Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise

HAL Id: hal-01872354

https://hal.archives-ouvertes.fr/hal-01872354

Preprint submitted on 12 Sep 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise

Ying Hu, Shanjian Tang

To cite this version:

Ying Hu, Shanjian Tang. Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise. 2018. �hal-01872354�

Stochastic LQ and Associated Riccati equation of PDEs Driven by State- and Control-Dependent White Noise

Ying Hu ^∗ Shanjian Tang ^† September 12, 2018

Abstract

The optimal stochastic control problem with a quadratic cost functional for linear par- tial differential equations (PDEs) driven by a state- and control-dependent white noise is formulated and studied. Both finite- and infinite-time horizons are considered. The multi- plicative white noise dynamics of the system give rise to a new phenomenon of singularity to the associated Riccati equation and even to the Lyapunov equation. Well-posedness of both Riccati equation and Lyapunov equation are obtained for the first time. The linear feedback coefficient of the optimal control turns out to be singular and expressed in terms of the solution of the associated Riccati equation. The null controllability is shown to be equivalent to the existence of the solution to Riccati equation with the singular terminal value. Finally, the controlled Anderson model is addressed as an illustrating example.

Keywords: linear quadratic optimal stochastic control, multiplicative space-time white noise, stochastic partial differential equation, Riccati equation, null controllability, singular termi- nal condition.

Mathematics Subject Classification (2010): 93E20, 60H15.

Short title: LQ control of white noise-driven PDEs.

∗IRMAR, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, and School of Mathematical Sciences, Fudan University, Shanghai 200433, China. Partially supported by Lebesgue Center of Mathematics

“Investissements d’avenir” program-ANR-11-LABX-0020-01, by ANR CAESARS (Grant No. 15-CE05-0024) and by ANR MFG (Grant No. 16-CE40-0015-01). email: ying.hu@univ-rennes1.fr

†Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China. Partially supported by National Science Foundation of China (Grant No. 11631004) and Science and Technology Commission of Shanghai Municipality (Grant No. 14XD1400400). email: sjtang@fudan.edu.cn

1 Introduction

In this paper, we consider the following stochastic evolutionary equation driven by both state- and control-dependent white noise:

dX_t = (AXt+B_tu_t)dt+ X∞ j=1

(Cj(t)Xt+D_j(t)ut)dβ_t^j, X₀ =x∈H

whereAis the infinitesimal generator of a strongly continuous semigroupe^tAof linear operators, B, C_j,and D_j are some bounded operators, and W is a cylindrical Wiener process in a Hilbert space H, with {β^j(t) := hW(t), eji, j = 1,2, . . .} being independent Brownian motions for an orthonormal basis {ej, j= 1,2, . . .}of H. The cost functional is

J(x, u) =E Z T

0

[hQtXt, Xti+hRtut, uti]dt+E[hGXT, XTi],

where Q, G, and R are some bounded operators. The optimal control problem is to find a U-valued adapted square-integrable process u in a feedback form (via the associated Riccati equation) such that J(x, u) is the minimal value of the cost functional J(x,·). More precise formulation will be given in the next section.

The general theory of linear quadratic optimal control (the so-called LQ theory) of Kalman [15]

paved one mile stone in the deterministic optimal control theory. The general stochastic exten- sion in a Euclidean space was given by Wonham [21] for the deterministic coefficients, and was further developed by Bismut [1] for the random coefficients. Subsequently, it was further studied by Peng [17] and Tang [19], and its theory is now rather complete.

Ichikawa [13, 14] considered the infinite-dimensional extension of Kalman’s LQ theory under the following setting: H is an infinite-dimensional Hilbert space and C is a bounded linear operator. Da Prato and Ichikawa [6] studied the infinite-dimensional LQ problem for the case of D= 0, self-adjointA, and unbounded coefficientB. The infinite dimensional case with stochastic coefficients driven by the so-called colored noise (where C is a Hilbert-Schmidt operator) is referred to Guatteri and Tessitore [10]. To our best knowledge, all the above-mentioned papers are restricted within the case when the linear SPDEs are driven by the so-called colored noise, which excludes the celebrated Anderson model. In this paper, we address the infinite dimensional stochastic LQ problem driven by an infinite number of Brownian motions (the so-called space- time white noise).

The introduction of the space-time white noise leads to the difficulty that the infinite sumP∞

i=1C_i^∗(s)P_sC_i(s) appears in both associated Lyapunov equation (3.9) and Riccati equa- tion (4.3), and thus challenges the solvability of both equations. To overcome this difficulty for Lyapunov equation (3.9), we introduce the representation via the solution of forward SPDE to establish an estimate of the sum, and for more details, see our Proposition 3.3 and its proof.

It is conventional to study the Riccati equation via the quasi-linearization method. While in

our context of the space-time white noise, the coefficients of these quasi-linearized equations become singular in the sense that these coefficients explode at both ends (time 0 and timeT).

Some fine estimates are applied to deduce the monotonicity and convergence of solutions of quasi-linearized equations. For more details, see our Theorem 4.4 and its proof. Finally, due to the space-time white noise in our context, the conventional Yosida’s approximation could not be applied to get the energy equality, and to attack the new difficulty, a new truncation is carefully constructed to deduce the energy equality and thus the feedback law of the optimal control. For more details, see our Theorem 5.2 and its proof.

We note that Anderson model has been widely studied in the litterature, and for more details, see Carmona and Molchanov [3], Conus, Joseph, and Khoshnevisan [4], and the references therein. We also emphasize that our results succeed at inclusion of the controlled Anderson SPDE. See Section 8.

The paper is organized as follows. In Section 2, we give the precise formulation of our quadratic optimal stochastic control problem for linear partial differential equations driven by a white noise. In Section 3, we study well-posedness of Lyapunov equations. In Section 4, we study the associated Riccati equation. In Section 5, we characterize the optimal control as a feedback form via the solution of Riccati equation. In Section 6, we address the infinite-horizon LQ control problem for the case of time-invariant coefficients. We show that when the system is stabilizable, the associated algebraic Riccati equation has a unique solution, and is again used to synthesize the optimal control into a feedback form. In Section 7, the null controllability is proved to be equivalent to the existence of solution of Riccati equation with the singular terminal condition. Finally in Section 8, we give examples for the controlled Anderson model.

2 Formulation of the linear quadratic optimal control

LetH, U be two separable Hilbert spaces. By S(H), we denote the space of all self-adjoint and bounded linear operators on Hand byS⁺(H) we denote the set of all non-negative operators in S(H). Moreover, if I ⊂R⁺ is an interval (bounded or unbounded), we denote by Cs(I;S(H)) (resp. C_s(I;S⁺(H))) the set of all mapsf :I → S(H) (resp. f :I → S⁺(H)) such thatf(·) is strongly continuous inH.

Consider the following stochastic evolutionary equation driven by both state- and control- dependent white noise:

dX_t= (AXt+B_tu_t)dt+ X∞ j=1

(Cj(t)Xt+D_j(t)ut)dβ_t^j, X₀ =x∈H, (2.1) which has the following mild form:

X_t=e^Atx+ Z t

0

e^A(t−s)B_su_sds+ Z t

0

X∞ j=1

e^A(t−s)(C_j(s)X_s+D_j(s)u_s)dβ_s^j. (2.2)

Here, A is the infinitesimal generator of a strongly continuous semigroup e^tA of linear opera- tors, B ∈ L^∞(0, T;L(U, H)), C_j ∈ L^∞(0, T;L(H)), D_j ∈ L^∞(0, T;L(U, H)) with the standard assumption that for someα∈(0,¹₂) and c >0,

X∞ j=1

|e^AtC_j(s)x|²_H ≤ct^−2α|x|²_H, t >0. (2.3) W is a cylindrical Wiener process in H, {β^j(t) := hW(t), eji, j = 1,2, . . .} are independent Brownian motions, with{e_j, j= 1,2, . . .}being an orthonormal basis of H. The cost functional is

J(x, u) =E Z T

0

[hQtX_t, X_ti+hRtu_t, u_ti]dt+E[hGXT, X_Ti], u∈L²_F(0, T;U) (2.4) where Q ∈ L^∞(0, T;S⁺(H)), G ∈ S⁺(H), and R ∈ L^∞(0, T;S⁺(U)) is strictly positive in the following sense: there is a positive number δ such that R ≥ δI_U. Throughout the paper, we assume that for anyv∈U, there is a constant c >0 such that

X∞ j=1

|Djv|²_H ≤c|v|²_U. (2.5)

The optimal control problem is to minimize J(x,·) among all the controls inL²_F(0, T;U).

Remark 2.1 Condition (2.5) means that D= (D₁, D₂, . . .) is a Hilbert-Schmidt operator. It is still open how to replace this condition with a condition like (2.3).

Lemma 2.2 For u∈ L²_F(0, T;U), the system (2.1) has a unique mild solution X in the space C_F([0, T];L²(Ω, H)) such that for someC >0,

sup

0≤t≤TE[|Xt|²] ≤ C

|x|²+E Z T

0

|us|²ds

. (2.6)

Proof. The existence and uniqueness of the mild solution can be found in [8]. We now derive the desired estimate for the solution. From (2.2), (2.3), and (2.5), we have

E[|Xt|²] ≤ C

|x|²+E Z t

0

|us|²ds

+CE Z t

0



 X∞ j=1

|e^A(t−s)C_j(s)Xs|²+ X∞ j=1

|e^A(t−s)D_j(s)us|²



 ds

≤ C

|x|²+E Z t

0

|us|²ds

+CE Z t

0

X∞ j=1

(t−s)^−2α|Xs|²_Hds.

Using an extended Gronwall’s inequality (see, e.g. [12]), we have the desired estimate.

3 Lyapunov equation: existence and uniqueness of solutions

We first give results on Lyapunov equation, which will be needed in the study of Riccati equation.

3.1 Forward SDE

LetA₀,Cb_j ∈L^∞(0, T;L(H)) with j= 1,2, . . ..

Assume that for some number c >0,

|A₀(s)|_L(H)≤c(T−s)^−α, s∈[0, T), (3.1)

and X∞

j=1

|e^AtCb_j(s)x|²_H ≤c t^−2α+ (T−s)^−2α

|x|²_H, (t, s)∈(0,∞)×[0, T). (3.2) Remark 3.1 Assumptions (3.1) and (3.2) are introduced to study the quasi-linearized sequence of Lyapunov equations for the original nonlinear Riccati equation. Note that both assumptions admit explosion at timeT.

Consider the following forward evolution equation: given the initial data (t, x), dY_s= (A+A₀(s))Y_sds+

X∞ i=1

Cb_i(s)Y_sdβ_sⁱ, s∈(t, T]. (3.3) Lemma 3.2 Let Assumptions (3.1)and (3.2)hold true. There is a unique mild solution to (3.3) satisfying

sup

t≤s≤TE|Ys|²_H ≤C|x|² for a positive constant C.

Proof. First we prove the uniqueness. Consider two solutions Y¹ and Y². Define ∆Y :=

Y¹−Y².We have

∆Y_s= Z s

t

e^A(s−r)A₀(r)∆Y_rdr+ X∞

i=1

Z s

t

e^A(s−r)Cb_i(r)∆Y_rdβ_rⁱ, s∈[t, T] and

E[|∆Y_s|²] ≤ 2E

Z s

t

e^A(s−r)A₀(r)∆Y_rdr

2

+ 2E

X∞ i=1

Z s

t

e^A(s−r)Cb_i(r)∆Y_rdβ_rⁱ

2

≤ C Z s

t

(s−r)^−2α+ (T−r)^−2α

E[|∆Yr|²]dr

≤ 2C Z s

t

(s−r)^−2αE[|∆Yr|²]dr. (3.4)

Thus,E[|∆Yr|²] = 0, and the uniqueness is proved.

Then we prove the existence. Define by Picard’s iteration: Y⁰≡0, and forn≥0, Y_sⁿ⁺¹ = e^A(s−t)x+

Z s

t

e^A(s−r)A₀(r)Y_rⁿdr+ X∞ i=1

Z s

t

e^A(s−r)Cb_i(r)Y_rⁿdβ_rⁱ. Thus, we have

E

|Y_sⁿ⁺¹|²

≤ 3|e^A(s−t)x|²+ 2C Z s

t

(s−r)^−2αE

|Y_rⁿ|² dr.

Denote byγ the solution of the following integral equation:

γ_s= 3|e^A(s−t)x|²+ 2C Z s

t

(s−r)^−2αγ_rdr, s∈(t, T]. (3.5) By recurrence, we haveE

|Y_sⁿ|²

≤γ_s,fors∈[t, T].

Now we show that {Yⁿ, n≥0} is a Cauchy sequence in C_F([t, T];L²(Ω, H)). We have Y_s^n+k+1−Y_sⁿ⁺¹ =

Z s

t

e^A(s−r)A₀(r)

Y_r^n+k−Y_rⁿ dr +

X∞ i=1

Z s

t

e^A(s−r)Cb_i(r)

Y_r^n+k−Y_rⁿ dβ_rⁱ,

Eh

|Y_s^n+k+1−Y_sⁿ⁺¹|²i

≤ 2C Z s

t

(s−r)^−2αEh

|Y_r^n+k−Y_rⁿ|²i dr.

Define

φs = lim sup

n

sup

k

sup

t≤r≤sEh

|Y_r^n+k+1−Y_rⁿ⁺¹|²i . We have

sup

t≤r≤sEh

|Y_r^n+k+1−Y_rⁿ⁺¹|²i

≤ 2C Z s

t

(s−r)^−2αEh

|Y_r^n+k−Y_rⁿ|²i

dr, (3.6)

φs ≤ 2C Z s

t

(s−r)^−2αφrdr. (3.7)

This shows thatφ= 0 and{Yⁿ}is a Cauchy sequence inC_F([0, T];L²(Ω, H)), and the existence of solution is proved.

3.2 Lyapunov equation

LetG∈ S(H) and f ∈L¹(0, T;S(H)). Assume that for α∈(0,¹₂),

|f(s)|_L(H)≤c(T−s)^−2α. (3.8)

Consider the following form of Lyapunov equation







P_t^′+A^∗P_t+P_tA+A^∗₀(t)Pt+P_tA₀(t) + X∞

i=1

b

C_i^∗(t)PtCb_i(t) +f_t= 0, t∈[0, T);

P_T =G.

(3.9)

We look for a mild solution:

P_t = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t)f_se^A(s−t)ds (3.10)

+ Z T

t

e^A∗(s−t)

"

A^∗₀(s)Ps+P_sA₀(s) + X∞ i=1

b

C_i^∗(s)PsCb_i(s)

#

e^A(s−t)ds.

Using Yosida’s approximation, we can prove that the following Lyapunov equation (associ- ated to a finite number of Brownian motions)

P_tⁿ = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t)f_se^A(s−t)ds (3.11)

+ Z T

t

e^A∗(s−t)

"

A^∗₀(s)P_sⁿ+P_sⁿA₀(s) + Xn

i=1

b

C_i^∗(s)P_sⁿCb_i(s)

#

e^A(s−t)ds,

has a unique solution Pⁿ∈Cs([0, T],S(H)) (see, e.g. Da Prato [5]).

Proposition 3.3 Let Assumptions (3.1), (3.2)and (3.8) hold true. Then, Pⁿ converges weakly to a bounded solution P ∈ C_s([0, T];S(H)) of (3.10) satisfying the estimate for some positive constant C,

X∞ i=1

hCbi(s)^∗PsCbi(s)x, xi ≤C(T −s)^−2α|x|², s∈[0, T).

Moreover, we have the following representation of P: hPtx, xi=E

hGY_T^t,x, Y_T^t,xi+ Z T

t

hfsY_s^t,x, Y_s^t,xids

, t∈[0, T] (3.12) where Y^t,x is the mild solution to (3.3).

Proof. For each intergern, let Y^n,t,x be the mild solution of dY_s= [A+A₀(s)]Y_sds+

Xn

i=1

b

C_i(s)Ysdβ_sⁱ, s∈(t, T]; Y_t=x.

We have the following representation:

hP_tⁿx, xi=E

hGY_T^n,t,x, Y_T^n,t,xi+ Z T

t

hfsY_s^n,t,x, Y_s^n,t,xids

. (3.13)

Since

n→+∞lim sup

t≤s≤TE[|Y_s^n,t,x−Y_s^t,x|²] = 0

where Y^t,x is the mild solution to (3.3), there exists P_t ∈ S(H) such that P_tⁿ converges to P_t weakly and we have by passing to the limit in (3.13) the desired representation (3.12).

Setzi =Cbi(t)x.

Let us estimate P∞

i=1|hPtz_i, z_ii|. We have

|hP_tz_i, z_ii| = E

hGY_T^t,zi, Y_T^t,zii+ Z T

t

hf_sY_s^t,zi, Y_s^t,ziids

≤ CE[||Y_T^t,zi||²] +C Z T

t

(T−s)^−2αE[||Y_s^t,zi||²]ds. (3.14) AsY^t,zi is the mild solution of the following equation

Y_s^t,zi =e^A(s−t)z_i+ Z s

t

e^A(s−r)A₀(r)Y_r^t,zidr+ Z s

t

e^A(s−r) X∞

i=1

b

C_i(r)Y_r^t,zidβ_rⁱ, we have

E[||Y_s^t,zi||²] ≤ C||e^A(s−t)z_i||²+C Z s

t

(s−r)^−2α+ (T −r)^−2α

E[||Y_r^t,zi||²]dr

≤ C||e^A(s−t)zi||²+ 2C Z s

t

(s−r)^−2αE[||Y_r^t,zi||²]dr. (3.15) Note that

Xn

i=1

||e^A(s−t)zi||² = Xn

i=1

||e^A(s−t)Cbi(t)x||² ≤C (s−t)^−2α+ (T −t)^−2α

||x||² ≤2C(s−t)^−2α||x||². Finally, we get from (3.15) that

Xn

i=1

E[||Y_s^t,zi||²]≤C(s−t)^−2α||x||²+C Z s

t

(s−r)^−2α Xn

i=1

E[||Y_r^t,zi||²]dr.

By the generalized Gronwall’s inequality (see Henry [12]), we have Xn

i=1

E[||Y_s^t,zi||²] ≤ C(s−t)^−2α||x||², and then letting n→ ∞, we have

X∞ i=1

E[||Y_s^t,zi||²]≤C(s−t)^−2α||x||². (3.16) Furthermore from (3.14) and (3.16), we have

X∞ i=1

hCbi(t)^∗PtCbi(t)x, xi = X∞

i=1

|hPtzi, zii|

≤ C(T −t)^−2α+C Z T

t

(T−s)^−2α(s−t)^−2α|x|²_Hds

= C(T −t)^−2α+C Z 1

0

(T−t)^−2α(1−r)^−2α(T −t)^−2αr^−2α(T −t)dr

= C(T −t)^−2α+C(T−t)^1−4α ≤C(1 +T^1−2α)(T−t)^−2α.

Passing to the limit in (3.11) by letting n→ ∞, we prove that P is the solution to (3.10).

Theorem 3.4 There exists a unique solutionP ∈C_s([0, T],S(H)) for (3.9) such that X∞

i=1

hCb_i(s)^∗P_sCb_i(s)x, xi ≤C(T −s)^−2α|x|².

Proof. The existence of solution is already proved in the preceding proposition. Now we prove the uniqueness.

Let ˜P be a solution, then it satisfies the following truncated Riccati equation:

P˜_t = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t) Xn

i=1

b

C_i^∗(s) ˜P_sCb_i(s)e^A(s−t)ds +

Z T

t

e^A∗(s−t)(fs+ X∞ i=n+1

Cb_i^∗(s) ˜PsCbi(s))e^A(s−t)ds. (3.17)

We have from (3.13) the following representation hP˜_tx, xi=E

"

hGY_T^n,t,x, Y_T^n,t,xi+ Z T

t

h(fs+ X∞ i=n+1

b

C_i^∗(s) ˜P_sCb_i(s))Y_s^n,t,x, Y_s^n,t,xids

# .

By passing to the limit, we deduce hP˜tx, xi=E

hGY_T^t,x, Y_T^t,xi+ Z T

t

hfsY_s^t,x, Y_s^t,xids

, from which we deduce the uniqueness.

From (3.12), we deduce also the following a priori estimate.

Proposition 3.5 Let P ∈ Cs([0, T],S(H)) be the unique solution, then the following a priori estimate holds:

|Pt| ≤C(|G|_L(H)+ Z T

t

|fs|_L(H)ds).

4 Riccati equation: existence and uniqueness of solutions

In this section, we study the Riccati equation associated to the linear-quadratic optimal control problem (2.2) and (2.4). Let us first state a lemma which will be used later.

Lemma 4.1 Let assumption (2.5) hold true. For P ∈ S⁺(H) such that for any x∈H, X∞

j=1

hC_j^∗(t)P Cj(t)x, xi<∞.

Then, PN

j=1D_j^∗(t)P C_j(t) converges strongly, whose limit is denoted by P∞

j=1D^∗_j(t)P C_j(t) and satisfies the following estimate:

X∞ j=1

D_j^∗(t)P Cj(t)x _U ≤C



 X∞ j=1

hC_j^∗(t)P Cj(t)x, xi





1 2

, x∈H

for some constant C >0.

Proof. In view of Assumption (2.5),

XN

j=M+1

D^∗_j(t)P Cj(t)x

_U = sup

|y|U≤1

* _N X

j=M+1

D_j^∗(t)P Cj(t)x, y +

≤ sup

|y|U≤1



 XN

j=M+1

|P¹²Cj(t)x|²_H





1

2 

 XN

j=M+1

|P¹²Dj(t)y|²_H





1 2

≤ C



 XN

j=M+1

|P¹²Cj(t)x|²_H





1 2

= C



 XN

j=M+1

hC_j^∗(t)P Cj(t)x, xi





1 2

.

Hence the sequencePN

j=1D^∗_j(t)P Cj(t)x is a Cauchy one, and we have the desired result.

Define for P ∈ S⁺(H),

Λ(t, P) :=Rt+ X∞ i=1

D^∗_i(t)P Di(t).

Since Λ(t, P)≥δI_U, we see that Λ(t, P) has an inverse Λ(t, P)⁻¹ ≤ ¹_δI_U. Define for s∈[0, T] and P ∈ S⁺(H),

λ(s, P) :=−Λ(s, P)⁻¹ B_s^∗P +X

i

D_i^∗(s)P C_i(s)

! , and for P ∈ S⁺(H) such that P∞

j=1hC_j^∗(t)P Cj(t)x, xi<∞for each x∈H, Bˆ(t, P) := −B_tΛ(t, P)⁻¹



B^∗_tP +X

j

D_j^∗(t)P C_j(t)



=B_tλ(t, P),

Cˆ_i(t, P) := C_i(t)−D_i(t)Λ(t, P)⁻¹



B_t^∗P+X

j

D_j^∗(t)P C_j(t)



=C_i(t) +D_i(t)λ(t, P).

We have

Lemma 4.2 ForP ∈C_s([0, T],S⁺(H)) such that

| X∞

i=1

C_i^∗(s)PsC_i(s)|_L(H)≤C(T −s)^−2α,

we have X

i

|e^AtCb_i(s, P_s)x|² ≤c t^−2α+ (T −s)^−2α

|x|²,

|Qs+λ^∗(s, Ps)Rsλ(s, Ps)|_L(H)≤c(T−s)^−2α,

|B(s, Pb s)|_L(H)≤c(T−s)^−α.

Proof. The third inequality is obvious. We now prove the first inequality.

X

i

|e^AtCb_i(s, Ps)x|²

≤ 2X

i

|e^AtC_i(s)x|²+ 2X

i

e^AtD_i(s)Λ(s, Ps)⁻¹



B_s^∗P_s+X

j

D^∗_j(s)PsC_j(s)



x

2

≤ 2ct^−2α|x|²+ 2c|e^At|_L(H)

Λ(s, P_s)⁻¹



B_s^∗P_s+X

j

D_j^∗(s)P_sC_j(s)



x

2

≤ 2ct^−2α|x|²+C|x|²+C

*X

j

C_j^∗(s)PsC_j(s)x, x +

≤ c t^−2α+ (T −s)^−2α

|x|². (4.1)

It remains to prove the second inequality. We have for each x∈H, sinceRs ≤Λ(s, Ps), hλ^∗(s, Ps)Rsλ(s, Ps)x, xi

≤

*

(P_sB_s+X

i

C_i^∗(s)P_sD_i(s))Λ(s, P_s)⁻¹(B_s^∗P_s+X

i

D_i^∗(s)P_sC_i(s))x, x +

≤ 2

PsBsΛ(s, Ps)⁻¹B_s^∗Psx, x + 2

*X

j

C_j^∗(s)PsDj(s)Λ(s, Ps)⁻¹X

i

D^∗_i(s)PsCi(s)x, x +

≤ 2

P_sB_sΛ(s, P_s)⁻¹B_s^∗P_sx, x + 2

*

Λ(s, P_s)⁻¹X

i

D_i^∗(s)P_sC_i(s)x,X

i

D_i^∗(s)P_sC_i(s)x +

≤ c(T−s)^−2α|x|²_H.

Let us consider the general Riccati equation:

Pt = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t) X∞ i=1

C_i^∗(s)PsCi(s)e^A(s−t)ds +

Z T

t

e^A∗(s−t)(Qs−λ^∗(s, Ps)Λ(s, Ps)λ(s, Ps))e^A(s−t)ds, t∈[0, T]. (4.2)

It is equivalent to the following form:

P_t = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t)

Bˆ^∗(s, Ps)Ps+P_sBˆ(s, Ps)

e^A(s−t)ds +

Z T

t

e^A∗(s−t) X∞ i=1

Cˆ_i^∗(s, Ps)PsCˆ_i(s, Ps)e^A(s−t)ds

+ Z T

t

e^A∗(s−t)(Qs+λ^∗(s, Ps)Rsλ(s, Ps))e^A(s−t)ds, t∈[0, T]. (4.3) Our existence proof will make use of the following quasi-linearized sequence {P^N} defined by the following Lyapunov equations: P⁰ ≡0, and

P_t^N+1 = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t)

Bˆ^∗(s, P_s^N)P_s^N+1+P_s^N+1B(s, Pˆ _s^N)

e^A(s−t)ds +

Z T

t

e^A∗(s−t) X∞ i=1

Cˆ_i^∗(s, P_s^N)P_s^N+1Cˆi(s, P_s^N)e^A(s−t)ds

+ Z T

t

e^A∗(s−t) Q_s+λ^∗(s, P_s^N)Rsλ(s, P_s^N)

e^A(s−t)ds, N = 0,1, . . . . (4.4) Note that if P^N ∈ C_s([0, T],S⁺(H)) and satisfies the inequality for a positive constant c which might depend onN:

|X

i

C_i^∗P_s^NC_i(s)|_L(H)≤c(T −s)^−2α,

then we see from Lemma 4.2 and Theorem 3.4 that the preceding Lyapunov equation (4.4) has a unique solutionP^N+1 ∈Cs([0, T],S⁺(H)) satisfying also the last inequality. Since obviously P⁰ satisfies the last inequality, we can define by induction a sequenceP^N+1 satisfying Lyapunov equation (4.4) forN ≥0 .

Lemma 4.3 The sequence {P_t^N, N ≥1} is a non-increasing sequence of self-adjoint operators for each t∈[0, T].

Proof. Now we show thatP_t^N ≥P_t^N+1 forN ≥1.

Define ∆P_t^N :=P_t^N −P_t^N+1, t∈[0, T]. We have

∆P_t^N = Z T

t

e^A∗(s−t)

Bˆ^∗(s, P_s^N−1)P_s^N +P_s^NB(s, Pˆ _s^N−1)

e^A(s−t)ds +

Z T

t

e^A∗(s−t) X∞ i=1

Cˆ_i^∗(s, P_s^N−1)P_s^NCˆ_i(s, P_s^N−1)e^A(s−t)ds

+ Z T

t

e^A∗(s−t)λ^∗(s, P_s^N−1)R_sλ(s, P_s^N−1)e^A(s−t)ds

− Z T

t

e^A∗(s−t)

Bˆ^∗(s, P_s^N)P_s^N+1+P_s^N+1B(s, Pˆ _s^N)

e^A(s−t)ds

− Z T

t

e^A∗(s−t) X∞ i=1

Cˆ_i^∗(s, P_s^N)P_s^N+1Cˆi(s, P_s^N)e^A(s−t)ds

− Z T

t

e^A∗(s−t)λ^∗(s, P_s^N)Rsλ(s, P_s^N)e^A(s−t)ds. (4.5) Define forK ∈ L(H, U) and P ∈ S⁺(H) such thatP∞

i=1hC_i^∗(s)P C_i(s)x, xi<∞, F(s, K, P) := (BsK)^∗P+P B_sK+

X∞ i=1

[Ci(s) +D_i(s)K]^∗P[Ci(s) +D_i(s)K] +K^∗R_sK. (4.6) We have for K∈L(H, U),

F(s, K, P) =F(s, λ(s, P), P) + [K−λ(s, P)]^∗Λ(s, P) [K−λ(s, P)]≥F(s, λ(s, P), P).

Equality (4.5) can be written into the following form:

∆P_t^N = Z T

t

e^A∗(s−t)

Bˆ^∗(s, P_s^N)∆P_s^N+ ∆P_s^NB(s, Pˆ _s^N)

e^A(s−t)ds +

Z T

t

e^A∗(s−t) X∞ i=1

Cˆ_i(s, P_s^N)∆P_s^NCˆ_i(s, P_s^N)e^A(s−t)ds

+ Z T

t

e^A∗(s−t)

F(s, λ(s, P_s^N−1), P_s^N)−F(s, λ(s, P_s^N), P_s^N)

e^A(s−t)ds. (4.7) Note that F(s, λ(s, P_s^N−1), P_s^N)−F(s, λ(s, P_s^N), P_s^N) ∈ S⁺(H) for each s ∈ [0, T]. Therefore, we have from the representation theorem that ∆P_s^N ≥0.

We have the following theorem.

Theorem 4.4 The Riccati equation (4.2)has a unique solutionP ∈C_s([0, T];S⁺(H))such that

|X

i

C_i^∗P_sC_i(s)|_L(H) ≤C(T −s)^−2α, s∈[0, T].

Proof. First, we see from the last lemma that P^N is a nondecreasing sequence of self- adjoint operators. Moreover, we see from (3.12) that each P^N is non-negative. Using the monotone sequence theorem (see Kantorovich and Akilov [16, Theorem 1, p. 169]), we see that P_t^N converges strongly to a non-negative self-adjoint operator, denoted byPt, which also satisfies the last inequality.

AsP_t^N converges strongly toP_t, noting the following

Λ(t, P_t^N)⁻¹−Λ(t, Pt)⁻¹ = Λ(t, P_t^N)⁻¹ Λ(t, Pt)−Λ(t, P_t^N)

Λ(t, Pt)⁻¹

= Λ(t, P_t^N)⁻¹ X∞ i=1

D^∗_i(t)(Pt−P_t^N)Di(t)

!

Λ(t, Pt)⁻¹, we see that Λ(t, P_t^N)⁻¹ converges strongly to Λ(t, Pt)⁻¹.

In view of our assumption (2.5), we have

X∞ j=1

D_j^∗(t)P_t^NCj(t)x− X∞ j=1

D^∗_j(t)PtCj(t)x _U

= sup

|y|U≤1

* _∞ X

j=1

D_j^∗(t)P_t^NC_j(t)x− X∞ j=1

D_j^∗(t)PtC_j(t)x, y +

= sup

|y|U≤1

X∞ j=1

D(P_t^N −P_t)¹²C_j(t)x,(P_t^N −P_t)¹²D_j(t)yE

≤ C



 X∞ j=1

(P_t^N −P_t)¹²C_j(t)x²





1 2

= C



 X∞ j=1

C_j^∗(t)(P_t^N−P_t)Cj(t)x, x



1 2

.

SincehC_j^∗(t)(P_t^N −P_t)C_j(t)x, xi ≤ hC_j^∗(t)(P_t¹−P_t)C_j(t)x, xi and X∞

j=1

h(C_j^∗(t)(P_t¹−P_t)Cj(t)x, xi<∞, by the Dominated Convergence Theorem, we see thatP∞

j=1D_j^∗(t)P_t^NC_j(t) converges strongly to P∞

j=1D_j^∗(t)PtC_j(t). Therefore, the non-homogeneous term in the Lyapunov equation of P^N+1 converges strongly.

By passing to the strong limit in the Lyapunov equation (4.4), we conclude that P is a solution.

Finally, we show the uniqueness. Let Pe be another solution of Riccati equation (4.2) such that

|X

i

C_i^∗(s)Pe_sC_i(s)|_L(H)≤C(T−s)^−2α, s∈[0, T).

DefineδP :=P −Pe. Then proceeding identically as in the last lemma, we have δP_t =

Z T

t

e^A∗(s−t)

Bˆ^∗(s, P_s)δP_s+δP_sB(s, Pˆ _s)

e^A(s−t)ds +

Z T

t

e^A∗(s−t) X∞

i=1

Cˆ_i(s, Ps)δPsCˆ_i(s, Ps)e^A(s−t)ds

+ Z T

t

e^A∗(s−t)h

F(s, λ(s, P_s),Pe_s)−F(s, λ(s,Pe_s),Pe_s)i

e^A(s−t)ds. (4.8) Since F(s, λ(s, Ps), Ps)−F(s, λ(s,Pe_s), Ps) is non-negative, we have δP ≥ 0. By symmetry, we also haveδP ≤0. Hence, we have δP ≡0.

5 Optimal feedback control

In this section, we study the linear quadratic optimal control problem (2.1) and (2.4).

Note that Itˆo’s formula could not be applied to systems driven by a white noise. To overcome the difficulty, we truncate the white noise by a finite number of Brownian motions.

Define X^N to be the unique solution of the following truncated state equation:

dX_t^N = (AX_t^N +Btut)dt+ XN

j=1

Cj(t)X_t^N +Dj(t)ut

dβ^j_t, (5.1) X₀^N = x∈H.

We denote by P^N the solution of the following truncated Lyapunov equation:

P_t^N = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t)

Bˆ^∗(s, Ps)P_s^N +P_s^NB(s, Pˆ s)

e^A(s−t)ds

+ Z T

t

e^A∗(s−t) XN

i=1

Cˆ_i^∗(s, Ps)P_s^NCˆ_i(s, Ps)e^A(s−t)ds (5.2)

+ Z T

t

e^A∗(s−t)(Qs+λ^∗(s, Ps)Rsλ(s, Ps))e^A(s−t)ds, t∈[0, T], whereP is the unique solution of the Riccati equation (4.2).

We have

Lemma 5.1 For t ∈ [0, T], P_t^N is non-decreasing, and is bounded from above, and strongly converges toP_t.

Proof. In view of (5.2), we see from the representation thatP_t^N is nonnegative for each N ≥1.

Furthermore, we have P_t^N+1−P_t^N =

Z T

t

e^A∗(s−t)

Bˆ^∗(s, Ps)(P_s^N+1−P_s^N) + (P_s^N+1−P_s^N) ˆB(s, Ps)

e^A(s−t)ds +

Z T

t

e^A∗(s−t) XN

j=1

C_j^∗(s)(P_s^N+1−P_s^N)C_j(s)e^A(s−t)ds +

Z T

t

e^A∗(s−t)Cb_N+1^∗ (s, Ps)P_s^N+1Cb_N+1(s, Ps)e^A(s−t)ds.

From the representation of the solution of Lyapunov equation, it is clear that P^N is non- decreasing. Using the same argument to consider the equation ofP−P^N, we see thatP_t^N ≤P_t. Hence there is a bounded P ≤ P such that P^N strongly converges to P which satisfies the Lyapunov equation:

P_t = e^A∗(T−t)Ge^A(T−t)+ Z T

t

e^A∗(s−t)

Bˆ^∗(s, Ps)Ps+P_sB(s, Pˆ s)

e^A(s−t)ds +

Z T

t

e^A∗(s−t) X∞ i=1

Cˆ_i^∗(s, Ps)PsCˆ_i(s, Ps)e^A(s−t)ds (5.3)

+ Z T

t

e^A∗(s−t)(Qs+λ^∗(s, Ps)Rsλ(s, Ps))e^A(s−t)ds, t∈[0, T], and

|X

i

C_i^∗PsCi(s)|_L(H)≤C(T−s)^−2α, s∈[0, T].

Since P (as a solution to the Riccati equation) is also a solution to the preceding Lyapunov equation with the non-homogeneous term being f_s := Q_s+λ^∗(s, P_s)R_sλ(s, P_s), we conclude from the uniqueness of the solution to the Lyapunov equation thatP =P.

Theorem 5.2 The cost functional has the following representation : J(x, u) =hP₀x, xi+E

Z T 0

hΛ(s, Ps)(us−λ(s, Ps)Xs), us−λ(s, Ps)Xsids

. The following feedback form:

ut=λ(t, Pt)Xt, t∈[0, T], (5.4) withX being the solution of the associated feedback system, is admissible and optimal.

Proof. We have the duality between the truncated state equation and the truncated Lyapunov equation by Yosida approximation ofA:

E

hGX_T^N, X_T^Ni+ Z T

0

h(Qs+λ^∗(s, Ps)Rsλ(s, Ps))X_s^N, X_s^Ni

ds

= hP₀^Nx, xi+ 2E Z T

0

hP_s^NX_s^N, B_su_sids

−E Z T

0

*

 bB^∗(s, Ps)P_s^N +P_s^NBb(s, Ps) + XN

j=1

b

C_j^∗(s, Ps)P_s^NCb_j(s, Ps)



X_s^N, X_s^N +

ds

+E Z T

0

XN

j=1

P_s^N(C_j(s)X_s^N +D_j(s)u_s), C_j(s)X_s^N +D_j(s)u_s ds.

SettingN → ∞, noting the following limit

N→∞lim sup

0≤t≤TE

|X_t^N −Xt|²

= 0, and Lemma 5.1, we have

E

hGXT, X_Ti+ Z T

0

h(Qs+λ^∗(s, Ps)Rsλ(s, Ps))Xs, X_si

ds

= hP₀x, xi+ 2E Z T

0

hPsXs, Bsusids

−E Z T

0

*

 bB^∗(s, P_s)P_s+P_sBb(s, P_s) + X∞ j=1

Cb_j^∗(s, P_s)P_sCb_j(s, P_s)



X_s, X_s +

ds

+E Z T

0

X∞ j=1

hPs(Cj(s)Xs+Dj(s)us), Cj(s)Xs+Dj(s)usids.