c 2009 Society for Industrial and Applied Mathematics 

SIAM J. CONTROL OPTIM. Vol. 48, No. 3, pp. 1473–1488

AN LQ PROBLEM FOR THE HEAT EQUATION ON THE HALFLINE WITH DIRICHLET BOUNDARY CONTROL AND NOISE∗ G. FABBRI† AND B. GOLDYS† Abstract. We study a linear quadratic problem for a system governed by the heat equation on a halfline with boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated nonstochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem. Key words. heat equation, Dirichlet boundary conditions, boundary noise, boundary control, weighted L2 space, analytic semigroup, stochastic convolution, linear quadratic control problem, Riccati equation AMS subject classifications. 35G15, 37L55, 49N10 DOI. 10.1137/070711529

1. Introduction. In this paper we are concerned with a linear quadratic control problem for a heat equation on the halfline [0, ∞) with Dirichlet boundary control and boundary noise. More precisely, for fixed 0 ≤ τ < T , we deal with the equation ⎧ ∂ ∂2 t ∈ [τ, T ], ξ > 0, ⎨ ∂t y(t, ξ) = ∂ξ 2 y(t, ξ), ˙ (t), t ∈ [τ, T ], (1) y(t, 0) = u(t) + W ⎩ ξ > 0, y(τ, ξ) = x0 (ξ), where W is a one-dimensional Brownian motion and u is a square-integrable control. Let us recall that a deterministic boundary control problem ⎧ ∂ ∂2 t ∈ [τ, T ], ξ > 0, ⎨ ∂t z(t, ξ) = ∂ξ 2 z(t, ξ), (2) z(t, 0) = u(t), t ∈ [τ, T ], ⎩ z(τ, ξ) = x0 (ξ), ξ > 0, is well understood; see, for example, [3], [13]. Denoting by A0 the Dirichlet Laplacian in L2 (0, ∞) and by D the Dirichlet map (defined as Dλ0 in (10) below), we can rewrite (2) in the form  z(t) = etA0 x0 + (λ0 − A0 )

t

e(t−s)A0 Du(s)ds ,

0

and it is easy to show that z(t) ∈ L2 (0, ∞) for all t ≥ 0. Therefore, the process  (3) X(t) = etA0 x0 + (λ0 − A0 )

t

 e(t−s)A0 Du(s)ds + (λ0 − A0 )

0

t

e(t−s)A0 DdW (s)

0

∗ Received

by the editors December 20, 2007; accepted for publication (in revised form) January 28, 2009; published electronically April 15, 2009. This work was supported by ARC Discovery project DP0558539. http://www.siam.org/journals/sicon/48-3/71152.html † School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia (gieffe79@gmail. com, [email protected]). 1473

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G. FABBRI AND B. GOLDYS

seems to be a good candidate for a solution to (1). However, it was shown in [7] that the process X is not L2 -valued. More precisely, it was shown that the solution to (1) considered on a finite interval and for u = 0, when rewritten in the form (3), is well defined in a negative Sobolev space H −α for α > 12 only. It is easy to see that the same conclusion holds in the case of halfline. Then it was shown in [2] (see also [4]) that the process  X can be defined pointwise on (0, ∞) and it takes values in a weighted space L2 0, ∞; ξ 1+θ dξ . This fact was used to study some properties of the process X (in fact in the aforementioned papers more general nonlinear equations are studied), but the problem is not reformulated as a stochastic evolution equation in  L2 0, ∞; ξ 1+θ dξ , and therefore advantages of using the weighted space are somewhat limited. We introduce the weighted spaces Hρ = L2 ((0, ∞); ρ(ξ)dξ), following the idea of Krylov [12], where for θ ∈ (0, 1) we have   ρ(ξ) = ξ 1+θ or ρ(ξ) = min 1, ξ 1+θ , ξ ≥ 0. It was proved in [11] and [12] that the Dirichlet Laplacian A0 defined on L2 (0, ∞)   extends to a generator A of an analytic semigroup etA on Hρ . We will show that the Dirichlet map takes values in dom ((−A)α ) for a certain α > 12 and therefore (3), when considered in Hρ , can be given a form  t  t e(t−s)A (λ0 − A) Du(s)ds + e(t−s)A (λ0 − A0 ) DdW (s); X(t) = etA x0 + 0

0

that is, we will study a controlled evolution equation  dx(t) = (Ax(t) + Bu(t)) dt + B dW (t), (4) x(τ ) = x0 ∈ Hρ for B = (λ0 − A)D. This fact is a starting point for our analysis of the linear quadratic control problem (1). We will demonstrate that the control problem (4) when considered in the space Hρ can be solved using by now classical techniques presented, for example, in [3]. Let us emphasize that while the focus of this paper is on the most interesting case of boundary control and boundary noise a more general control problem  dx(t) = (Ax(t) + Bu1 (t) + v(t)) dt + B dW (t) + dW1 (t), (5) x(τ ) = x0 ∈ Hρ with spatially distributed noise W1 and control v might be easily considered using the same technique. Let us note that if the boundary conditions are of Neumann type, then the analogue of (1) has a solution in L2 (0, ∞) and has been studied intensely (also for more general parabolic equations with boundary noise); see, for example, [7], [17], [8], [9]. We study the linear quadratic problem characterized by the cost functional 

(6)

T

J(τ, x0 , u) = E τ

|Cx(t)|2Y + |u(t)|2R dt + Gx(T ), x(T )Hρ

and governed by a state equation of the form (4). The operator C that appears in (6) is in L(Hρ ; Y ) for a certain Hilbert space Y , and G ∈ L(Hρ ; Hρ ) is symmetric and positive. The direct solution of the Riccati equation related to a linear quadratic problem

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HEAT EQUATION WITH BOUNDARY CONTROL AND NOISE

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driven by a stochastic equation different from ours was studied in the Neumann case (nonweighted setting) in [10] (see also [1] and [5] for the control inside the domain case (α = 1)). Our approach is different from the one used in the aforementioned works since we directly use the solution of the Riccati equation for the “associated” deterministic problem. The deterministic linear quadratic problem associated with ours is that characterized by the state equation x(t) ˙ = Ax(t) + Bu(t) and the functional  (7) τ

T

  |Cx|2Y + |u|2R dt + Gx(T ), x(T )Hρ .

It is well known (see [3] and section 3 below) that the solution to the linear quadratic problem given above is determined by the operator-valued function P : [0, T ] → L (Hρ , Hρ ) which solves the so-called Riccati equation   P (t) = −A∗ P (t) − P (t)A∗ − C ∗ C + P (t)ABB ∗ A∗ P (t), (8) P (T ) = G. Such a problem has been intensely studied (see [3] and [14] and the references therein). We will refer in particular to the direct solution approach and will use the formalism introduced in section 2.2 of [3]. We show that the Riccati equation (8) has a unique solution P in the space Cs,α ([0, T ]; Σ(H)) (see Definition 3.2). Let us note that in the deterministic case the minimum of the cost functional (7) is given by P (τ )x0 , x0 H . In the study of the problem with boundary noise some of the tools and the results of the deterministic case, such as the properties of the elements of Cs,α ([0, T ]; Σ (Hρ )) and the solution of (8), are still useful. It is possible to express the value function and the optimal feedback in terms of P . A term due to the noise appears in the expression of the minimal cost, and we have that (Theorem 3.7) (9) V (τ, x0 ) = inf J(τ, x0 , u) u∈Uτ



T

= P (τ )x0 , x0  + τ

1 ((λ0 − A)D(1)), P (s)((λ0 − A)D(1))H ds. 2

2. The heat equation in Hρ . 2.1. Notation. We will work in a weighted space Hρ = L2 ([0, ∞); ρ(ξ)dξ), where either ρ(ξ) = ξ 1+θ ∧ 1 or ρ(ξ) = ξ 1+θ for some θ ∈ (0, 1) and ξ ≥ 0. All the results proved in what follows are valid for both weights, and therefore, in order to simplify notation we will use the same notation H = Hρ for both weights. Let us recall that f ∈ H if and only if  ∞ f 2 (ξ)ρ(ξ) dξ < ∞ 0

and H is a Hilbert space with the scalar product  ∞ φ(ξ)ψ(ξ)ρ(ξ) dξ φ, ψH =

∀φ, ψ ∈ H.

0

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G. FABBRI AND B. GOLDYS

Given λ > 0, the Dirichlet map Dλ is defined as follows:  (λ − ∂x2 )φ(ξ) = 0 (10) Dλ (a) = φ ⇐⇒ φ(0) = a

∀ξ > 0,

so Dλ (a) = aψλ , where  (11)

ψλ : R+ → R, √

e− λξ . ψλ : ξ →

Clearly ψλ ∈ H. It is well known that for every x0 ∈ L2 (0, ∞) the solution y to the heat equation with zero Dirichlet boundary condition ⎧ ∂ ∂2 t > 0, ξ > 0, ⎨ ∂t y(t, ξ) = ∂ξ 2 y(t, ξ), y(t, 0) = 0, t ≥ 0, ⎩ y(0, ξ) = x0 (ξ), ξ > 0, is given by the following well-known expression:  ∞ (12) y(t, ξ) = k(t, ξ, η)x0 (η) dη, 0

where (13)

1 k(t, ξ, η) = √ 4πt

 2 2 − (ξ−η) − (ξ+η) 4t 4t −e e ,

η, ξ ≥ 0.

This formula defines the corresponding heat semigroup T (t)x0 = y(t) in L2 (0, ∞). It is also well known that (T (t)) is a symmetric C0 -semigroup of contractions on L2 (0, ∞). 2.2. Properties of the heat semigroup on H. Proposition 2.1. For each of the weights ρ(ξ) above, the heat semi considered  group (T (t)) extends to a bounded C0 -semigroup etA t≥0 on H with the generator   A : dom(A) → H. The semigroup etA t≥0 is analytic.   Proof. The case ρ(ξ) = ξ 1+θ : H = L2 [0, ∞), ξ 1+θ dξ . Let f ∈ L2 (0, ∞). Then by Theorem 2.5 in [12] there exists C > 0 independent of f and such that

tA

e f ≤ C|f |H , t ≥ 0. H Since L2 (0, ∞) is dense in H, etA can be extended to H and the strong continuity follows by standard arguments. Let A0 be the generator of (T (t)) in L2 (0, ∞) and let D = dom (A0 ) ∩ H ⊂ H. Clearly etA D ⊂ D,

t ≥ 0,

  and D is dense in H. Therefore D is a core for the generator A of etA in H. If f ∈ D, then AetA f =

∂2 T (t)f ∂ξ 2

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HEAT EQUATION WITH BOUNDARY CONTROL AND NOISE

1477

and again by Theorem 2.5 in [12] we have

2

∂

≤ C |f |H . T (t)f

∂ξ 2

t H Since D is a core for the generator A in H, the above estimate can be extended to any f ∈ H and therefore

tA

Ae f ≤ C |f |H , H t

f ∈ H.

The last inequality is equivalent to theanalyticity of the semigroup (eta ) in H. The case ρ(ξ) = 1 ∧ ξ 1+θ : H = L2 [0, ∞), 1 ∧ ξ 1+θ dξ . Let x ∈ C0∞ (0, ∞) and t ≤ T . Then the functions x1 = xI[0,1] and x2 are in 2 L (0, ∞) and Hρ for both weights ρ. It follows that |(T (t)x|H ≤ |T (t)(χ[0,1] x)|H + |T (t)(χ(1,+∞) x)|H ≤ |T (t)(χ[0,1] x)|L21+θ + |T (t)(χ(1,+∞) x)|L2 (0,+∞)

(14)

ξ

≤ C|x|H for a certain C > 0. The fact that C does not depend on t ≤ T is a consequence of the C0 -property of Tt on L2ξ1+θ (shown in the first part of the proof) and on L2 (0, ∞).   Therefore (T (t)) has an extension to a semigroup etA on H and the C0-property follows by standard arguments. Similar arguments yield analyticity of etA . Lemma 2.2. Assume that λ > 0 and r > 0. Then  1 θ α ψλ ∈ dom((r − A) ) ∀α ∈ 0, + . 2 4   In particular Dλ ∈ L(R; dom((λ − A)α )) for all α ∈ 0, 12 + θ4 . Proof. We consider the case of ρ(ξ) = ξ 1+θ only. The other case may be proved by similar, if somewhat simpler, arguments. Note first that if ψλ ∈ (dom(A), H)2,σ , then ψλ ∈ dom ((r − A)α ) for all α ∈ (0, 1 − σ);1 see, for example, Theorem 11.5.1 in [16]. Hence the claim will follow if we show that ψλ ∈ (dom(A), H)2,σ for 1 1 θ − <σ< . 2 4 2

(15)

By Theorem 10.1 of [15], ψλ ∈ (dom(A), H)2,σ if and only if 

∞

(16) 0

  2 t2σ−3 etA − I ψλ H dt < ∞,

and taking into account (15) it is enough to show that  (17)

I := 0

1 (dom(A), H) 2,σ

1

2 t2σ−3 (etA − I)ψλ H < ∞.

denotes the real interpolation space.

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1478

G. FABBRI AND B. GOLDYS

To show (17) we will use (12) and (13) and the definition of ψλ . Denoting by N the cumulative distribution function of the standard normal distribution, we obtain 

1

I=

 t2σ−3

0

∞

2

  ξ 1+θ etA − I ψλ (ξ) dξdt

0

2 (ξ−η)2  ∞ − (ξ+η)2 4t e− 4t e −λη −λη λξ √ √ = t ξ dη − dη − e dξdt e e 4pt 4pt 0 0 0 0    1  ∞ √ ξ 2σ−3 1+θ −λξ λ2 t = t ξ e N √ −λ t e 2t 0 0   2 √ 2 ξ −eλξ eλ t 1 − N √ + λ t − eλξ dξdt 2t ≤ 2 (I1 + I2 + I3 ) , 

1



2σ−3



∞

∞

1+θ

where I1 , I2 , and I3 are, respectively, 2  2   ξ √ λ t e − 1 N √ − λ 2t t ξ dξ dt, I1 := e 2t 0 0 2    1  +∞ √ ξ 2σ−3 1+θ −λξ I2 := t ξ dξ dt, e N √ − λ 2t − 1 2t 0 0 2    1  +∞ √ ξ 2σ−3 1+θ λξ λ2 t t ξ dξ dt. e e 1 − N √ + λ 2t I3 := 2t 0 0 

1





+∞

2σ−3

−λξ

1+θ

2

2

Since for t ∈ [0, 1] we have |eλ t − 1| ≤ (eλ − 1)t we find that I1 converges for every σ > 0. I3 can be estimated, using that the standard estimate 2

1 e−s /2 √ s 2π

(1 − N(s)) ≤ as follows: 

1

I3 ≤ 0

 t2σ−3



≤ C1

+∞

0 1

0





t2σ−2

2

ξ 1+θ e2λξ e2λ

+∞

t 2t −ξ 2 /(2t) e ξ2

ξ −1+θ e2λξ−ξ

2

/(2t)

dξ dt

dξ dt

0

1

= C1

 t2σ−2

0

 ≤ C1

0 1

t 0

+∞

2σ−2+ θ2

θ−1

√

2

y −1+θ t 2 e2λy t−y /2 t1/2 dξ dt  +∞ 2 dt y −1+θ e2λy−y /2 dξ < ∞, 0

where the finiteness of the first term follows from (15). The estimate for I2 can be obtained in a similar way. 2.3. Properties of the solution of the state equation. Let W be a real Brownian motion on a probability space (Ω, F , P) and let (Ft ) denote the natural filtration of W . We need to give a rigorous meaning to (1). To this end we will

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HEAT EQUATION WITH BOUNDARY CONTROL AND NOISE

assume in what follows that

 and α ∈

λ0 > 0

1 1 θ , + 2 2 4

1479



are fixed. We will denote by D the operator Dλ0 ∈ L(R; D((λ0 − A)α )) and ψλ0 = Dλ0 (1). By Proposition 2.1 the semigroup etA is analytic, and therefore for any γ≥0 (18)

(λ0 − A)γ etA H ≤ Mγ t−γ

∀t ∈ (0, T ];

see, for example, [18, Theorem 6.13, page 75]. By Lemma 2.2 the operator B = (λ0 − A)D : R → Hα−1 is bounded.2 Moreover, for t > 0 the operator AetA Dλ0 = (λ0 − A)

1−α tA

e

α

(λ0 − A) Dλ0 : R → H

is bounded as well. We will write etA B = AetA Dλ0 . Now, we reformulate (1), still formally, as a stochastic evolution equation in H:  dx(t) = (Ax(t) + Bu(t)) dt + B dW (t), (19) x(τ ) = x0 ∈ H, 2 (τ, T ; R) of progressively measurable where the control u is chosen in the set MW processes endowed with the norm

 u2M 2 = E W

T

|u(t)|2 dt < ∞.

τ

The next two results show that we can give a meaning to (19). Theorem 2.3. For all γ < 2α − 1 the following hold: (i) The operator t → etA B : R → H is bounded for each t > 0 and the function t → etA Ba is continuous for every a ∈ R. (ii) 

T

(20) 0

 2 s−γ (λ0 − A) esA ψλ0 H ds < ∞.

(iii) For every T > τ ≥ 0 the process  WA (t) =

t

e(t−s)A B dW (s),

t ∈ [τ, T ],

τ

is well defined, belongs to C([τ, T ]; L2 (Ω; H)), and has continuous trajectories in H. Proof. (i) It follows immediately from the definition of B and Lemma 2.2 since (21)

etA Ba = a(λ0 − A)1−α etA (λ0 − A)α ψλ0 ,

a ∈ R.

2 For β > 0 the space H−β is defined as a completion of H with respect to the norm |x| −β = | (λ0 − A)−β x|.

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1480

G. FABBRI AND B. GOLDYS

(ii) By (21) and (18) we have for α ∈

1

1 2, 2

+

θ 4



2   (λ0 − A) esA ψλ0 2 =

(λ0 − A)1−α esA (λ0 − A)α ψλ0

≤ HS

C 2 |(λ0 − A)α ψλ0 | s2(1−α)

and the estimate (20) follows immediately for a certain γ < 2α − 1. (iii) Using (20) with γ = 0 we find immediately that, for every t ≥ 0, WA (t) is well defined and (see, for example, [6, Proposition 4.5, page 91])  (22)

t

2

E |WA (t)|H =

τ

|esA ((λ − A)D)|2HS ds < ∞.

Such an estimate also gives, through standard arguments, the mean square continuity. The continuity follows from (20) for γ > 0 using a factorization argument as in [7, Theorem 2.3, page 174]. 2 Lemma 2.4. Let T > 0 be fixed, λ > 0, and u ∈ MW (τ, T ; R). Then the process 

t

I(t) =

t ≤ T,

e(t−s)A Bu(s) ds,

τ 2 (τ, T ; H), and there exists C > 0 such that is well defined, I ∈ MW

EI2M 2 ≤ Cu2M 2 . W

W

Moreover, I is in C(τ, T ; L2 (Ω, H)) and has continuous trajectories. Proof. The first part of the lemma follow from (21) by standard arguments. The mean square continuity and continuity of I follow from (18) and the H¨ older inequality (since α > 1/2) in the expression  t  (λ0 − A)1−α e(t−s)A [(λ0 − A)α Du(s)] ds, I(t) = τ

and the claim follows. 2 Definition 2.5. Let u ∈ MW . An H-valued predictable process x, defined on [0, T ], is called a mild solution of (19) if

 T

P

|x(s)|2 ds < ∞ = 1

τ

and  x(t) = e

(t−τ )A

x0 +



t

e τ

(t−s)A

Bu(s) ds +

t

e(t−s)A B dW (s).

τ

Theorem 2.6. Equation (19) has a unique mild solution x ∈ C(τ, T ; L2 (Ω, H)). Moreover, x has continuous trajectories P-a.s. If u = 0, then (19) defines a Markov process in H. Proof. The  t properties of the stochastic convolution term come from Theorem 2.3 and those of τ e(t−s)A Bu(s) from Lemma 2.4. The Markov property can be proved with standard arguments (see, for example, [6, Theorem 9.8, page 249]).

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HEAT EQUATION WITH BOUNDARY CONTROL AND NOISE

1481

def

2.4. The approximating equation. Let In = (n(n − A)−1 )2 . We will approximate x using def

xn = In x.

(23) We have that

C([τ,T ];L2 (Ω,H))

xn −−−−−−−−−−−→ x.

(24)

We use it to obtain more regularity and to guarantee the existence of a strong solution and then to be able to apply Ito’s rule (Proposition 3.6). From Theorem 2.6 we know that xn ∈ C([τ, T ]; L2 (Ω, dom(A2 ))). We have Bn := In B ∈ L(R; dom(A)) and 2 then Bn u ∈ MW (τ, T ; dom(A)). Furthermore, xn satisfies the following stochastic differential equation:  dxn (t) = (Axn (t) + Bn u(t)) dt + Bn dW (t), (25) xn (τ ) = In x0 in the strong (and then mild) sense (see [6, section 6.1]). So we have  t  t  t (26) xn (t) = In x0 + Axn (s) ds + Bn u(s) ds + Bn dW (s). τ

τ

τ

3. The linear quadratic problem. Let us recall that we work under the assumption 1 1 θ <α< + . 2 2 4 We consider another Hilbert space Y , an operator C ∈ L(H; Y ), and a symmetric and positive G ∈ L(H; H). For a fixed T > 0 we define the set of the admissible controls 2 as Uτ = MW (τ, T ; R). We consider the linear quadratic optimal control problem governed by (19) and the quadratic cost functional (to be minimized)

 T   2 2 |Cx(t)|Y + |u(t)|R dt + Gx(T ), x(T ) . (27) J(τ, x0 , u) := E τ

The value function of the problem is V (τ, x0 ) := inf J(τ, x0 , u). u∈Uτ

We now consider the “associated” deterministic linear quadratic problem. It is characterized by the state equation  x(t) ˙ = Ax(t) + Bu(t), (28) x(τ ) = x0 , by the set of admissible controls UDET := L2 (τ, T ; R), and by the functional  JDET (τ, x0 , u) := τ

T

  |Cx(t)|2Y + |u(t)|2R dt + Gx(T ), x(T ) .

In what follows we will use the following notation.

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1482

G. FABBRI AND B. GOLDYS

Notation 3.1. Σ(H) = {T ∈ L(H; H) : T hermitian} Σ+ (H) = {T ∈ Σ(H) : T x, x ≥ 0 for all x ∈ H} Cs ([0, T ]; Σ(H)) = {F : [0, T ] → Σ(H) : F strongly continuous} Note that (see [3, page 137]) for every P ∈ Cs ([0, T ]; Σ(H)) sup P (t) < ∞.

(29)

t∈[0,T ]

The Riccati equation formally associated with the deterministic control problem (28) has the form   P (t) = −A∗ P (t) − P (t)A∗ − C ∗ C + P (t)ABB ∗ A∗ P (t), (30) P (T ) = G, but the concept of solution to this equation requires a rigorous definition. We start with some notation. Definition 3.2. We denote by Cs,α ([0, T ]; Σ(H)) the set of all P ∈ Cs ([0, T ]; Σ(H)) such that (i) P (t)x ∈ D((λ0 − A∗ )1−α ) (ii) (iii)

def

∀x ∈ H, ∀t ∈ [0, T ),

∗ 1−α

P(t) ∈ C([0, T ); L(H)), VP (t) = (λ0 − A ) lim (T − t)1−α VP (t)x = 0 ∀x ∈ H. t→T −

Given P ∈ Cs,α ([0, T ]; Σ(H)), the norm |P |α is defined as |P |α = sup P (t) + sup (T − t)(1−α) (λ0 − A∗ )1−α P (t). def

t∈[0,T )

t∈[0,T )

It can be proved (see [3, page 205]) that Cs,α ([0, T ]; Σ(H)), endowed with the def

norm | · |α , is a Banach space. We will use the notation E = (λ0 − A)α D ∈ L(R; H). Note that if |P |α < ∞, then (since α > 1/2) |P |L2 (0,T ;L(H)) < ∞.

(31)

Definition 3.3. We say that P ∈ Cs,α ([0, T ]; Σ(H)) is a weak solution of the Riccati equation (30) if for all x, y ∈ dom(A) and all t ∈ (0, T ) (32)  d ∗ ∗ dt P (t)x, y = − P (t)x, Ay − P (t)Ax, y − Cx, Cy + E VP (t)x, E VP (t)y , P (T ) = G. We now recall the existence and uniqueness theorem for (32). Theorem 3.4. (i) The Riccati equation (32) has a unique weak solution in P in Cs,α ([0, T ]; Σ+ (H)). (ii) P ∈ Cs,α ([0, T ]; Σ+ (H)) is a weak solution of (30) if and only if it solves the following mild equation: ∗

(33) P (t) = e(T −t)A Ge(T −t)A +



T

∗

e(s−t)A C ∗ Ce(s−t)A ds

t

 + t

T

∗

e(s−t)A VP∗ (s)EE ∗ VP (s)e(s−t)A ds.

Proof. See [3, Theorem 2.1, page 207] for the proof of (i) and [3, Proposition 2.1, page 206] for (ii).

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HEAT EQUATION WITH BOUNDARY CONTROL AND NOISE

1483

3.1. Dynamic programming. Lemma 3.5. We have that  T ((λ0 − A)ψλ0 ), P (t)((λ0 − A)ψλ0 )H dt < ∞. 0

Proof. We use the fact that P satisfies the mild equation (33). We have that  T (34) ((λ0 − A)ψλ0 ), P (t)((λ0 − A)ψλ0 )H dt 0



T

E ∗ (λ0 − A∗ )1−α P (t)(λ0 − A)1−α E(1) dt

= 0

def



=

 +  +

0 T

= I1 + I2 + I3 T

∗

E ∗ (λ0 − A∗ )1−α e(T −t)A Ge(T −t)A (λ0 − A)1−α E(1) dt  

0 T

T

E ∗ (λ0 − A∗ )1−α

E ∗ (λ0 − A∗ )1−α



0

∗

e(s−t)A C ∗ Ce(s−t)A ds (λ0 − A)1−α E(1) dt

t T

t

∗

e(s−t)A VP∗ (s)EE ∗ VP (s)e(s−t)A ds(λ0 − A)1−α E(1) dt.

For I1 we have to only check the T , and it follows from the fact  integrability for t →∗  that α > 1/2 and from (18):  (λ0 − A∗ )1−α e(T −t)A  ≤ M1−α (T − t)1−α . For I2 we proceed in a similar way: we can write I2 as  T T 

2 

I2 =

C (λ0 − A)1−α e(s−t)A E(1) ds dt, 0

t

and we can conclude as for I1 , using (18). For I3 we can observe that  T T

2 ∗

∗ I3 =

E (λ0 − A∗ )1−α e(s−t)A VP∗ (s)E(1) ds dt. 0

t

Note that from (ii) of Definition 3.2 and from the finiteness of the norm |P |α we know that C1 (35) VP∗ (s) ≤ (T − s)1−α and ∗

E ∗ (λ0 − A∗ )1−α e(s−t)A  ≤

C2 . (s − t)1−α

The claim follows by straightforward computations. 2 Proposition 3.6. If u ∈ MW (τ, T ; R) is a control and x is the related trajectory, then

 T 2 2 (36) E Gx(T ), x(T ) + |Cx(t)|Y + |u(t)|R dt τ



T

= P (τ )x(τ ), x(τ ) + E  + τ

τ T

|u(t) + E ∗ VP (t)x(t)|2R

1 (λ0 − A)ψλ0 , P (t)((λ0 − A)ψλ0 )H dt. 2

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1484

G. FABBRI AND B. GOLDYS

Proof. We will perform the following steps: we first approximate x using xn T d defined in (23), then we compute τ 0 dt P (t)xn (t), xn (t) dt using Ito’s formula, and eventually we consider to the limit n → ∞ and then T0 → T . Let L(dom(A); H) be the space of bounded operators from dom(A) endowed with the graph norm to H. Note that since P ∈ C([τ, T ); L(H; H)) it is a fortiori an element of C([τ, T ); L(dom(A); H)). Consider T0 < T and the following function (dom(A) is endowed with the graph norm):  Φ : [τ, T0 ] × dom(A) → R, Φ : (t, x) → P (t)x, xH . Note that in the definition of Φ we use the scalar product of H and not of dom(A). Φ is twice continuously differentiable with locally bounded derivatives in x on [τ, T0 ]× dom(A). Moreover we have that ∂x Φ(t, x) = P (t)x and ∂x2 Φ(t, x)(y, z) = 2 P (t)y, zH . The first derivative in t is also continuous and locally bounded on [τ, T0 ] × dom(A). Invoking (32) we have that d dt

P (t)x, yH = − P (t)x, AyH − P (t)Ax, y − Cx, CyH + E ∗ VP (t)x, E ∗ VP (t)yH .

Such an expression can be discontinuous for t = T only (this is the reason why we have considered T0 < T ). We have already observed that xn satisfy the integral equation (26) also in dom(A), and then we can use Ito’s rule (see [6, page 105]): we have that  (37)

T0

P (T0 )xn (T0 ), xn (T0 ) = P (τ )xn (τ ), xn (τ ) −  −2  +2

T0

τ T0

 P (t)xn (t), Axn (t) dt +

τ T0

τ

E ∗ VP (t)xn (t), E ∗ VP (t)xn (t)R dt 

VP (t)xn (t), In (λ0 − A)α Du(t) dt + 2

τ

Cxn (t), Cxn (t)Y dt

T0

P (t)xn (t), Axn (t) dt

τ



T0

+2

VP (t)xn (t), In (λ0 − A)α D dW (t) dt

τ



T0

+ τ

1 ((λ0 − A)In ψλ0 ), P (t)((λ0 − A)In ψλ0 )H dt. 2

By simplifying the terms P (t)xn (t), Axn (t), adding and subtracting |u(t)|2R and T 2 0 0 u(t), E ∗ VP (t)xn (t)R inside the integral, and taking the expectation, we find that

 T0

(38) E P (T0 )xn (T0 ), xn (T0 ) + τ

|Cxn (t)|2Y + |u(t)|2R dt 

= P (τ )xn (τ ), xn (τ ) + E τ



T0

+ 2E

T0

|u(t) + E

∗

VP (t)xn (t)|2R

VP (t)xn (t), (In − I)(λ0 − A)α Du(t) dt

τ

 + τ

T0

1 ((λ0 − A)In ψλ0 ), P (t)((λ0 − A)In ψλ0 )H dt. 2

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1485

HEAT EQUATION WITH BOUNDARY CONTROL AND NOISE C([τ,T0 ];L2 (Ω,H))

We now want to pass with n → ∞. Since by (24) we have xn −−−−−−−−−−−− → x, it is n→∞ clear that    T0 2 |Cxn (t)|Y dt lim E P (T0 ) xn (T0 ) , xn (T0 ) + n→∞

τ





= E P (T0 ) x (T0 ) , x (T0 ) +

T0

τ

 |Cx(t)|2Y

dt

and lim P (τ )xn (τ ), xn (τ ) = P (τ )x(τ ), x(τ ) .

n→∞

Since VP ∈ C([τ, T0 ] ; L(H)) and (λ0 − A)α D = E is bounded, the dominated convergence yields   T0

lim E

n→∞

|u(t) + E

τ



T0

VP (t)xn (t)|2R  =E

T0

τ

dt + 2

α

VP (t)xn (t), (In − I) (λ0 − A) Du(t) dt

τ

2

|u(t) + E VP (t)x(t)|R dt.

Finally, using the arguments similar to those in the proof of Lemma 3.5 we obtain  T0 1 (λ0 − A) In ψλ0 , P (t) (λ0 − A) In ψλ0 H dt lim E n→∞ 2 τ  T0 1 =E (λ0 − A) ψλ0 , P (t) (λ0 − A) ψλ0 H dt, 2 τ and therefore, putting together the above results, we obtain

 T0 (39) E P (T0 )x(T0 ), x(T0 ) + |Cx(t)|2Y + |u(t)|2R dt τ



= P (τ )x(τ ), x(τ ) + E  + τ

T0

|u(t) + E

τ T0

∗

VP (t)x(t)|2R

1 ((λ0 − A)ψλ0 ), P (t)((λ0 − A)ψλ0 )H dt. 2

Now we pass to the limit in T0 ↑ T in (39). To show the convergence of the left-hand side of (39) it is enough to invoke monotone convergence and to show that (40)

lim E P (T0 ) x (T0 ) , x (T0 ) = E P (T ) x (T ) , x (T ) .

T0 →T

To this end note that (41)

P (T )x(T ), x(T ) − P (T0 ) x (T0 ) , x (T0 ) = (P (T ) − P (T0 )) x(T ), x(T ) + P (T0 )(x(T ) − x(T0 )), x(T ) + P (T0 )x(T0 ), x(T ) − x(T0 ) .

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1486

G. FABBRI AND B. GOLDYS

Then the strong continuity of P at T yields lim (P (T ) − P (T0 )) x(T ), x(T ) = 0;

T0 →T

hence by (29), the fact that x ∈ C([τ, T ]; L2 (Ω; H)), and dominated convergence we obtain lim E (P (T ) − P (T0 )) x(T ), x(T ) = 0.

T0 →T

Again, since x ∈ C([τ, T ]; L2 (Ω; H)), we find that  1/2  1/2 2 2 |E P (T0 )(x(T ) − x(T0 )), x(T )| ≤ sup P (t) sup E|x(t)| E |x(T ) − x (T0 )| t≤T

t≤T

and therefore lim E P (T0 )(x(T ) − x(T0 )), x(T ) = 0.

T0 →T

By the same arguments we obtain lim E P (T0 )x(T0 ), x(T ) − x(T0 ) = 0,

T0 →T

and therefore we obtain the convergence of the left-hand side of (39). To prove convergence of the second term in the right-hand side of (39) it is enough to show 2 (τ, T ; H). Indeed, invoking (35) we have that that VP x ∈ MW  T  T 2 E |VP (s)x(s)| ds ≤ C3 VP (s)2 E|x(s)|2 ds 0 0 (42)  T ≤ C4 |x|2C([τ,T ];L2 (Ω;H))

VP (s)2 ds < ∞.

0

The convergence for the third term of the right-hand side of (39) for T0 → T follows from Lemma 3.5. Theorem 3.7. Let τ ∈ [0, T ] and x0 be in H. Then there exists a unique optimal pair (u∗ , x∗ ) at (τ, x0 ). The optimal control u∗ is given by the feedback formula u∗ (t) = −E ∗ VP (t)x∗ (t),

(43)

and the value function of the problem is  T 1 V (τ, x0 ) = P (τ )x0 , x0  + ((λ0 − A)ψλ0 ), P (s)((λ0 − A)ψλ0 )H ds. τ 2 Proof. We begin proving that the equation (44) x∗ (t) = e(t−τ )A x0 −



t

(λ0 − A)1−α e(t−s)A EE ∗ VP (s)x∗ (s) ds +

τ



t

e(t−s)A B dW (s)

τ

has a unique solution and is in C([τ, T ]; L (Ω; H)). Consider the mapping ⎧ φ → Ψ(φ), ⎪ ⎪  t ⎪ ⎪ ⎨ Ψ(φ) = e(t−τ )A x0 − (λ0 − A)1−α e(t−s)A EE ∗ VP (s)φ(s) ds τ  t ⎪ ⎪ ⎪ ⎪ ⎩ e(t−s)A B dW (s). + 2

τ

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HEAT EQUATION WITH BOUNDARY CONTROL AND NOISE

1487

We want to prove that Ψ(φ) defines a contraction on C([τ, t]; L2 (Ω; H)) if we choose t small enough. Consider ψ and φ in C([τ, T ]; L2 (Ω; H)):   2 E |(Ψ(ψ) − Ψ(φ))(t)| (45) 

2



t 1−α (t−s)A ∗ (λ0 − A) EE VP (s)(ψ − φ)(s) ds

e = E

τ    t 1 2 |(ψ − φ)(s)| ds ≤ C1 E |VP |2L2 (0,T ;L(H)) 2(1−α) τ (t − s)  t 1 ≤ C2 |(ψ − φ)(s)|2C([τ,t];L2 (Ω;H)) ds, 2(1−α) (t − s) τ where the constants C1 and C2 do not depend on t. So if t is small enough, Ψ is a contraction on C([τ, t]; L2 (Ω; H)). Similar estimates (together with the fact that WA ∈ C(τ, T ; L2 (Ω; H)) prove that the image of Ψ is in C([τ, t]; L2 (Ω; H)). Proceeding by iterations (we can choose a uniform step) we have the existence and uniqueness of the solution of (44) on C([τ, T ]; L2 (Ω; H)). We will now prove that u defined by (43) is the optimal control. Its admissibility 2 (that is, u∗ ∈ MW (τ, T ; R)) can be proved using the same argument we used in (42). 2 Now we observe that Proposition 3.6 implies, for every u ∈ MW (τ, T ; R),  T 1 ((λ0 − A)ψλ0 ), P (t)((λ0 − A)ψλ0 )H dt (46) J(τ, x0 , u) ≥ P (τ )x0 , x0  + τ 2 and the couple (u∗ , x∗ ) satisfies (47)

J(τ, x0 , u∗ ) = P (τ )x0 , x0  +

 τ

T

1 ((λ0 − A)ψλ0 ), P (t)((λ0 − A)ψλ0 )H dt, 2

so it is optimal. If (ˆ u, x ˆ) is another optimal couple, then by (46) and (47) we have that  T 1 J(τ, x0 , u ˆ) = P (τ )x0 , x0  + ((λ0 − A)ψλ0 ), P (t)((λ0 − A)ψλ0 )H dt, τ 2 and then (39) yields |ˆ u(t) + E ∗ VP (t)ˆ x(t)| = 0 dt ⊗ P-a.e. Then x ˆ satisfies (44), but the solution to (44) is unique by the solution of Theorem ˆ and 2.6, and finally we can choose continuous versions of xˆ and uˆ such that x∗ = x u∗ = uˆ. REFERENCES [1] N. U. Ahmed, Stochastic control on Hilbert space for linear evolution equations with random operator-valued coefficients, SIAM J. Control Optim., 19 (1981), pp. 401–430. ` s and S. Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise [2] E. Alo boundary conditions, Ann. Inst. H. Poincar´e Probab. Statist., 38 (2002), pp. 125–154. [3] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter, Representation and Control of Infinite-Dimensional Systems, Vol. II, Systems Control Found. Appl., Birkh¨ auser Boston, Boston, MA, 1993.

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1488

G. FABBRI AND B. GOLDYS

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