Optimal control of stochastic differential equations with dynamical
  boundary conditions
Optimal control of stochastic differential equations
with dynamical boundary conditions
Stefano BONACCORSI∗, Fulvia CONFORTOLA†, Elisa MASTROGIACOMO
Dipartimento di Matematica, Università di Trento,
via Sommarive 14, 38050 Povo (Trento), Italia
In this paper we investigate the optimal control problem for a class of stochastic Cauchy
evolution problem with non standard boundary dynamic and control. The model is
composed by an infinite dimensional dynamical system coupled with a finite dimensional
dynamics, which describes the boundary conditions of the internal system. In other
terms, we are concerned with non standard boundary conditions, as the value at the
boundary is governed by a different stochastic differential equation.
Keywords: Stochastic differential equations in infinite dimensions, dynamical bound-
ary conditions, optimal control
1991 MSC :
1 Setting of the problem
Our model is a one dimensional semilinear diffusion equation in a confined system,
where interactions with extremal points cannot be disregarded. The extremal points
have a mass and the boundary potential evolves with a specific dynamic. Stochas-
ticity enters through fluctuations and random perturbations both in the inside as
on the boundaries; in particular, in our model we assume that the control process
is perturbed by a noisy term.
There is a growing literature concerning such problems; we shall mention the
paper [2] where a problem in a domain O ⊂ Rn is concerned; the authors cite as an
example an SPDE with stochastic perturbations which appears in connection with
random fluctuations of the atmospheric pressure field. As opposite to ours, however,
that paper is not concerned with control problems. Quite recently, the authors
became aware of the paper [1] where a different application to some generalized
Lamb model is proposed.
The internal dynamic is described by a stochastic evolution problem in the unit
∗stefano.bonaccorsi@unitn.it
†Current address: fulvia.confortola@unimib.it
http://arxiv.org/abs/0704.0524v1
2 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
interval D = [0, 1]
∂tu(t, x) = ∂
xu(t, x) + f(t, x, u(t, x)) + g(t, x, u(t, x))Ẇ (t, x) (1)
which we write as an abstract evolution problem on the space L2(0, 1)
du(t) = Amu(t) + F (t, u(t)) dt+G(t, u(t)) dW (t), (2)
where the leading operator is Am = ∂
x with domain D(Am) = H
2(0, 1). We assume
that f and g are real valued mappings, defined on [0, T ]× [0, 1]× R, which verify
some boundedness and Lipschitz continuity assumptions.
The boundary dynamic is governed by a finite dimensional system which follows
a (ordinary, two dimensional) stochastic differential equation
∂tvi(t) = −bivi(t) + ∂νu(t, i) + hi(t)V̇i(t), i = 0, 1
where bi are positive numbers and hi(t) are bounded, measurable functions; ∂ν is
the normal derivative on the boundary, and coincides with (−1)i∂x for i = 0, 1. For
notational semplicity, we introduce the 2× 2 diagonal matrices B = diag(−b−0, b1)
and h(t) = diag(h0(t), h1(t)). There is a constraint
Lu = v
which we interpret as the operator evaluating boundary conditions; the system is
coupled by the presence, in the second equation, of a feedback term C that is an
unbounded operator
∂xu(0)
−∂xu(1)
The idea is to write the problem in abstract form for the vector u =
the space X = L2(0, 1)× R2, that is
du = Au(t) + F(t,u(t)) dt +G(t,u(t)) dW(t)
u(0) =
Our main concern is to study spectral properties of the matrix operator
on the domain
D(A) = {u ∈ D(Am)× R2 : Lu = v}.
Theorem 1. A is the infinitesimal generator of a strongly continuous, analytic
semigroup of contractions etA, self-adjoint and compact.
Control of stochastic differential equations with dynamical boundary conditions 3
We shall prove the above theorem in Section 2. Further, we shall prove that
A is a self-adjoint operator with compact resolvent, which implies that the gener-
ated semigroup is Hilbert-Schmidt. Moreover, we can characterize the complete,
orthonormal system of eigenfunctions associated to A.
Let us fix a complete probability space (Ω,F, {Ft},P); on this space we de-
fine W (t), that is a space-time Wiener process taking values in X and V (t) =
(V1(t), V2(t)), that is a R
2-valued Wiener process, such that W (t, x) and V (t) are
independent.
As a corollary to Theorem 1, using standard results for infinite dimensional
stochastic differential equations, compare [3, Theorem 7.4], we obtain the following
existence result
Theorem 2. For any initial condition
∈ X×R2 there exists a unique process
u ∈ L2F (0, T ;X × R2) such that
u(t) = etA
e(t−s)AF(u(s)) ds+
e(t−s)AG(u(s)) dW(s)
that is by definition a mild solution of (3).
The abstract semigroup setting we propose in this paper allows to obtain an
optimal control synthesis for the above evolution problem with boundary control
and noise. This means that we assume a boundary dynamics of the form:
∂tv(t) = bv(t)− ∂νu(t, ·) + h(t)[z(t) + V̇ (t)] (4)
where z(t) is the control process and takes values in a given subset of R2.
As before, we can write the system – defined by the internal evolution problem
(1) and the dynamical boundary conditions described by (4) – in the following
abstract form
duzt = Au
t dt+ F(t,u
t ) dt+G(t,u
t )[Pzt dt+ dWt]
ut0 = u0.
P : R2 → X denote the immersion of the boundary space in the product space
X = L2(0, 1)× R2.
The aim is to choose a control process z, within a set of admissible controls, in
such way to minimize a cost functional of the form
J(t0, u0, z) = E
λ(s,uzs , zs)) ds+ Eφ(u
T ) (6)
where λ and φ are given real functions. In our setting, altough the control lives in a
finite dimensional space, we obtain an abstract optimal control problem in infinite
dimensions. Such type of problems has been exhaustively studied by Fuhrman and
Tessitore in [8]. The control problem is understood in the usual weak sense (see [7]).
We prove that if f and g are sufficiently regular then the abstract control problem,
under suitable assumptions on λ and φ, can be solved and we can characterize
optimal controls by a feedback law (see Theorem 17 and compare Theorem 7.2 in
[8]).
4 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
Theorem 3. In our assumptions, there exists an admissible control {z̄t, t ∈ [0, T ]}
taking values in a bounded subset of R2, such that the closed loop equation:
duτ = Auτ dτ +G(τ,uτ )PΓ(τ,uτ ,G(τ,uτ )
∗∇xv(τ,uτ )) dτ
+ F(τ,uτ ) dτ +G(τ,uτ ) dWτ , τ ∈ [t0, T ],
ut0 = u0 ∈ X.
admits a solution and the couple (z,u) is optimal for the control problem.
Stochastic boundary value problems are already present in the literature, see the
paper [11] and the references therein; in those papers, the approach to the solution
of the system is more similar to that in [2]. We also need to mention the paper [5]
for a one dimensional case where the boundary values are set equal to a white noise
mapping.
2 Generation properties
Let X = L2(0, 1) be the Hilbert space of square integrable real valued functions
defined on D = [0, 1] and X = X × R2. In this section we consider the following
initial-boundary value problem on the space X
u(t) = Amu(t)
v(t) = Lu(t)
v(t) = Bv(t)− Cu(t)
u(0) = u0 ∈ X, v(0) = v0 ∈ R2.
In the above equation, Am is an unbounded operator with maximal domain
Am = ∂
x, D(Am) = H
2(0, 1);
B is a diagonal matrix with negative entries (−b0,−b1).
Let C : D(C) ⊂ X → ∂X the feedback operator, defined on D(C) = H1(0, 1)
∂xu(0)
−∂xu(1)
The boundary evaluation operator L is the mapping L : X → R2 given by
Its inverse is the Dirichlet mapping D
λ : R
2 → D(Am)
λ φ = u(x) ∈ D(Am) :
(λI −Am)u(x) = 0,
Lu = φ.
As proposed in [10], we define a mild solution of (8) a function u ∈ C([0, T ];X)
such that
u(t) = u0 +Am
u(s) ds, t ∈ [0, T ]
v(t) = v0 +B
v(s) ds+ C
u(s) ds.
Control of stochastic differential equations with dynamical boundary conditions 5
In order to use semigroup theory to study equation (8), we consider a matrix
operator describing the evolution with feedback on the boundary
on the domain
D(A) = {u ∈ D(Am)× R2 : Lu = v}.
Then a mild solution for equation (8) exists if and only if A is the generator of a
strongly continuous semigroup.
The above definition of the domain D(A) puts in evidence the relation between
the first and the second component of the vector u. There is a different characteri-
zation that is sometimes useful in the applications.
Let us define the operator A0 as A0 = Am on D(A0) = {u ∈ D(Am) : Lu = 0}.
We can then write the domain of A as
D(A) = {u ∈ D(Am)× ∂X : u−DA,L0 v ∈ D(A0)}.
The operator A can be decomposed as the product
I −DA,L0
Then, according to Engel [6], A is called a one-sided K-coupled matrix-valued
operator.
Proof of Theorem 1
In this section we apply form theory in order to prove generation property of the
operator A, compare the monograph [13].
Proposition 4. A is the infinitesimal generator of a strongly continuous, analytic
semigroup of contractions, self-adjoint and compact.
We will give the proof in two steps. First of all we will consider the following
form:
a(u,v) =
u′(x)v′(x) dx + b0 u(0) v(0) + b1 u(1) v(1)
on the domain
u = (u, α) ∈ H1(0, 1)× R2 | u(0) = α0, u(1) = α1
and we will show that it is densely defined, closed, positive, symmetric and continue.
Moreover, the operator associated with the form a is (A, D(A)) defined above.
According to [13], this implies that the operator A is self-adjoint and generates a
contraction semigroup etA on X that is analytic of angle π
. Then we will show the
self-adjointness and the compactness of the semigroup etA. To see this, we will refer
to [9].
Let us begin with the properties of the form a.
6 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
Lemma 5. The form a is densely defined, closed, positive, symmetric and continue.
Proof. By assumption, since b0 and b1 are positive real numbers, it follows that in
particular a is symmetric and positive.
It is clear that V is a linear subspace of X. Observe that V is dense in X if any
u ∈ X can be approximated with elements of V . Consider (u, α) ∈ L2[0, 1] × R2.
Since C∞c [0, 1] is dense in L
2(0, 1) it follows that for all ε > 0 there exists v ∈
C∞c [0, 1] such that
|u− v|L2[0,1] ≤
Now let ρ0(x) be a symmetric function in C
c (R) with support in Bε(0), ρ0(0) = 1
ρ0(x) dx = ε/3. Finally, let ρ1(x) = ρ0(x−1). Then, if we define the function
ρ = v + α0 ρ0
[0,1]
+ α1 ρ1
[0,1]
, we have:
|u− ρ|L2[0,1] ≤ |u− v|L2[0,1] + |α0ρ0|L2[0,1] + |α1ρ1|L2[0,1] ≤
≤ max {1, α0, α1} ε.
Morever, ρ(0) = α0 and ρ(1) = α1. Thus
|(u, α)− (ρ, ρ(0), ρ(1))|
for a suitable M . This shows that V is dense in X.
In order to check closedness and continuity of a, observe first that the norm
induced by a on the space V is equivalent to the norm given by the inner product
(u,v)V =
[u′(x)v′(x) + u(x)v(x)] dx+ u(1)v(1) + u(0)v(0).
In fact, if we set b = b0 + b1, we have
‖u‖a =
a(u,u) + ‖u‖2V
so that
‖u‖2a ≤ 2 ‖u‖
H1(0,1) + 2b
u(0)2 + u(1)2
≤ max {2, 2b} ‖u‖2V .
Now observe that V becomes a Hilbert space when equipped with the inner product
defined above since V is a closed subspace of H1(0, 1)× R2. Then a is closed.
Finally, a is continuous. To see this, take u,v ∈ V ; then
|a(u,v)| ≤
|u′(x)v′(x)| dx+ b [|u(0)| |v(0)|+ |u(1)| |v(1)|]
≤ ‖u‖H1(0,1) ‖v‖H1(0,1) + b [|u(0)| |v(0)|+ |u(1)| |v(1)|]
≤ ‖u‖V ‖v‖V ≤M ‖u‖a ‖v‖a
by the Cauchy-Schwartz inequality.
Control of stochastic differential equations with dynamical boundary conditions 7
Lemma 6. The operator associated with a is (A, D(A)) defined above.
Proof. Denote by (C, D(C)) the operator associated with a. By definition, C is given
D(C) = {f ∈ V | ∃g ∈ X s.t. a(f ,g) = (g,h)X∀h ∈ V }
Cf = −g.
Let us first show that A ⊂ C. Take f ∈ D(A). Then for all h ∈ V
a(f ,h) =
f ′(x)h′(x) dx + b0f(0)h(0) + b1f(1)h(1)
= f ′(x)h(x)|10 −
f ′′(x)h(x) dx + b0f(0)h(0) + b1f(1)h(1)
= f ′(1)h(1)− f ′(0)h(0)−
f ′′(x)h(x) dx + b0f(0)h(0) + b1f(1)h(1).
At the same time, if we set α = (f(0), f(1)), β = (h(0), h(1)), we have
(Af ,h) = (Af, h)L2(0,1) + (Cf +Bα, β)R2 =
f ′′(x)h(x) dx + f ′(0)h(0)− f ′(1)h(1)
− b0f(0)h(0)− b1f(1)h(1) = −a(f ,g).
The last equality shows that A ⊂ C.
To check the converse inclusion C ⊂ A take f ∈ D(C). By definition, there exists
g ∈ X such that
a(f ,h) = (g,h)X, ∀h ∈ V
that is,
f ′(x)h′(x) dx =
g(x)h(x) dx.
Now choose h = (h, α) ∈ V such that the function h belongs to H10 (0, 1) (the
existence of such a function is ensured by the continuous embedding of H10 (0, 1)in
H1(0, 1)). Then by the last equality we cand derive that f ′ ∈ H1(0, 1) and g
is the weak derivative of f ′: it follows that f ′ ∈ H1(0, 1) and we conclude that
f ∈ H2(0, 1). Integrating by parts as in the proof of the first inclusion we see that
a(f ,h) =
f ′(x)h′(x) dx + b0f(0)h(0) + b1f(1)h(1)
= f ′(x)h(x)|10 −
f ′′(x)h(x) dx + b0f(0)h(0) + b1f(1)h(1)
= (−Af ,h) = (g,h), ∀h ∈ V.
This implies that Af = −g, and the proof is complete.
8 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
Corollary 7. The operator (A, D(A)) is self-adjoint and dissipative. Moreover it
has compact resolvent.
Proof. The self-adjointness of A follows by [13] (Proposition 1.24) and he dissipativ-
ity is obsvious. Since D(A) ⊂ H2(0, 1)×R2, the operator A has compact resolvent
and the claim follows.
Taking into account the above corollary, it follows that A generates a contraction
semigroup (etA)t≥0 on X that is analytic of angle π/2 and self-adjoint. Finally, by
[9, Corollary XIX.6.3] we obtain that etA is compact for all t > 0.
Thus we have just proved Proposition 4.
Remark 1. By the Spectral Theorem [9, Chapter XIX, Corollary 6.3] it follows
that there exists an orthonormal basis {en}n∈N of X and a sequence {λn}n∈N of real
negative numbers λn ≤ 0, such that en ∈ D(A), Aen = λnen and lim
λn = −∞.
Moreover, A is given by
λn(u, en)en, u ∈ D(A)
etAu =
eλnt(u, en)en, u ∈ X.
2.1 Spectral properties of the matrix operator
We shall now apply Theorem 2.5 in Engel[6] in order to describe the spectrum of
A. According to that result
σ(A) ⊆ σ(A0) ∪ σ(B) ∪ S (9)
where
S = {λ ∈ ρ(A0) ∩ ρ(B) : Det(F (λ)) = 0}. (10)
The matrix F (λ) is defined as
F (λ) = I − (λ−B)LλKλR(λ,B)
where the operators Lλ and Kλ are given by
Lλ = −BR(λ,B)R(0, B)C, Kλ = −A0R(λ,A0)DA,L0 .
Notice that the matrix F (λ) can also be written as
F (λ) = I + CA0R(λ,A0)D
0 R(λ,B).
Remark 2. In case when the feedback operator matrix C is identically zero, the
above construction implies that S = ∅.
Control of stochastic differential equations with dynamical boundary conditions 9
Determining the set S
In the following, we construct explicitly the set S. The idea is to construct the
matrix F (λ) and compute its determinant.
We have to distinguish two cases. If λ < 0 we have
Det(F (λ)) = 1 +
−λcos(
λ+ b0
λ+ b1
(λ+ b0)(λ + b1)
We note that the equation Det(F (λ)) = 0 has infinite solutions {λj}j∈N and
every λj belongs to the interval (−π2(j + 1)2,−π2j2).
Each λj is eigenvalue of the operator A corresponding to the eigenfunction φj =
(ej(x), ej(0), ej(1)) where
ej(x) =
−λjBj
b0 + λj
−λjx+Bj sin
−λjx.
for a normalizing constant 0 < Bj <
If λ > 0 then
Det(F (λ)) = 1 +
1 + e2
− 1 + e2
b0 + λ
b1 + λ
(b0 + λ) (b1 + λ)
We note that Det(F (λ)) > 0 for every λ > 0. This means that there are not
elements λ strictly positive in S. Moreover the eigenvalues of A in S are all negative.
Remark 3. It is possible to verify directly with some computation that the eigen-
values of A are not eigenvalues of A.
Further, the same happens in general with the eigenvalues of B, except in case
b0 and b1 satisfy an explicit relation. In any case, also if b0 and b1 happen to belong
to σ(A), they are in a finite number and do not affect its behaviour.
Therefore, with no loss of generality, in the following we may and do assume
that all the eigenvalues of A are contained in S.
Theorem 8. In the above assumptions the semigroup etA is Hilbert-Schmidt, that
|etAφi|2L2(0,1)×R2 <∞ (11)
for any orthonormal basis {φi} of L2(0, 1)× R2.
Proof. In order to prove that the semigroup etA is Hilbert-Schmidt, it is enough
verify the (11) for an orthonormal basis. Let {φi} the orthonormal sequence of
eigenfunctions of the operator A described in Remark 1. Then
|etAφi|2L2(0,1)×R2 =
e2tλi
10 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
where λi are the eigenvalues of the operator A. By (9) it follows that
e2tλi ≤
i:λi∈σ(A)
e2tλi +
i: λi∈σ(B)
e2tλi +
i:λi∈S
e2tλi .
But, by Remark 3 we have that
e2tλi ≤
i: λi∈σ(B)
e2tλi +
i:λi∈S
e2tλi
and the first of the last two series is a finite sum and the second one converges since
the eigenvalues λi in S are asymptotic to −π2i2.
3 The abstract problem
In this section we are concerned with problem (3): we introduce the relevant assump-
tions and we formulate the main existence and uniqueness result for its solution.
Let W = (W,V ) be the Wiener process taking values in = L2(0, 1) × R2. We
denote {Ft, t ∈ [0, T ]} the natural filtration of W, augmented with the family N
of P-null sets of FT :
Ft = σ(W(s) : s ∈ [0, t]) ∨N.
The filtration {Ft} satisfies the usual conditions.
Define F : [0, T ]× X → X for every u =
F(t,u) = F
F (t, u)
where F (t, u)(ξ) = f(t, ξ, u(ξ)).
Let G be the mapping [0, T ]×X → L(X,X) such that, for u =
and y =
in X,
G1(t, u) y
G2(t, v) η
where
(G1(t, u) y)(ξ) = g(t, ξ, u(ξ))y(ξ) and (G2(t, v) · η) = h(t) η;
we stress that h is a diagonal matrix.
Therefore, we are concerned with the following abstract problem
dut = Aut dt+ F(t,ut) dt+G(t,ut)dWt
ut0 = u0
on which we formulate the following assumptions.
Control of stochastic differential equations with dynamical boundary conditions 11
Assumption 9.
(i) f : [0, T ] × [0, 1] × R → R, is a measurable mapping, bounded and Lipschitz
continuous in the last component
|f(t, x, u)| ≤ K, |f(t, x, u)− f(t, x, v)| ≤ L|u− v|.
for every t ∈ [0, T ], x ∈ [0, 1], u, v ∈ R.
(ii) g : [0, T ]× [0, 1]× R → R, is a measurable mapping such that
|g(t, x, u)| ≤ K, |g(t, x, u)− g(t, x, v)| ≤ L|u− v|
for every t ∈ [0, T ], x ∈ [0, 1], u, v ∈ R.
(iii) h : [0, T ] →M(2, 2) is a bounded measurable mapping verifying |h(t)| ≤ K for
every t ∈ [0, T ].
The existence and uniqueness of the solution to (12) is a standard result in the
literature, see for instance the monograph [3]. In order to apply the known results,
we shall verify that the nonlinear coefficients F and G satisfy suitable Lipschitz
continuous conditions. That will be enough to prove the existence of a mild solution
which is a process ut adapted to the filtration Ft satisfying the following integral
equation
ut = e
e(t−s)AF(s,us) ds+
e(t−s)AG(s,us) dWs. (13)
Proposition 10. Under Assumptions 9(i)–(iii), the following hold:
1. the mapping F : X → X is measurable and satisfies, for some constant L > 0,
|F(t,u)− F(t,v)|X ≤ L|u− v|X u,v ∈ X.
2. G is a mapping [0, T ]× X → L(X) such that
a. for every v ∈ X the map G(·, ·)v : [0, T ]× X → X is measurable,
b. esAG(t,u) ∈ L2(X) for every s > 0, t ∈ [0, T ] and u ∈ X, and
c. for every s > 0, t ∈ [0, T ] and u.v ∈ X we have
|esAG(t,u)|L2(X) ≤ L s
−1/4 (1 + |u|X), (14)
|esAG(t,u)− esAG(t,v)|L2(X) ≤ L s
−1/4|u− v|X, (15)
|G(t,u)|L(X) ≤ L (1 + |u|X), (16)
for a constant L > 0.
Proof. 1. We have, for u =
and v =
|F(t,u)− F(t,v)|X = |F (t, u)− F (t, v)|X ≤ L|u− v|X ≤ L|u− v|X.
12 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
2. Condition (16) follows from the definition of G and the Assumptions 9 (ii)-(iii)
on g and h.
Now we prove condition (14). Let {φk}k∈N be an orthonormal basis in X.
|esAG(t,u)|2L2(X) =
| < esAG(t,u)φj , φk > |2X
| < G(t,u)φj , esAφk > |2X
≤ |G(t,u)|2L(X) |esA|2L2(X) ≤ L
2(1 + |u|2
)|esA|2L2(X).
Using Theorem 8,
|esA|2L2(X) ≈
e−2sn
where f(t) ≈ g(t) means that f(s)/g(s) = O(1) as s→ 0; this verifies (14).
In order to prove the last statement (15), we take the orthonormal basis
{φk}k∈N consisting of eigenvectors of A (see Remark 1). We recall that φk =
(ek(x), ek(0), ek(1)) where
ek(x) = Bk
b0 + λk
−λkx+Bk sin
−λkx.
We have
|esAG(t,u)− esAG(t,v)|2L2(X) =
| < esA[G(t,u)−G(t,v)]φj , φk > |2X
| < G(t,u)−G(t,v)φj , esAφk > |2X =
e2sλk |G(t,u)−G(t,v)φk|2.
But, for u =
and v =
, by the definition of the operator G, we have
|G(t,u)−G(t,v)φk|2X =
|g(t, x, u(x))− g(t, x, v(x))|2|ek(x)|2dx
K2|u(x)− v(x)|2dx ≤ K2|u− v|2
since the function g is Lipschitz and |ek(x)| ≤ Bk is uniformly bounded in k.
Consequently
|esAG(t,u)− esAG(t,v)|L2(X) ≤ {
e2tλk}1/2K|u− v|X
≤ |esA|L2(X)K|u− v|X
which concludes the proof.
Control of stochastic differential equations with dynamical boundary conditions 13
Proposition 11. Under the assumptions 9 for every p ∈ [2,∞) there exists a
unique process u ∈ Lp(Ω;C([0, T ];X)) solution of (12).
Proof. We can apply Theorem 5.3.1 in [4]. In fact by Proposition 4 the operator
A generates a strongly continuous semigroup {etA} of bounded linear operators in
the Hilbert space X. Moreover, for this theorem to apply we need to verify that
coefficients F and G satisfy conditions (14)—(16), which follows from Proposition
4 Stochastic control problem
After some preliminaries, in this section we are concerned with an abstract control
problem in infinite dimensions. We settle the problem in the framework of weak
control problems (see [7]).
We aim to control the evolution of the system by the boundary. This means
that we assume a boundary dynamic of the form:
∂tv(t) = bv(t)− ∂νu(t, ·) + h(t)[z(t) + V̇ (t)] (17)
where z(t) is the control process. We require that z ∈ L2(Ω× [0, T ];R2).
As in the previous section we can write the system
∂tu(t, x) = ∂
xu(t, x) + f(t, x, u(t, x)) + g(t, x, u(t, x))Ẇ (t, x)
∂tv(t) = bv(t)− ∂νu(t, ·) + h(t)[z(t) + V̇ (t)]
in the following abstract form
duzt = Au
t dt+ F(t,u
t ) dt+G(t,u
t )[Pzt dt+ dWt] ut0 = u0 (19)
where P : R2 → X is the immersion of the boundary space in the product space
X = X×R2. Equation (19), in the framework of stochastic optimal control problem,
is called the controlled state equation associated to an admissible control system.
We recall that, in general, fixed t0 ≥ 0 and u0 ∈ X, an admissible control system
(a.c.s) is given by (Ω,F, {Ft}t≥0,P, {Wt}t≥0, z) where
• (Ω,F,P) is a probability space,
• {Ft}t≥0 is a filtration in it, satisfying the usual conditions,
• {Wt}t≥0 is a Wiener process with values in X and adapted to the filtration
{Ft}t≥0,
• z is a process with values in a space K, predictable with respect to the fil-
tration {Ft}t≥0 and satisfies the constraint: z(t) ∈ Z, P-a.s., for almost every
t ∈ [t0, T ], where Z is a suitable domain of K.
14 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
In our case the space K coincide with R2.
To each a.c.s. we associate the mild solution uz of state equation the mild so-
lution uz ∈ C([t0, T ];L2(Ω;X)) of the state equation. We introduce the functional
J(t0, u0, z) = E
λ(s,uzs , zs)) ds+ Eφ(u
T ) (20)
We consider the problem of minimizing the functional J over all admissible
control systems (which is known in the literature as the weak formulation of the
control problem); any a.c.s. that minimize J -if it exsts- is called optimal for the
control problem.
We define in classical way the Hamiltonian function relative to the above problem
ψ : [0, T ]× X× X → R
setting
ψ(t,u,w) = inf
{λ(t,u, z)+ < w, P z >} (21)
and we define he following set
Γ(t,u,w) = {z ∈ Z : λ(t,u, z)+ < w, P z >= ψ(t,u, z)}
We consider the Hamilton-Jacobi-Bellman equation associated to the control
problem
∂v(t, x)
+ Lt[v(t, ·)](x) = ψ(t, x, v(t, x),G(t, x)∗∇xv(t, x)),
t ∈ [0, T ], x ∈ X,
v(T, x) = Φ(x).
where the operator Lt is defined by
Lt[φ](x) =
Trace
G(t, x)G(x)
∗∇2φ(x)
+ < Ax,∇φ(x) > .
Under suitable assumptions, if we let v denote the unique solution of (22) then we
have J(t, x, z) ≥ v(t, x) and the equality holds if and only if the following feedback
law is verified by z and uzσ:
z(σ) = Γ(σ,uzσ,G(σ,u
∗∇xv(σ,uzσ)).
Thus, we can characterize optimal controls by a feedback law.
This class of stochastic control problems, in infinite dimensional setting, has
been studied by Fuhrman and Tessitore [8] (We refer to Theorem 7.2 in that paper
for precise statements and additional results).
In order to characterize optimal controls by a feedback law we have to require
that the abstract operators F and G satisfy further regularity conditions.
We will prove that, under suitable assumptions on the functions f and g in the
problem (18), the abstract operators fit the required conditions.
Control of stochastic differential equations with dynamical boundary conditions 15
We impose that the operators F and G are Gâteaux differentiable. This notion
of differentiability is weaker than the differentiability in the Fréchet sense.
We recall that for a mapping F : X → V , where X and V denote Banach spaces,
the directional derivative at point x ∈ X in the direction h ∈ X is defined as
∇F (x;h) = lim
F (x+ sh)− F (x)
whenever the limit exists in the topology of V . F is called Gâteaux differentiable
at point x if it has directional derivative in every direction at point x and there
exists an element of L(X,V ), denoted ∇F (x) and called Gâteaux derivative, such
that ∇F (x;h) = ∇F (x)h for every h ∈ X .
Definition 12. We say that a mapping F : X → V belongs to the class G1(X ;V )
if it is continuous, Gâteaux differentiable on X, and ∇F : X → L(X,V ) is strongly
continuous.
The last requirement of the definition means that for every h ∈ X the map
∇F (·)h : X → V is continuous. Note that ∇F : X → L(X,V ) is not continuous
in general if L(X,V ) is endowed with the norm operator topology; clearly, if this
happens then F is Fréchet differentiable on X . Membership of a map in G1(X,V )
may be conveniently checked as shown in the following lemma.
Lemma 13. A map F : X → V belongs to G1(X,V ) provided the following condi-
tions hold:
i) the directional derivatives ∇F (x;h) exist at every point x ∈ X and in every
direction h ∈ X;
ii) for every h, the mapping ∇F (·;h) : X → V is continuous;
iii) for every x, the mapping h 7→ ∇F (x;h) is continuous from X to V .
When F depends on additional arguments, the previous definitions and proper-
ties have obvious generalizations.
The following assumptions are necessary in order to provide Gâteaux differen-
tiability for the coefficients of the abstract formulation.
Assumption 14. For a.a. t ∈ [0, T ], ξ ∈ [0, 1] the functions f(t, ξ, ·) and g(t, ξ, ·)
belong to the class C1(R).
Proposition 15. Under assumptions 9 and 14, for every s > 0, t ∈ [0, T ],
F(t, ·) ∈ G1(X,X), esAG(t, ·) ∈ G1(X, L2(X)).
Proof. The first statement is an immediate consequence of the fact that f(t, ξ, ·) ∈
C1(R,R). In order to prove that esAG(t, ·) belongs to the class G1(X, L2(X)) we use
the continuous differentiability of g and an argument similar to that used in the
proof of Proposition 10.
We note that, for u =
and v =
, the gradient operator∇
esAG(t,u)
is an Hilbert Schmidt operator that maps
7→ esA
gu(t, ·, u(·))w(·)v(·)
= esA (∇u(G(t,u)v)(w))
16 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
In fact, we have
esAG(t,u+ rv) − esAG(t,u)
−∇esAG(t,u)v
L2(X)
= lim
esAG(t,u+ rv) − esAG(t,u)
φj − esA (∇u(G(t,u)v)φj) , φk >
= lim
G(t,u+ rv) −G(t,u)
−∇uG(t,u)v
φj , e
sAφk >
= lim
e2sλk
G(t,u+ rv) −G(t,u)
−∇uG(t,u)v
= lim
e2sλk
g(t, u(ξ) + rv(ξ)) − g(t, u(ξ))
ek(ξ)− gu(t, u(ξ))v(ξ)ek(ξ)
≤ c lim
e2sλk
g(t, u(ξ) + rv(ξ)) − g(t, u(ξ))
− gu(t, u(ξ))v(ξ)
= c lim
e2sλk
gu(t, u(ξ) + αrv(ξ)) − gu(t, u(ξ))
dα v(ξ)
and, by dominated convergence, this limit is equal to zero. In similar way we can
prove the points (ii)− (iii) of Lemma 13 to obtain the thesis.
In order to prove the main result of this section we require the following hypoth-
esis.
Assumption 16.
(i) λ is measurable and for a.e. t ∈ [0, T ], for all u,u′ ∈ X, z ∈ Z
|λ(t,u, z)− λ(t,u′, z)| ≤ C|1 + u+ u′|m|u− u′|
|λ(t, 0, z)| ≤ C
for suitable C ∈ R+, m ∈ N;
(ii) Z is a Borel and bounded subset of R2;
(iii) Φ ∈ G1(X,R) and, for every σ ∈ [0, T ], ψ(σ, ·, ·) ∈ G1,1(X × X,R);
(iv) for every t ∈ [0, T ], u,w,h ∈ X
ψ(t,u,w)h| + |∇
φ(u)h| ≤ L|h|(1 + |u|)m;
(v) for all t ∈ [0, T ], for all u ∈ X and w ∈ X there exists a unique Γ(t,u,w) ∈ Z
that realizes the minimum in (21). Namely
λ(t,u,Γ(t,u,w))+ < w, PΓ(t,u,w) >= ψ(t,u,w)
Control of stochastic differential equations with dynamical boundary conditions 17
Theorem 17. Suppose that assumptions 9, 14 and 16 hold. For all a.c.s. we have
J(t0, u0, z) ≥ v(t0, u0) and the equality holds if and only if the following feedback
law is verified by z and uz:
z(σ) = Γ(σ,uzσ, G(σ,u
∗∇xv(σ,uzσ)), P− a.s. for a.a. σ ∈ [t0, T ]. (23)
Finally there exists at least an a.c.s. for which (23) holds. In such a system the
closed loop equation:
duτ = Auτ dτ +G(τ,uτ )PΓ(τ,uτ ,G(τ,uτ )
∗∇xv(τ,uτ )) dτ
+ F(τ,uτ ) dτ +G(τ,uτ ) dWτ , τ ∈ [t0, T ],
ut0 = u0 ∈ X.
admits a solution and if z(σ) = Γ(σ,uσ, G(σ,uσ)
∗∇xv(σ,uσ)) then the couple (z,u)
is optimal for the control problem.
Proof. By Proposition 4 we know that A generates a strongly continuous semigroup
of linear operators etA on X. The assumption 9 ensures that the statements in
Proposition 10 hold. Moreover the assumption 14 guarantees that the results in
Proposition 15 are true. Finally these conditions together with the assumption 16
allow us to apply Theorem 7.2 in [8] and to perform the synthesis of the optimal
control.
References
1. M. Bertini, D. Noja, A. Posilicano, Dynamics and Lax-Phillips scattering for gen-
eralized Lamb models, J. Phys. A: Math. Gen. 39 (2006), 15173–15195
2. Igor Chueshov, Björn Schmalfuss, Parabolic stochastic partial differential equations
with dynamical boundary conditions, Differential Integral Equations 17 (2004), no.
7-8, 751–780.
3. Giuseppe Da Prato, Jerzy Zabczyk, Stochastic equations in infinite dimen-
sions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University
Press, Cambridge, 1992.
4. G. Da Prato, J. Zabczyk, Ergodicity for infinite-dimensional systems, Lon-
don Mathematical Society Lecture Notes Series, 229, Cambridge University Press,
1996.
5. A. Debussche, M. Fuhrman, G. Tessitore, Optimal Control of a Stochastic Heat
Equation with Boundary-noise and Boundary-control, to appear in ESAIM Con-
trol, Optimisation and Calculus of Variations.
6. K.-J. Engel, Spectral theory and generator property for one-sided coupled operator
matrices, Semigroup Forum 58 (1999), 267–295.
7. W. H. Fleming, H. M. Soner, Controlled Markov processes and viscosity
solutions, Springer-Verlag, 1993.
8. M. Fuhrman, G. Tessitore, Non linear Kolmogorov equations in infinite dimen-
sional spaces: the backward stochastic differential equations approach and appli-
cations to optimal control, Ann. Probab. 30 (2002), no. 3: 1397-1465.
9. Israel Gohberg, Seymour Goldberg, Marinus A. Kaashoek, Classes of linear oper-
ators, Vol. I, Birkhser Verlag, Basel, 1990. Operator Theory: Advances and Applica-
tions, 49.
18 S. Bonaccorsi, F. Confortola, E. Mastrogiacomo
10. Marjeta Kramar, Delio Mugnolo, Rainer Nagel, Semigroups for initial-boundary
value problems.In: Evolution equations: applications to physics, industry,
life sciences and economics (Levico Terme, 2000), 275–292, Progr. Nonlinear
Differential Equations Appl., 55, Birkhuser, Basel, 2003.
11. Bohdan Maslowski, Stability of semilinear equations with boundary and pointwise
noise, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 55–93.
12. D. Mugnolo, Asymptotics of semigroups generated by operator matrices, Ulmer
seminare 10 (2005), 299–311.
13. El Maati Ouhabaz, Analysis of heat equations on domains, London Math-
ematical Society Monographs Series, 31. Princeton University Press, Princeton, NJ,
2005.
	Setting of the problem
	Generation properties
	Spectral properties of the matrix operator
	The abstract problem
	Stochastic control problem
