×𝑛 matrices with real elements; 𝐴^𝑇: the transpose of the matrix 𝐴; 𝐴≥0(𝐴>0): 𝐴 is positive semidefinite (positive definite) real matrix; 𝐼: identity matrix; ‖𝑥‖∶=𝑥^𝑇𝑥=(∑^𝑛_𝑖=1|𝑥_𝑖|²)^1/2 for 𝑛-dimensional vector 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇; ‖𝐴‖∶=max_𝑥∈𝑅^𝑛_{,‖𝑥‖=1}‖𝐴𝑥‖ for 𝐴∈𝑅^𝑛×𝑛; 𝑁(⋅)>(≥0): 𝑁(𝑡)>(≥)0 for a.s. 𝑡∈𝑅⁺; 𝑀(⋅)>(≥)𝑁(⋅): 𝑀(⋅)−𝑁(⋅)>(≥)0; 𝑋: a given Hilbert space; 𝐿²_ℱ𝑅⁺;𝑋=∶𝑓∶𝑅⁺×Ω⟶𝑋isanℱ_𝑡−adaptedprocesssuchthat𝔼^∞₀𝑓(𝑡)²;𝒮𝑑𝑡<∞₂=∶𝑣_𝑡,0≤𝑡≤∞,isanℱ_𝑡−adaptedprocesssuchthat𝔼sup_{0≤𝑡≤∞}𝑣_𝑡²;𝐿<∞₂=∶𝜉,𝜉isavector-valuedℱ_∞−measurablerandomvariablesuchthat𝔼𝜉²<∞. (1)

We consider the infinite horizon BSDE: 𝑥_𝑡=𝜉+^∞_𝑡𝑓𝑠,𝑥_𝑠,𝑧_𝑠𝑑𝑠−^∞_𝑡𝑧_𝑠𝑑𝑊_𝑠,𝑡∈[0,∞]; (2)(𝑥,𝑧) take value in 𝑅^𝑛×𝑅^𝑛, 𝜉∈𝐿², and 𝑓 is a map from Ω×[0,∞]×𝑅^𝑛×𝑅^𝑛 onto 𝑅^𝑛 which satisfies the following.(H2.1)For all (𝑥,𝑧)∈𝑅^𝑛×𝑅^𝑛, 𝑓(⋅,𝑥,𝑧) is progressively measurable and 𝔼^∞₀𝑓(𝑠,0,0)𝑑𝑠²<∞. (3)(H2.2)There exist two positive deterministic functions 𝑢₁(𝑡) and 𝑢₂(𝑡) such that, for all (𝑥_𝑖,𝑧_𝑖)∈𝑅^𝑛×𝑅^𝑛, 𝑖=1,2, 𝑓𝑡,𝑥₁,𝑧₁−𝑓𝑡,𝑥₂,𝑧₂≤𝑢₁(𝑡)𝑥₁−𝑥₂+𝑢₂(𝑡)𝑧₁−𝑧₂,𝑡∈[0,∞_∞0),𝑢₁(𝑡)𝑑𝑡<∞,^∞₀𝑢²₂(𝑡)𝑑𝑡<∞. (4)Then we have the following.

Theorem 1 (see Wu [14]). There exists a unique solution (𝑥,𝑧)∈𝒮²×𝐿²_ℱ(𝑅⁺;𝑅^𝑛) satisfying the BSDE (2).
Let us again consider a function 𝐹, which will be in the sequel the generator of the BSDE, defined on Ω×[0,∞]×𝑆^𝑛×𝑆^𝑛, with values in 𝑆^𝑛, such that the process (𝐹(𝑡,𝑦,𝑧))_{𝑡∈[0,∞]} is a progressively measurable process for each (𝑦,𝑧)∈𝑆^𝑛×𝑆^𝑛.
Along the line of Chen and Wang [13] or Wu [14] combined with that in Peng [19] for matrixed-valued BSDEs result in finite horizon, we get the following existence and uniqueness theorem for infinite horizon matrix-valued BSDEs.

Theorem 2. Suppose that 𝐹 satisfies the following.(H2.1′)For all (𝑦,𝑧)∈𝑆^𝑛×𝑆^𝑛, 𝐹(⋅,𝑦,𝑧) is progressively measurable and 𝔼^∞₀‖𝐹(𝑠,0,0)‖𝑑𝑠²<∞. (5)(H2.2′)There exist two positive deterministic functions 𝑢₁(𝑡) and 𝑢₂(𝑡) such that, for all (𝑦_𝑖,𝑧_𝑖)∈𝑅^𝑛×𝑅^𝑛, 𝑖=1,2, 𝐹𝑡,𝑦₁,𝑧₁−𝐹𝑡,𝑦₂,𝑧₂≤𝑢₁(𝑡)𝑦₁−𝑦₂+𝑢₂(𝑡)𝑧₁−𝑧₂,𝑡∈[0,∞), (6)and ∫^∞₀𝑢₁(𝑡)𝑑𝑡<∞, ∫^∞₀𝑢²₂(𝑡)𝑑𝑡<∞, 𝜉 is a given 𝑆^𝑛-valued random variable, and 𝜉∈𝐿². Then, the following matrix-valued infinite horizon BSDE 𝑌_𝑡=𝜉+^∞_𝑡𝐹𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝑠−^∞_𝑡𝑍_𝑠𝑑𝑊_𝑠 (7) admits a unique solution (𝑌,𝑍)∈𝒮²×𝐿²_ℱ(𝑅⁺;𝑅^𝑛).

3. Problem Statement

Now, we consider the following stochastic control system governed by an infinite horizon linear BSDE: 𝑥(𝑡)=𝜉−^∞_𝑡[𝐴(𝑠)𝑥(𝑠)(𝑠)𝑢(𝑠)(𝑠)𝑑(𝑠)(𝑠)𝑧(𝑠−+𝐵+𝐶+𝐷)]𝑑𝑠_∞𝑡𝑧(𝑠)𝑑𝑊(𝑠). (8)𝑍∈ℛ^𝑛𝑍 is the penalty output, and the energy of the output signal 𝑍 is given by ‖𝑍‖²₂=𝑥^𝑇₀𝐻𝑥₀+𝔼^∞₀𝑥^𝑇_𝑡𝑄_𝑡𝑥_𝑡+𝑧^𝑇_𝑡𝑆_𝑡𝑧_𝑡+𝑢^𝑇_𝑡𝑢_𝑡𝑑𝑡, (9) where 𝐻 is a nonnegative symmetric constant matrix and 𝑄_𝑡(𝜔) and 𝑆_𝑡(𝜔) are nonnegative symmetric bounded progressively measurable matrix-valued processes. 𝑢 and 𝑑 stand for the control input and exogenous disturbance signal, respectively. The energy of the disturbances is ‖𝑑‖²₂=𝔼^∞₀𝑑^𝑇_𝑡𝑑_𝑡𝑑𝑡. (10)

Later, we will state assumptions on the coefficients 𝐴(⋅), 𝐵(⋅), 𝐶(⋅), 𝐷(⋅), 𝑄(⋅), 𝑆(⋅) so as to guarantee the existence of a unique solution pair (𝑥(⋅),𝑧(⋅))∈𝒮²×𝐿²_ℱ(𝑅⁺;𝑅^𝑛) of BSDE (8) for any 𝑢∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑢), 𝑑∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑), and 𝜉∈𝐿². We refer to such a four-tuple (𝑥(⋅),𝑧(⋅);𝑢(⋅),𝑑(⋅)) as an admissible triple.

Now, we first define the infinite horizon backward stochastic 𝐻₂/𝐻_∞ control as follows.

Definition 3 (backward stochastic 𝐻₂/𝐻_∞ control). For given 𝛾>0 and 𝑑∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑), find, if possible, a control 𝑢=𝑢^∗∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑢), such that (i)the trajectory of the closed-loop system (8) with 𝜉=0 satisfies ‖𝑍‖²₂≤𝛾²‖𝑑‖²₂,∀𝑑≠0∈𝐿²_ℱ𝑅⁺;𝑅^𝑛𝑑and (11) (ii)when the worst case disturbance ([4]) 𝑑^∗∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑), if existing, is implemented in (8), 𝑢^∗ minimizes the quadratic performance ‖𝑍‖²₂ simultaneously.
If we define 𝐽₁(𝑢,𝑑)=‖𝑍‖²₂−𝛾²‖𝑑‖²₂,𝐽₂(𝑢,𝑑)=‖𝑍‖²₂ (12) then the mixed 𝐻₂/𝐻_∞ control problem is equivalent to find the Nash equilibria (𝑢^∗,𝑑^∗) defined as 𝐽₁𝑢^∗,𝑑^∗≥𝐽₁𝑢^∗,𝑑,∀𝑑∈𝐿²_ℱ𝑅⁺;𝑅^𝑛𝑑,𝐽₂𝑢,𝑑^∗≥𝐽₂𝑢^∗,𝑑^∗,∀𝑢∈𝐿²_ℱ𝑅⁺;𝑅^𝑛𝑢,𝐽₁𝑢^∗,𝑑≤0,∀𝑑≠0∈𝐿²_ℱ𝑅⁺;𝑅^𝑛𝑑,𝜉=0. (13)(14)(15) Obviously, inequality (15) is associated with the 𝐻_∞ performance. The first Nash inequality (13) is to keep that 𝑑^∗ is the worst case disturbance, while the second one (14) is related with the 𝐻₂ performance. Clearly, if the Nash equilibria (𝑢^∗,𝑑^∗) exist and satisfy inequality (15), then 𝑢^∗ is our desired 𝐻₂/𝐻_∞ controller and 𝑑^∗ is the worst case disturbance. In this case, we also say that the infinite horizon backward stochastic 𝐻₂/𝐻_∞ control admits a solution (𝑢^∗,𝑑^∗).
Throughout this paper, we assume the following.(A1) All matrices mentioned in this paper are bounded progressively measurable processes.(A2)𝔼^∞₀‖𝐴(𝑡𝔼)‖𝑑𝑡<∞,_∞0‖𝐷(𝑡)‖²𝑑𝑡<∞. (16)(A3)𝛾>0,𝑄≥0,𝔼^∞₀‖𝑄(𝑡)‖𝑑𝑡<∞,𝐵𝐵^𝑇(⋅)>𝐶𝐶^𝑇(⋅)𝛾²,𝔼^∞₀𝐵(𝑡)𝐵(𝑡)^𝑇−𝐶(𝑡)𝐶(𝑡)^𝑇𝛾²𝑑𝑡<∞. (17)
From Theorem 2, we obtain that assumption (A2) is sufficient to guarantee the existence of a unique solution pair (𝑥(⋅),𝑧(⋅))∈𝒮²×𝐿²_ℱ(𝑅⁺;𝑅^𝑛) of BSDE (8) for any 𝑢∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑢) and 𝑑∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑).

4. The Necessary and Sufficient Condition

In this section, we give a necessary and sufficient condition for the existence of a unique solution to the backward stochastic 𝐻₂/𝐻_∞ control problem. We begin our presentation with some preliminaries.

Consider the following uncontrolled stochastic perturbed system: 𝑑𝑥_𝑡=𝐴(𝑡)𝑥_𝑡+𝐶(𝑡)𝑑_𝑡+𝐷(𝑡)𝑧_𝑡𝑑𝑡+𝑧_𝑡(𝑡),𝑥(∞)[).𝑑𝐵=𝜉,𝑡∈0,∞ (18) Let 𝑍 be the to-be-controlled output. For any 0<𝑇<∞, define the perturbation operator 𝕃∶𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑)→𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑍) as 𝕃(𝑑)=𝑍|_𝑥∞=0,𝑡≥0,𝑑∈𝐿²_ℱ𝑅⁺;𝑅^𝑛𝑑, (19) with its norm ‖𝕃‖₂∶=sup_{𝑑∈𝐿2ℱ(𝑅+;𝑅𝑛𝑑),𝑑≠0,𝑥∞=0}‖𝕃(𝑑)‖₂‖𝑑‖₂=sup_{𝑑∈𝐿2ℱ(𝑅+;𝑅𝑛𝑑),𝑑≠0,𝑥∞=0}‖𝑍‖₂‖𝑑‖₂, (20) where ‖𝑍‖²₂=𝑥^𝑇₀𝐻𝑥₀+𝔼^∞₀𝑥^𝑇_𝑡𝑄_𝑡𝑥_𝑡+𝑦^𝑇_𝑡𝑆_𝑡𝑦_𝑡𝑑𝑡. (21) Obviously, 𝕃 is a nonlinear operator.

Definition 4. Let 𝛾>0; system (18) is said to have 𝕃₂-gain less than or equal to 𝛾 if for any nonzero 𝑑∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑), ‖𝕃‖₂≤𝛾.

Proposition 5. For system (18) and given disturbance attenuation 𝛾>0, if there exists a function 𝐏(⋅), satisfying the following SDE (the variables 𝑡 and 𝜔 are suppressed): 𝑑𝑃=−𝐴^𝑇𝑃−𝑃𝐴−𝑄−𝑃𝐶𝐶^𝑇𝑃𝛾²𝑑𝑡−𝐷^𝑇(𝑡),(0)[),𝑃𝑑𝑊𝑃+𝑆≤0,𝑃=−𝐻,𝑡∈0,∞ (22) then ‖𝕃‖₂≤𝛾.

Proof. It only needs to note the following identity: ‖𝑍‖²₂−𝛾²‖𝑑‖²₂=‖𝑍‖²₂−𝛾²‖𝑑‖²₂+𝔼^∞₀𝑑𝑥^𝑇𝑃𝑥−𝑥^𝑇₀𝐻𝑥₀=−𝛾²𝐶𝑣−^𝑇𝑃𝑥𝛾²²₂+𝔼^∞₀𝑦^𝑇(𝑃+𝑆)𝑦𝑑𝑡≤0. (23)

The following theorem is a necessary and sufficient condition for the existence of a unique solution to the infinite horizon backward stochastic 𝐻₂/𝐻_∞ control problem.

Theorem 6. For system (8), the backward stochastic 𝐻₂/𝐻_∞ control problem admits a solution if and only if the corresponding uncontrolled system (18) has 𝕃₂-gain less than or equal to 𝛾. Moreover, if the backward stochastic 𝐻₂/𝐻_∞ control problem admits a solution, then the solution is unique with 𝑢^∗=𝐵^𝑇𝑝^∗2,𝑑^∗𝐶=−^𝑇𝑝^∗2𝛾², (24) where (𝑝^∗,𝑥^∗,𝑧^∗) is the solution of the following FBSDE: 𝑑𝑝^∗_𝑡=2𝑄𝑥^∗_𝑡−𝐴^𝑇𝑝^∗_𝑡𝑑𝑡+2𝑆𝑧^∗_𝑡−𝐷^𝑇𝑝^∗_𝑡𝑑𝐵(𝑡),𝑑𝑥^∗_𝑡=𝐴𝑥^∗_𝑡+𝐵𝐵^𝑇𝑝^∗_𝑡2−𝐶𝐶^𝑇𝑝^∗_𝑡2𝛾²+𝐷𝑧^∗_𝑡𝑑𝑡+𝑧^∗_𝑡(𝑡),𝑝𝑑𝐵_∗0=2𝐻𝑥^∗₀,𝑥^∗_∞=𝜉,𝑡∈[0,∞). (25)

Proof. (1) The Sufficient Condition. To show that the backward stochastic 𝐻₂/𝐻_∞ control problem admits a unique solution (𝑢^∗,𝑑^∗) if the corresponding uncontrolled system (18) has 𝕃₂-gain less than or equal to 𝛾, we will show that (𝑢^∗,𝑑^∗) is a solution firstly.
Look at the above FBSDE; from [16], the FBSDE (25) has a unique solution (𝑝^∗_𝑡,𝑥^∗_𝑡,𝑧^∗_𝑡). Now, we try to prove that 𝑑^∗ is the worst case disturbance. For any given 𝑑∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑), suppose that 𝑥^𝑑 is the trajectory corresponding to (𝑢^∗,𝑑)∈𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑢)×𝐿²_ℱ(𝑅⁺;𝑅^𝑛𝑑). It is easy to see the trajectory corresponding to 0,𝑑^∗−𝑑 (26) is 𝑥^∗−𝑥^𝑑 with initial state 𝑥^∗₀−𝑥^𝑑₀ and terminal state 0. Hence, (𝑥^∗−𝑥^𝑑,𝑧^∗−𝑧^𝑑) is the solution corresponding to 𝑑^∗−𝑑 for system (18) with terminal state 0. Since system (18) has 𝕃₂-gain less than or equal to 𝛾, then 𝔼^∞₀−𝑥^∗−𝑥^𝑑𝑇𝑄𝑥^∗−𝑥^𝑑−𝑧^∗−𝑧^𝑑𝑇𝑆𝑧^∗−𝑧^𝑑+𝛾²𝑑^∗−𝑑^𝑇𝑑^∗−𝑥−𝑑𝑑𝑡_∗0−𝑥^𝑑₀^𝑇𝐻𝑥^∗₀−𝑥^𝑑₀≥0,𝐽₂𝑢^∗,𝑑^∗−𝐽₂𝑢^∗,𝑑=𝔼^∞₀𝑥^∗𝑇𝑄𝑥^∗−𝑥^𝑑𝑇𝑄𝑥^𝑑+𝑧^∗𝑇𝑆𝑧^∗−𝑧^𝑑𝑇𝑆𝑧^𝑑−𝛾²𝑑^∗𝑇𝑑^∗+𝛾²𝑑^𝑇𝑑𝑑𝑡+𝑥₀^∗𝑇𝐻𝑥^∗₀−𝑥₀^𝑑𝑇𝐻𝑥^𝑑₀=𝔼^∞₀𝑥^∗−𝑥^𝑑𝑇𝑄𝑥^∗−𝑥^𝑑−2𝑥^𝑑𝑇𝑄𝑥^𝑑−𝑥^∗+𝑧^∗−𝑧^𝑑𝑇𝑆𝑧^∗−𝑧^𝑑−2𝑧^𝑑𝑇𝑆𝑧^𝑑−𝑧^∗−𝛾²𝑑^∗𝑇𝑑^∗+𝛾²𝑑^𝑇𝑑+𝑥𝑑𝑡_∗0−𝑥^𝑑₀^𝑇𝐻𝑥^∗₀−𝑥^𝑑₀−2𝑥₀^𝑑𝑇𝐻𝑥^𝑑₀−𝑥^∗₀. (27)(28) Applying Itô’s formula to 𝑝^∗𝑇(𝑥^𝑑−𝑥^∗), −2𝑥₀^∗𝑇𝐻𝑥^𝑑₀−𝑥^∗₀=𝔼^∞₀𝑑𝑝^∗𝑇𝑥^𝑑−𝑥^∗=𝔼^∞₀2𝑥^∗𝑇𝑄𝑥^𝑑−𝑥^∗+2𝑧^∗𝑇𝑆𝑧^𝑑−𝑧^∗+𝐶^𝑇𝑝^∗𝑇𝑑−𝑑^∗𝑑𝑡=𝔼^∞₀2𝑥^∗𝑇𝑄𝑥^𝑑−𝑥^∗+2𝑧^∗𝑇𝑆𝑧^𝑑−𝑧^∗−2𝛾²𝑣^∗𝑇𝑑−𝑑^∗𝑑𝑡. (29) Substituting 2𝑥₀^∗𝑇𝐻(𝑥^𝑑₀−𝑥^∗₀) into (28), we get 𝐽₂𝑢^∗,𝑑^∗−𝐽₂𝑢^∗,𝑑=𝔼^∞₀−𝑥^∗−𝑥^𝑑𝑇𝑄𝑥^∗−𝑥^𝑑−𝑧^∗−𝑧^𝑑𝑇𝑆𝑧^∗−𝑧^𝑑+𝛾²𝑑^∗−𝑑^𝑇𝑑^∗−𝑥−𝑑𝑑𝑡_∗0−𝑥^𝑑₀^𝑇𝐻𝑥^∗₀−𝑥^𝑑₀. (30) From (27), then 𝐽₂𝑢^∗,𝑑^∗−𝐽₂𝑢^∗,𝑑≥0. (31) So 𝑑^∗ is the worst case disturbance. Moreover, for 𝑥_∞=0, the FBSDE (25) admits a unique solution (𝑝^∗,𝑥^∗,𝑧^∗)=(0,0,0); then 𝐽₂𝑢^∗,𝑑≤𝐽₂𝑢^∗,𝑑^∗=0. (32)
Hence, 𝑢^∗ restrains the exogenous disturbance. In the following, we will show that 𝑢^∗ also minimizes that cost function when the worst case disturbance 𝑑^∗ is implemented into system (8).
For any 𝑢∈𝐿²_ℱ(ℝ^𝑛𝑢), let 𝑥^𝑢_𝑡 be the trajectory of the system (8) corresponding to (𝑢,𝑑^∗). Let us first consider 𝐽₁𝑢^∗,𝑑^∗−𝐽₁𝑢,𝑑^∗=𝐼₁, (33) where 𝐼₁=−𝔼^∞₀𝑥^∗𝑇𝑄(𝑡)𝑥^∗−𝑥^𝑢𝑇𝑄𝑥^𝑢+𝑧^∗𝑇𝑆(𝑡)𝑧^∗−𝑧^𝑢𝑇𝑆𝑧^𝑢+𝑢^∗𝑇𝑢^∗−𝑢^𝑇𝑢𝑑𝑡+𝑥₀^∗𝑇𝐻𝑥^∗₀−𝑥₀^𝑢𝑇𝐻𝑥^𝑢₀=𝔼^∞₀𝑥^∗−𝑥^𝑢𝑇𝑄𝑥^∗−𝑥^𝑢+𝑧^∗−𝑧^𝑢𝑇𝑆𝑧^∗−𝑧^𝑢+𝑢^∗−𝑢^𝑇𝑢^∗−𝑢+2𝑥^∗𝑇𝑄(𝑡)𝑥^𝑢−𝑥^∗+2𝑦^∗𝑇𝑆(𝑡)𝑧^𝑢−𝑧^∗+2𝑢^∗𝑇𝑢−𝑢^∗+𝑥𝑑𝑡_∗0−𝑥^𝑢₀^𝑇𝐻𝑥^∗₀−𝑥^𝑢₀+2𝑥₀^∗𝑇𝐻𝑥^𝑢₀−𝑥^∗₀. (34) From 𝑝^∗₀=2𝐻𝑥^∗₀, we use Itô’s formula to 𝑝_𝑡^∗𝑇(𝑥^𝑢_𝑡−𝑥^∗_𝑡) and get 2𝑥₀^∗𝑇𝐻𝑥^𝑢₀−𝑥^∗₀=−𝔼^∞₀2𝑥^∗𝑇𝑄(𝑡)𝑥^𝑢−𝑥^∗+2𝑧^∗𝑇𝑆𝑧^𝑢−𝑧^∗+2𝑢^∗𝑇𝑢−𝑢^∗𝑑𝑡. (35) Then because of 𝑄, 𝑆, and 𝐻 being nonnegative, we have 𝐽₁𝑢^∗,𝑣^∗−𝐽₁𝑢,𝑣^∗=𝐼₁≥0. (36) Therefore, 𝑑^∗ minimizes the cost function when the worst case disturbance 𝑑^∗ is implemented into system (8).
So, (𝑢^∗,𝑑^∗)=(𝐵^𝑇𝑝^∗/2,−𝐶^𝑇𝑝^∗/2𝛾²) is a solution of the backward stochastic 𝐻₂/𝐻_∞ control problem.
We are now in a position to prove the uniqueness of the solution. Assume that the backward stochastic 𝐻₂/𝐻_∞ control has a solution (𝑢¹,𝑑¹), (𝑥¹,𝑧¹) is the corresponding solution for (8), and 𝑝¹ is the solution of the following BSDE: 𝑑𝑝¹=2𝑄𝑥¹−𝐴^𝑇𝑝¹𝑑𝑡+2𝑆𝑧¹−𝐷^𝑇𝑝¹(𝑡),𝑝𝑑𝐵₁₀=2𝐻𝑥¹₀. (37) Implementing 𝑑¹, having inf_{𝑢∈𝐿2ℱ(𝑅𝑛𝑢)}𝐽₁𝑢,𝑑¹, (38) is a standard LQ optimal control problem. By uniqueness, 𝑢¹=𝐵^𝑇𝑝¹/2.
Let 𝑥 be the trajectory corresponding to (𝑢¹,𝑑)=(𝑢¹,−𝐶^𝑇𝑝¹/𝛾²); then 0≥𝐽₂𝑢¹,𝑑−𝐽₂𝑢¹,𝑑¹=𝔼^∞₀𝑥^𝑇𝑄𝑥−𝑥^1𝑇𝑄𝑥¹+𝑧^𝑇𝑆𝑧−𝑧^1𝑇𝑆𝑧¹−𝛾²𝑑^𝑇𝑑+𝛾²𝑑^1𝑇𝑑¹𝑑𝑡+𝑥^𝑇₀𝐻𝑥₀−𝑥₀^1𝑇𝐻𝑥¹₀=𝔼^∞₀𝑥¹−𝑥^𝑇𝑄𝑥¹−𝑥−2𝑥^1𝑇𝑄𝑥¹−𝑥+𝑧¹−𝑧^𝑇𝑆𝑧¹−𝑧−2𝑧^1𝑇𝑆𝑧¹−𝑧+𝛾²𝑑^1𝑇𝑑¹−𝛾²𝑑^𝑇𝑑+𝑥𝑑𝑡₁₀−𝑥₀^𝑇𝐻𝑥¹₀−𝑥₀−2𝑥₀^1𝑇𝐻𝑥¹₀−𝑥₀. (39) Applying Itô’s formula to 𝑝^1𝑇(𝑥¹−𝑥), 2𝑥₀^1𝑇𝐻𝑥¹₀−𝑥₀=𝔼^∞₀𝑑𝑝^1𝑇𝑥¹−𝑥=𝔼^∞₀2𝑥^1𝑇𝑄𝑥¹+𝐶−𝑥_𝑇𝑧^1𝑇𝑑¹−𝑑+2𝑧^1𝑇𝑆𝑧¹−𝑧𝑑𝑡=𝔼^∞₀2𝑥^1𝑇𝑄𝑥¹−𝑥+2𝑧^1𝑇𝑆𝑧¹−𝑧−2𝛾²𝑑^𝑇𝑑¹−𝑑𝑑𝑡. (40) Substituting −2𝑥₀^1𝑇𝐻(𝑥¹₀−𝑥₀) into (39), then 0≥𝐽₂𝛾,𝑥₀;𝑢¹,𝑑−𝐽₂𝛾,𝑥₀;𝑢¹,𝑑¹=𝔼^∞₀𝑥¹−𝑥^𝑇𝑄𝑥¹+𝑧−𝑥₁−𝑧^𝑇𝑆𝑧¹−𝑧+𝛾²𝑧−𝑧^1𝑇𝑧−𝑧¹𝑑𝑡. (41) Because of 𝑄, 𝑆, and 𝑀 being nonnegative, we get 𝑑¹=𝑣=𝐶^𝑇𝑧¹/𝛾².
Therefore, (𝑢¹,𝑑¹)=(𝑢^∗,𝑑^∗).
(2) The Necessary Condition. Here we assume that a solution exists; then from the uniqueness of the solution, we get that (𝑢^∗,𝑑^∗) is the unique solution and we will show that system (18) has 𝕃₂-gain less than or equal to 𝛾.
For 𝑥_𝑇=0, the FBSDE (25) has a unique solution (𝑝^∗,𝑥^∗,𝑧^∗)=(0,0,0); then (𝑢^∗,𝑑^∗)=(0,0) and 𝐽₂𝑢^∗,𝑑≤𝐽₂𝑢^∗,𝑑^∗=0,∀𝑑∈𝐿²_ℱℛ^𝑛𝑑. (42) Therefore, system (18) has 𝕃₂-gain less than or equal to 𝛾.

5. The Linear Feedback Solution

The main result of this section gives the equivalent linear feedback solution. For the purpose of this section the coefficients 𝐴_𝑡, 𝐵_𝑡, 𝐶_𝑡, 𝐷_𝑡, 𝐸_𝑡, 𝑄_𝑡, and 𝑆_𝑡 are assumed deterministic functions; (18) has 𝕃₂-gain less than or equal to 𝛾.

Let (𝑝,𝑥,𝑧) be the solution of (25); we first give the relations between 𝑝, 𝑥, and 𝑧 using the undetermined coefficients method. Now, we introduce the following generalized matrix-valued Riccati equation (the variables 𝑡 are suppressed): 𝐾̇−𝐴𝐾−𝐾𝐴^𝑇+2𝐾𝑄𝐾−𝐵𝐵^𝑇2+𝐶𝐶^𝑇2𝛾²+𝐷(𝐼−2𝐾𝑆)⁻¹𝐾𝐷^𝑇=0,𝐾(∞)=0. (43) Similar to the line developed by Lim and Zhou [6], we can prove that (43) admits a unique solution 𝐾(⋅). Letting 𝐾(⋅) be the solution to (43), we define the following equations: 𝑑ℎ=𝐴ℎ−2𝐾𝑄ℎ+𝐷(𝐼−2𝐾𝑆)⁻¹𝜂(𝑡),ℎ(∞)𝑑𝑡+𝜂𝑑𝐵=𝜉. (44) Equation (44) is a linear BSDE and admits a unique solution (ℎ,𝜂).

Theorem 7. Suppose that (𝑝(⋅),𝑥(⋅),𝑧(⋅)), 𝐾(⋅), and (ℎ(⋅),𝜂(⋅)) are the solutions of (25), (43), and (44), respectively; then the following relations are satisfied: 𝑥(𝑡)(𝑡)𝑝(𝑡)(𝑡),𝑧(𝑡)=()=𝐾+ℎ𝐼−2𝐾(𝑡)𝑆(𝑡)⁻¹𝜂(𝑡)−𝐾(𝑡)𝐷(𝑡)^𝑇𝑝(𝑡),𝑥(0)=(𝐼−2𝐾(0)𝐻)⁻¹ℎ(0). (45)

Proof. Let 𝑥(𝑡)=𝐾(𝑡)𝑝(𝑡)+ℎ(𝑡). We apply Itô’s formula to 𝑥(𝑡), 𝐾(𝑡)𝑝(𝑡)+ℎ(𝑡), respectively, and it is easy to check that 𝐾(𝑡) and ℎ(𝑡) satisfy (43) and (44), respectively.

From Theorem 7, we know that 𝑥(⋅) can be written to the functions of 𝐾(⋅), 𝑝(⋅), and ℎ(⋅). Now we would like to derive the feedback solution using the undetermined coefficients method. First, we introduce the generalized matrix-valued Riccati equation and a linear SDE: Σ̇+Σ𝐴+𝐴^𝑇Σ+Σ𝐵𝐵^𝑇2−𝐶𝐶^𝑇2𝛾²−𝐷(𝐼−2𝐾𝑆)⁻¹𝐾𝐷^𝑇Σ(0)×Σ−2𝑄=0,=2𝐻,=𝑑𝑟−𝐴^𝑇𝑟−Σ𝐵𝐵^𝑇𝑟2+Σ𝐶𝐶^𝑇𝑟2𝛾²+Σ𝐷(𝐼−2𝐾𝑆)⁻¹𝐾𝐷^𝑇𝑟−Σ𝐷(𝐼−2𝐾𝑆)⁻¹𝜂+)𝑑𝑡(2𝑆−Σ)(𝐼−2𝐾𝑆⁻¹𝜂−𝐾𝐷^𝑇(𝐼−Σ𝐾)⁻¹(Σℎ+𝑟)−𝐷^𝑇(𝐼−Σ𝐾)⁻¹((𝑡),𝑟(0)Σℎ+𝑟)𝑑𝐵=0. (46)(47) Similar to the line developed by Lim and Zhou [6], we can prove that (46) admits a unique solution Σ(⋅). Equation (47) is a linear SDE and has a unique solution 𝑟(⋅).

Theorem 8. The backward stochastic 𝐻₂/𝐻_∞ control problem has a feedback solution (𝑢^∗,𝑣^∗), 𝑢^∗=𝐵^𝑇(Σ𝑥+𝑟)2,𝑣^∗𝐶=−^𝑇(Σ𝑥+𝑟)2𝛾². (48)

Proof. Let 𝑝(𝑡)=Σ(𝑡)𝑥(𝑡)+𝑟(𝑡). We apply Itô’s formula to 𝑝(𝑡) and Σ(𝑡)𝑥(𝑡)+𝑟(𝑡), respectively, and it is easy to check that Σ(𝑡) and 𝑟(𝑡) satisfy (46) and (47).

Remark 9. From Theorem 8, we see that the solution involves an additional random nonhomogeneous term 𝑟(⋅). This addition disqualifies (48) from a feedback control of the current state, contrary to the deterministic or stochastic forward 𝐻₂/𝐻_∞ (see [4, 5]) cases. The reason is because 𝑟(⋅) depends on (ℎ(⋅),𝜂(⋅)), which in turn depends on 𝜉, the terminal condition of part of the state variable, 𝑥(⋅). This is one of the major distinctive features of the backward stochastic 𝐻₂/𝐻_∞ problem.
Finally, it is important to recognize that the expressions for the backward stochastic 𝐻₂/𝐻_∞ control, as presented in Theorems 6 and 8, are equivalent expressions of the same process; that is, this does not contradict the uniqueness of the solution.
We present an example to illustrate the above theoretical results as follows.

Example 10. Consider the backward stochastic 𝐻₂/𝐻_∞ control problem of the following one-dimensional system: 𝑥(𝑡)=𝜉−^∞_𝑡2𝑒^−𝑠𝑥(𝑠)+√4𝑒^−𝑠(𝑠)+√+2𝑢𝑒^−𝑠+1𝑑(𝑠)+2𝑒^−𝑠/2𝑧(𝑠−)𝑑𝑠_∞𝑡𝑧(𝑠)𝑑𝑊(𝑠), (49) with controlled output energy ‖𝑍‖²₂=𝔼^∞₀𝑒^−𝑡𝑥²_𝑡2+𝑢²_𝑡𝑑𝑡. (50) If we take 𝛾=√2/2, then the Riccati equation (43) specializes to 𝐾(𝑡)=^∞_𝑡𝑒^−𝑠𝐾(𝑠)²−1𝑑𝑠. (51) Solving it yields 𝐾(𝑡)=(1−𝑒^2𝑒−𝑡)/(1+𝑒^2𝑒−𝑡). Equation (44) specializes to ℎ(𝑡)=𝜉−^∞_𝑡2𝑒^−𝑠−𝑒^−𝑠1−𝑒^2𝑒−𝑠1+𝑒^2𝑒−𝑠ℎ(𝑠)+2𝑒^−𝑠/2𝜂(𝑡−)𝑑𝑡_∞𝑡𝜂(𝑡)𝑑𝑊(𝑡). (52) Then, from Theorem 7, we get a unique solution 𝑢^∗,𝑑^∗=2𝑒^−𝑡+12⋅1+𝑒^2𝑒−𝑡1−𝑒^2𝑒−𝑡⋅[𝑥(𝑡)(𝑡,−√−ℎ)]𝑒^−𝑡+1⋅1+𝑒^2𝑒−𝑡1−𝑒^2𝑒−𝑡⋅[𝑥(𝑡)−ℎ(𝑡)], (53) of the backward 𝐻₂/𝐻_∞ control problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation (11221061 and 61174092), 111 project (B12023), the National Science Fund for Distinguished Young Scholars of China (no. 11125102), the Science and Technology Project of Shandong Province (2013GRC32201), and the Doctoral Foundation of University of Jinan (XBS1213).