Infinite Horizon Optimal Control of Stochastic Delay Evolution Equations in Hilbert Spaces

Abstract

The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamics is governed by a stochastic delay evolution equation in Hilbert spaces. The existence and uniqueness of the optimal control are obtained by means of associated infinite horizon backward stochastic differential equations without assuming the Gâteaux differentiability of the drift coefficient and the diffusion coefficient. An optimal control problem of stochastic delay partial differential equations is also given as an example to illustrate our results.

1. Introduction

In this paper, we consider a controlled stochastic evolution equation of the following form: 𝑑𝑋^𝑢(𝑠)=𝐴𝑋^𝑢(𝑠)𝑑𝑠+𝐹𝑠,𝑋^𝑢_𝑠𝑑𝑠+𝐺𝑠,𝑋^𝑢_𝑠𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑠+𝐺𝑠,𝑋^𝑢_𝑠(𝑠),𝑋𝑑𝑊𝑠≥𝑡,_𝑢𝑡=𝑥, (1) where 𝑋^𝑢_𝑠(𝑙)=𝑋^𝑢(𝑠+𝑙),𝑙∈[−𝜏,0],𝑥∈𝐶([−𝜏,0],𝐻). (2)𝑢 is the control process in a measurable space   (𝑈,𝒰),and 𝑊 is a cylindrical Wiener process in a Hilbert space Ξ.𝐴 is the generator of a strongly continuous semigroup of bounded linear operator in another Hilbert space 𝐻, and the coefficients 𝐹 and 𝐺, defined on [0,∞)×𝐶([−𝜏,0],𝐻), are assumed to satisfy Lipschitz conditions with respect to appropriate norms. We introduce the cost function 𝐽(𝑢)=𝐸^∞_𝑡𝑒^−𝜆𝑠𝑔𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑠. (3) Here, 𝑔 is a given real function, 𝜆 is large enough, and the control problem is understood in the weak sense. We wish to minimize the cost function over all admissible controls.

The particular form of the control system is essential for our results, but it covers numerous interesting cases. For example, in the particular cases 𝑈=𝐻 and 𝑅(𝑡,𝑥,𝑢)=𝑢, the term 𝑢(𝑡)𝑑𝑡+𝑑𝑊(𝑡) in the state equation can be considered as a control affected by noise.

The stochastic optimal control problem was considered in 1977 by Bismut [1]. The optimal control problem for stochastic partial differential equations in the framework of a compact control state space has been studied in [2–5]. Buckdahn and Raşcanu [6] considered an optimal control problem for a semilinear parabolic stochastic differential equation with a nonlinear diffusion coefficient, and the existence of a quasioptimal (nonrelaxed) control is showed without assuming convexity of the coefficients. In [7–11], the authors provided a direct (classical or mild) solution of the Hamilton-Jacobi-Bellman equation for the value function, which is then used to prove that the optimal control is related to the corresponding optimal trajectory by a feedback law. In Gozzi [10, 11], the existence and uniqueness of a mild solution of the associated Hamilton-Jacobi-Bellman equation are proved, when the diffusion term only satisfies weak nondegeneracy conditions. The proofs are based on the corresponding regularity properties of the transition semigroup of the associated Ornstein-Uhlenbeck process.

The main tools for the control problem are techniques from the theory of backward stochastic differential equations (BSDEs) in the sense of Pardoux and Peng, first considered in the nonlinear case in [12]; see [13, 14] as general references. BSDEs have been successfully applied to control problems; see, for example, [15, 16] and we also refer the reader to [17–20]. Fuhrman and Tessiture [19] considered the optimal control problem for stochastic differential equation in the strong form, assuming Lipschitz conditions and allowing degeneracy of the diffusion coefficient, under some structural constraint on the state equation. Existence of an optimal control for stochastic systems in infinite dimensional spaces also has been obtained in [21–27]. In [21], Fuhrman and Tessitore showed the regularity with respect to parameters and the regularity in the Malliavin spaces for the solution of the backward-forward system and defined the feedback law by Malliavin calculus. Finally, the optimal control is obtained by the feedback. Appealing to the Malliavin calculus, compared with Fuhrman et al. [23], the existence of optimal control for stochastic differential equations with delay is proved by the feedback law. Fuhrman and Tessiture [24] dealt with an infinite horizon optimal control problem for the stochastic evolution equation in Hilbert space, and the optimal control is showed by means of infinite horizon backward stochastic differential equation in infinite dimensional spaces and Malliavin calculus. In Masiero [25], the infinite horizon optimal control problem for stochastic evolution equation is also studied by means of the Hamilton-Jacobi-Bellman equation. In Fuhrman [26], a class of optimal control problems governed by stochastic evolution equations in Hilbert spaces which includes state constraints is considered, and the optimal control is obtained by the Fleming logarithmic transformation. Masiero [27] studied stochastic evolution equations evolving in a Banach space where 𝐺 is a constant and characterized the optimal control via a feedback law by avoiding use of Malliavin calculus. Since there is a lack of regularity of 𝐹 and 𝐺, Malliavin calculus is not available in this case; the method in [27] also cannot be used as 𝐺 is not a constant, but we can prove a theorem similar to [26, Proposition 3.2], which will be used to define the feedback law.

In the present paper, we study the infinite horizon optimal control problem for stochastic delay evolution equations in Hilbert spaces, and by using Theorem 10, the optimal control is obtained. Since we do not relate the optimal feedback law with the gradient of the value function and do not consider the associated Hamilton-Jacobi-Bellman equation, we can drop the Gâteaux differentiability of the drift term and the diffusion term.

The plan of the paper is as follows. In the next section, some notations are fixed, and the stochastic delay evolution equations are considered with an infinite horizon; in particular, continuous dependence on initial value (𝑡,𝑥) is proved. In Section 3, we give the proof of Theorem 10, which is the key of many subsequent results. The addressed optimal control problem is considered, and the fundamental relation between the optimal control problem and BSDEs is established in Section 4. Section 5 is devoted to proving the existence and uniqueness of optimal control in the weak sense. Finally, an application is given in Section 6.

2. Preliminaries

We list some notations that are used in this paper. We use the symbol |⋅| to denote the norm in a Banach space 𝐹, with a subscript if necessary. Let Ξ, 𝐻, and 𝐾 denote real separable Hilbert spaces, with scalar products (⋅,⋅)_Ξ, (⋅,⋅)_𝐻, and (⋅,⋅)_𝐾, respectively. For fixed 𝜏>0, 𝒞=𝐶([−𝜏,0],𝐻) denotes the space of continuous functions from [−𝜏,0] to 𝐻, endowed with the usual norm |𝑓|_𝐶=sup_{𝜃∈[−𝜏,0]}|𝑓(𝜃)|_𝐻. Let Ξ^∗ denote the dual space of Ξ, with scalar product (⋅,⋅)_Ξ^∗, and let 𝐿(Ξ,𝐻) denote the space of all bounded linear operators from Ξ into 𝐻; the subspace of Hilbert-Schmidt operators, with the Hilbert-Schmidt norm, is denoted by 𝐿₂(Ξ,𝐻).

Let (Ω,ℱ,𝑃) be a complete space with a filtration {ℱ_𝑡}_𝑡≥0 which satisfies the usual condition. By a cylindrical Wiener process with values in a Hilbert space Ξ, defined on (Ω,ℱ,𝑃), we mean a family {𝑊(𝑡),𝑡≥0} of linear mappings Ξ→𝐿²(Ω) such that for every 𝜉,𝜂∈Ξ, {𝑊(𝑡)𝜉,𝑡≥0} is a real Wiener process and 𝐸(𝑊(𝑡)𝜉⋅𝑊(𝑡)𝜂)=(𝜉,𝜂)_Ξ. In the following, {𝑊(𝑡),𝑡≥0} is a cylindrical Wiener process adapted to the filtration {ℱ_𝑡}_𝑡≥0.

In this section and the next section, {ℱ_𝑡}_𝑡≥0 will denote the natural filtration of 𝑊, augmented with the family 𝒩 of 𝑃-null of ℱ. The filtration {ℱ_𝑡,𝑡≥0} satisfies the usual conditions. For [𝑎,𝑏],[𝑎,∞)⊂[0,∞), we also use the following notations: ℱ_[𝑎,𝑏](𝑊(𝑠)(𝑎)[ℱ=𝜎−𝑊∶𝑠∈𝑎,𝑏])∨𝒩,_[𝑎,∞)=𝜎(𝑊(𝑠)−𝑊(𝑎)∶𝑠∈[a,∞))∨𝒩. (4) By 𝒫 we denote the predictable 𝜎-algebra, and by ℬ(Λ) we denote the Borel 𝜎-algebra of any topological space Λ.

Similar to [24], we define several classes of stochastic processes with values in a Banach space 𝐹 as follows. (i)𝐿²_𝒫(Ω×[𝑡,∞);𝐹) denotes the space of equivalence classes of processes 𝑌∈𝐿²(Ω×[𝑡,∞);𝐹), admitting a predictable version. 𝐿²_𝒫(Ω×[𝑡,∞);𝐹) is endowed with the norm |𝑌|²=𝐸^∞_𝑡|𝑌(𝑠)|²𝑑𝑠. (5)(ii)𝐿^𝑝_𝒫(Ω;𝐿^𝑞_𝛽([𝑡,∞);𝐹)), defined for 𝛽∈𝑅 and 𝑝,𝑞∈[1,∞), denotes the space of equivalence classes of processes {𝑌(𝑠),𝑠≥𝑡}, with values in 𝐹, such that the norm |𝑌|^𝑝=𝐸^∞_𝑡𝑒^𝑞𝛽𝑠|𝑌(𝑠)|^𝑞𝑑𝑠^𝑝/𝑞 (6) is finite and 𝑌 admits a predictable version. (iii)𝒦^𝑝_𝛽(𝑡) denotes the space 𝐿^𝑝_𝒫(Ω;𝐿²_𝛽([𝑡,∞);𝐹))×𝐿^𝑝_𝒫(Ω;𝐿²_𝛽([𝑡,∞);𝐿₂(Ξ,𝐹))). The norm of an element (𝑌,𝑍)∈𝒦^𝑝_𝛽 is |(𝑌,𝑍)|=|𝑌|+|𝑍|. Here, 𝐹 is a Hilbert space.(iv)𝐿^𝑝_𝒫(Ω;𝐶([𝑡,𝑇];𝐹)), defined for 𝑇>𝑡≥0 and 𝑝∈[1,∞), denotes the space of predictable processes {𝑌(𝑠),𝑠∈[𝑡,𝑇]} with continuous paths in 𝐹, such that the norm |𝑌|^𝑝=𝐸sup_{[]𝑠∈𝑡,𝑇}|𝑌(𝑠)|^𝑝 (7)is finite. Elements of 𝐿^𝑝_𝒫(Ω;𝐶([𝑡,𝑇];𝐹)) are identified up to indistinguishability. (v)𝐿^𝑞_𝒫(Ω;𝐶_𝜂([𝑡,∞);𝐹)), defined for 𝜂∈𝑅 and 𝑞∈[1,∞), denotes the space of predictable processes {𝑌(𝑠),𝑠≥𝑡} with continuous paths in 𝐹, such that the norm |𝑌|^𝑞=𝐸sup_𝑠≥𝑡𝑒^𝜂𝑞𝑠|𝑌(𝑠)|^𝑞 (8) is finite. Elements of 𝐿^𝑞_𝒫(Ω;𝐶_𝜂(𝐹)) are identified up to indistinguishability. (vi)Finally, for 𝜂∈𝑅 and 𝑞∈[1,∞), we defined ℋ^q_𝜂(𝑡) as the space 𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂([𝑡,∞);𝐹))∩𝐿^𝑞_𝒫(Ω;𝐶_𝜂([𝑡,∞);𝐹)), endowed with the norm |𝑌|_ℋ^𝑞_𝜂=|𝑌|_𝐿^𝑞_𝒫(Ω;𝐿^𝑞_{𝜂([𝑡,∞);𝐹))}+|𝑌|_𝐿^𝑞_{𝒫(Ω;𝐶𝜂([𝑡,∞);𝐹))}. (9)For simplicity, we denote 𝐿^𝑝_𝒫(Ω;𝐿^𝑞_𝛽([0,∞);𝐹)), 𝐿^𝑞_𝒫(Ω;𝐶_𝜂([0,∞);𝐹)), ℋ^𝑞_𝜂(0), and 𝒦^𝑝_𝛽(0) by 𝐿^𝑝_𝒫(Ω;𝐿^𝑞_𝛽(𝐹)), 𝐿^𝑞_𝒫(Ω;𝐶_𝜂(𝐹)), ℋ^𝑞_𝜂, and 𝒦^𝑝_𝛽, respectively.

Now, for every fixed 𝑡≥0, we consider the following stochastic delay evolution equation: 𝑑𝑋(𝑠)=𝐴𝑋(𝑠)𝑑𝑠+𝐹𝑡,𝑋_𝑠𝑑𝑠+𝐺𝑠,𝑋_𝑠(𝑠),[),𝑋𝑑𝑊𝑠∈𝑡,∞_𝑡=𝑥∈𝒞. (10) We make the following assumptions.

Hypothesis 1. (i) The operator 𝐴 is the generator of a strongly continuous semigroup {𝑒^𝑡𝐴,𝑡≥0} of bounded linear operators in the Hilbert space 𝐻. We denote by 𝑀 and 𝜔 two constants such that |𝑒^𝑡𝐴|≤𝑀𝑒^𝜔𝑡, for 𝑡≥0.
(ii) The mapping 𝐹: [0,∞)×𝒞→𝐻 is measurable and satisfies, for some constant 𝐿>0 and 0≤𝜃<1, 𝑒^𝑠𝐴𝐹(𝑡,𝑥)≤𝐿𝑒^𝜔𝑠𝑠^−𝜃1+|𝑥|_𝐶,𝑒^𝑠𝐴𝐹(𝑡,𝑥)−𝑒^𝑠𝐴𝐹𝑡,𝑦≤𝐿𝑒^𝜔𝑠𝑠^−𝜃𝑥−𝑦_𝐶,𝑠>0,𝑡∈[0,+∞),𝑥,𝑦∈𝒞. (11)
(iii) 𝐺 is a mapping [0,∞)×𝒞→𝐿(Ξ,𝐻) such that for every 𝑣∈Ξ, the map 𝐺𝑣: [0,∞)×𝒞→𝐻 is measurable, 𝑒^𝑠𝐴𝐺(𝑡,𝑥)∈𝐿₂(Ξ,𝐻) for every s>0, 𝑡∈[0,∞) and 𝑥∈𝒞, and 𝑒^𝑠𝐴𝐺(𝑡,𝑥)_{𝐿2(Ξ,𝐻)}≤𝐿𝑒^𝜔𝑠𝑠^−𝛾1+|𝑥|_𝐶,𝑒^𝑠𝐴𝐺(𝑡,𝑥)−𝑒^𝑠𝐴𝐺𝑡,𝑦_{𝐿2(Ξ,𝐻)}≤𝐿𝑒^𝜔𝑠𝑠^−𝛾𝑥−𝑦_𝐶,𝑠>0,𝑡∈[0,+∞),𝑥,𝑦∈𝒞, (12) for some constants 𝐿>0 and 𝛾∈[0,1/2).
We say that 𝑋 is a mild solution of (10) if it is a continuous, {ℱ_𝑡}_𝑡≥0-predictable process with values in 𝐻, and it satisfies 𝑃-a.s., 𝑋(𝑠)=𝑒^{(𝑠−𝑡)𝐴}𝑥(0)+^𝑠_𝑡𝑒^{(𝑠−𝜎)𝐴}𝐹𝜎,𝑋_𝜎+𝑑𝜎_𝑠𝑡𝑒^{(𝑠−𝜎)𝐴}𝐺𝜎,𝑋_𝜎(𝜎)[),𝑋𝑑𝑊,𝑠∈𝑡,∞_𝑡=𝑥∈𝒞. (13) To stress dependence on initial data, we denote the solution by 𝑋(𝑠,𝑡,𝑥). Note that 𝑋(𝑠,𝑡,𝑥) is ℱ_[𝑡,𝑠] measurable, hence, independent of ℱ_𝑡.
We first recall a well-known result on solvability of (10) on bounded interval.

Theorem 1. Assume that Hypothesis 1 holds. Then, for all 𝑞∈[2,∞) and 𝑇>0, there exists a unique process 𝑋∈𝐿^𝑞_𝒫(Ω,𝐶([𝑡,𝑇];𝐻)) as mild solution of (10). Moreover, 𝐸sup_{𝑠∈[𝑡,𝑇]}|𝑋(𝑠)|^𝑞≤𝐶1+|𝑥|_𝐶^𝑞, (14)

for some constant C depending only on 𝑞,𝛾,𝜃,T,𝜏,L,𝜔, and M.

By Theorem 1 and the arbitrariness of  𝑇 in its statement, the solution is defined for every 𝑠≥𝑡. We have the following result.

Theorem 2. Assume that Hypothesis 1 holds and the process 𝑋(⋅,𝑡,𝑥) is mild solution of (10) with initial value (𝑡,𝑥)∈[0,∞)×𝒞. Then, for every 𝑞∈[1,∞), there exists a constant 𝜂(𝑞) such that the process 𝑋_⋅(𝑡,𝑥)∈ℋ^𝑞_𝜂(𝑞)(𝑡). Moreover, for a suitable constant 𝐶>0, one has 𝐸sup_s≥t𝑒^{𝜂(𝑞)𝑞𝑠}𝑋_𝑠^𝑞_𝐶+𝐸^∞_𝑡𝑒^{𝜂(𝑞)𝑞𝑠}𝑋_𝑠^𝑞_𝐶𝑑𝑠≤𝐶1+|𝑥|_𝐶^𝑞, (15) with the constant 𝜂(𝑞) depending only on 𝑞,𝛾,𝜃,𝜏,𝐿,𝜔, and 𝑀.

Proof. We define a mapping Φ from ℋ^𝑞_𝜂(𝑡)×[0,∞)×𝒞 to ℋ^𝑞_𝜂(𝑡) by the formula Φ𝑋_⋅,𝑡,𝑥_𝑠(𝑙)=𝑒^{(𝑠+𝑙−𝑡)𝐴}𝑥(0)+_𝑡^𝑠+𝑙𝑒^{(𝑠+𝑙−𝜎)𝐴}𝐹𝜎,𝑋_𝜎+𝑑𝜎_{𝑡𝑠+𝑙}𝑒^{(𝑠+𝑙−𝜎)𝐴}𝐺𝜎,𝑋_𝜎(𝜎),[)[]Φ𝑋𝑑𝑊𝑠∈𝑡,∞,𝑙∈−𝜏,0,𝑠+𝑙≥𝑡,_⋅,𝑡,𝑥_𝑠(𝑙)(),[)[]=𝑥𝑠+𝑙−𝑡𝑠∈𝑡,∞,𝑙∈−𝜏,0,𝑠+𝑙<𝑡. (16) We are going to show that, provided 𝜂 is suitably chosen, Φ(⋅,𝑡,𝑥) is well defined and that it is a contraction in ℋ^𝑞_𝜂(𝑡); that is, there exists 𝑐<1 such that Φ𝑋¹_⋅,𝑡,𝑥−Φ𝑋¹_⋅,𝑡,𝑥_{ℋ𝑞𝜂(𝑡)}≤𝑐𝑋¹_⋅−𝑋²_{⋅ℋ𝑞𝜂(𝑡)},𝑋¹_⋅,𝑋²_⋅∈ℋ^𝑞_𝜂(𝑡). (17) For simplicity, we set 𝑡=0, and we treat only the case 𝐹=0, the general case being handled in a similar way. We will use the so called factorization method; see [28, Theorem 5.2.5]. Let us take 𝑞>1 and 𝛼∈(0,1) such that 1/𝑞<𝛼<(1/2)−𝛾,andlet𝑐_𝛼⁻¹=∫^s_𝜎(𝑠−𝑟)^𝛼−1(𝑟−𝜎)^−𝛼𝑑𝑟.
By the stochastic Fubini theorem, Φ𝑋_⋅,0,𝑥_𝑠(𝑙)=𝑒^{(𝑠+𝑙)𝐴}𝑥(0)+𝑐_𝛼0^𝑠+𝑙_𝜎^𝑠+𝑙(𝑠+𝑙−𝑟)^𝛼−1(𝑟−𝜎)^−𝛼×𝑒^{(𝑠+𝑙−𝑟)𝐴}𝑒^{(𝑟−𝜎)𝐴}𝑑𝑟𝐺𝜎,𝑋_𝜎𝑑𝑊(𝜎)=𝑒^{(𝑠+𝑙)𝐴}𝑥(0)+Φ^′𝑋_𝑠(𝑙),[)[]Φ𝑋𝑠∈0,∞,𝑙∈−𝜏,0,𝑠+𝑙≥0,_⋅,0,𝑥_𝑠(𝑙)(),[)[]=𝑥𝑠+𝑙𝑠∈0,∞,𝑙∈−𝜏,0,𝑠+𝑙<0, (18) where Φ^′𝑋_⋅𝑠(𝑙)=𝑐_𝛼0^𝑠+𝑙(𝑠+𝑙−𝑟)^𝛼−1𝑒^{(𝑠+𝑙−𝑟)𝐴}𝑌(𝑟)𝑌(𝑟)=𝑑𝑟,_𝑟0(𝑟−𝜎)^−𝛼𝑒^{(𝑟−𝜎)𝐴}𝐺𝜎,𝑋_𝜎𝑑𝑊(𝜎). (19) Since sup_{−𝜏≤𝑙≤0}|𝑒^{(𝑠+𝑙)𝐴}𝑥(0)|≤𝑀𝑒^𝜔𝑠|𝑥|_𝐶, the process 𝑒^{(𝑠+⋅)𝐴}𝑥(0), 𝑠≥0, belongs to ℋ^𝑞_𝜂 provided 𝜔+𝜂<0. Next, we estimate Φ^′(𝑋_⋅), where Φ^′𝑋_⋅𝑠(𝑙)≤𝑐_𝛼0^𝑠+𝑙(𝑠+𝑙−𝑟)^𝛼−1𝑀𝑒^{(𝑠+𝑙−𝑟)𝜔}|𝑌(𝑟)|𝑑𝑟, (20) setting 𝑞^′=𝑞/(𝑞−1), so that 𝑒^𝑞𝜂𝑠Φ^′𝑋_⋅𝑠^𝑞≤𝑐^𝑞_𝛼𝑀^𝑞sup_{−𝜏≤𝑙≤0}𝑒^𝑞𝜂𝑠₀^𝑠+𝑙(𝑠+𝑙−𝑟)^𝛼−1𝑒^{𝜔(𝑠+𝑙−𝑟)}|𝑌(𝑟)|𝑑𝑟^𝑞≤𝑐^𝑞_𝛼𝑀^𝑞sup_{−𝜏≤𝑙≤00}^𝑠+𝑙(𝑠+𝑙−𝑟)^𝛼−1𝑒^{((𝜔+𝜂)/𝑞′)(𝑠+𝑙−𝑟)}×𝑒^{((𝜔+𝜂)/𝑞)(𝑠−𝑟)}𝑒^𝜂𝑟|𝑌(𝑟)|𝑑𝑟^𝑞≤𝑐^𝑞_𝛼𝑀^𝑞sup_{−𝜏≤𝑙≤00}^𝑠+𝑙𝑒^{(𝜂+𝜔)(𝑠+𝑙−𝑟)}(𝑠+𝑙−𝑟)^{(𝛼−1)𝑞′}𝑑𝑟^𝑞/𝑞′×₀^𝑠+𝑙𝑒^{(𝜂+𝜔)(𝑠−𝑟)}𝑒^𝑞𝜂𝑟|𝑌(𝑟)|^𝑞𝑑𝑟≤𝑐^𝑞_𝛼𝑀^𝑞𝑠₀𝑒^{(𝜂+𝜔)𝑟}𝑟^{𝑞′(𝛼−1)}𝑑𝑟^𝑞/𝑞′×^𝑠₀𝑒^{(𝜂+𝜔)(𝑠−𝑟)}𝑒^𝑞𝜂𝑟|𝑌(𝑟)|^𝑞𝑑𝑟. (21) Applying the Young inequality for convolutions, we have ^∞₀𝑒^𝑞𝜂𝑠Φ^′𝑋_⋅𝑠^𝑞𝑑𝑠≤𝑐^𝑞_𝛼𝑀^𝑞∞₀𝑒^{(𝜂+𝜔)𝑠}𝑠^{𝑞′(𝛼−1)}𝑑𝑠^𝑞/𝑞′×^∞₀𝑒^{(𝜂+𝜔)𝑠}𝑑𝑠^∞₀𝑒^𝑞𝜂𝑠|𝑌(𝑠)|^𝑞𝑑𝑠, (22) and we conclude that Φ^′(𝑋_⋅)_{𝐿𝑞𝒫(Ω;𝐿𝑞𝜂(𝒞))}≤𝑐_𝛼𝑀|𝑌|_𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂(𝐻))×^∞₀𝑒^{(𝜂+𝜔)𝑠}𝑠^{𝑞′(𝛼−1)}𝑑𝑠^1/𝑞′×^∞₀𝑒^{(𝜂+𝜔)𝑠}𝑑𝑠^1/𝑞. (23) If we start again from (20) and apply the Hölder inequality, we obtain 𝑒^{𝜂(𝑠+𝑙)}Φ^′𝑋_⋅𝑠(𝑙)≤𝑐_𝛼𝑀₀^𝑠+𝑙𝑟^{(𝛼−1)𝑞′}𝑒^{(𝜔+𝜂)𝑟𝑞′}𝑑𝑟^1/𝑞′×₀^𝑠+𝑙𝑒^𝜂𝑟𝑞|𝑌(𝑟)|^𝑞𝑑𝑟^1/𝑞,𝑒^𝜂𝑠Φ^′𝑋_⋅𝑠≤𝑐_𝛼𝑀^𝑠₀𝑟^{(𝛼−1)𝑞′}𝑒^{(𝜔+𝜂)𝑟𝑞′}𝑑𝑟^1/𝑞′×^𝑠₀𝑒^𝜂𝑟𝑞|𝑌(𝑟)|^𝑞𝑑𝑟^1/𝑞. (24) So, we conclude that Φ^′𝑋_{⋅𝐿𝑞𝒫(Ω;𝐶𝜂(𝒞))}≤𝑐_𝛼𝑀|𝑌|_𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂(𝐻))×^∞₀𝑟^{(𝛼−1)𝑞′}𝑒^{(𝜔+𝜂)𝑟𝑞′}𝑑𝑟^1/𝑞′. (25) On the other hand, by the Burkholder-Davis-Gundy inequalities, for some constant 𝑐_𝑞 depending only on 𝑞, we have 𝐸|𝑌(𝑟)|^𝑞≤𝑐_𝑞𝐸^𝑟₀(𝑟−𝜎)^−2𝛼𝑒^{(𝑟−𝜎)𝐴}𝐺𝜎,𝑋_𝜎²_{𝐿2()Ξ,𝐻}𝑑𝜎^𝑞/2≤𝐿^𝑞𝑐_𝑞𝐸×^𝑟₀(𝑟−𝜎)^{−2𝛼−2𝛾}𝑒^{2𝜔(𝑟−𝜎)}1+𝑋_𝜎²_𝐶𝑑𝜎^𝑞/2, (26) which implies that 𝐸|𝑌(𝑟)|^𝑞2/𝑞≤𝐿²𝑐_𝑞^2/𝑞𝑟₀(𝑟−𝜎)^{−2𝛼−2𝛾}×𝑒^{2𝜔(𝑟−𝜎)}𝐸1+𝑋_𝜎𝐶^𝑞2/𝑞𝑑𝜎, (27) so that 𝑒^2𝜂𝑟𝐸|𝑌(𝑟)|^𝑞2/𝑞≤𝐶₁^𝑟₀(𝑟−𝜎)^{−2𝛼−2𝛾}𝑒^{2(𝜔+𝜂)(𝑟−𝜎)}𝑒^2𝜂𝜎𝑑𝜎+𝐶₂^𝑟₀(𝑟−𝜎)^{−2𝛼−2𝛾}𝑒^{2(𝜔+𝜂)(𝑟−𝜎)}×𝑒^2𝜂𝜎𝐸𝑋_𝜎^𝑞_𝐶^2/𝑞𝑑𝜎, (28) for suitable constants 𝐶₁,𝐶₂. Applying the Young inequality for convolutions, we obtain ^∞₀𝑒^𝑞𝜂𝑟𝐸|𝑌(𝑟)|^𝑞𝑑𝑠≤𝐶₁^∞₀𝑠^{−2𝛼−2𝛾}𝑒^{2(𝜔+𝜂)𝑠}𝑑𝑠^𝑞/2∞₀𝑒^𝑞𝜂𝑠𝑑𝑠+𝐶₂^∞₀𝑠^{−2𝛼−2𝛾}𝑒^{2(𝜔+𝜂)𝑠}𝑑𝑠^𝑞/2×^∞₀𝑒^𝑞𝜂𝑠𝐸𝑋_𝑠^𝑞_𝐶𝑑𝑠. (29) This shows that |𝑌|_𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂(𝐻)) is finite provided we assume that 𝜂<0 and 𝜔+𝜂<0; so, the map is well defined.
If 𝑋¹_⋅,𝑋²_⋅ are processes belonging to ℋ^𝑞_𝜂 and 𝑌¹,𝑌² are defined accordingly, the entirely analogous passages show that 𝑌¹−𝑌²_{𝐿𝑞𝒫(Ω;𝐿𝑞𝜂(𝐻))}≤𝐿𝑐_𝛼^1/𝑞𝑋¹_⋅−𝑋²_{⋅𝐿𝑞𝒫(Ω;𝐿𝑞𝜂(𝒞))}×^∞₀𝑠^{−2𝛼−2𝛾}𝑒^{2(𝜔+𝜂)𝑠}𝑑𝑠^1/2. (30) Recalling the inequalities (23) and (25) and noting that the map Y→Φ^′(X_⋅) is linear, we obtain an explicit expression for the constant 𝑐 in (17), and it is immediate to verify that 𝑐<1 provided 𝜂<0 is chosen sufficiently large. We fix such a value of 𝜂(𝑞). The first result is a consequence of the contraction principle. The estimate (15) also follows from the contraction property of Φ(⋅,𝑡,𝑥).

For investigating the dependence of the solution 𝑋(𝑠,𝑡,𝑥) on the initial data 𝑥 and 𝑡, we reformulate (13) as an equation on [0,∞). We set 𝑆(𝑠)=𝑒^𝑠𝐴,for𝑠≥0,𝑆(𝑠)=𝐼,for𝑠<0, (31) and we consider the equation 𝑋(𝑠)()𝑥(∨(+=𝑆𝑠−𝑡((0∧𝑠−𝑡))−𝜏))_𝑠0𝐼_[𝑡,∞)(𝜎)𝑆(𝑠−𝜎)𝐹𝜎,𝑋_𝜎+𝑑𝜎_𝑠0𝐼_[𝑡,∞)(𝜎)𝑆(𝑠−𝜎)×𝐺𝜎,𝑋_𝜎(𝜎)[),𝑋𝑑𝑊,𝑠∈0,∞₀(𝜃)=𝑥((−𝑡+𝜃)∨(−𝜏)),𝜃∈[−𝜏,0]. (32) Under the assumptions of Hypothesis 1, by Theorem 2, it is easy to prove that equation (32) has a unique solution 𝑋 and 𝑋_⋅∈ℋ^𝑞_𝜂(𝑞) for every 𝑞∈[2,∞). It clearly satisfies 𝑋(𝑠)=𝑥((𝑠−𝑡)∨(−𝜏)) for 𝑠∈[−𝜏,𝑡), and its restriction to the time internal [𝑡,∞) is the unique mild solution of (10). From now on, we denote by 𝑋(𝑠,𝑡,𝑥),𝑠∈[0,∞), the solution of (32).

We need the following parameter-depending contraction principle, which is stated in the following lemma and proved in [29, Theorems 10.1 and 10.2].

Lemma 3 (Parameter Depending Contraction Principle). Let 𝐵,𝐷 denote Banach spaces. Let ℎ∶𝐵×𝐷→𝐵 be a continuous mapping satisfying ℎ𝑥₁,𝑦−ℎ𝑥₂,𝑦≤𝛼𝑥₁−𝑥₂, (33) for some 𝛼∈[0,1) and every 𝑥₁,𝑥₂∈𝐵,y∈𝐷. Let 𝜙(𝑦) denote the unique fixed point of the mapping ℎ(⋅,𝑦)∶𝐵→𝐵. Then, 𝜙∶𝐷→𝐵 is continuous.

Theorem 4. Assume that Hypothesis 1 holds true. Then, for every 𝑞∈[1,∞), the map (𝑡,𝑥)→𝑋_⋅(𝑡,𝑥) is continuous from [0,∞)×𝒞 to ℋ^q_𝜂(q).

Proof. Clearly, it is enough to prove the claim for 𝑞 large. Let us consider the map Φ defined in the proof of Theorem 2. In our present notation, Φ can be seen as a mapping from ℋ^𝑞_𝜂×[0,∞)×𝒞 to ℋ^𝑞_𝜂 as follows: Φ𝑋_⋅,𝑡,𝑥_𝑠(𝑙)()𝑥(0)+=𝑆𝑠+𝑙−𝑡_0𝑠+𝑙𝐼_[𝑡,∞)(𝜎)𝑆(𝑠+𝑙−𝜎)𝐹𝜎,𝑋_𝜎+𝑑𝜎_0𝑠+𝑙𝐼_[𝑡,∞)(𝜎)𝑆(𝑠+𝑙−𝜎)×𝐺𝜎,𝑋_𝜎(𝜎),[)[]Φ𝑋𝑑𝑊𝑠∈0,∞,𝑙∈−𝜏,0,𝑠+𝑙≥𝑡,_⋅,𝑡,𝑥_𝑠(𝑙))∨(,[)[]=𝑥((𝑠+𝑙−𝑡−𝜏))𝑠∈0,∞,𝑙∈−𝜏,0,𝑠+𝑙≤𝑡. (34) By the arguments of the proof of Theorem 2, Φ(⋅,𝑡,𝑥) is a contraction in ℋ^𝑞_𝜂 uniformly with respect to 𝑡,𝑥. The process 𝑋_⋅(𝑡,𝑥) is the unique fixed point of Φ(⋅,𝑡,𝑥). So, by the parameter-depending contraction principle (Lemma 3), it suffices to show that Φ is continuous from ℋ^𝑞_𝜂×[0,∞)×𝒞 to ℋ^𝑞_𝜂. From the contraction property of Φ(⋅,𝑡,𝑥) mentioned earlier, we have that Φ(⋅,𝑡,𝑥) is continuous, uniformly in 𝑡, 𝑥. Moreover, for fixed 𝑋_⋅, it is easy to verify that Φ(𝑋_⋅,⋅,⋅) is continuous from [0,∞)×𝒞 to ℋ^𝑞_𝜂. The proof is finished.

Remark 5. By similar passages, we can show that, for fixed 𝑡, Theorem 4 still holds true for 𝑞 large enough if the spaces [0,∞)×𝒞 and ℋ^𝑞_𝜂 are replaced by the spaces 𝐿^𝑞(Ω,𝒞,ℱ_𝑡) and ℋ^𝑞_𝜂(𝑡) respectively, where 𝐿^𝑞(Ω,𝒞,ℱ_𝑡) denotes that the space of ℱ_𝑡-measurable function with value in 𝒞, such that the norm |𝑥|^𝑞=𝐸|𝑥|^𝑞_𝐶, (35) is finite.

3. The Backward-Forward System

In this section, we consider the system of stochastic differential equations, 𝑃-a.s., 𝑋(𝑠)=𝑒^{(𝑠−𝑡)𝐴}𝑥(0)+^𝑠_𝑡𝑒^{(𝑠−𝜎)𝐴}𝐹𝜎,𝑋_𝜎+𝑑𝜎_𝑠𝑡𝑒^{(𝑠−𝜎)𝐴}𝐺𝜎,𝑋_𝜎(𝜎)[),𝑋𝑑𝑊,𝑠∈𝑡,∞_𝑡𝑌(𝑠)(𝑇)+=𝑥∈𝒞,−𝑌^𝑇_𝑠𝑍(𝜎)𝑑𝑊(𝜎)+𝜆^𝑇_𝑠𝑌(𝜎)=𝑑𝜎_𝑇𝑠𝜓𝜎,𝑋_𝜎,𝑌(𝜎),𝑍(𝜎)𝑑𝜎,0≤𝑠≤𝑇<∞, (36) for 𝑠 varying on the time interval [𝑡,∞)⊂[0,∞). As in Section 2, we extend the domain of the solution setting 𝑋(𝑠,𝑡,𝑥)=𝑥((𝑠−𝑡)∨(−𝜏)) for 𝑠∈[−𝜏,𝑡).

We make the following assumptions.

Hypothesis 2. The mapping 𝜓∶[0,∞)×𝒞×𝐾×𝐿₂(Ξ,𝐾)→𝐾 is Borel measurable such that, for all 𝑡∈[0,∞), 𝜓(𝑡,⋅)∶𝒞×𝐾×𝐿₂(Ξ,𝐾)→𝐾 is continuous, and for some 𝐿_𝑦,𝐿_𝑧>0, 𝜇∈𝑅, and 𝑚≥1, 𝜓𝑠,𝑥,𝑦₁,𝑧₁−𝜓𝑠,𝑥,𝑦₂,𝑧₂≤𝐿_𝑦𝑦₁−𝑦₂+𝐿_𝑧𝑧₁−𝑧₂,𝜓𝑠,𝑥,𝑦,𝑧≤𝐿1+|𝑥|^𝑚_𝐶+𝑦+|𝑧|,𝜓𝑠,𝑥,𝑦₁,𝑧−𝜓𝑠,𝑥,𝑦₂,𝑧,𝑦₁−𝑦_2𝐾≥𝜇𝑦₁−𝑦₂², (37) for every 𝑠∈[0,∞),𝑥∈𝒞, 𝑦,𝑦₁,𝑦₂∈𝐾, 𝑧,𝑧₁,and𝑧₂∈𝐿₂(Ξ,𝐾).
We note that the third inequality in (37) follows from the first one taking 𝜇=−𝐿_𝑦 but that the third inequality may also hold for different values of 𝜇.
Firstly, we consider the backward stochastic differential equation 𝑌(𝑠)−𝑌(𝑇)+^𝑇_𝑠𝑍(𝜎)𝑑𝑊(𝜎)+𝜆^𝑇_𝑠𝑌(𝜎)=𝑑𝜎_𝑇𝑠𝜓𝜎,𝑋_𝜎,𝑌(𝜎),𝑍(𝜎)𝑑𝜎,0≤𝑠≤𝑇<∞. (38)𝐾 is a Hilbert space, the mapping 𝜓∶[0,∞)×𝒞×𝐾×𝐿₂(Ξ,𝐾)→𝐾 is a given measurable function, 𝑋_⋅ is a predictable process with values in another Banach space 𝒞, and 𝜆 is a real number.

Theorem 6. Assume that Hypothesis 2 holds. Let 𝑝>2 and 𝛿<0 be given, and choose 𝛿𝑞≥𝑚𝑝,𝜂>𝑚. (39) Then, the following hold. (i)For 𝑋_⋅∈𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂(𝒞)) and 𝜆>−(𝛿+𝜇−(𝐿²_𝑧/2)), (38) has a unique solution in 𝒦^p_𝛿 that will be denoted by (𝑌(𝑋_⋅)(𝑠),𝑍(𝑋_⋅)(𝑠)),𝑠≥0. (ii)The estimate 𝐸sup_𝑠≥0𝑌𝑋_⋅(𝑠)^𝑝𝑒^𝑝𝛿𝑠+𝐸^∞₀𝑒^2𝛿𝜎𝑌(𝑋_⋅)(𝜎)²𝑑𝜎^𝑝/2+𝐸^∞₀𝑒^2𝛿𝜎𝑍𝑋_⋅(𝜎)²𝑑𝜎^𝑝/2≤𝑐1+𝑋_⋅^𝑚_𝐿^𝑞_𝒫Ω;𝐿^𝑞_𝜂(𝒞)^𝑝 (40) holds for a suitable constant 𝑐. In particular, 𝑌(𝑋_⋅)∈𝐿^𝑝_𝒫(Ω;𝐶_𝛿(𝐾)).(iii)The map 𝑋_⋅→(𝑌(𝑋_⋅),𝑍(𝑋_⋅)) is continuous from 𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂(𝒞)) to 𝒦^𝑝_𝛿, and 𝑋_⋅→𝑌(𝑋_⋅) is continuous from 𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂(𝒞)) to 𝐿^𝑝_𝒫(Ω;𝐶_𝛿(𝐾)). (iv)The statements of points (i), (ii), and (iii) still hold true if the space 𝐿^𝑞_𝒫(Ω;𝐿^𝑞_𝜂(𝒞)) is replaced by the space 𝐿^𝑞_𝒫(Ω;𝐶_𝜂(𝒞)).

Proof. The theorem is very similar to Proposition 3.11 in [24]. The only minor difference is that the mapping 𝜓∶[0,∞)×𝒞×𝐾×𝐿₂(Ξ,𝐾)→𝐾 is a given measurable function, while in [24], the measurable function 𝜓 is from 𝐻×𝐾×𝐿₂(Ξ,𝐾) to 𝐾; however, the same arguments apply.

Theorem 7. Assume that Hypothesis 1 holds and that Hypothesis 2 holds true in the particular case 𝐾=𝑅. Then, for every 𝑝>2,𝑞,𝛿<0 satisfying (39) with 𝜂=𝜂(𝑞), and for every 𝜆>𝜆^′=−(𝛿+𝜇−(𝐿²_𝑧/2)), there exists a unique solution in ℋ^𝑞_𝜂(𝑞)×𝒦^𝑝_𝛿 of (36) that will be denoted by (𝑋(⋅,𝑡,𝑥),𝑌(⋅,𝑡,𝑥),𝑍(⋅,𝑡,𝑥)). Moreover, 𝑌(⋅,𝑡,𝑥)∈𝐿^𝑝_𝒫(Ω;𝐶_𝛿(𝑅)). The map (𝑡,𝑥)→(𝑌(⋅,𝑡,𝑥),𝑍(⋅,𝑡,𝑥)) is continuous from [0,∞)×𝒞 to 𝒦^𝑝_𝛿, and the map (𝑡,𝑥)→𝑌(⋅,𝑡,𝑥) is continuous from [0,∞)×𝒞 to 𝐿^𝑝_𝒫(Ω;𝐶_𝛿(𝑅)).

Proof. We first notice that the system is decoupled; the first does not contain the solution (𝑌,𝑍) of the second one. Therefore, under the assumption of Hypothesis 1 by Theorem 2, there exists a unique solution 𝑋(⋅,𝑡,𝑥) and 𝑋_⋅(𝑡,𝑥)∈ℋ^𝑞_𝜂(𝑞) of the first equation. Moreover, from Theorem 4, it follows that the map (𝑡,𝑥)→𝑋_⋅(𝑡,𝑥) is continuous from [0,∞)×𝒞 to ℋ^𝑞_𝜂(𝑞).
Let 𝐾=𝑅; from Theorem 6, we have that there exists a unique solution (𝑌(⋅,𝑡,𝑥),𝑍(⋅,𝑡,𝑥))∈𝒦^𝑝_𝛿 of the second equation, and the map 𝑋_⋅→(𝑌(𝑋_⋅),𝑍(𝑋_⋅)) is continuous from ℋ^𝑞_𝜂(𝑞) to 𝒦^𝑝_𝛿 while X_⋅→(Y(X_⋅)) is continuous from ℋ^𝑞_𝜂(𝑞) to 𝐿^𝑝_𝒫(Ω;𝐶_𝛿(𝑅)). We have proved that (𝑋(⋅,𝑡,𝑥),𝑌(⋅,𝑡,𝑥),𝑍(⋅,𝑡,𝑥))∈ℋ^𝑞_𝜂(𝑞)×𝒦^𝑝_𝛿 is the unique solution of (36), and the other assertions follow from composition.

Remark 8. From Remark 5, by similar passages, we can show that for fixed 𝑡 and for 𝑞 large enough, under the assumptions of Theorem 7, the map 𝑥→(𝑌(⋅,𝑡,𝑥),𝑍(⋅,𝑡,𝑥)) is continuous from 𝐿^𝑞(Ω,𝒞,ℱ_𝑡) to 𝒦^𝑝_𝛿(𝑡).
We also remark that the process 𝑋(⋅,𝑡,𝑥) is ℱ_[𝑡,∞) measurable, since 𝒞 is separable Banach space, we have that 𝑋_⋅(𝑡,𝑥) is ℱ_[𝑡,∞) measurable; So that 𝑌(𝑡) is measurable with respect to both ℱ_[𝑡,∞) and ℱ_𝑡, it follows that 𝑌(𝑡) is deterministic.
For later use, we notice three useful identities; for 𝑡≤𝑠<∞, the equality, 𝑃-a.s., 𝑋_𝑙𝑠,𝑋_𝑠(𝑡,𝑥)=𝑋_𝑙(𝑡,𝑥),𝑙∈[𝑠,∞) (41) is a consequence of the uniqueness of the solution of (13). Since the solution of the backward equation is uniquely determined on an interval [𝑠,∞) by the values of the process 𝑋_⋅ on the same interval, for 𝑡≤𝑠<∞, we have, 𝑃-a.s., 𝑌𝑙,𝑠,𝑋_𝑠()()[),𝑍𝑡,𝑥=𝑌𝑙,𝑡,𝑥,for𝑙∈𝑠,∞𝑙,𝑠,𝑋_𝑠(𝑡,𝑥)=𝑍(𝑙,𝑡,𝑥),fora.a.𝑙∈[𝑠,∞). (42)

Lemma 9 (see [30]). Let 𝐸 be a metric space with metric 𝑑, and let 𝑓∶Ω→𝐸 be strongly measurable. Then, there exists a sequence 𝑓_𝑛,𝑛∈𝑁, of simple 𝐸-valued functions (i.e., 𝑓_𝑛 is ℱ/ℬ(E) measurable and takes only a finite number of values) such that for arbitrary 𝜔∈Ω, the sequence 𝑑(𝑓_𝑛(𝜔),𝑓(𝜔)),𝑛∈𝑁, is monotonically decreasing to zero.

Let now 𝑓∈𝐿^𝑞(Ω,𝒞). By Lemma 9 we get the existence of a sequence of simple function 𝑓_𝑛,𝑛∈𝑁, such that 𝑓_𝑛(𝜔)−𝑓(𝜔)↓0forall𝜔∈Ωas𝑛→∞. (43) Hence, 𝑓_𝑛→𝑓 in |⋅|_𝐿^𝑞_(Ω,𝒞) by Lebesgue's dominated convergence theorem.

We are now in a position of showing the main result in this section.

Theorem 10. Assume that Hypothesis 1 holds true and that Hypothesis 2 holds in the particular case 𝐾=𝑅. Then, there exist two Borel measurable deterministic functions 𝜐∶[t,∞)×𝒞→𝑅 and 𝜁∶[𝑡,∞)×𝒞→Ξ^∗=𝐿(Ξ,𝑅)=𝐿₂(Ξ,𝑅), such that for 𝑡∈[0,∞) and 𝑥∈𝒞, the solution (𝑋(𝑡,𝑥),𝑌(𝑡,𝑥),𝑍(𝑡,𝑥)) of (36) satisfies 𝑌(𝑠,𝑡,𝑥)=𝜐𝑠,𝑋_𝑠(𝑡,𝑥),𝑍(𝑠,𝑡,𝑥)=𝜁𝑠,𝑋_𝑠(),[).𝑡,𝑥𝑃-𝑎.𝑠.,𝑓𝑜𝑟𝑎.𝑎.𝑠∈𝑡,∞ (44)

Proof. We apply the techniques introduced in [26, Proposition 3.2]. Let {𝑒_𝑖} be a basis of Ξ^∗, and let us define 𝑍^𝑖,𝑁=((𝑍,𝑒_𝑖)_Ξ^∗∧𝑁)∨(−𝑁). Then, for every 0≤𝑡₁<𝑡₂<∞,Δ>0, and 𝑥₁,𝑥₂∈𝒞, we have that 𝐸^𝑡1_𝑡^+Δ₁𝑍^𝑖,𝑁𝑠,𝑡₁,𝑥₁𝑑𝑠−𝐸^𝑡2_𝑡^+Δ₂𝑍^𝑖,𝑁𝑠,𝑡₂,𝑥₂𝑑𝑠≤𝐸^𝑡2_𝑡1𝑍^𝑖,𝑁𝑠,𝑡₁,𝑥₁𝑑𝑠+𝐸^𝑡1_𝑡^+Δ₂𝑍^𝑖,𝑁𝑠,𝑡₁,𝑥₁−𝑍^𝑖,𝑁𝑠,𝑡₂,𝑥₂𝑑𝑠+𝐸^𝑡2_𝑡^+Δ_1+Δ𝑍^𝑖,𝑁𝑠,𝑡₂,𝑥₂𝑡𝑑𝑠≤2𝑁₂−𝑡₁+Δ^1/2𝑒^{−𝛿(𝑡1+Δ)}×𝐸^∞₀𝑒^2𝛿𝑠𝑍^𝑖,𝑁𝑠,𝑡₁,𝑥₁−𝑍^𝑖,𝑁𝑠,𝑡₂,𝑥₂²𝑑𝑠^𝑝/21/𝑝≤2𝑁𝑡₂−𝑡₁+Δ^1/2𝑒^{−𝛿(𝑡1+Δ)}×𝐸^∞₀𝑒^2𝛿𝑠𝑍𝑠,𝑡₁,𝑥₁−𝑍𝑠,𝑡₂,𝑥₂²𝑑𝑠^𝑝/21/𝑝. (45) From Theorem 7, we have that the map (𝑡,𝑥)→∫_𝑡^𝑡+Δ𝑍^𝑖,𝑁(𝑠,𝑡,𝑥)𝑑𝑠 is continuous from [0,∞)×𝒞 to 𝑅. By Remark 8, we also have that, for fixed 𝑡, the map 𝑥→𝐸∫_𝑡^𝑡+Δ𝐸𝑍^𝑖,𝑁(𝑠,𝑡,𝑥)𝑑𝑠 is continuous from 𝐿^𝑞(Ω,𝒞,ℱ_𝑡) to 𝑅 for 𝑞 large enough. Let us define 𝜁^𝑖,𝑁(𝑡,𝑥)=liminf_𝑛→∞𝑛𝐸_𝑡^{𝑡+(1/𝑛)}𝑍^𝑖,𝑁()[)𝑠,𝑡,𝑥𝑑𝑠,𝑡∈0,∞,𝑥∈𝒞. (46) It is clear that 𝜁^𝑖,𝑁∶[0,∞)×𝒞→𝑅 is a Borel function.
We fix 𝑥 and 0≤𝑡≤𝑠<∞. For 𝑙∈[𝑠,∞), we denote 𝐸[𝑍^𝑖,𝑁(𝑙,𝑠,𝑦)]|_{𝑦=𝑋𝑠(𝑡,𝑥)}, the random variable obtained by composing 𝑋_𝑠(𝑡,𝑥) with the map 𝑦→𝐸[𝑍^𝑖,𝑁(𝑙,𝑠,𝑦)].
By Lemma 9, there exists a sequence of 𝒞-valued ℱ_𝑠-measurable simple functions 𝑓_𝑚∶Ω→𝒞,𝑓_𝑚=^𝑁𝑚_𝑘=1ℎ_𝑘^(𝑚)𝐼_{{𝑓𝑚=ℎ𝑘}^(𝑚)_},𝑁_𝑚∈𝑁, (47) where ℎ₁^(𝑚),…,ℎ_𝑚^(𝑚) are pairwise distinct and Ω=⋃^𝑁𝑚_𝑘=1{𝑓_𝑚=ℎ_𝑘^(𝑚)}, such that 𝑓_𝑚(𝜔)−𝑋_𝑠(𝜔)↓0forall𝜔∈Ωas𝑛→∞. (48) For any 𝐵∈ℱ_𝑠, we have _𝐵𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁()=𝑙,𝑡,𝑥𝑑𝑙𝑑𝑃_{𝐵𝑠𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,𝑋_𝑠𝑑𝑙𝑑𝑃=𝐸𝐼_𝐵𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,𝑋_𝑠𝑑𝑙=lim_𝑚→∞𝐸𝐼_𝐵𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,𝑓_𝑚𝑑𝑙=lim^𝑁_𝑚→∞^𝑚_𝑘=1𝐸𝐼_𝐵𝐼_{{𝑓𝑚=ℎ𝑘}^(𝑚)_}𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,ℎ_𝑘^(𝑚)𝑑𝑙=lim_𝑚→∞𝐸𝐼_𝐵^𝑁𝑚_𝑘=1𝐼_{{𝑓𝑚=ℎ𝑘}^(𝑚)_}𝐸_𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,ℎ_𝑘^(𝑚)𝑑𝑙=lim_𝑚→∞𝐸𝐼_𝐵𝐸_𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,𝑦𝑑𝑙∣_{𝑦=𝑓𝑚}=lim_{𝑚→∞𝐵}𝐸_𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,𝑦𝑑𝑙∣_{𝑦=𝑓𝑚}𝑑𝑃=_𝐵𝐸_𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,𝑦𝑑𝑙∣_{𝑦=𝑋𝑠}𝑑𝑃. (49) and we get that 𝜁^𝑖,𝑁𝑠,𝑋_𝑠(𝑡,𝑥)=liminf_𝑛→∞𝑛×𝐸_𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁𝑙,𝑠,𝑦𝑑𝑙∣_{𝑦=𝑋𝑠(𝑡,𝑥)}=liminf_𝑛→∞𝑛𝐸_𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁(𝑙,𝑡,𝑥)𝑑𝑙ℱ_𝑠,𝑃-a.s. (50) Fix 𝑡 and 𝑥. Recalling that |𝑍^𝑖,𝑁|≤𝑁, by the Lebesgue theorem on differentiation, it follows that 𝑃-a.s. lim_𝑛→∞𝑛_𝑠^{𝑠+(1/𝑛)}𝑍^𝑖,𝑁(𝑙,𝑡,𝑥)𝑑𝑙=𝑍^𝑖,𝑁(),[).𝑠,𝑡,𝑥fora.a.𝑠∈𝑡,∞ (51) By the boundedness of 𝑍^𝑖,𝑁, applying the dominated convergence theorem, we get that 𝜁^𝑖,𝑁𝑠,𝑋_𝑠(𝑡,𝑥)=𝐸𝑍^𝑖,𝑁(𝑠,𝑡,𝑥)ℱ_𝑠=𝑍^𝑖,𝑁(),[).𝑠,𝑡,𝑥𝑃-a.s.,fora.a.𝑠∈𝑡,∞ (52) Now, we have proved that for every 𝑡,𝑥, 𝜁^𝑖,𝑁𝑠,𝑋_𝑠(𝑡,𝑥)=𝑍^𝑖,𝑁(),[),𝑠,𝑡,𝑥𝑃-a.s.fora.a.𝑠∈𝑡,∞ (53) for every 𝑖,𝑁. Let 𝐶⊂[0,∞)×𝒞 denote the set of pairs (𝑡,𝑥) such that lim_𝑁→∞𝜁^𝑖,𝑁(𝑡,𝑥) exists and the series ∑^∞_𝑖=1(lim_𝑁→∞𝜁^𝑖,𝑁(𝑡,𝑥))𝑒_𝑖 converges in Ξ^∗. We define 𝜁(𝑡,𝑥)=^∞_𝑖=1lim_𝑁→∞𝜁^𝑖,𝑁(𝑡,𝑥)𝑒_𝑖,()𝜁()()𝑡,𝑥∈𝐶,𝑡,𝑥=0,𝑡,𝑥∉𝐶. (54) Since 𝑍 satisfies 𝑍(𝜔,𝑠,𝑡,𝑥)=^∞_𝑖=1lim_𝑁→∞𝑍^𝑖,𝑁(𝜔,𝑠,𝑡,𝑥)𝑒_𝑖, (55) for every 𝜔,𝑠,𝑡,𝑥. From (53), it follows that for every 𝑡,𝑥, we have (𝑠,𝑋_𝑠(𝜔,𝑡,𝑥))∈𝐶, 𝑃-a.s., for almost all 𝑠∈[𝑡,∞), and 𝑍(𝑠,𝑡,𝑥)=𝜁(𝑠,𝑋_𝑠(𝑡,𝑥))𝑃-a.s., for a.a. 𝑠∈[𝑡,∞).
We define 𝜐(𝑡,𝑥)=𝑌(𝑡,𝑡,𝑥); since 𝑌(𝑡,𝑡,𝑥) is deterministic, so the map (𝑡,𝑥)→𝜐(𝑡,𝑥) can be written as a composition 𝜐(𝑡,𝑥)=Γ₃(Γ₂(𝑡,Γ₁(𝑡,𝑥))) with Γ₁∶[0,∞)×𝒞→𝐿^𝑝_𝒫Ω;𝐶_𝛿(𝑅,Γ₁)()(),Γ𝑡,𝑥=𝑌⋅,𝑡,𝑥₂∶[0,∞)×𝐿^𝑝_𝒫Ω;𝐶_𝛿(𝑅)→𝐿^𝑝(),ΓΩ,𝑅₂()(𝑡),Γ𝑡,𝑉=𝑉₃∶𝐿^𝑝(Ω,𝑅)→𝑅,Γ₃𝜉=𝐸𝜉. (56) From Theorem 7, it follows that Γ₁ is continuous. By |𝑉(𝑡)−𝑈(𝑠)|_𝐿^𝑝_(Ω,𝑅)≤|𝑉(𝑡)−𝑉(𝑠)|_𝐿^𝑝_(Ω,𝑅)+𝑒^{−𝛿𝑝𝑠}|𝑉−𝑈|_{|𝐿𝑝𝒫𝛿(Ω;𝐶(𝑅))}, (57) we have that Γ₂ is continuous. It is clear that Γ₃ is continuous. Then, the map (𝑡,𝑥)→𝜐(𝑡,𝑥) is continuous from [0,∞)×𝒞 to 𝑅; therefore, 𝜐(𝑡,𝑥)is a Borel measurable function. From uniqueness of the solution of (36), it follows that 𝑌(𝑠,𝑡,𝑥)=𝜐(𝑠,𝑋_𝑠(𝑡,𝑥)), 𝑃-a.s., for a.a. 𝑠∈[𝑡,∞).

4. The Fundamental Relation

Let (Ω,ℱ,𝑃) be a given complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions. {𝑊(𝑡),𝑡≥0} is a cylindrical Wiener process in Ξ with respect to {ℱ_𝑡}_𝑡≥0. We will say that an {ℱ}_𝑡≥0-predictable process 𝑢 with values in a given measurable space (𝑈,𝒰) is an admissible control. The function 𝑅∶[0,∞)×𝒞×𝑈→Ξ is measurable and bounded. We consider the following controlled state equation: 𝑑𝑋^𝑢(𝑠)=𝐴𝑋^𝑢(𝑠)𝑑𝑠+𝐹𝑠,𝑋^𝑢_𝑠𝑑𝑠+𝐺𝑠,𝑋^𝑢_𝑠𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑠+𝐺𝑠,𝑋^𝑢_𝑠(𝑠),[),𝑋𝑑𝑊𝑠∈𝑡,∞_𝑢𝑡=𝑥. (58)

Here, we assume that there exists a mild solution of (58) which will be denoted by 𝑋^𝑢(𝑠,𝑡,𝑥) or simply by 𝑋^𝑢(𝑠). We consider a cost function of the form: 𝐽(𝑢)=𝐸^∞_𝑡𝑒^−𝜆𝑠𝑔𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑠. (59) Here, 𝑔 is function on [0,∞)×𝒞×𝑈 with real values. Our purpose is to minimize the function 𝐽 over all admissible controls.

We define in a classical way the Hamiltonian function relative to the previous problem; for all 𝑡∈[0,∞),𝑥∈𝒞,and𝑧∈Ξ^∗, 𝜓(𝑡,𝑥,𝑧)=inf𝑔(𝑡,𝑥,𝑢)+𝑧𝑅(𝑡,𝑥,𝑢)∶𝑢∈𝑈. (60) and the corresponding, possibly empty, set of minimizers Γ(𝑡,𝑥,𝑧)=𝑢∈𝑈,𝑔(𝑡,𝑥,𝑢)+𝑧𝑅(𝑡,𝑥,𝑢)=𝜓(𝑡,𝑥,𝑧). (61)

We are now ready to formulate the assumptions we need.

Hypothesis 3. (i) 𝐴, 𝐹, and G verify Hypothesis 1.
(ii) (𝑈,𝒰) is a measurable space. The map 𝑔∶[0,∞)×𝒞×𝑈→𝑅 is continuous and satisfies |𝑔(𝑡,𝑥,𝑢)|≤𝐾_𝑔(1+|𝑥|^𝑚𝑔_𝐶) for suitable constants 𝐾_𝑔>0, 𝑚_𝑔>0 and all 𝑥∈𝒞,𝑢∈𝑈. The map 𝑅∶[0,∞)×𝒞×𝑈→Ξ is measurable, and |𝑅(𝑡,𝑠,𝑢)|≤𝐿_𝑅 for a suitable constant 𝐾_𝑅>0 and all 𝑥∈𝒞,𝑢∈𝑈, and𝑧∈Ξ^∗.

(iii) The Hamiltonian 𝜓 defined in (60) satisfies the requirements of Hypothesis 2 (with 𝐾=𝑅).

(iv) We fix here 𝑝>2, q and 𝛿<0 satisfying (39) with 𝜂=𝜂(𝑞) and such that 𝑞>𝑚_𝑔.

We are in a position to prove the main result of this section.

Theorem 11. Assume that Hypothesis 3 holds, and suppose that 𝜆 verifies 𝐿𝜆>−𝛿−𝜇+^2𝑧2∨𝐿−𝛿+^2𝑅2∨𝐿𝑝−1_2𝑅𝑚_𝑔2𝑞−𝑚_𝑔−𝜂𝑞𝑚_𝑔. (62) Let 𝜐,𝜁 denote the function in the statement of Theorem 10. Then, for every admissible control 𝑢 and for the corresponding trajectory 𝑋 starting at (𝑡,𝑥), one has 𝐽(𝑢)=𝜐(𝑡,𝑥)+𝐸^∞_𝑡𝑒^−𝜆𝑠−𝜓𝑠,𝑋^𝑢_𝑠,𝜁𝑠,𝑋^𝑢_𝑠+𝜁𝑠,𝑋^𝑢_𝑠×𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)+𝑔𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑠. (63)

Proof. Consider (58) in the probability space (Ω,ℱ,𝑃) with filtration {ℱ_𝑡}_𝑡≥0 and with an {ℱ_𝑡}_𝑡≥0-cylindrical Wiener process {𝑊(𝑡),𝑡≥0}. Let us define 𝑊^𝑢(𝑠)=𝑊(𝑠)+^𝑠_𝑡∧𝑠𝑅𝜎,𝑋^𝑢_𝜎(𝜎)[),𝜌(𝑇),𝑢𝑑𝜎,𝑠∈0,∞=exp^𝑇_𝑡−𝑅^∗𝑠,𝑋^𝑢_𝑠(𝑠)(𝑠)−1,𝑢𝑑𝑊2^𝑇_𝑡𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)²𝑑𝑠. (64) Let 𝑃^𝑢 be the unique probability on ℱ_[0,∞) such that 𝑃^𝑢∣_ℱ𝑇=𝜌(𝑇)𝑃∣_ℱ𝑇. (65) We notice that under 𝑃^𝑢, the process 𝑊^𝑢 is a Wiener process. Let us denote by {ℱ^𝑢_𝑡}_𝑡≥0 the filtration generated by 𝑊^𝑢, and completed in the usual way. Relatively to 𝑊^𝑢 (58) can be rewritten as 𝑑𝑋^𝑢(𝑠)=𝐴𝑋^𝑢(𝑠)𝑑𝑠+𝐹𝑠,𝑋^𝑢_𝑠𝑑𝑠+𝐺𝑠,𝑋^𝑢_𝑠𝑑𝑊^𝑢(𝑠)[),𝑋,𝑠∈𝑡,∞_𝑢𝑡=𝑥. (66) In the space (Ω,ℱ_[0,∞),{ℱ^𝑢_𝑡}_𝑡≥0,𝑃^𝑢), we consider the following system of forward-backward equations: 𝑋^𝑢(𝑠)=𝑒^{(𝑠−𝑡)𝐴}𝑥(0)+^𝑠_𝑡𝑒^{(𝑠−𝜎)𝐴}𝐹𝜎,𝑋^𝑢_𝜎+𝑑𝜎_𝑠𝑡𝑒^{(𝑠−𝜎)𝐴}𝐺𝜎,𝑋^𝑢_𝜎𝑑𝑊^𝑢(𝜎)[),𝑋,𝑠∈𝑡,∞_𝑢𝑡𝑌=𝑥∈𝒞,_𝑢(𝑠)−𝑌^𝑢(𝑇)+^𝑇_𝑠𝑍^𝑢(𝜎)𝑑𝑊^𝑢(𝜎)+𝜆^𝑇_𝑠𝑌^𝑢(𝜎)=𝑑𝜎_𝑇𝑠𝜓𝜎,𝑋^𝑢_𝜎,𝑍^𝑢(𝜎)𝑑𝜎,0≤𝑠≤𝑇<∞. (67) Applying the Itô formula to 𝑒^−𝜆𝑠𝑌^𝑢(𝑠) and writing the backward equation in (67) with respect to the process 𝑊, we get 𝑌^𝑢(𝑠)+^𝑇_𝑠𝑒^−𝜆𝜎𝑍^𝑢(𝜎)(𝜎)=𝑑𝑊_𝑇𝑠𝑒^−𝜆𝜎𝜓𝜎,𝑋^𝑢_𝜎,𝑍^𝑢(𝜎)−𝑍^𝑢(𝜎)𝑅𝜎,𝑋^𝑢_𝜎,𝑢(𝜎)𝑑𝜎+𝑒^−𝜆𝑇𝑌^𝑢(𝑇). (68) Recalling that 𝑅 is bounded, we get, for all 𝑟≥1 and some constant 𝐶, 𝐸^𝑢𝜌(𝑇)^−𝑟=𝐸^𝑢exp𝑟^𝑇_𝑡𝑅^∗𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑊^𝑢(𝑠)−12^𝑇_𝑡𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)²𝑑𝑠=𝐸^𝑢exp^𝑇_𝑡𝑟𝑅^∗𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑊^𝑢(𝑠)−12^𝑇_𝑡𝑟²𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)²𝑟()𝑑𝑠×exp𝑟−12^𝑇_𝑡𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)²𝑑𝑠≤𝑒^{(1/2)𝑟(𝑟−1)𝑇𝐿2𝑅}𝐸^𝑢×exp^𝑇_𝑡2𝑅^∗𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑊^𝑢(𝑠)−12^𝑇_𝑡4𝑅𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)²𝑑𝑠=𝑒^{(1/2)𝑟(𝑟−1)𝑇𝐿2𝑅}. (69) It follows that 𝐸^𝑇_𝑡|𝑒^−𝜆𝑠𝑍^𝑢(𝑠)|²𝑑𝑠^1/2=𝐸^𝑢𝑇_𝑡𝑒^−𝜆𝑠𝑍^𝑢(𝑠)²𝑑𝑠^1/2𝜌⁻¹≤𝐸^𝑢𝑇_𝑡𝑒^−𝜆𝑠𝑍^𝑢(𝑠)²𝑑𝑠^1/2×𝐸^𝑢𝜌^−21/2<∞. (70) We conclude that the stochastic integral in (68) has zero expectation. If we set 𝑠=𝑡 in (68) and we take expectation with respect to 𝑃, we obtain 𝑒^−𝜆𝑇𝐸𝑌^𝑢(𝑇)−𝑌^𝑢(𝑡)=𝐸^𝑇_𝑡𝑒^−𝜆𝜎−𝜓𝜎,𝑋^𝑢_𝜎,𝑍^𝑢(𝜎)+𝑍^𝑢(𝜎)𝑅𝜎,𝑋^𝑢_𝜎,𝑢(𝜎)𝑑𝜎. (71) By Theorem 7, 𝑌^𝑢(⋅,𝑡,𝑥)∈𝐿^𝑝_𝒫(Ω;𝐶_𝛿(𝑅)), so that 𝐸^𝑢|𝑌(𝑇,𝑡,𝑥)|^𝑝≤𝐶exp−𝑝𝛿𝑇. (72) By the Hölder inequality, we have that for suitable constant 𝐶>0, 𝐸|𝑌(𝑇,𝑡,𝑥)|=𝐸^𝑢𝜌⁻¹(𝑇𝑌(𝜌)|𝑇,𝑡,𝑥)|≤𝐸^{−𝑝/(𝑝−1)(𝑝−1)/𝑝}𝐸|𝑌(𝑇,𝑡,𝑥)|^𝑝1/𝑝≤𝐶𝑒^{((𝐿2𝑅/2(𝑝−1))−𝛿))𝑇}. (73) From Theorem 2, we obtain 𝐸^𝑢sup_𝑠≥𝑡𝑒^𝜂𝑞𝑠|𝑋^𝑢_𝑠|^𝑞<∞; by the similar process, we get that 𝐸𝑋^𝑢_𝑇^𝑚𝑔≤𝐶𝑒^{(𝐿2𝑅𝑚𝑔(2𝑞−2𝑚𝑔)−1−𝜂(𝑞)𝑚𝑔)𝑇}, (74) for suitable constant 𝐶>0 and 𝐸^∞_𝑡𝑒^−𝜆𝜎𝑔𝜎,𝑋^𝑢_𝜎,𝑢(𝜎)𝑑𝜎<∞. (75) Since 𝑌^𝑢(𝑡,𝑡,𝑥)=𝜐(𝑡,𝑥) and 𝑍^𝑢(𝑠,𝑡,𝑥)=𝜁(𝑠,𝑋^𝑢_𝑠(𝑡,𝑥)),𝑃-a.s., for a.a. 𝑠∈[𝑡,∞), we have that 𝑒^−𝜆𝑇𝐸𝑌^𝑢(𝑇)−𝑣(𝑡,𝑥)=𝐸^𝑇_𝑡𝑒^−𝜆𝜎−𝜓𝜎,𝑋^𝑢_𝜎,𝜁𝜎,𝑋^𝑢_𝜎+𝜁𝜎,𝑋^𝑢_𝜎𝑅𝜎,𝑋^𝑢_𝜎,𝑢(𝜎)𝑑𝜎. (76) Thus, adding and subtracting 𝐸∫^∞_𝑡𝑒^−𝜆𝜎𝑔(𝜎,𝑋^𝑢_𝜎,𝑢(𝜎))𝑑𝜎 and letting 𝑇→∞, we conclude that 𝐽(𝑢)=𝜐(𝑡,𝑥)+𝐸^∞_𝑡𝑒^−𝜆𝑠−𝜓𝑠,𝑋^𝑢_𝑠,𝜁𝑠,𝑋^𝑢_𝑠+𝜁𝑠,𝑋^𝑢_𝑠𝑅×𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)+𝑔𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑠. (77) The proof is finished.

We immediately deduce the following consequences.

Theorem 12. Let 𝑡∈[0,∞) and 𝑥∈𝒞 be fixed, assume that the set-valued map Γ has nonempty values and it admits a measurable selection Γ₀∶[0,∞)×𝒞×Ξ^∗→𝑈, and assume that a control 𝑢(⋅) satisfies 𝑢(𝑠)=Γ₀𝑠,𝑋^𝑢_𝑠,𝜁𝑠,𝑋^𝑢_𝑠,𝑃−𝑎.𝑠.,𝑓𝑜𝑟𝑎𝑙𝑚𝑜𝑠𝑡𝑒𝑣𝑒𝑟𝑦𝑠∈[𝑡,∞). (78) Then, 𝐽(𝑡,𝑥,𝑢)=𝜐(𝑡,𝑥), and the pair (𝑢(⋅),𝑋) is optimal for the control problem starting from 𝑥 at time 𝑡.
Such a control can be shown to exist if there exists a solution for the so-called closed-loop equation as follows: 𝑑𝑋(𝑠)=𝐴𝑋(𝑠)𝑑𝑠+𝐹𝑠,𝑋_𝑠𝑑𝑠+𝐺𝑠,𝑋_𝑠×𝑅𝑠,𝑋_𝑠,Γ₀𝑠,𝑋_𝑠,𝜁𝑠,𝑋_𝑠(𝑠,[),𝑋𝑑𝑠+𝑑𝑊))𝑠∈𝑡,∞_𝑡=𝑥, (79) since in this case, we can define an optimal control setting 𝑢(𝑠)=Γ₀𝑠,𝑋_𝑠,𝜁𝑠,𝑋_𝑠. (80) However, under the present assumptions, we cannot guarantee that the closed-loop equation has a solution in the mild sense. To circumvent this difficulty, we will revert to a weak formulation of the optimal control problem.

5. Existence of Optimal Control

We formulate the optimal control problem in the weak sense following the approach of [31]. The main advantage is that we will be able to solve the closed-loop equation in a weak sense, and, hence, to find an optimal control, even if the feedback law is nonsmooth.

We call (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃,𝑊) an admissible setup, if (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) is a filtered probability space satisfying the usual conditions, and 𝑊 is a cylindrical 𝑃-Wiener process with values in Ξ, with respect to the filtration {ℱ_𝑡}_𝑡≥0.

By an admissible control system, we mean (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃,𝑊,𝑢,𝑋^𝑢), where (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃,𝑊) is an admissible setup, 𝑢 is an ℱ_𝑡-predictable process with values in 𝑈, and 𝑋^𝑢 is a mild solution of (58). An admissible control system will be briefly denoted by (𝑊,𝑢,𝑋^𝑢) in the following. Our purpose is to minimize the cost functional 𝐽(𝑢)=𝐸^∞_𝑡𝑒^−𝜆𝑠𝑔𝑠,𝑋^𝑢_𝑠,𝑢(𝑠)𝑑𝑠, (81) over all the admissible control system.

Our main result in this section is based on the solvability of the closed-loop equation 𝑑𝑋(𝑠)=𝐴𝑋(𝑠)𝑑𝑠+𝐹𝑠,𝑋_𝑠𝑑𝑠+𝐺𝑠,𝑋_𝑠×𝑅𝑠,𝑋_𝑠,Γ₀𝑠,𝑋_𝑠,𝜁𝑠,𝑋_𝑠(𝑠,[),𝑋𝑑𝑠+𝑑𝑊))𝑠∈𝑡,∞_𝑡=𝑥. (82) In the following sense, we say that 𝑋 is a weak solution of (82) if there exists an admissible setup (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃,𝑊) and an ℱ_𝑡-adapted continuous process 𝑋(𝑡) with values in 𝐻, which solves the equation in the mild sense; namely, 𝑃-a.s., 𝑋(𝑠)=𝑒^{(𝑠−𝑡)𝐴}𝑥(0)+^𝑠_𝑡𝑒^{(𝑠−𝜎)𝐴}𝐹𝜎,𝑋_𝜎+𝑑𝜎_𝑠𝑡𝑒^{(𝑠−𝜎)𝐴}𝐺𝜎,𝑋_𝜎𝑅×𝜎,𝑋_𝜎,Γ₀𝜎,𝑋_𝜎,𝜁𝜎,𝑋_𝜎+𝑑𝜎_𝑠𝑡𝑒^{(𝑠−𝜎)𝐴}𝐺𝜎,𝑋_𝜎𝑑𝑊_𝜎[),𝑋,𝑠∈𝑡,∞_𝑡=𝑥. (83)(84)

Theorem 13. Assume that Hypothesis 3 holds. Then, there exists a weak solution of the closed-loop equation (82) which is unique in law.

Proof (uniqueness). Let 𝑋 be a weak solution of (82) in an admissible setup (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃,𝑊). We define 𝜌(𝑇)=exp^𝑇_𝑡−𝑅^∗𝜎,𝑋_𝜎,Γ₀𝜎,𝑋_𝜎,𝜁𝜎,𝑋_𝜎(𝜎)−1𝑑𝑊2^𝑇_𝑡𝑅𝜎,𝑋_𝜎,Γ₀𝜎,𝑋_𝜎,𝜁𝜎,𝑋_𝜎²𝑑𝜎. (85) Since 𝑅 is bounded, the Girsanov theorem ensures that there exists a probability measure 𝑃⁰ such that the process 𝑊⁰(𝑠)=𝑊(𝑠)+^𝑠_𝑡∧𝑠𝑅𝜎,𝑋_𝜎,Γ₀𝜎,𝑋_𝜎,𝜁𝜎,𝑋_𝜎[),𝑑𝜎,𝑠∈0,∞ (86) is a 𝑃⁰-Wiener process and 𝑃⁰∣_ℱ𝑇=𝜌(𝑇)𝑃∣_ℱ𝑇. (87) Let us denote by {ℱ⁰_𝑡}_𝑡≥0 the filtration generated by 𝑊⁰ and completed in the usual way. In (Ω,ℱ_[0,∞),{ℱ⁰_𝑡}_𝑡≥0,𝑃⁰), 𝑋 is a mild solution of 𝑑𝑋(𝑠)=𝐴𝑋(𝑠)𝑑𝑠+𝐹𝑡,𝑋_𝑠𝑑𝑠+𝐺𝑠,𝑋_𝑠𝑑𝑊⁰(𝑠)[),𝑋,𝑠∈𝑡,∞_𝑡𝜌(𝑇)=𝑥,=exp^𝑇_𝑡−𝑅^∗𝜎,𝑋_𝜎,Γ₀𝜎,𝑋_𝜎,𝜁𝜎,𝑋_𝜎𝑑𝑊⁰(𝜎)+12^𝑇_𝑡𝑅𝜎,𝑋_𝜎,Γ₀𝜎,𝑋_𝜎,𝜁𝜎,𝑋_𝜎²𝑑𝜎. (88) By Hypothesis 3, the joint law of 𝑋 and 𝑊⁰ is uniquely determined by 𝐴,𝐹,𝐺, and 𝑥. Taking into account the last displayed formula, we conclude that the joint law of 𝑋 and 𝜌(𝑇) under 𝑃⁰ is also uniquely determined, and consequently so is the law of 𝑋 under 𝑃. This completes the proof of the uniqueness part.

Proof (existence). Let (Ω,ℱ,𝑃) be a given complete probability space. {𝑊(𝑡),𝑡≥0} is a cylindrical Wiener process on (Ω,ℱ,𝑃) with values in Ξ, and {ℱ_𝑡}_𝑡≥0 is the natural filtration of {𝑊(𝑡),𝑡≥0}, augmented with the family of 𝑃-null sets. Let 𝑋(⋅) be the mild solution of 𝑑𝑋(𝑠)=𝐴𝑋(𝑠)𝑑𝑠+𝐹𝑠,𝑋_𝑠𝑑𝑠+𝐺𝑠,𝑋_𝑠(𝑠)[),𝑋𝑑𝑊,𝑠∈𝑡,∞_𝑡=𝑥, (89) and by the Girsanov theorem, let 𝑃¹ be the probability on Ω under which 𝑊¹(𝑠)=𝑊(𝑠)−^𝑠_𝑡∧𝑠𝑅𝜎,𝑋_𝜎,Γ₀𝜎,𝑋_𝜎,𝜁𝜎,𝑋_𝜎𝑑𝜎 (90) is a Wiener process (notice that 𝑅 is bounded). Then, 𝑋 is the weak solution of (82) relatively to the probability 𝑃¹ and the Wiener process 𝑊¹.

Now, we can state the main result of this section.

Corollary 14. Assume that Hypothesis 3 holds true and 𝜆 verifies (62) Also, assume that the set-valued map Γ has nonempty values and it admits a measurable selection Γ₀∶[0,∞)×𝒞×Ξ^∗→𝑈. Then, for every 𝑡∈[0,∞) and x∈𝒞 and for all admissible control system (𝑊,𝑢,𝑋^𝑢), one has 𝐽(𝑢,𝑡,𝑥)≥𝜐(𝑡,𝑥), (91) and the equality holds if 𝑢(𝑠)=Γ₀𝑠,𝑋^𝑢_𝑠,𝜁𝑠,𝑋^𝑢_𝑠,𝑃−𝑎.𝑠.𝑓𝑜𝑟𝑎𝑙𝑚𝑜𝑠𝑡𝑒𝑣𝑒𝑟𝑦𝑠∈[𝑡,∞). (92)

Moreover, from Theorem 13, it follows that the closed-loop equation (82) admits a weak solution (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃,𝑊,𝑋) which is unique in law, and setting 𝑢(𝑠)=Γ₀𝑠,𝑋_𝑠,𝜁𝑠,𝑋_𝑠, (93) we obtain an optimal admissible control system (𝑊,𝑢,𝑋).

6. Applications

In this section, we present a simple application of the previous results. We consider the stochastic delay partial differential equation in the bounded domain 𝐵⊂𝑅^𝑛 with smooth boundary 𝜕𝐵 as follows: 𝑑𝑧^𝑢(𝑡,𝜉)=Δ𝑧^𝑢(𝑡,𝜉)𝑑𝑡+𝑓𝑡,𝑧^𝑢_𝑡(𝜉+)𝑑𝑡_𝑑𝑖=1𝑔_𝑖𝑡,𝑧^𝑢_𝑡(𝜉)𝑟^𝑖(𝜉)𝑢^𝑖(𝑡)𝑑𝑡+𝑑𝑊^𝑖(𝑡,𝑧_𝑢0)()()[],𝑧𝜃,𝜉=𝑥𝜃,𝜉,𝜉∈𝐵,𝜃∈−1,0_𝑢(𝑡,𝜉)=0,𝑡∈[0,∞),𝜉∈𝜕𝐵. (94) Here, 𝑊=(𝑊¹,𝑊²,…,𝑊^𝑑) is a standard Wiener process in 𝑅^𝑑, and the functions 𝑓∶[0,+∞)×𝐶([−1,0],𝑅)→𝑅 and 𝑔_𝑖∶[0,+∞)×𝐶([−1,0],𝑅)→𝑅 are Lipschitz continuous and bounded. Setting 𝑈 as a bounded subset of 𝑅^𝑑, Ξ=𝑅^𝑑, 𝐻=𝐿²(𝐵), and 𝑥∈𝐶([−1,0],𝐻). We define 𝐹 and 𝐺 as following: 𝐹(𝜉)((𝜉,(𝐺()𝑧𝜉)=𝑡,𝑥)(=𝑓𝑡,𝑥))𝑡,𝑥)(^𝑑_𝑖=1𝑔_𝑖(𝑡,𝑥(𝜉))𝑧^𝑖(𝜉),𝜉∈𝐵,𝑥∈𝐶([−1,0],𝐻),𝑧∈𝐿(Ξ,𝐻), (95) and let 𝐴 denote the Laplace operator Δ in 𝐿²(𝐵); with domain 𝑊^2,2(𝐵)⋂𝑊₀^1,2(𝐵) then, (94) has the form (58) and Hypothesis 1 holds.

Let us consider the optimal control problem associated with the cost 𝐽(𝑢)=𝐸^∞₀𝑒^−𝜆𝑡_𝐵𝜎𝜉,𝑧^𝑢_𝑡(𝜉)𝑑𝜉+𝑢²(𝑡)𝑑𝑡, (96) where 𝜆 verifies (62) and 𝜎∶𝐶([−1,0],𝑅)×𝑈→[0,∞) is a bounded measurable function. Define 𝑔∶𝐶([−1,0],𝐻)×𝑈→[0,∞) and 𝑅∶𝐶([−1,0],𝐻)×𝑈→Ξ by 𝑔(𝑦,𝑢)=∫_𝐵𝜎(𝑡,𝑦(𝜉),𝑢)𝑑𝜉+𝑢² and 𝑅(𝑦,𝑢)=(∫_𝐵𝑟¹(𝜉)𝑢¹𝑑𝜉,∫_𝐵𝑟²(𝜉)𝑢²𝑑𝜉,…,∫_𝐵𝑟^𝑑(𝜉)𝑢^𝑑𝑑𝜉) for 𝑦∈𝐶([−1,0],𝐻), 𝑢=(𝑢¹,𝑢²,…,𝑢^𝑑)∈𝑈, respectively. It can be easily verified that Hypothesis 3 holds true, and the set-valued map Γ has nonempty values and it admits a measurable selection Γ₀∶[0,∞)×𝒞×Ξ^∗→𝑈. Then, the closed-loop equation (82) admits a weak solution (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃,𝑊,𝑢,𝑧_⋅(⋅)), and setting 𝑢(𝑠)=Γ₀𝑠,𝑧_𝑠(⋅),𝜁𝑠,𝑧_𝑠(⋅), (97) we obtain an optimal admissible control system (𝑊,𝑢,𝑧(⋅)).