Advances in Differential Equations
Volume 10, Number 12 (2005), 1389–1436
OPTIMAL BOUNDARY CONTROL AND
RICCATI THEORY FOR ABSTRACT DYNAMICS
MOTIVATED BY HYBRID SYSTEMS OF PDES
Paolo Acquistapace
Università di Pisa, Dipartimento di Matematica
Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Francesca Bucci
Università di Firenze, Dipartimento di Matematica Applicata
Via S. Marta 3, 50139 Firenze, Italy
Irena Lasiecka
University of Virginia, Department of Mathematics
P. O. Box 400137, Charlottesville, VA 22904-4137
(Submitted by: Giuseppe Da Prato)
Abstract. We study the quadratic optimal control problem over a finite
time horizon for a class of abstract systems with non analytic underlying semigroup etA and unbounded control operator B. It is assumed
∗
that a suitable decomposition of the operator B ∗ etA is valid, where
only one component satisfies a ‘singular estimate’, whereas for the other
component specific regularity properties hold. Under these conditions,
we prove well posedness of the associated differential Riccati equation,
and in particular that the gain operator is bounded on a dense set. In
spite of the unifying abstract framework used, the prime motivation (and
application) of the resulting theory of linear-quadratic problems comes
from optimal boundary control of a thermoelastic system with clamped
boundary conditions. The non-trivial trace regularity estimate showing
that this PDE mixed problem fits into the distinct class of models under
examination—for which we have developed the present, novel optimal
control theory—is established, as well.
1. Introduction
In the development of the optimal control theory for (linear) Partial Differential Equations (PDEs) over the last four decades, some major steps
can be singled out. A first extension of the finite-dimensional results to
Accepted for publication: June 2005.
AMS Subject Classifications: 49N10; 49J20, 49N35, 74K20, 74F05.
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Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
the infinite-dimensional context was carried out with the study of mixed
(initial/boundary-value) problems for PDEs with distributed control, where
the corresponding abstract dynamics y ! = Ay + Bu yields a bounded control operator B. Next, motivated by the modeling of parabolic equations
with boundary control, systems with unbounded control operators B have
arisen, making the analysis of the corresponding Riccati equations much
more challenging. As is well known, the substantial new difficulties have
been overcome by exploiting analyticity of the free dynamics operator etA .
A drastic change of scenario occurs in the case of hyperbolic flows. Justified
by relevant PDE illustrations of hyperbolic (and hyperbolic-like) problems,
less restrictive hypotheses on the unboundedness of B are necessary, resulting in a general unboundedness of the gain operator B ∗ P which appears in
the associated Algebraic Riccati equations, and in a lack of well posedness
of the Differential Riccati equations, unless a certain degree of smoothing of
the observation operator is assumed. It is beyond the scope of the present
paper to give a comprehensive account of the literature on the linear quadratic problem in an infinite-dimensional context. The reader is referred to
the advanced monographs [7] and [19], and their numerous references, for an
exhaustive treatment of the aforementioned theories. Still, we are pleased
to recall explicitly the pioneering book [5] and the treatise [22], and some
of the most valuable contributions to the development of the Riccati theory
for a single PDE with boundary control. Namely, (i) the former works [10],
[12, 13], [17], [11] for parabolic dynamics (see also [2, 3] for non-autonomous
systems); and (ii) [14]—a cornerstone of the theory pertaining to hyperboliclike dynamics—along with the subsequent refinements of [9], [23, 24]).
In recent years, new PDE models stem from modern technological applications, such as structural acoustic problems1, thermo-elastic systems, and
more generally systems of coupled PDEs of different type. It is natural that
the two basic classes of evolutions for which a theory of the corresponding optimal control problems was available in the literature—the classes of
parabolic-like and hyperbolic-like dynamics—were no longer suitable to describe many of those PDE models. Thus, motivated by the analysis of a
structural acoustic system, which comprised a hyperbolic equation strongly
coupled with a parabolic-like one, with the work [4] a novel class of abstract
models (of the same form y ! = Ay+Bu) has been identified, characterized by
a ‘singular estimate’ for the operator etA B near t = 0. This kind of estimate
1These systems of interconnected PDEs arise in the modeling of noise reduction prob-
lems within mechanical structures.
Optimal boundary control and Riccati theory
1391
reveals that although the overall dynamics is not analytic, the smoothing
properties of the analytic component of the system have somewhat propagated into the entire system. Under this abstract condition assumed on
etA B—which captures an intrinsic feature of the coupled dynamics—a satisfactory optimal control theory is valid, which constitutes an extension of the
analytic one (see [15], [20], [21], [16]). To date, it has been shown that this
theory covers many relevant systems of PDEs, including various structural
acoustic models, thermoelastic problems, ‘sandwich beams’ problems, etc.
(see, e.g., [9], [21]).
However, several important systems still do not fit exactly into the above
class, though they are close. An example is given by an established thermoelastic plate model with clamped boundary conditions2, subject to Dirichlet
controls acting on the boundary ([8]). The precise mathematical description of the mixed PDE problem will be given in Section 3. It is in response
to this demand that an extension of the ‘singular estimates model’ (briefly,
the ‘third’ class) has attracted considerable attention. In this new frame∗
work, the singular estimate does not need to hold for B ∗ etA , but only for
a component of it. A prime illustration (as well as motivation) for this new
class of evolutions is the aforementioned thermoelastic problem. Then, as
usual and as it should be, the theory developed—namely, the assumptions
imposed on the abstract dynamics, and the corresponding results—meets
the requirements of concrete physical applications.
The goal of this paper is to provide a complete Riccati theory for a new
class of systems extending the third class (hence, also the analytic class).
It is important to emphasize that while the new class is motivated by significant PDE models—one may say systems of PDEs which display a hyperbolic/parabolic coupling, yet with an overall hyperbolic character—the
mathematical aspects of the theory and the corresponding results are new
and different from the canonical, by now, singular case. Indeed, the most
notable differences are that the gain operator is no longer bounded on the
state space—rather, on a dense subspace—and the optimal control function
is no longer continuous in time, unlike the case of parabolic-like dynamics
or systems yielding singular estimates.
1.1. Orientation. The plan of the paper is the following. In the next Section 2 we introduce the abstract dynamics under consideration, along with
2From a physical point of view, clamped boundary conditions are the most natural
ones.
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Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
the corresponding assumptions, and we give the precise statement of the paper’s main result (Theorem 2.3). Its long proof will occupy several sections
(namely, Sections 4 through 6, in addition to Appendix B). Hence, Section 2
includes as well a series of focused observations regarding the model and the
key steps of the proof.
In Section 3 we describe the PDE mixed problem motivating and illustrating the abstract theory. The nontrivial boundary regularity estimate,
which is crucial in order to prove that this thermoelastic problem fits into
the abstract framework, is shown here.
The proof of Theorem 2.3 is started with the analysis of the regularity of
the operator L which maps the boundary input to the interior solution (and
of its adjoint operator L∗ , as well). The corresponding results are collected
in Appendix B.
Section 4 is focused on the optimal pair, and includes in particular preliminary information concerning the properties of the evolution map Φ(t, s).
Section 5 contains the fundamental result that the gain operator B ∗ P (t)
is well defined and bounded on a dense subset of the state space (Proposition 5.3).
In Section 6, seeking to derive that the operator P (t)—initially defined
in terms of the optimal state—does satisfy the Differential Riccati equation
on D(A), we ascertain the differentiability of the evolution map Φ(t, s) on
D(A).
1.2. Notation. The notation used in the paper is largely standard within
the literature: in particular, the one pertaining to function spaces. We
just make a few remarks, for the reader’s convenience. Let H and X be
a Hilbert and a Banach space, respectively. The symbol (·, ·)H denotes the
inner product in H (the sub-script will be omitted, when H is clear), whereas
!·, ·"X represents the pairing of X ∗ and X.
We employ the symbol D(A), and not simply D(A), to denote the domain
of a closed operator A: the reason is that the letter D naturally stands for
the Dirichlet mapping.
By χE we mean the characteristic function of the set E. If q is a Sobolev
q
exponent (q #= 1), we shall write q ! in lieu of q−1
. Regarding the estimates,
we often employ the letter c (cT , C) to denote any constant, whose value
may therefore change from line to line in a given computation.
Optimal boundary control and Riccati theory
1393
2. Assumptions and main result
We consider the abstract control system
!
y ! (t) = Ay(t) + Bu(t) ,
y(0) = y0
t ∈ [0, T ] ,
(2.1)
with associated cost functional
" T
#
$
%Ry(t)%2Z + %u(t)%2U dt
J(u, y) =
(2.2)
0
to be minimized over all u ∈ L2 (0, T ; U ), where y is the solution of (2.1) due
to u. The following hypotheses on the model will be assumed throughout
the paper.
Hypothesis 2.1. T is a positive number, Y , U and Z are separable complex
Hilbert spaces, R is a bounded linear operator from Y to Z; A : D(A) ⊆
Y → Y is a closed linear operator, with A−1 ∈ L(Y ), which generates a
strongly continuous semigroup {etA }t≥0 on Y . Finally, B : U → [D(A∗ )]! is
a bounded linear operator.
∗
Hypothesis 2.2. For each t ∈ [0, T ], the operator B ∗ etA can be represented
as
∗
B ∗ etA x = F (t)x + G(t)x,
t ≥ 0, x ∈ D(A∗ ),
(2.3)
where F (t) : Y → U , and G(t) : D(A∗ ) → U , t > 0 are bounded linear
operators satisfying the following assumptions:
(i) there is γ ∈ ( 12 , 1) such that
%F (t)%L(Y,U ) ≤ ct−γ
∀t ∈ (0, T ];
(2.4)
(ii) the operator G(·) belongs to L(Y, Lp (0, T ; U )) for all p ∈ [1, ∞[, with
%G(·)%L(Y,Lp (0,T ;U )) ≤ cp < ∞
∀p ∈ [1, ∞[;
(2.5)
(iii) there is ε > 0 such that:
(a) the operator G(·)A∗ −ε belongs to L(Y, C([0, T ], U )), and in particular
%A−ε G(t)∗ %L(U,Y ) ≤ c < ∞
∀t ∈ [0, T ];
(b) the operator R∗ R belongs to L(D(Aε ), D(A∗ ε )), i.e.,
%A∗ ε R∗ RA−ε %L(Y ) ≤ c < ∞;
(2.6)
(2.7)
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Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
(c) there is q ∈ (1, 2) (depending, in general, on ε) such that
∗
the operator B ∗ e·A R∗ RAε has an extension belonging to
L(Y, Lq (0, T ; U )).
It is a standard matter to verify that under Hypotheses 2.1 and 2.2, for
fixed s ∈ [0, T ), the problem of minimizing the functional
" T
#
$
Js (u, y) =
%Ry(t)%2Z + %u(t)%2U dt
(2.8)
s
among the functions u ∈ L2 (s, T ; U ) and y ∈ L2 (s, T ; Y ) satisfying the mild
form of the state equation
! !
y (t) = Ay(t) + Bu(t), t ∈ [s, T ]
(2.9)
y(s) = x ∈ Y
has a unique solution (ŷ(·, s; x), û(·, t; x)). The mild form of (2.9) reads as
y(t, s; x) = e(t−s)A x + (Ls u)(t),
where the operator Ls is defined by
" t
(Ls u)(t) =
e(t−s)A Bu(σ) dσ
s
t ∈ [s, T ],
∀u ∈ L2 (s, T ; U ).
We note that the adjoint of Ls is the operator
" T
∗
∗
B ∗ e(τ −t)A y(τ ) dτ
∀y ∈ L2 (s, T ; Y );
(Ls y)(t) =
(2.10)
(2.11)
(2.12)
t
in particular it does not depend on s, or, more precisely, L∗t y = (L∗s y)|(t,T )
for t ∈ (s, T ]. The regularity properties of Ls and L∗s will be analyzed in
Appendix B.
Let us state our main result.
Theorem 2.3. With reference to the control problem (2.1)–(2.2), under
Hypotheses 2.1 and 2.2 the following statements are valid for each s ∈ [0, T ).
(i) For each x ∈ Y the optimal pair (û(·, s; x), ŷ(·, s; x)) satisfies
%
ŷ(·, s; x) ∈ C([s, T ], Y ), û(·, s; x) ∈
Lp (s, T ; U ).
(2.13)
1≤p<∞
(ii) The operator Φ(t, s) ∈ L(Y ), defined for x ∈ Y by
Φ(t, s)x = ŷ(t, s; x) = e(t−s)A x + [Ls û(·, s; x)](t), s ≤ t ≤ T,
(2.14)
Optimal boundary control and Riccati theory
1395
is an evolution operator; i.e.,
Φ(t, s) = Φ(t, τ )Φ(τ, s)
Φ(t, t) = IY ,
for s ≤ τ ≤ t ≤ T.
(iii) For each t ∈ [0, T ] the operator P (t) ∈ L(Y ), defined by
P (t)x =
"
T
∗
e(τ −t)A R∗ RΦ(τ, t)x dτ,
t
x ∈ Y,
(2.15)
is self-adjoint and positive; it belongs to L(Y, C([0, T ], Y )) and is
such that
(P (s)x, x)Y = Js (û(·, s; x), ŷ(·, s; x))
∀s ∈ [0, T ].
(iv) The gain operator B ∗ P (t) belongs to L(D(Aε ), C([0, T ], U )) and the
optimal pair satisfies for s ≤ t ≤ T
û(t, s; x) = −B ∗ P (t)ŷ(t, s; x)
∀x ∈ Y.
(2.16)
(v) The operator Φ(t, s) defined in (2.14) satisfies for s < t ≤ T :
1
∂Φ
(t, s)x = −Φ(t, s)(A − BB ∗ P (t))x ∈ L γ (s, T ; [D(A∗ ε )]! )
∂s
(2.17)
for all x ∈ D(A), and
∂Φ
(t, s)x = (A − BB ∗ P (t))Φ(t, s)x ∈ C([s, T ], [D(A∗ )]! )
∂t
(2.18)
for all x ∈ D(Aε ).
(vi) The operator P (t) satisfies the following differential Riccati equation
in [0, T ) :
(P ! (t)x, y)Y + (P (t)x, Ay)Y + (P (t)Ax, y)Y
+(R∗ Rx, y)Y − (B ∗ P (t)x, B ∗ P (t)y)Y = 0
∀ x, y ∈ D(A).
(2.19)
The proof of Theorem 2.3 will be split into several sections. This section
provides for the reader’s convenience a series of focused observations. We
first sketch out a comparison with the previous general classes of (linear)
evolutions considered in the literature, and next we give some outlining
ideas for the proof of Theorem 2.3.
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Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
2.1. A comparison with previous abstract models. Let us briefly recall
the major classes of abstract systems (of the form (2.1)) introduced so far
for the representation of mixed problems for PDEs with boundary/point
control, and the essential results pertaining to the corresponding Riccati
theories. In each case, the basic assumptions in Hypothesis 2.1 will be taken
for granted.
First class: (Parabolic-like dynamics). Under the assumptions
• etA is an analytic semigroup,
• there exists r ∈ [0, 1[ such that A−r B is bounded,
one has that (−A∗ )1−$ P is bounded, and hence the key property that B ∗ P (t)
is bounded on the state space Y (besides further regularity properties for
the Riccati operator P (t)). This ensures well posedness of the corresponding
differential Riccati equations.
Second class: (Hyperbolic-like dynamics). This class is characterized
by the following ‘trace regularity assumption’, also known as admissibility
condition:
" T
∗
•
||B ∗ etA x||2U dt ≤ CT ||x||2Y , ∀x ∈ D(A∗ ).
0
In this case it is known that B ∗ P (t) may not be densely defined, unless the
observation operator R has some smoothing property. If T = ∞, B ∗ P is
defined only as a suitable extension Be∗ P .
Third class: (Systems which yield singular estimates). The sole assumption
• there exists γ ∈ [0, 1[ such that %etA B%L(U,Y ) ≤ C t−γ near t = 0,
ensures that B ∗ P (t) is bounded on Y . The motivation for introducing this
model has come from coupled systems of hyperbolic/parabolic PDEs.
The present class: We make reference to systems characterized by the
assumptions listed in Hypothesis 2.2. For these models Theorem 2.3 establishes the following novel, distinct result: B ∗ P (t) is densely defined and
bounded on the space D(Aε ). Consequently, the differential Riccati equation
is well posed, as desired.
Notice that if G = 0 this model reduces to the preceding one, whereas
if F = 0 one may find some reminiscence of the second (hyperbolic) class.
Indeed, more precisely,
(1) Hypothesis 2.2(ii) is stronger than the admissibility condition of the
hyperbolic case, since the latter requires only p = 2.
Optimal boundary control and Riccati theory
1397
(2) As for Hypothesis 2.2(iii), item (a) always holds true with ε = 1.
However, a “small” ε is worth having, in order that the most challenging condition (iii)(c) is satisfied.
(3) It is important to emphasize that the assumption (iii)(b) just requires
that the operator R∗ R ‘maintains regularity’; in particular, it allows
R to be the identity.
(4) Finally, a distinctive feature of condition (iii)(c) is the correlation between the parameters ε and q. Notice that admissibility corresponds
to ε = 0 and q = 2, which are limiting points for the parameters’
range.
2.2. Outlining ideas for the proof of Theorem 2.3. The approach of
our analysis is of variational type, and it follows the abstract operator treatment of the quadratic optimal control problem for PDEs—and of the related
Differential/Algebraic Riccati equations—developed in the last twenty-five
years by the authors of [19]. Thus, the general (by now classical) programme
to be carried out consists of the following, somewhat standard, steps. (i)
First, the optimal pair {û, ŷ} is characterized only in terms of the data of
the problem (see Remark 4.2); (ii) next, an operator P (t) is constructed
in terms of the optimal state (see formula (5.1)), and hence of the original
data of the problem; (iii) finally, the operator P (t) is shown to satisfy the
differential Riccati equation.
However, at a technical level, new difficulties arise due to the distinct
assumptions on the abstract dynamics. This specifically pertains to the
third step, by far the most complex.
We recall that a prime theoretical issue concerning the solving of optimal
control problems with quadratic cost functional for PDE mixed problems
is to investigate the regularity of the operator L which maps the boundary
input to the interior solution (and of its adjoint operator L∗ , as well). This
preliminary analysis, reported in Appendix B, leads to the detailed conclusions of Proposition B.3 and of Proposition B.4, and it culminates in the
statements of Proposition B.5—which pinpoints the regularity of the operator R∗ RL—and its Corollary B.7. In turn, Proposition B.5 is critically used
in order to show Proposition 5.3, which establishes that the gain operator
B ∗ P (t) is well defined and bounded on D(Aε ); that is the crux of the matter
of the whole theory.
Finally, in order to derive that the operator P (t) defined in (5.1) satisfies
the Differential Riccati equation on D(A), one needs to differentiate strongly
the evolution map Φ(t, s) on D(A) (where Φ(t, s)x = ŷ(t, s; x)). This is a
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Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
challenging point, as well. We recall that in the case of systems which yield
singular estimates for the operator eAt B, one has, as a consequence, that
the same degree of singularity is transferred to the operator Φ(t, s)B, and
this property is critical in deriving the Differential Riccati equation. Of
course this does not hold in the present case. Section 6 is entirely devoted to
showing—by using direct and rigorous proofs—differentiability of Φ(t, s)x
with respect to both variables, when x ∈ D(A), resulting in Theorem 6.1
and Theorem 6.2, respectively (the latter being much more intricate and
lengthy).
3. A physically motivated PDE illustration
In this section we introduce a thermoelastic problem which constitutes a
significant PDE application of the abstract theory developed in the paper.
By showing this illustration at the very outset, we aim to facilitate the PDE
interpretation of the various assumptions listed in Hypothesis 2.2.
Let Ω be a bounded domain of R2 with smooth boundary Γ, and let T >
0. We consider the following model of a thermoelastic plate with clamped
boundary conditions, in the variables w(t, x) (vertical displacement) and
θ(t, x) (temperature):

wtt − ρ∆wtt + ∆2 w + ∆θ = 0
in Q := (0, T ] × Ω




θ
−
∆θ
−
∆w
=
0
in Q

t
 t
(3.1)
on Σ := (0, T ] × Γ
w = ∂w
∂ν = 0



θ=u
on Σ



0
1
0
w(0, ·) = w , wt (0, ·) = w , θ(0, ·) = θ in Ω,
where ρ > 0, so that the elastic equation is of Kirchhoff type. The dynamics
of the plate is influenced by a thermal control u acting on the temperature
at the boundary.
We associate to (3.1) the following quadratic cost functional to be minimized
over all u ∈ L2 (0, T ; L2 (Γ)):
" T"
" T"
2
2
2
(|∆w| +|∇wt | +|θ| ) dx dt+
|u|2 ds dt . (3.2)
J(w, wt , θ; u) =
0
Ω
0
Γ
We want to rewrite problem (3.1) in the abstract form (2.1). To this purpose we start by introducing the realization of the Laplacian with Dirichlet
boundary conditions
AD f = −∆f,
D(AD ) = H 2 (Ω) ∩ H01 (Ω),
(3.3)
Optimal boundary control and Riccati theory
1399
and the realization of the Bilaplacian with clamped boundary conditions
Denoting by D :
Af = ∆2 f,
L2 (Γ)
→
L2 (Ω)
h = Dg
and setting
D(A) = H 4 (Ω) ∩ H02 (Ω).
the Dirichlet map; i.e.
*
∆h = 0 in Ω
⇐⇒
h=g
on Γ,
M = I + ρAD , D(M) = D(AD ),
we may rewrite problem (3.1) as


 Mwtt + Aw − AD θ = −AD Du,
θt + AD θ + AD wt = AD Du

 w(0) = w0 , w (0) = w1 , θ(0) = θ0 .
t
(3.4)
(3.5)
(3.6)
(3.7)
Now let us rearrange the left-hand side of (3.7). Consider the operator A
given by


0
I
0
0
M−1 AD  ,
(3.8)
A :=  −M−1 A
0
−AD
−AD
with domain
D(A) := D(A3/4 ) × [D(A1/2 ) ∩ D(AD )] × D(AD ) :
(3.9)
it is proved in [18] that it generates a strongly continuous contraction semigroup {etA }t≥0 in the space
Y := D(A1/2 ) × D(M1/2 ) × L2 (Ω) = H02 (Ω) × H01 (Ω) × L2 (Ω).
We also have
A−1

−A−1 AD −A−1 M −A−1
 ∈ L(Y ).
I
0
0
=
−1
−I
0
−AD

Moreover, if we set U := L2 (Γ), and define the operator


0
B :=  −M−1 AD D  ∈ L(U, [D(A∗ )]! ),
AD D
then it is easy to see that

0
A−1 B =  0  ∈ L(U, Y ).
−D

(3.10)
(3.11)
(3.12)
(3.13)
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Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Using the operators (3.8) and (3.12), problem (3.7) reduces to the form
(2.1) with respect to the variable y = (w, wt , θ), with initial datum y0 =
(w0 , w1 , θ0 ). In this way we have verified Hypothesis 2.1.
In order to verify Hypothesis 2.2, we note that the adjoint A∗ of A is given
by


0
−I
0
A∗ =  M−1 A 0 −M−1 AD  ,
(3.14)
0
AD
−AD
with domain
D(A∗ ) ≡ D(A) = [H 3 (Ω) ∩ H02 (Ω)] × H02 (Ω) × [H 2 (Ω) ∩ H01 (Ω)].
Next, recalling the well-known property
/
3
∂g //
∗
D AD g =
∀g ∈ H 2 +δ (Ω) ∩ H01 (Ω),
∂ν /Γ
it follows that


/
y1
/
∂
∗
∗

y
(y3 − y2 )//
B y=B
=
2
∂ν
Γ
y3
∀δ > 0,
(3.15)
(3.16)
∀y ∈ D(A∗ ).
(3.17)
We evaluate now the operator B ∗ etA . Following [8], we have
 0  

w
w(t)
∗
∗
etA z = etA  w1  =  −wt (t) 
θ(t)
θ0
(3.18)
∗
where (w(t), wt (t), θ(t)) is the solution of the uncontrolled version of system
(3.7) with initial datum (w0 , −w1 , θ0 ). Hence, by (3.17) and the boundary
conditions in (3.1),
/
∂
∂θ(t) //
∗
/
(3.19)
B ∗ etA z =
(wt (t) + θ(t))/ =
/ .
∂ν
∂ν Γ
Γ
It is shown in [8] that
∂θ(t) //
(3.20)
/ = F (t)z + G(t)z,
∂ν Γ
where F (t) and G(t) are suitable, explicit operators satisfying the following
estimates for every y ∈ Y :
3
%F (t)y%Y ≤ Cε t− 4 −ε %y%Y
∀ε ∈ (0, 14 ),
∀t ∈ (0, T ],
∀p ∈ [1, ∞).
%G(·)y%Lp (0,T ;U ) ≤ Cp %y%Y
Namely, (i) and (ii) of Hypothesis 2.2 hold true.
(3.21)
(3.22)
Optimal boundary control and Riccati theory
1401
Regarding Hypothesis 2.2(iii), the validity of the regularity assumption in
item (a), for arbitrarily small ε, was proved in [8], as well. Next, notice that
in the present case the observation operator R coincides with the identity
operator I, while D(A∗ ε ) ≡ D(Aε ) (see [8, Remark 2.5]). Consequently,
Hypothesis 2.2(iii)(b) is readily verified.
As for the most challenging Hypothesis 2.2(iii)(c), it will be satisfied provided
that we show that there exist q ∈ (1, 2) and K ≥ 0 such that
∗
%B ∗ e·A A∗ ε y%Lq (0,T ;U ) ≤ K%y%Y
∀y ∈ D(A∗ ε ) .
(3.23)
Indeed, combining (3.23) with Hypothesis 2.2(iii)(b) gives, a fortiori,
∗
∗
%B ∗ e·A R∗ RAε y%Lq (0,T ;U ) = %B ∗ e·A A∗ ε A∗ −ε R∗ RAε y%Lq (0,T ;U )
≤ K%A∗ −ε R∗ RAε y%Y ≤ K%R∗ R%L(D(Aε ),D(A∗ ε )) %y%Y
(3.24)
∀y ∈ D(A∗ ε ) .
Set now z = A∗ ε−1 y: as, by (3.19),
d tA∗
∂θt (t) //
(3.25)
e z=
/ ;
dt
∂ν Γ
in order to verify Hypothesis 2.2(iii)(c) it will be sufficient to prove that
0
0
0 ∗ d ·A∗ 0
≤ C%z%D(A∗ 1−ε )
∀z ∈ D(A∗ ) .
(3.26)
e z0 q
0B
dt
L (0,T ;U )
In view of the second equality in (3.25), the above estimate is equivalent (in
PDE terms) to a boundary regularity estimate, which has been established
in [1, Theorem 1.1] where, more precisely, it is shown that
0 ∂θ 0
0 t0
0
0
∂ν Lq (0,T ;L2 (Γ))
1
2
≤ Cq,ε %w0 %H 3−ε (Ω)∩H02 (Ω) + %w1 %H 2−ε (Ω) + %θ0 %H 2−2ε (Ω)∩H 1 (Ω)
∗
B ∗ etA A∗ ε y = B ∗
0
0
0
for all ε ∈ (0,
and q ∈
Thus, the control problem
(3.1)–(3.2) fits into the abstract theory of the present paper.
1
4)
4
(1, min{ 87 , 3+4ε
}).
4. The optimal pair
Let us consider the optimal control problem (2.9)–(2.8) over the interval
[s, T ], i.e., with s as initial time. Throughout the paper, we will make
critical use of several regularity results relative to the operator Ls and to
its adjoint L∗s (defined by (2.11) and (2.12), respectively). For the sake of
a more focused exposition, this fundamental analysis—usually performed at
the outset—is postponed to Appendix B. So our first result establishes the
distinct regularity properties of the optimal pair of problem (2.9)–(2.8).
1402
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Proposition 4.1. Under Hypotheses 2.1 and 2.2, for the optimal control
problem (2.8)–(2.9) there is a unique optimal pair (û(·, s; x), ŷ(·, s; x)), such
that
%
Lp (s, T ; U ) ;
ŷ(·, s; x) ∈ C([s, T ], Y ), û(·, s; x) ∈
p<∞
moreover,
%ŷ(·, s; x)%C([s,T ],Y ) + %û(·, s; x)%Lp (s,T ;Y ) ≤ cp,T %x%Y
∀p ∈ [1, ∞). (4.1)
Proof. As remarked in the Introduction, it is a standard matter to show
existence and uniqueness of the optimal pair in L2 (s, T ; Y ) × L2 (s, T ; U );
moreover, since
ŷ(t, s; x) = e(t−s)A x + (Ls û(·, s : x))(t),
t ∈ [s, T ],
(4.2)
by Proposition B.3(ii) we immediately get ŷ(·, s; x) ∈ Lp (s, T ; Y ) with p =
2
2γ−1 .
Next, writing down the Euler equation Js! (ŷ(·, s; x), û(·, s; x)) = 0, and
taking into account (4.2), we find the equation
Λs û(·, s; x) + Ns x = 0,
(4.3)
where
Λs u = u + L∗s R∗ RLs u,
Ns x = L∗s R∗ Re(·−s)A x,
By (4.2) we may rewrite (4.3) as
u ∈ L2 (s, T ; U ),
x ∈ Y.
û(·, s; x) = −L∗s R∗ Rŷ(·, s; x).
(4.4)
(4.5)
(4.6)
We apply now a classical bootstrap process: we set
pn
pn+1 =
, 0 ≤ n < N,
p0 = 2,
1 − (1 − γ)pn
1
; such an integer
where N ∈ N is the first positive integer such that pN ≥ 1−γ
exists because
(1 − γ)pn
4(1 − γ)
>
pn+1 − pn = pn
> 0.
1 − (1 − γ)pn
2γ − 1
Now we use repeatedly Proposition B.3 and Proposition B.4: starting from
the fact that û(·, s; x) ∈ Lp0 (s, T ; U ) and ŷ(·, s; x) ∈ Lp1 (s, T ; Y ), we deduce
as a first step
Ns x, L∗s R∗ RLs û(·, s; x), Λs û(·, s; x) ∈ Lp2 (s, T ; U ),
Optimal boundary control and Riccati theory
1403
which in turn gives û(·, s; x) ∈ Lp2 (s, T ; U ) as well as, by (4.2), ŷ(·, s; x) ∈
Lp3 (s, T ; Y ). Thus, after n steps, we get
û(·, s; x) ∈ Lp2n (s, T ; U ),
ŷ(·, s; x) ∈ Lp2n+1 (s, T ; Y ).
This process stops as soon as 2n or 2n+1 equals N : indeed, if N = 2n we get
at the n-th step û(·, s; x) ∈ LpN (s, T : U ) and ŷ(·, s; x) ∈ C([s, T ], Y ), so that
in the next step we obtain û(·, s; x) ∈ Lp (s, T ; U ) for 3
all p < ∞; if N = 2n+1
we find directly, at the (n + 1)-th step, û(·, s; x) ∈ 1≤p<∞ Lp (s, T ; U ) and
ŷ(·, s; x) ∈ C([s, T ], Y ).
!
Remark 4.2. It is clear that the dependence on x of the optimal pair is
linear. More precisely, we observe that the operator Λs given by (4.4) is
invertible in L2 (s, T ; Y ) with %Λ−1
s %L(L2 (s,T ;Y )) ≤ 1, since it has the form
I + U with U bounded, self-adjoint and positive. Hence we can rewrite (4.3)
as
û(·, s; x) = −Λ−1
(4.7)
s Ns x,
and, by (4.2),
ŷ(·, s; x) = e(·−s)A x − Ls Λ−1
s Ns x.
(4.8)
The result of Remark 4.2 can be improved. In fact we have:
Proposition 4.3. Under Hypotheses 2.1 and 2.2, let Λs be the operator
introduced in (4.4). Then
(i) Λs ∈ L(Lq (s, t; U )),
(ii) Λs is bijective,
where q is the exponent appearing in Hypothesis 2.2(iii)(c).
Proof. By Proposition B.3 and Proposition B.4, the operator Λs = I +
L∗s R∗ RLs is well defined in Lq (s, T ; U ) and clearly its range is contained
#
in Lq (s, T ; U ). In addition, its restriction to Lq (s, T ; U ) coincides with its
#
#
adjoint Λ∗s and we have Λ∗s (Lq (s, T ; U )) ⊆ Lq (s, T ; U ) too. Now we observe
#
#
that the operator Λ∗s : Lq (s, T ; U ) → Lq (s, T ; U ) is injective (since Λ∗s u =
#
0 with u ∈ Lq (s, T ; U ) ⊆ L2 (s, T ; U ) implies u = 0) and surjective (the
#
equation Λ∗s u = v ∈ Lq (s, T ; U ) ⊆ L2 (s, T ; U ) has a solution u ∈ L2 (s, T ; U );
by repeated application of Proposition B.3 and Proposition B.4 we get u ∈
#
Lq (s, T ; U )). Hence, by classical functional analysis, Λs : Lq (s, T ; U ) →
q
L (s, T ; U ) is injective with range R(Λs ) closed in Lq (s, T ; U ). But, since
ker Λ∗s = {0},
R(Λs ) = {f ∈ Lq (s, T ; U ) : !f, g"Lq‘ (s,T ;U ) = 0 ∀g ∈ ker Λ∗s } = Lq (s, T ; U ),
1404
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
so that R(Λs ) = Lq (s, T ; U ); i.e., Λs : Lq (s, T ; U ) → Lq (s, T ; U ) is bijective.
!
Define now for 0 ≤ s ≤ t ≤ T , according to Remark 4.2,
Φ(t, s)x = ŷ(t, s; x),
Ψ(t, s)x = û(t, s; x).
(4.9)
In order to prove that (t, s) 3→ Φ(t, s) is an evolution operator, we will need
the following lemma, also of independent use in the sequel. Since the proofs
of the next two statements are pretty routine, they will be omitted.
Lemma 4.4. Under Hypotheses 2.1 and 2.2, let Ls and Λs be the opera∗ ∗
tors defined in (2.11) and (4.4). Then the operator I − Ls Λ−1
s Ls R R has an
2
∗
∗
inverse in L(L (s, T ; Y )), given by I + Ls Ls R R.
Proposition 4.5. Under Hypotheses 2.1 and 2.2, let Φ(t, s) and Ψ(t, s) be
defined by (4.9). Then for 0 ≤ s < r < t < T we have
Φ(t, r)Φ(r, s)x = Φ(t, s)x,
Ψ(t, r)Φ(r, s)x = Ψ(t, s)x
∀x ∈ Y.
We know from Proposition 4.1 that t 3→ Φ(t, s)x belongs to C([s, T ], Y ) for
all x ∈ Y . In the next statement we describe the regularity of s 3→ Φ(t, s)x.
Proposition 4.6. Under Hypotheses 2.1 and 2.2, let Φ(t, s) be defined by
(4.9). Then s 3→ Φ(t, s)x belongs to C([0, t], Y ) for all x ∈ Y .
Proof. We remark firstly that by (4.1) we have
%Φ(t, s)%L(Y ) ≤ cT
Hence, if s < t and h ∈ [0, t − s), we get
for 0 ≤ s ≤ t ≤ T.
(4.10)
%Φ(t, s + h)x − Φ(t, s)x%Y ≤ %Φ(t, s + h)%L(Y ) %x − Φ(s + h, s)x%Y
≤ cT %x − Φ(s + h, s)x%Y → 0 as h → 0+ .
Next, if h ∈ [0, s) we can write, choosing p >
B.3(iv) and Proposition B.4(ii),
1
1−γ
and applying Proposition
%Φ(t, s − h)x − Φ(t, s)x%Y ≤ %Φ(t, s)%L(Y ) %Φ(s, s − h)x − x%Y
#
$
≤ cT %ehA x − x%Y + %[Ls−h L∗s−h R∗ RΦ(·, s − h)x](s)%Y
#
$
≤ cT %ehA x − x%Y + %L∗s−h R∗ RΦ(·, s − h)x%Lp (s−h,s;Y )
#
$
≤ cT %ehA x − x%Y + %Φ(·, s − h)x%Lr (s−h,s;Y )
#
1$
≤ cT %ehA x − x%Y + %x%Y h r → 0 as h → 0+ ,
where r =
p
1+(1−γ)p .
!
Optimal boundary control and Riccati theory
1405
We now aim to show that the closed operator Φ(t, s)B admits a bounded
extension on the space D(A∗ ε ). In order to do this, the following preparatory
result is needed.
Proposition 4.7. Under Hypotheses 2.1 and 2.2, let Λs be the operator
∗ ∗
(·−s)A B has a
introduced in (4.4). Then the closed operator Ls Λ−1
s Ls R Re
q
bounded extension belonging to L(U, L 1−(1−γ)q (s, T ; Y )), where q and γ are
the exponents introduced in Hypothesis 2.2(iii).
Proof. Let u ∈ U . By Lemma A.2 we have
e(·−s)A Bu ∈ Lp (s, T ; [D(A∗ ε )]! )
Hence, by Corollary B.7,
∀p ∈ [1, γ1 ) .
L∗s R∗ Re(·−s)A Bu ∈ Lq (s, T ; U ) .
Thus, Proposition 4.3 gives
∗ ∗
(·−s)A
Bu ∈ Lq (s, T ; U )
Λ−1
s Ls R Re
!
and finally Proposition B.3 yields the result.
Corollary 4.8. Under Hypotheses 2.1 and 2.2, let Φ(t, s) be defined by (4.9).
Then
∀p ∈ [1, γ1 ) .
Φ(·, s)B ∈ L(U, Lp (s, T ; [D(A∗ ε )]! ))
Proof. By (4.8) and (4.5) we have
∗ ∗
(·−s)A
x
Φ(·, s)x = e(·−s)A x − Ls Λ−1
s Ls R Re
∀x ∈ Y.
(4.11)
Now choose y = Bu, with u ∈ U , so that y ∈ [D(A∗ )]! . Then for almost all
t ∈ [s, T ] we have Φ(t, s)Bu ∈ [D(A∗ )]! , since the right member belongs to
that space: in fact, by Lemma A.2 and Proposition 4.7 we have
4
#
∀p ∈ 1, γ1 ,
e(·−s)A Bu ∈ Lp (s, T ; [D(A∗ ε )]! ))
q
and since
∗ ∗
(·−s)A
Ls Λ−1
Bu ∈ L 1−(1−γ)q (s, T ; [D(A∗ ε )]! ) ,
s Ls R Re
q
1−(1−γ)q
> γ1 , we immediately get the result.
!
Corollary 4.9. Under Hypotheses 2.1 and 2.2, let Φ(t, s) be defined by (4.9).
Then
Ls L∗s R∗ RΦ(·, s)B ∈ L(U, Lp (s, T ; [D(A∗ ε )]! ))
Proof. It follows by equation (4.11) and Corollary 4.8.
∀p ∈ [1, γ1 ) .
!
1406
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
5. The Riccati operator
Let us define the Riccati operator of the control problem (2.2)-(2.1) as
follows:
" T
∗
P (t)x =
e(σ−t)A R∗ RΦ(σ, t)x dσ, t ∈ [0, T ], x ∈ Y.
(5.1)
t
By (4.10) it is clear that
%P (t)%L(Y ) ≤ cT (T − t)
∀t ∈ [0, T ].
(5.2)
Proposition 5.1. Under Hypotheses 2.1 and 2.2, let P (t) be defined by (5.1).
Then P (t) = P (t)∗ ≥ 0 and
(P (t)x, x)Y = Jt (ŷ(·, t; x), û(·, t; x))
∀t ∈ [0, T ],
where Jt is defined by (2.8).
∀x ∈ Y,
Proof. We have for each x ∈ Y
" T
(P (t)x, x)Y =
(RΦ(σ, t)x, Re(σ−t)A x)Z dσ
t
=
"
t
+
=
"
T
"
t
T
t
=
"
t
(RΦ(σ, t)x, RΦ(σ, t)x)Z dσ
T
T
(RΦ(σ, t)x, R[Lt L∗t R∗ RΦ(·, t)x](σ))Z dσ
%Rŷ(σ, t; x)%2Z dσ
+
%Rŷ(σ, t; x)%2Z dσ +
"
T
t
"
t
T
%[L∗t R∗ RΦ(·, t)x](σ)%2U dσ
%û(σ, t; x)%2U dσ
= Jt (ŷ(·, t; x), û(·, t; x)) ≥ 0,
so that P (t) ≥ 0. More generally we get, proceeding as before,
" T
" T
(Rŷ(σ, t; x), Rŷ(σ, t; z))Z dσ +
(û(σ, t; x), û(σ, t; z))U dσ
(P (t)x, z)Y =
= (x, P (t)z)Y
t
t
∀x, z ∈ Y,
so that P (t) is selfadjoint.
!
Remark 5.2. An easy application of the dominated convergence theorem
shows that P (·)x ∈ C([0, T ], Y ) and supt∈[0,T ] %P (t)x%Y ≤ cT %x%Y for all
x ∈ Y ; i.e. P ∈ L(Y, C([0, T ], Y )).
Optimal boundary control and Riccati theory
1407
Our next result states well posedness and regularity of the gain operator
B ∗ P (t).
Proposition 5.3. Under Hypotheses 2.1 and 2.2, let P (t) be defined by (5.1).
Then B ∗ P (·) ∈ L(D(Aε ), C([0, T ], U )).
Proof. For x ∈ D(Aε ), using (2.3) and (4.2), we give a meaning to the
expression
" T
∗
B ∗ P (t)x =
B ∗ e(σ−t)A R∗ RΦ(σ, t)x dσ
(5.3)
t
by splitting it as follows:
" T −t
"
F (σ)R∗ RΦ(σ + t, t)x dσ +
B ∗ P (t)x =
+
0
T −t
"
T −t
G(σ)R∗ ReσA x dσ
0
G(σ)R∗ R[Lt û(·, t; x)](σ + t) dσ = I + II + III.
0
Now, by (2.4),
%I%U ≤ c
"
0
T −t
σ −γ %x%Y dσ ≤ cT %x%Y ,
and by Lemmas A.3 and A.4
%II%U ≤ cT %R∗ Re·A x%C([0,T ],D(A∗ ε )) ≤ cT %x%D(Aε ) ;
finally, by Proposition B.5,
%III%U ≤ cT %R∗ RLt û(·, t; x)%C([t,T ],D(A∗ ε ))
≤ cT %û(·, t; x)%Lq# (t,T ;U ) ≤ cT %x%Y .
This shows that
%B ∗ P (·)x%L∞ (0,T ;U ) ≤ cT %x%D(Aε ) .
(5.4)
Let us prove continuity of
at t0 ∈ [0, T ].
If t ≥ t0 , we have by Proposition 4.5 and (2.12),
B ∗ P (·)x
B ∗ P (t)x − B ∗ P (t0 )x =
" T
"
∗ (σ−t)A∗ ∗
=
B e
R RΦ(σ, t)x dσ −
t
=
"
t
T
t0
T
(σ−t)A∗
B∗e
∗
B ∗ e(σ−t0 )A R∗ RΦ(σ, t0 )x dσ
R∗ RΦ(σ, t)(x − Φ(t, t0 )x)dσ
+[L∗0 R∗ RΦ(·, t0 )x](t) − [L∗0 R∗ RΦ(·, t0 )x](t0 )
1408
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
=
"
t
T
"
[F (σ − t) + G(σ − t)]R∗ Re(σ−t)A (x − Φ(t, t0 )x)dσ
T
[F (σ − t) + G(σ − t)]R∗ R[Lt û(·, t; x − Φ(t, t0 )x)](σ)dσ
t
+[L∗0 R∗ Re(·−t0 )A x](t) − [L∗0 R∗ Re(·−t0 )A x](t0 )
+[L∗0 R∗ R[Lt0 û(·, t0 ; x)]](t) − [L∗0 R∗ R[Lt0 û(·, t0 ; x)]](t0 )
+
= (I1 + I2 ) + (I3 + I4 ) + I5 + I6 .
By (2.4) and Proposition B.3(iv)
" T
(σ − t)−γ %x − Φ(t, t0 )x%Y dσ = o(1)
%I1 %U + %I3 %U ≤ c
t
as t → t+
0.
Next,
%I2 %U ≤
+
"
"
T −t
0
T −t
0
%G(σ)R∗ ReσA (x − e(t−t0 )A x)%U dσ+
%G(σ)R∗ ReσA [Lt0 û(·, t0 ; x)](t)%U dσ.
The first term is estimated by Lemmas A.3 and A.4:
" T −t
%G(σ)R∗ ReσA (x − e(t−t0 )A x)%U dσ
0
≤ c%x − e(t−t0 )A x%D(Aε ) = o(1)
as t → t+
0;
the second term is estimated by Lemma A.3, Remark B.6 and (4.1):
" T −t
%G(σ)R∗ ReσA [Lt0 û(·, t0 ; x)](t)%U dσ ≤
0
≤ c(T − t)%R∗ Re·A [Lt0 û(·, t0 ; x)](t)%C([0,T −t],D(A∗ ε )) ≤
≤ c%û(·, t0 ; x)%Lq# (t0 ,t;U ) = o(1)
as t → t+
0.
Similarly, I4 is estimated using Lemma A.3 and Proposition B.5:
" T −t
%I4 %U ≤
%G(σ)R∗ R[Lt û(·, t; x − Φ(t, t0 )x)](σ)%U dσ
0
≤ c%R∗ RLt û(·, t; x − Φ(t, t0 )x)%C([t,T ],D(A∗ ε ))
≤ c%û(·, t; x − Φ(t, t0 )x)%Lq# (t,T ;U )
≤ c%x − Φ(t, t0 )x%Y = o(1)
as t → t+
0.
Optimal boundary control and Riccati theory
1409
Finally, by Proposition B.4(iv) and Lemma A.4, %I5 %U = o(1) as t → t+
0,
whereas by Propositions B.4(iv) and B.5 we get %I6 %U = o(1) as t → t+
0.
Summing up, we have proved that
%B ∗ P (t)x − B ∗ P (t0 )x%U → 0
If, instead, t < t0 , we write
B ∗ P (t)x − B ∗ P (t0 )x =
+
+
"
T
t0
T
"
t0
(σ−t)A∗
B∗e
"
t0
as t → t+
0.
(5.5)
∗
B ∗ e(σ−t)A R∗ RΦ(σ, t)x dσ
t
R∗ RΦ(σ, t0 )x[Φ(t0 , t)x − x]dσ
∗
∗
B ∗ e(σ−t0 )A [e(t0 −t)A − 1]R∗ RΦ(σ, t0 )x dσ.
5t
∗
We introduce now the additional term t 0 B ∗ e(σ−t)A R∗ Rx dσ, obtaining
" t0
∗
∗
∗
B ∗ e(σ−t)A R∗ R[Φ(σ, t)x − x]dσ
B P (t)x − B P (t0 )x =
+
"
t
T
t0
∗
B ∗ e(σ−t)A R∗ R[Φ(σ, t0 )x − x]dσ + [L∗0 yt0 (t) − L∗0 yt0 (t0 )]
= I + II + III,
where
yt0 (r) =
*
R∗ RΦ(r, t0 )x if r ∈ [t0 , T ]
R∗ Rx
if r ∈ [0, t0 ] .
Note that yt0 ∈ C([0, T ], Y ) ∩ L∞ (0, T ; D(A∗ ε )): indeed
!
6
4
A∗ ε R∗ R e(r−t0 )A x + [Lt0 û(·, t0 ; x)](r) if r ∈ [t0 , T ]
∗ε
A yt0 (r) =
A∗ ε R∗ Rx
if r ∈ [0, t0 ],
so that Lemma A.4, Proposition B.5 and (4.1) imply
%A∗ ε yt0 (r)%Y ≤ c%x%D(A∗ ε ) + c%R∗ RLt0 û(·, t0 ; x)%C([t0 ,T ],D(A∗ ε ))
≤ c%x%D(A∗ ε ) + c%û(·, t0 ; x)%Lq# (0,T ;U ) ≤ c%x%D(A∗ ε ) .
Thus we get, by Proposition B.4(iv), %III%U = o(1) as t → t−
0 . On the other
hand, by (2.4), Lemma A.3, Lemma A.4 and Proposition B.5, we have
0 " t0
0
0
0
%I%U = 0
[F (σ − t) + G(σ − t)]R∗ R(Φ(σ, t)x − x)dσ 0
t
U
1410
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
"
≤c
"
+
t
t0
t
t0
−γ
(σ − t)
%Φ(σ, t)x − x%Y dσ +
"
t0
t
%G(σ − t)R∗ R[e(σ−t)A x − x]%U dσ
%G(σ − t)R∗ R[Lt û(·, t; x)](σ)%U dσ
≤ c%x%Y (t0 − t)1−γ + %x%D(Aε ) [o(1) + c(t0 − t)] = o(1) as t → t−
0,
and, using also (4.1) and (A.3),
0" T
0
0
0
[F (σ − t) + G(σ − t)]R∗ RΦ(σ, t0 )(Φ(t0 , t)x − x)dσ 0
%II%U = 0
U
t0
≤ c%Φ(t0 , t)x − x%Y
0" T
0
0
0
+0
G(σ − t)R∗ Re(σ−t0 )A (Φ(t0 , t)x − x)dσ 0
t0
T
0"
0
+0
t0
U
0
0
G(σ − t)R∗ RLt0 û(·, t0 ; Φ(t0 , t)x − x)]dσ 0
= o(1) + c
"
U
T
t0
(σ − t0 )−ε %Φ(t0 , t)x − x%Y dσ
as t → t−
0.
+ c%û(·, t0 ; Φ(t0 , t)x − x)%Lq# (t0 ,T ;U ) = o(1)
This proves that
%B ∗ P (t)x − B ∗ P (t0 )x%U → 0
Summing up, we have obtained
lim %B ∗ P (t)x − B ∗ P (t0 )x%U = 0
t→t0
as t → t−
0.
(5.6)
∀x ∈ D(Aε ),
!
and the proof is complete.
Corollary 5.4. Under Hypotheses 2.1 and 2.2, let P (t) and Φ(t, s) be defined
by (5.1) and (4.9). Then we have
(i) if x ∈ Y , then
%
P (·)Φ(·, s)x ∈ C([s, T ], Y ),
B ∗ P (·)Φ(·, s)x ∈
Lp (s, T ; U );
1≤p<∞
(ii) if x ∈
D(Aε ),
then
B ∗ P (·)Φ(·, s)x
∈ C([s, T ], U ) .
Proof. (i) The first result follows trivially by Remark 5.2 and Proposition
4.1. The second one follows by remarking that
B ∗ P (·)Φ(·, s)x = L∗s R∗ RΦ(·, s)x
(5.7)
Optimal boundary control and Riccati theory
1411
and applying Proposition B.4 (iii).
(ii) Let x ∈ D(Aε ). We write
B ∗ P (·)Φ(·, s)x = L∗s R∗ Re(·−s)A x + L∗s R∗ Rû(·, s; x) .
The first term on the right-hand side belongs to C([s, T ], U ), because we
have e(·−s)A x ∈ C([s, T ], D(Aε )) and we can use Hypothesis 2.2(iii)(b) as
well as Proposition B.4(iv). The second term is in C([s, T ], U ) in view of
Propositions 4.1, B.5 and B.3(iv). The estimate
%B ∗ P (·)Φ(·, s)x%C([s,T ],U ) ≤ cT %x%D(Aε )
also follows.
∀x ∈ D(Aε )
(5.8)
!
Corollary 5.5. Under Hypotheses 2.1 and 2.2, let P (t) be defined by (5.1).
Then for each x ∈ Y we have
−B ∗ P (t)Φ(t, s)x = Ψ(t, s)x = û(t, s; x)
for 0 ≤ s < t < T ;
(5.9)
in particular, for each t ∈ (0, T ] the function s 3→ û(t, s; x) is in C([0, t[, U )
if x ∈ Y , and is in C([0, t], U ) if x ∈ D(Aε ).
Proof. If x ∈ Y , using (5.7) and (4.6) we immediately get (5.9). The
continuity properties of s 3→ û(t, s; x) follow by the properties of Φ(t, s)x
(Proposition 4.6).
!
As a consequence of the previous results, we show that the operator P (·)
solves the Riccati integral equation.
Corollary 5.6. Under Hypotheses 2.1 and 2.2, let P (t) be defined by (5.1).
Then for all x, y ∈ Y and t ∈ [0, T ] we have
" T
(RΦ(σ, t)x, RΦ(σ, t)y)Z dσ
(P (t)x, y)Y =
t
(5.10)
"
T
+
(B ∗ P (σ)Φ(σ, t)x, B ∗ P (σ)Φ(σ, t)y)U dσ.
t
Proof. It follows by Proposition 5.1 and (5.9).
!
6. Differentiability of the evolution map Φ(t, s)
In this section we analyze the differentiability properties of the operator
Φ(t, s) defined by (4.9). The regularity results established in Theorems 6.1
and 6.2 below—together with boundedness of B ∗ P (t) on D(Aε ), already
proved in Proposition 5.3—enable us to conclude that P (t) satisfies the
1412
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Differential Riccati equation on D(A), thus completing the proof of Theorem 2.3.
Theorem 6.1. Under Hypotheses 2.1 and 2.2, let Φ(t, s) be defined by (4.9).
Then for each t ∈ (s, T ]
%
∂
Lp (s, T ; [D(A∗ )]! ) ∀x ∈ Y ;
∃ Φ(t, s)x = [A − BB ∗ P (t)]Φ(t, s)x ∈
∂t
1≤p<∞
moreover,
∂
Φ(t, s)x ∈ C([s, T ], [D(A∗ )]! )
∂t
Proof. We start from the equation
Φ(t, s)x = e(t−s)A x − [Ls L∗s R∗ RΦ(·, s)x](t),
∀x ∈ D(Aε ) .
t ∈ (s, T ],
x ∈ Y,
(6.1)
which is a consequence of (4.11) and Lemma 4.4. For h > 0 we write
Φ(t + h, s)x − Φ(t, s)x
h
"
e(t+h−s)A x − e(t−s)A x 1 t+h (t+h−σ)A
=
e
B[L∗s R∗ RΦ(·, s)x](σ)dσ
−
h
h t
" t (t+h−σ)A
e
− e(t−σ)A
−
B[L∗s R∗ RΦ(·, s)x](σ)dσ = I + II + III.
h
s
Now if y ∈ D(A∗ ) we have, as h → 0+ ,
∗
7
ehA − I 8
(I, y)Y = e(t−s)A x,
→ (e(t−s)A x, A∗ y)Y ,
y
h
Y
whereas, by (5.7),
"
1 t+h ∗
∗
(B P (σ)Φ(σ, s)x, B ∗ e(t+h−σ)A y)U dσ
(II, y)Y = −
h t
→ −(B ∗ P (t)Φ(t, s)x, B ∗ A∗ −1 A∗ y)U
=
(6.2)
(6.3)
((A−1 B)B ∗ P (t)Φ(t, s)x, A∗ y)Y ;
finally, by (2.11) and (5.7),
(III, y)Y
∗
7
ehA − I 8
− Ls [B ∗ P (·)Φ(·, s)x](t),
y
h
Y
→ −(Ls [B ∗ P (·)Φ(·, s)x](t), A∗ y)Y .
=
(6.4)
Optimal boundary control and Riccati theory
1413
By (6.2), (6.3) and (6.4) we get as h → 0+
7 Φ(t + h, s)x − Φ(t, s)x 8
→ (Φ(t, s)x − (A−1 B)B ∗ P (t)Φ(t, s)x, A∗ y)Y .
,y
h
Y
(6.5)
Now we turn to the limit as h → 0− . As
Φ(t + h, s)x − Φ(t, s)x
h
" t+h (t+h−σ)A
(t+h−s)A
x − e(t−s)A x
e
− e(t−σ)A
e
=
−
B[L∗s R∗ RΦ(·, s)x](σ)dσ
h
h
s
"
1 t (t−σ)A
e
B[L∗s R∗ RΦ(·, s)x](σ)dσ = I + II + III,
+
h t+h
we get as before
∗
7
I − e−hA 8
→ (e(t−s)A x, A∗ y)Y ,
y
(I, y)Y = e(t+h−s)A x,
h
Y
∗
7
I − e−hA 8
∗
y
(II, y)Y = − [Ls B P (·)Φ(·, s)x](t + h),
h
Y
→ −([Ls B ∗ P (·)Φ(·, s)x](t), A∗ y)Y ,
"
1 t
∗
(III, y)Y =
(B ∗ P (σ)Φ(σ, s)x, B ∗ e(t−σ)A y)Y dσ
h t+h
→ −(B ∗ P (t)Φ(t, s)x, (A−1 B)∗ A∗ y)Y
=
−((A−1 B)B ∗ P (t)Φ(t, s)x, A∗ y)Y ;
hence when h → 0− we obtain again (6.5). This proves that
∂
(Φ(t, s)x, y)Y = (Φ(t, s)x, A∗ y)Y − ((A−1 B)B ∗ P (t)Φ(t, s)x, y)Y
∂t
= !(A − BB ∗ P (t))Φ(t, s)x, y"D(A∗ ) ;
(6.6)
∂
i.e., the derivative ∂t
Φ(t, s)x exists as an element of [D(A∗ )]! . But Proposition 4.1 implies AΦ(·, s)x ∈ C([s, T ], [D(A∗ )]! ), and Corollary 5.4(i) gives
%
B B ∗ P (·)Φ(t, s)x ∈
Lp (s, T ; [D(A∗ )]! ) .
1≤p<∞
Moreover, if x ∈ D(Aε ) then B ∗ P (·)Φ(·, s)x ∈ C([s, T ], [D(A∗ )]! ) by Corollary 5.4(ii). This proves the result.
!
The proof of the differentiability of Φ(t, s) with respect to s is more involved and it requires a number of lemmas. The result is the following:
1414
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Theorem 6.2. Under Hypotheses 2.1 and 2.2, let Φ(t, s) be defined by (4.9).
Then for each s ∈ [0, t)
∃
1
∂
Φ(t, s)x = −Φ(t, s)[A − BB ∗ P (s)]x ∈ L γ (s, T ; [D(A∗ ε )]! )
∂s
∀x ∈ D(A) ,
where γ is the exponent appearing in (2.4).
Proof. We split the proof into five steps.
. We start from equation
Step 1. Splitting of the ratio Φ(t,s+h)x−Φ(t,s)x
h
(6.1). For h > 0 we can write
Φ(t, s + h)x − Φ(t, s)x
h
(t−s−h)A
x − e(t−s)A x 1
e
Φ(·, s + h) − Φ(·, s) 2
=
− Ls+h L∗s+h R∗ R
x (t)
h
h
Ls+h L∗s+h − Ls L∗s ∗
[R RΦ(·, s)x](t),
−
h
whereas for h < 0 we have
Φ(t, s + h)x − Φ(t, s)x
h
e(t−s−h)A x − e(t−s)A x Ls+h L∗s+h − Ls L∗s ∗
=
−
[R RΦ(·, s + h)x](t)
h
h
1
Φ(·, s + h) − Φ(·, s) 2
x (t).
− Ls L∗s R∗ R
h
Hence, we obtain for h > 0, recalling Lemma 4.4,
Φ(t, s + h)x − Φ(t, s)x
= [I + Ls+h L∗s+h R∗ R]−1
h
1 e(t−s−h)A x − e(t−s)A x L L∗ − Ls L∗
2
s+h s+h
s
×
−
[R∗ RΦ(·, s)x] (t)
h
h
∗
∗
L
R
R]
= [I − Ls+h Λ−1
s+h s+h
1 e(t−s−h)A x − e(t−s)A x L L∗ − Ls L∗
2
s+h s+h
s
×
−
[R∗ RΦ(·, s)x] (t);
h
h
Optimal boundary control and Riccati theory
1415
i.e.,
Φ(t, s + h)x − Φ(t, s)x
h
=
e(t−s−h)A x − e(t−s)A x Ls+h L∗s+h − Ls L∗s ∗
−
[R RΦ(·, s)x]
h
h
(t−s−h)A x − e(t−s)A x
∗
∗ e
−Ls+h Λ−1
L
R
R
s+h s+h
h
∗
∗
+Ls+h Λ−1
s+h Ls+h R R
Ls+h L∗s+h − Ls L∗s ∗
[R RΦ(·, s)x],
h
(6.7)
h > 0.
Similarly, for h < 0, using Lemma 4.4,
Φ(t, s + h)x − Φ(t, s)x
= [I + Ls L∗s R∗ R]−1
h
1 e(t−s−h)A x − e(t−s)A x L L∗ − Ls L∗
2
s+h s+h
s
×
−
[R∗ RΦ(·, s + h)x] (t)
h
h
∗ ∗
= [I − Ls Λ−1
L
R
R]
s
s
2
1 e(t−s−h)A x − e(t−s)A x L L∗ − Ls L∗
s+h s+h
s ∗
×
−
R RΦ(·, s + h)x] (t);
h
h
i.e.,
Φ(t, s + h)x − Φ(t, s)x
h
e(t−s−h)A x − e(t−s)A x Ls+h L∗s+h − Ls L∗s ∗
−
R RΦ(·, s + h)x(t)
h
h
1 e(t−s−h)A x − e(t−s)A x 2
−1 ∗ ∗
−Ls Λs Ls R R
h
1 L L∗ − Ls L∗
2
s+h s+h
s ∗
∗ ∗
+Ls Λ−1
L
R
R
RΦ(·,
s
+
h)x
(t), h < 0.
R
s
s
h
=
(6.8)
Both expressions in (6.7) and (6.8) contain four terms, whose limits as h →
0± must be evaluated.
Step 2. The limit of the first two terms in (6.7) and (6.8). First of all, for
the first term we clearly have, for each x ∈ D(A) and p ∈ [1, ∞):
lim
h→0±
e(·−s−h)A x − e(·−s)A x
= −Ae(·−s)A x in Lp (s, T ; Y ).
h
(6.9)
1416
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
The limit of the second term in (6.7) and (6.8) results from the following
claim:
Lemma 6.3. Under Hypotheses 2.1 and 2.2, for each x ∈ D(A) we have
−
Ls+h L∗s+h − Ls L∗s ∗
∗
[R RΦ(·, s)x] + e(·−s)A BB ∗ P (s)x
h
1
γ
in the space L (s +
δ ∈ (0, T − s), and
δ, T ; [D(A∗ ε )]! )
= [L
1
1−γ
as h → 0+
(6.10)
(s + δ, T ; D(A ))]! , for each
∗ε
Ls+h L∗s+h − Ls L∗s ∗
∗
[R RΦ(·, s + h)x] + e(·−s)A BB ∗ P (s)x as h → 0−
h
(6.11)
1
1
ε !
ε
∗
∗
!
γ
1−γ
(s, T ; D(A ))] .
in the space L (s, T ; [D(A )] ) = [L
−
Proof. For h > 0 and t ∈ [s + h, T ], we have
Ls+h L∗s+h − Ls L∗s ∗
[R RΦ(·, s)x](t)
h
" T
"
1 s+h (t−σ)A
∗
=
e
B
B ∗ e(τ −t)A R∗ RΦ(τ, s)x dτ dσ
h s
σ
" s+h
1
e(t−σ)A BB ∗ P (σ)Φ(σ, s)x dσ
=
h s
4
1 6
= Ls B ∗ P (·)Φ(·, s)x · χ[s,s+h] (·) (t).
h
Similarly, for h < 0 and t ∈ [s, T ] we have:
−
Ls+h L∗s+h − Ls L∗s ∗
[R RΦ(·, s + h)x](t)
h
" T
" s
1
∗
(t−σ)A
e
B
B ∗ e(τ −t)A R∗ RΦ(τ, s + h)x dτ dσ
=−
h s+h
σ
"
1 s (t−σ)A
=−
e
BB ∗ P (σ)Φ(σ, s + h)x dσ
h s+h
4
6
1
= − Ls+h B ∗ P (·)Φ(·, s + h)x · χ[s+h,s] (·) (t).
h
Now if h > 0 and δ > 0 we get by Proposition B.3(i) and Corollary 5.4(ii):
0
0 L L∗ − L L∗
s s
0
0 s+h s+h
∗
RΦ(·,
s)x]
[R
0 1/γ
0
h
L
(s+δ,T ;[D(A∗ ε )]# )
−
Optimal boundary control and Riccati theory
1417
01 6
40
0
0
= 0 Ls B ∗ P (·)Φ(·, s)x · χ[s,s+h] (·) 0 1/γ
h
L
(s+δ,T ;[D(A∗ ε )]# )
1
≤ cT %B ∗ P (·)Φ(·, s)x · χ[s,s+h] (·)%L1 (s,T ;U )
h
1
= cT %B ∗ P (·)Φ(·, s)x%L1 (s,s+h;U )
h
≤ cT %B ∗ P (·)Φ(·, s)x%L∞ (s,T ;U ) ≤ cT %x%D(Aε ) ;
similarly, for h < 0 we obtain
0
0 L L∗ − L L∗
s s
0
0 s+h s+h
∗
[R RΦ(·, s + h)x]0 1/γ
0
h
L
(s+h,T ;[D(A∗ ε )]# )
01
6
40
0
0
= 0 Ls+h B ∗ P (·)Φ(·, s + h)x · χ[s+h,s] (·) 0 1/γ
h
L
(s+h,T ;[D(A∗ ε )]# )
1
≤ cT
%B ∗ P (·)Φ(·, s + h)x · χ[s+h,s] (·)%L1 (s+h,T ;U )
|h|
1
= cT
%B ∗ P (·)Φ(·, s + h)x%L1 (s+h,s;U )
|h|
≤ cT %B ∗ P (·)Φ(·, s + h)x%L∞ (s+h,T ;U ) ≤ cT %x%D(Aε ) .
Hence,
0
0 L L∗ − L L∗
s s
0
0 s+h s+h
≤ cT
[R∗ RΦ(·, s)x]0 1/γ
0
h
L
(s+δ,T ;[D(A∗ ε )]# )
for h > 0,
(6.12)
0
0 L L∗ − L L∗
s s
0
0 s+h s+h
≤ cT for h < 0.
[R∗ RΦ(·, s + h)x]0 1/γ
0
h
L
(s,T ;[D(A∗ ε )]# )
(6.13)
#
1
Now fix p ∈ (1, γ1 ), so that p! > 1−γ
, and let ϕ ∈ Lp (s, T ; D(A∗ ε )). If h > 0
we have
/ " T 9 L L∗ − L L∗
s s
s+h s+h
/
−
[R∗ RΦ(·, s)x](t)
/
h
s+δ
/
:
/
−e(t−s)A BB ∗ P (s)x, ϕ(t)
dt
/
D(A∗ ε )
"
"
/ T 1 s+h 1
∗
/
=/
(B ∗ P (σ)Φ(σ, s)x, B ∗ e(t−σ)A ϕ(t))U
s+δ h s
/
2
∗
/
−(B ∗ P (s), B ∗ e(t−s)A ϕ(t))U dσdt/
1418
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
/"
/
≤/
"
/
1 s+h ∗
∗
/
(B P (σ)Φ(σ, s)x − B ∗ P (s)x, B ∗ e(t−σ)A ϕ(t))U dσdt/
h
s+δ
s
/ " T 1 " s+h
/
∗
∗
/
/
+/
(B ∗ P (s)x, B ∗ e(t−σ)A [I − e(σ−s)A ]ϕ(t))U dσ dt/
h
s+δ
s
= I1 + I2 .
T
In order to estimate I1 and I2 , we recall that σ 3→ B ∗ P (σ)Φ(σ, s)x is continuous in [s, T ] by Corollary 5.4(ii). Hence by (A.3)
" T " (s+h)∧t
1
∗
%B ∗ e(t−σ)A ϕ(t)%U dσdt
I1 = o(1)
s h s
"
"
cT
1 s+h T
≤ o(1)
%ϕ(t)%D(A∗ ε ) dtdσ
γ
h s
σ (t − σ)
"
"
81
dt
cT s+h 7 T
p
%ϕ%Lp# (s,T ;D(A∗ ε )) dσ
≤ o(1)
γp
h s
σ (t − σ)
as h → 0+ .
≤ o(1)cT %ϕ%Lp# (s,T ;D(A∗ ε )) = o(1)
In addition, using again (A.3),
"
"
cT
1 s+h T
∗
I2 ≤
%[I − e(σ−s)A ]ϕ(t)%D(A∗ ε ) dt dσ;
γ
h s
(t
−
σ)
σ
now the integrand goes to 0 as h → 0+ , and is dominated by the function
cT (t − σ)−γ %ϕ(t)%D(A∗ ε ) ,
(6.14)
whose integral with respect to the variables (t, σ) is finite. Thus we conclude
that
as h → 0+ .
(6.15)
I1 + I2 → 0
#
Similarly, for h < 0 and ϕ ∈ Lp (s, T ; D(A∗ ε )) we have
/ " T 9 L L∗ − L L∗
s s
s+h s+h
/
[R∗ RΦ(·, s + h)x](t)
−
/
h
s
/
:
/
−e(t−s)A BB ∗ P (s)x, ϕ(t)
dt
/
D(A∗ ε )
"
"
/ T 1 s 1
∗
/
=/
(B ∗ P (σ)Φ(σ, s + h)x, B ∗ e(t−σ)A ϕ(t))U
s h s+h
/
2
∗
/
−(B ∗ P (s), B ∗ e(t−s)A ϕ(t))U dσdt/
Optimal boundary control and Riccati theory
/"
/
≤/
s
T
/"
/
+/
1
h
"
/
∗
∗
/
(B ∗ P (σ)Φ(σ, s + h)x, B ∗ e(t−σ)A (I − eσ−s)A )ϕ(t))U dσdt/
s
s+h
" s
1
s h
= J1 + J2 .
T
/
∗
/
(B ∗ P (σ)Φ(σ, s + h)x − B ∗ P (s)x, B ∗ e(t−s)A ϕ(t))U dσdt/
s+h
We see that, as before,
"
J2 = o(1)
s
s+h
1
h
"
s
T
cT
%ϕ(t)%D(A∗ ε ) dt dσ
(t − s)γ
≤ o(1)cT %ϕ%Lp# (s,T ;D(A∗ ε )) = o(1)
whereas
J1 ≤
1419
1
h
"
s
s+h
"
T
s
as h → 0+ ,
cT
∗
%[I − e(s−σ)A ]ϕ(t)%D(A∗ ε ) dt dσ;
(t − σ)γ
the integrand goes to 0 as h → 0− , and is dominated by the function (6.14).
Thus
J1 + J2 → 0
as h → 0− .
(6.16)
ε
ε
∗
Now we recall that, since both D(A ) and D(A ) are separable (Remark
#
A.1(iv)), the space Lp (s, T ; [D(A∗ ε )]! ) is the dual of Lp (s, T ; D(A∗ ε )); hence
by (6.15) and (6.16) we conclude that for each p ∈ (1, γ1 )
Ls+h L∗s+h − Ls L∗s ∗
∗
[R RΦ(·, s)x] + e(·−s)A BB ∗ P (s)x
h
as h → 0+ in Lp (s + δ, T ; [D(A∗ ε )]! ) ∀δ > 0,
(6.17)
Ls+h L∗s+h − Ls L∗s ∗
∗
[R RΦ(·, s + h)x] + e(·−s)A BB ∗ P (s)x
h
as h → 0− in Lp (s, T ; [D(A∗ ε )]! ).
(6.18)
−
and
−
Now we want to improve the above convergences up to p = 1/γ. Set for
simplicity
Ls+h L∗s+h − Ls L∗s ∗
ϕh = −
[R RΦ(·, s)x].
h
Choose any sequence {hn } such that hn → 0+ : then, recalling (6.12), by
1
weak∗ compactness there are v ∈ L γ (s + δ, T ; [D(A∗ ε )]! ) and a subsequence
{hnk } ⊆ {hn } such that
∗
ϕhnk + v
in L1/γ (s + δ, T ; [D(A∗ ε )]! ) .
1420
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Taking into account (6.17), we also have as k → ∞
∗
ϕhnk + v
in Lp (s + δ, T ; [D(A∗ ε )]! ) ,
so that v(t) = e(t−s)A BB ∗ P (s)x a.e. in [s + δ, T ] as elements of [D(A∗ ε )]! ,
and finally
1
e(·−s)A BB ∗ P (s)x = v ∈ L γ (s + δ, T ; [D(A∗ ε )]! ) .
Moreover, by the arbitrariness of the sequence {hn }, we obtain as h → 0+
1
∗
ϕh + e(·−s)A BB ∗ P (s)x in L γ (s + δ, T ; [D(A∗ ε )]! ).
Proceeding quite similarly, setting
Ls+h L∗s+h − Ls L∗s ∗
ψh = −
[R RΦ(·, s + h)x],
h
we find as h → 0−
1
∗
ψh + e(·−s)A BB ∗ P (s)x in L γ (s, T ; [D(A∗ ε )]! ).
(6.19)
(6.20)
By (6.19) and (6.20) the proof of Lemma 6.3 is complete.
!
Step 3. Two auxiliary results. In the evaluation of the limits of the third
term in (6.7) and (6.8) we need two lemmas, the first of which improves
Proposition 4.3.
Lemma 6.4. Under Hypotheses 2.1 and 2.2, let Λs be the operator introduced
in (4.4). Then for each p ∈ [2, ∞) there is cp,T > 0 such that
%Λ−1
s %L(Lp (s,T ;U )) ≤ cp,T .
Proof. We already know, by Remark 4.2, that
%Λ−1
s %L(L2 (s,T ;U )) ≤ 1.
2
Let v ∈ Lp (s, T ; U ), p > 2; then there exists u = Λ−1
s v ∈ L (s, T ; U ),
∗
∗
and moreover u = v − Ls R RLs u. Hence, applying Proposition B.4(ii) and
Proposition B.3(ii), we get
%u%Lp (s,T ;U ) ≤ %v%Lp (s,T ;U ) + %L∗s R∗ RLs u%Lp (s,T ;U )
≤ %v%Lp (s,T ;U ) + cT %Ls u%Lp1 (s,T ;U ) ≤ %v%Lp (s,T ;U ) + cT %u%Lp2 (s,T ;U ) ,
where
p
,
p1 =
1 + (1 − γ)p
p2 =

p

 1+2(1−γ)p
if
p
1+2(1−γ)p
>1

1 + δ, δ > 0, if
p
1+2(1−γ)p
≤ 1.
(6.21)
Optimal boundary control and Riccati theory
1421
If p2 ≤ 2, we deduce
%u%Lp (s,T ;U ) ≤ %v%Lp (s,T ;U ) + cT %u%Lp2 (s,T ;U )
≤ %v%Lp (s,T ;U ) + cT %u%L2 (s,T ;U )
6
4
≤ cT %v%Lp (s,T ;U ) + %v%L2 (s,T ;U ) ≤ cT %v%Lp (s,T ;U ) .
If p2 > 2, we iterate the above argument:
%u%Lp (s,T ;U ) ≤ %v%Lp (s,T ;U ) + cT %u%Lp2 (s,T ;U )
6
4
≤ %v%Lp (s,T ;U ) + cT %v%Lp2 (s,T ;U ) + %L∗s R∗ RLs u%Lp2 (s,T ;U )
≤ cT %v%Lp (s,T ;U ) + cT %u%Lp3 (s,T ;U )
where
p3 =

p

 1+4(1−γ)p
if
p
1+4(1−γ)p
>1

1 + δ, δ > 0, if
p
1+4(1−γ)p
≤ 1.
After a finite number of steps, we fall into the first case, and we conclude
that
!
%u%Lp (s,T ;U ) ≤ cp,T %v%Lp (s,T ;U ) .
Lemma 6.5. Under Hypotheses 2.1 and 2.2, let Λs be the operator introduced
p
in (4.4). Then for all p ∈ [2, ∞[ and ϕ ∈ Lr∨2 (s, T ; U ), with r = 1+2(1−γ)p
,
we have
−1
+
%[Λ−1
s+h − Λs ]ϕ%Lp (s+h,T ;U ) → 0 as h → 0 ,
−1
%[Λ−1
s+h − Λs ]ϕ%Lp (s,T ;U ) → 0
as h → 0− .
Proof. Let h > 0. With p1 and p2 given by (6.21) we have, using Lemma
6.4, Proposition B.4(ii) and Corollary B.3,
−1
−1
−1
%[Λ−1
s+h − Λs ]ϕ%Lp (s+h,T ;U ) = %Λs+h [Λs − Λs+h ]Λs ϕ%Lp (s+h,T ;U )
≤ cT %[Λs − Λs+h ]Λ−1
s ϕ%Lp (s+h,T ;U )
= cT %L∗s R∗ R(Ls − Ls+h )Λ−1
s ϕ%Lp (s+h,T ;U )
≤ cT %[Ls − Ls+h ]Λ−1
s ϕ%Lp1 (s+h,T ;Y )
= cT %Ls [Λ−1
s ϕ · χ[s,s+h] ]%Lp1 (s+h,T ;Y )
so that
−1
≤ cT %Λ−1
s ϕ · χ[s,s+h] %Lp2 (s+h,T ;U ) = cT %Λs ϕ%Lp2 (s,s+h;U ) ,
−1
+
%[Λ−1
s+h − Λs ]ϕ%Lp (s+h,T ;U ) → 0 as h → 0 .
1422
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Let now h < 0. Similarly we have
−1
−1
−1
%[Λ−1
s+h − Λs ]ϕ%Lp (s,T ;U ) = %Λs [Λs − Λs+h ]Λs+h ϕ%Lp (s,T ;U )
≤ cT %[Λs − Λs+h ]Λ−1
s+h ϕ%Lp (s,T ;U )
= cT %L∗s R∗ R(Ls − Ls+h )Λ−1
s+h ϕ%Lp (s,T ;U )
≤ cT %[Ls − Ls+h ]Λ−1
s+h ϕ%Lp1 (s,T ;Y )
= cT %Ls+h [Λ−1
s+h ϕ · χ[s+h,s] ]%Lp1 (s,T ;Y )
−1
≤ cT %Λ−1
s+h ϕ · χ[s+h,s] %Lp2 (s+h,T ;U ) = cT %Λs+h ϕ%Lp2 (s+h,s;U ) ,
with p1 and p2 as above. If |h| is sufficiently small, so that
%L∗s+h R∗ RLs+h %L(Lp2 (s+h,s;U )) ≤
1
2
we get
Λ−1
s+h ϕ
= (I +
L∗s+h R∗ RLs+h )−1 ϕ
=
∞
;
(−1)k (Ls+h R∗ RLs+h )k ϕ;
k=0
hence
and consequently
%Λ−1
s+h ϕ%Lp2 (s+h,s;U ) ≤ 2%ϕ%Lp2 (s+h,s;U )
−1
%[Λ−1
s+h − Λs ]ϕ%Lp (s,T ;U ) → 0
as h → 0− .
This concludes the proof of Lemma 6.5.
!
Step 4. The limit of the third term in (6.7) and (6.8). For the third term
in (6.7) we can write
# e(·−s−h)A − e(·−s)A $
∗ ∗
(·−s)A
Ax](t)
x (t)− Ls Λ−1
s Ls R R[e
h
# e(·−s−h)A − e(·−s)A
$
∗
∗
x − e(·−s)A Ax (t)
= Ls+h Λ−1
s+h Ls+h R R
−h
$
#
−1
−1 ∗ ∗
+ Ls+h [Λs+h − Λs ]Ls R Re(·−s)A Ax (t)
∗
∗
− Ls+h Λ−1
s+h Ls+h R R
∗ ∗
(·−s)A
Ax](t);
+ [Ls+h − Ls ][Λ−1
s Ls R Re
hence, using Proposition B.3(iv), Lemma 6.4, Proposition B.4(iii), and
Lemma 6.5,
0
1 e(·−s−h)A − e(·−s)A 2
0
∗
∗
L
R
R
x
0 − Ls+h Λ−1
s+h s+h
h
Optimal boundary control and Riccati theory
1423
0
0
∗ ∗
(·−s)A
−Ls Λ−1
L
R
R[e
Ax]
0
s
s
L∞ (s+h,T ;Y )
0
1 e(·−s−h)A − e(·−s)A
20
0
0
∗
∗
(·−s)A
≤ 0Ls+h Λ−1
L
R
R
Ax
x
−
e
0 ∞
s+h
s+h
−h
L (s+h,T ;Y )
01
20
0
0
−1
−1 ∗ ∗
(·−s)A
+ 0 Ls+h [Λs+h − Λs ]Ls R Re
Ax 0 ∞
L (s+h,T ;Y )
0 1
20
0
0
∗ ∗
(·−s)A
+ 0Ls Λ−1
Ax · χ[s,s+h] 0 ∞
s Ls R Re
L (s+h,T ;Y )
0 e(·−s−h)A − e(·−s)A
0
0
0
≤ cT 0
x − e(·−s)A Ax0 1
−h
L 1−γ (s+h,T ;Y )
0
0
0
0
−1 ∗ ∗
(·−s)A
+cT 0[Λ−1
Ax0 p
s+h − Λs ]Ls R Re
L (s+h,T ;U )
0
0
0 −1 ∗ ∗ (·−s)A
0
+cT 0Λs Ls R Re
Ax · χ[s,s+h] 0 p
→ 0 as h → 0+ ,
L (s,T ;U )
where p >
1
1−γ ;
thus
∗
∗
−Ls+h Λ−1
s+h Ls+h R R
→
1 e(·−s−h)A − e(·−s)A 2
x (t)
h
∗ ∗
(·−s)A
Ls Λ−1
Ax
s Ls R Re
∞
in L (s + δ, T ; Y )
(6.22)
as h → 0
+
for every δ ∈ (0, T − s).
On the other hand, consider the third term in (6.8): for h < 0 we have,
similarly but more easily,
0
1 e(·−s−h)A − e(·−s)A 2
0
−1 ∗ ∗
−
L
Λ
L
R
R
x (t)
0
s s
s
h 0
0
∗ ∗
(·−s)A
Ax]0 ∞
− Ls Λ−1
s Ls R R[e
L (s,T ;Y )
i.e.,
0
1 e(·−s−h)A − e(·−s)A
20
0
0
∗ ∗
(·−s)A
= 0Ls Λ−1
L
R
R
Ax
x
−
e
0 ∞
s
s
−h
L (s,T ;Y )
0 e(·−s−h)A − e(·−s)A
0
0
0
≤ cT 0
→ 0 as h → 0+ ;
x − e(·−s)A Ax0 1
−h
L 1−γ (s,T ;Y )
∗ ∗
−Ls Λ−1
s Ls R R
→
1 e(·−s−h)A − e(·−s)A 2
x
h
∗ ∗
(·−s)A
Ax
Ls Λ−1
s Ls R Re
∞
in L (s, T ; Y )
(6.23)
−
as h → 0 .
1424
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Step 5. The limit of the fourth term in (6.7) and (6.8). Finally we consider
the last term in (6.7) and (6.8). Fix δ ∈ (0, T − s) and h ∈ (0, δ]. For each
1
z ∈ L 1−γ (s, T ; Y ) we have:
/" T 7
1 L L∗ − Ls L∗
2
s+h s+h
/
s ∗
−1
∗
∗
L
Λ
L
R
R
RΦ(·,
s)x
(t)
R
/
s+h s+h s+h
h
s+δ
/
8
/
∗ ∗
(·−s)A
∗
+[Ls Λ−1
L
R
Re
BB
P
(s)x](t),
z(t)
dt
/
s
s
Y
/ " T 71 L L∗ − L L∗
2
6
#
s s ∗
s+h s+h
/
∗
≤/
R RΦ(·, s)x (t), R∗ R Ls+h Λ−1
s+h Ls+h
h
s+δ
/
4 $ 8
/
∗
z
(t)
−R∗ RLs Λ−1
L
dt
/
s
s
Y
/ " T 71
2
/
Ls+h L∗s+h − Ls L∗s ∗
+//
R RΦ(·, s)x (t)
h
s+δ
/
4 $ 8
# ∗ 6
/
(t−s)A
∗
−1 ∗
BB P (s)x, R R Ls Λs Ls z (t) dt/ = T1 + T2 .
+e
Y
1
Concerning T1 , we note that its first factor is bounded in L γ (s, T ; [D(A∗ ε )]! )
by (6.12); its second factor can be rewritten as
# ∗ 6
4 $
∗
∗
−1 ∗
R R Ls+h Λ−1
L
−
R
RL
Λ
L
z
s
s
s
s+h
s+h
−1 ∗
∗
−1 ∗
= R∗ RLs+h [Λ−1
s+h − Λs ]Ls z + R RLs Λs Ls (z · χ[s,s+h] ),
and hence it is estimated, by Proposition B.5, Lemmas 6.5 and 6.4, and
Proposition B.4(iii), as follows:
0# ∗ 6
4 $0
∗
0 R R Ls+h Λ−1 L∗ − R∗ RLs Λ−1
0
s Ls z L1/(1−γ) (s+δ,T ;D(A∗ ε ))
s+h s+h
−1 ∗
∗
≤ cT %[Λ−1
s+h − Λs ]Ls z%Lq# (s+δ,T ;U ) + cT %Ls (z · χ[s,s+h] )%Lq# (s,T ;U )
≤ o(1) + %z%L1/(1−γ) (s,s+h;Y ) = o(1)
as h → 0+ .
Thus, we get T1 → 0 as h → 0+ .
∗
For the term T2 we simply remark that R∗ RLs Λ−1
s Ls z ∈ C([s + δ, T ], Y )
∗
by Propositions B.4(iii) and B.5; hence, by the weak convergence of the
first factor to 0 (Lemma 6.3), we get T2 → 0 as h → 0+ .
Thus, we have obtained, for each δ ∈ (0, T − s),
∗
∗
Ls+h Λ−1
s+h Ls+h R R
+
Ls+h L∗s+h − Ls L∗s ∗
[R RΦ(·, s)x]
h
−e(·−s)A BB ∗ P (s)x
1
γ
in L (s + δ, T ; Y ) as h →
(6.24)
0+ .
Optimal boundary control and Riccati theory
1425
On the other hand, for h < 0, similarly but more easily we have for each
1
z ∈ L 1−γ (s, T ; Y ):
/" T 7
1 L L∗ − Ls L∗
2
s+h s+h
/
s ∗
−1 ∗ ∗
L
Λ
L
R
R
RΦ(·,
s
+
h)x
(t)
R
/
s s
s
h
s
/
1
2
8
/
∗ ∗
(·−s)A
∗
+ Ls Λ−1
L
R
Re
BB
P
(s)x
dt
(t),
z(t)
/
s
s
Y
/ " T 71 L L∗ − L L∗
s s ∗
s+h s+h
/
=/
R RΦ(·, s + h)x
h
s
/
2
8
/
∗
+e(·−s)A BB ∗ P (s)x (t), [R∗ RLs Λ−1
L
z](t)
dt
/,
s
s
Y
and as before, by Lemma 6.3, Propositions B.5 and B.4(iii), and Lemma 6.4,
we obtain
∗ ∗
Ls Λ−1
s Ls R R
Ls+h L∗s+h − Ls L∗s ∗
[R RΦ(·, s + h)x]
h
+ −e(·−s)A BB ∗ P (s)x
1
γ
in L (s, T ; Y )
This concludes the proof of Theorem 6.2.
(6.25)
as h → 0+ .
!
Appendix A. Basic facts
This Appendix contains some preliminary results which follow directly
from Hypotheses 2.1 and 2.2.
Remarks A.1. (i) Since A−1 ∈ L(Y, D(A)), it is easily seen that A = A∗∗ .
Consequently, the operator A−1 B belongs to L(U, Y ), in view of the estimate
|(A−1 Bu, y)Y | = |((A∗∗ )−1 Bu, y)Y |
= |!Bu, A∗ −1 y"D(A∗ ) | ≤ %B%L(U,[D(A∗ )]# ) %u%U %y%Y
which holds for all u ∈ U and y ∈ Y . In particular
∗
∗
B ∗ etA = (A−1 B)∗ etA A∗ ∈ L(D(A∗ ), U ).
∗
Hence, G(t) = B ∗ etA −F (t) is an element of L(D(A∗ ), U ) for each t ∈ (0, T ],
as claimed in Hypothesis 2.2.
(ii) For each w ∈ Y define the element Aε w ∈ [D(A∗ ε )]! according to the
rule
!Aε w, x"D(A∗ ε ) = (w, A∗ ε x)Y
∀x ∈ D(A∗ ε );
1426
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
then it is easy to check that the following characterizations hold true:
!
[D(A∗ ε )]! 5 {z ∈ [D(A∗ )]! : z = Aε w, w ∈ Y },
!
%z%[D(A∗ ε )]# 5 %A−ε z%Y ,
[D(Aε )]! 5 {z ∈ [D(A)]! : z = A∗ ε v, v ∈ Y },
%v%[D(A∗ ε )]# 5 %A∗ −ε z%Y .
(A.1)
(A.2)
(iii) Conditions (2.5) and (2.6) are regularity assumptions on G(·): (2.6)
says in particular that the norm %G(t)%L(D(A∗ ε ),U ) is bounded in [0, T ].
(iv) The separability of Y implies that both D(A) and D(A∗ ) are separable
spaces with their respective graph norms: indeed, if the sequence {xn } ⊂ Y
is dense in Y , then it is easy to see that {n(n − A)−1 xn } is dense in D(A)
and {n(n − A∗ )−1 xn } is dense in D(A∗ ). The same property holds for the
spaces D(Aε ) and D(A∗ ε ).
The following lemmas illustrate further important consequences of the
assumptions imposed on the model.
Lemma A.2. Under assumptions 2.1 and 2.2, we have
and hence
%etA B%L(U,D(A∗ ε )# ) ≤ c t−γ
∗
%B ∗ etA %L(D(A∗ ε ),U ) ≤ c t−γ
∀t ∈ (0, T ],
∀t ∈ (0, T ].
(A.3)
Proof. If u ∈ U , by Remark A.1(i) one has for all z ∈ D(A∗ ε )
/
/ tA
/!e Bu, z"D(A∗ ε ) / ≤ %A−ε etA Bu%Y %z%D(A∗ ε ) ,
and on the other hand, for all y ∈ Y ,
/
/
/
/ −ε tA
/(A e Bu, y)Y / = /(u, B ∗ etA∗ A∗ −ε y)U /
/
/
= /(u, F (t)A∗ −ε y)U + (u, G(t)A∗ −ε y)U /
6
4
≤ c%u%U t−γ %A∗ −ε y%Y + %y%Y ≤ ct−γ %u%U %y%Y .
Hence,
%A−ε etA Bu%Y ≤ ct−γ %u%U
and the result follows.
∀u ∈ U,
!
Lemma A.3. Under assumptions 2.1 and 2.2, for each f ∈ C([0, T ], D(A∗ ε ))
we have G(·)f (·) ∈ C([0, T ], U ).
Optimal boundary control and Riccati theory
1427
Proof. Fix t0 ∈ [0, T ]; then by Hypothesis 2.2(iii)(a) we have
G(·)f (t0 ) = G(·)A∗ −ε [A∗ ε f (t0 )] ∈ C([0, T ], U ) ,
with
%G(·)f (t0 )%C([0,T ],U ) ≤ c%f (t0 )%D(A∗ ε ) ≤ c%f %c([0,T ],D(A∗ ε ) .
In particular, evaluating G(·)f (t0 ) at t = t0 we get
%G(t0 )f (t0 )%U ≤ c%f %C([0,T ],D(A∗ ε )) ,
and since t0 is arbitrary we obtain
f ∈ B(0, T ; D(A∗ ε ))
=⇒
G(·)f (·) ∈ B(0, T ; U ).
(A.4)
Let us now prove continuity of t 3→ G(t)f (t) at t0 . We have
%G(t)f (t) − G(t0 )f (t0 )%U
≤ %G(t)[f (t) − f (t0 )]%U + %[G(t) − G(t0 )]f (t0 )%U
≤ %G(·)[f (t) − f (t0 )]%C([0,T ],U ) + %G(t)f (t0 ) − G(t0 )f (t0 )%U
≤ c%f (t) − f (t0 )%D(A∗ ε ) + %G(t)f (t0 ) − G(t0 )f (t0 )%U
= o(1) as t → t0 .
!
Lemma A.4. Under Hypotheses 2.1 and 2.2, we have
R∗ R ∈ L([D(A∗ ε )]! , [D(Aε )]! ) .
Proof. We start from Remark A.1(ii): if z ∈ [D(A∗ ε )]! and w = A−ε z ∈ Y ,
we have
%R∗ Rz%[D(Aε )]# = %A∗ −ε R∗ Rz%Y = %A∗ −ε R∗ RAε w%Y
≤ %A∗ −ε R∗ RAε %L(Y ) %w%Y = %[A∗ −ε R∗ RAε ]∗ %L(Y ) %z%[D(A∗ ε )]#
= %R∗ R%L(D(Aε ),D(A∗ ε )) %z%[D(A∗ ε )]# .
!
Lemma A.5. Under Hypotheses 2.1 and 2.2, we have
∗
t 3→ B ∗ etA R∗ R ∈ L([D(A∗ ε )]! , Lq (0, T ; U )) .
Proof. If z ∈ [D(A∗ ε )]! and w = A−ε z ∈ Y , we have by Hypothesis
2.2(iii)(c)
81/q 7 " T
81/q
7" T
∗
q
∗ tA∗ ∗
%B e R Rz%U dt
=
%B ∗ etA R∗ RAε w%qU dt
0
0
∗ ·A∗
≤ %B e
∗
ε
R RA %L(Y,Lq (0,T ;U ) %w%Y ≤ c%z%[D(A∗ ε )]# .
!
1428
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
Lemma A.6. Under Hypotheses 2.1 and 2.2, we have
G∗ ∈ L(L1 (0, T ; U ), [D(A∗ ε )]! ).
#
Proof. If x ∈ Y we have for all p ∈ (1, ∞) and ϕ ∈ Lp (0, T ; U ):
" T
∗
(G(t)x, ϕ(t))U dt.
(x, G ϕ)Y =
0
Now if x ∈ D(A∗ ε ) and y = A∗ ε x, by Hypothesis 2.2(iii)(a) we may write
for all ϕ ∈ L1 (0, T ; U ):
/" T
/
/
/
∗
(G(t)x, ϕ(t))U dt/
|(x, G ϕ)Y | = /
0
/"
/
=/
0
T
/ /"
/ /
(G(t)A∗ −ε y, ϕ(t))U dt/ = /
0
T
/
/
(y, A−ε G(t)∗ ϕ(t))U dt/
≤ c%y%Y sup %A−ε G(t)∗ %L(U,Y ) %ϕ%L1 (0,T ;U ) .
This shows that
t∈[0,T ]
%G∗ ϕ%
D(A∗ ε )#
≤ c%ϕ%L1 (0,T ;U ) .
Appendix B. Regularity analysis of the operators Ls and L∗s
This Appendix is focused on the regularity analysis of the operator Ls (and
of its adjoint L∗s ) defined by (2.11) ((2.12), respectively). It is important to
emphasize that while the basic regularity of Ls and L∗s is a prerequisite of the
whole theory and hence is needed throughout the paper, it will be specifically
the conclusion of Proposition B.5 that will play a major role in the proof of
boundedness of the gain operator B ∗ P (t) on D(Aε ) (see Proposition 5.3).
In view of the decomposition (2.3), it is natural to rewrite Ls as the sum
Ls1 + Ls2 , where

" t


F (t − σ)∗ u(σ) dσ ,
 Ls1 u(t) :=
(B.1)
u ∈ Lp (s, T ; U ) .
"s t


 Ls2 u(t) :=
G(t − σ)∗ u(σ) dσ ,
s
Because of the distinct assumptions imposed on F (t) and G(t), a separate
analysis of the operators Ls1 and Ls2 is necessary. The corresponding boundedness (continuity) results are stated in Proposition B.1 and Proposition B.2
below, respectively. For the sake of brevity, and since the component F (t)
∗
of B ∗ eA t satisfies a singular estimate, the proof of Proposition B.1—which
pertains to the operator Ls1 —will be omitted (the reader is referred, e.g., to
[21]).
Optimal boundary control and Riccati theory
1429
Proposition B.1. Under Hypotheses 2.1 and 2.2, let Ls1 be the operator
defined by (B.1). Then we have
1
(i) if 1 ≤ p < 1−γ
, then Ls1 ∈ L(Lp (s, T ; U ), Lq (s, T ; Y )) with q =
p
1−(1−γ)p ;
1
(ii) if p = 1−γ
, then Ls1 ∈ L(Lp (s, T ; U ), Lq (s, T ; Y )) for all q ∈ [1, ∞[;
1
, then Ls1 ∈ L(Lp (s, T ; U ), C([s, T ], Y )).
(iii) if p > 1−γ
Moreover, in all cases the norm of Ls1 does not depend on s.
Proposition B.2. Under Hypotheses 2.1 and 2.2, let Ls2 be the operator
defined by (B.1). Then we have
(i) if p = 1, then Ls2 ∈ L(L1 (s, T ; U ), C([s, T ], [D(A∗ ε )]! ));
(ii) if p > 1, then Ls2 ∈ L(Lp (s, T ; U ), C([s, T ], Y )).
Moreover, in all cases the norm of Ls does not depend on s.
Proof. We note that, in principle, Ls2 u(t) just belongs to [D(A∗ )]! when u
is U -valued. Let us start by proving the second assertion. Indeed, for each
ϕ ∈ L1 (s, T ; D(A∗ )) we have, using Hypothesis 2.2(iii)(a):
/
/" T
/
/
!Ls2 u(t), ϕ(t)"D(A∗ ) /
/
s
/
/" T 9" t
:
/
/
G(t − σ)∗ u(σ)dσ, ϕ(t)
dt
=/
/
∗
D(A )
s
s
" T" t
|(u(σ), G(t − σ)ϕ(t))U |dσdt
≤
s
≤
"
s
T
s
≤ cp# ,T
%u%Lp (s,t;U ) %G(t − ·)ϕ(t)%Lp# (s,t;U ) dt
"
T
s
L1 (s, T ; D(A∗ ))
As
L∞ (s, T ; Y ), and
%u%Lp (s,t;U ) %ϕ(t)%Y dt ≤ cp# ,T %u%Lp (s,T ;U ) %ϕ%L1 (s,T ;Y ) .
is dense in L1 (s, T ; Y ), we deduce Ls2 u ∈ [L1 (s, T ; Y )]! =
(B.2)
%Ls2 u%L∞ (s,T ;Y ) ≤ cp# ,T %u%Lp (s,T ;U ) .
Next, we claim that Ls2 u is continuous when u is continuous. Indeed, fix
t0 ∈ (s, T ]; if t ∈ [s, t0 [ we have for each x ∈ D(A∗ ):
|!Ls2 u(t) − Ls2 u(t0 ), x"D(A∗ ) | =
/9 " t−s
:
/
=/
G(q)∗ u(t − q)dq, x
0
−
∗
D(A )
9"
0
t0 −s
G(q)∗ u(t0 − q)dq, x
:
D(A∗ )
/
/
/
1430
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
/"
/
=/
0
t−s
/"
/
+/
/
/
(u(t − q) − u(t0 − q), G(q)x)U dq /
t0 −s
t−s
≤
sup
|σ−τ |≤|t−t0 |
/
/
(u(t0 − q), G(q)x)U dq /
%u(σ) − u(τ )%U
+%u%C([s,T ],U )
"
≤ cT %G(·)x%L2 (s,T ;U )
≤ cT %x%Y
7
sup
t0 −s
t−s
7
|σ−τ |≤|t−t0 |
"
t−s
0
%G(q)x%U dq
%G(q)x%U dq
sup
|σ−τ |≤|t−t0 |
%u(σ) − u(τ )%U +
%u(σ) − u(τ )%U +
√
t0 − t
8
√
t0 − t
8
and the last member tends to 0 as t → t−
0 . Similarly one gets for t0 ∈ [s, T [
and t ∈ (t0 , T ]
|!Ls2 u(t) − Ls2 u(t0 ), x"D(A∗ ) | ≤ cT %x%Y ω(t − t0 )
with limr→0+ ω(r) = 0. By density we deduce
%Ls2 u(t) − Ls2 u(t0 )%Y → 0
as t → t0 ,
so that our claim is proved.
Finally, using (B.2), an easy density argument yields that
Ls2 u ∈ C([s, T ], Y )
whenever u ∈
with p > 1. The second assertion is proved.
The first statement can be proved quite similarly: if u ∈ L1 (s, T ; U ), then
for each w ∈ D(A∗ ε ) we have, using Remark A.1(ii)-(iii):
9" t
:
G(t − σ)∗ u(σ) dσ, w
!Ls2 u(t), w"D(A∗ ε ) =
D(A∗ ε )
s
" t
=
(u(σ), (G(t − σ)A∗ −ε )A∗ ε w)U dσ
Lp (s, T ; U )
s
≤
"
s
T
%u(σ)%U dσ sup %G(·)A∗ −ε (A∗ ε w)%U ≤ c%u%L1 (s,T ;U ) %w%D(A∗ ε ) .
[0,T ]
!
The continuity of Ls2 u follows as before when u ∈ C([s, T ], U ).
Summing up, we obtain the following regularity statements about the
operator Ls .
Optimal boundary control and Riccati theory
1431
Proposition B.3. Under Hypotheses 2.1 and 2.2, let Ls be the operator
defined by (2.11). We have
1
(i) if p = 1, then Ls ∈ L(L1 (s, T ; U ), L γ (s, T ; D(A∗ ε )! ));
1
(ii) if 1 < p < 1−γ
, then Ls ∈ L(Lp (s, T ; U ), Lq (s, T ; Y )) with q =
p
1−(1−γ)p ;
1
(iii) if p = 1−γ
, then Ls ∈ L(Lp (s, T ; U ), Lq (s, T ; Y )) for all q ∈ [1, ∞[;
1
(iv) if p > 1−γ , then Ls ∈ L(Lp (s, T ; U ), C([s, T ], Y )).
Moreover, in all cases the norm of Ls does not depend on s.
Proof. Every statement follows combining the results of Propositions B.1
and B.2.
!
Regarding the adjoint operator L∗s , the following boundedness results are
valid.
Proposition B.4. Under Hypotheses 2.1 and 2.2, let Ls be the operator
defined by (2.11). For its adjoint L∗s , defined by (2.12), we have
(i) if p = 1, then L∗s ∈ L(L1 (s, T ; Y ), Lr (s, T ; U )) for all r ∈ [1, γ1 );
1
(ii) if p ∈ (1, 1−γ
), then L∗s ∈ L(Lp (s, T ; Y ), Lr (s, T ; U )) with
p
r = 1−(1−γ)p ;
1
(iii) if p ≥ 1−γ
, then L∗s ∈ L(Lp (s, T ; Y ), Lr (s, T ; U )) for all r ∈ [1, ∞);
(iv) if p =
1
1−γ ,
1
then L∗s ∈ L(L 1−γ (s, T ; D(A∗ ε )), C([s, T ], U )).
Moreover, in all cases the norm of Ls does not depend on s.
Proof. It follows by duality, using Proposition B.3. Namely:
1
, for every v ∈
(i) Fix r ∈ (1, γ1 ) and let y ∈ L1 (s, T ; Y ). As r! > 1−γ
#
Lr (s, T ; U ) we can write, by Corollary B.3(iii),
/" T
/ /" T
/
/
/ /
/
∗
(Ls y(t), v(t))U dt/ = /
(y(t), Ls v(t))Y dt/
/
s
s
≤ %y%L1 (s,T ;Y ) %Ls v%L∞ (s,T ;Y ) ≤ cr,T %y%L1 (s,T ;Y ) %v%Lr# (s,T ;Y ) ,
so that L∗s y ∈ Lr (s, T ; U ), and
%L∗s y%Lr (s,T ;U ) ≤ cq,T %y%L1 (s,T ;Y ) .
1
) and y ∈ Lp (s, T ; Y ). Then p! >
(ii) Now let p ∈ (1, 1−γ
r=
p
)!
( 1−(1−γ)p
=
p#
1+(1−γ)p#
we have r ∈ (1,
1
1−γ )
and
p!
=
1
γ and
r
1−(1−γ)r ;
defining
thus, as
1432
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
before, for every v ∈ Lr (s, T ; U ) we get by Corollary B.3(ii)
/" T
/ /" T
/
/
/ /
/
∗
(Ls y(t), v(t))U dt/ = /
(y(t), Ls v(t))Y dt/
/
s
s
≤ %y%Lp (s,T ;Y ) %Ls v%Lp# (s,T ;Y ) ≤ cp,T %y%Lp (s,T ;Y ) %v%Lr (s,T ;Y ) ,
#
and consequently L∗s y ∈ Lr (s, T ; Y ) and
%L∗s y%Lr# (s,T ;U ) ≤ cp,T %y%Lp (s,T ;Y ) .
As r! =
p
1−(1−γ)p ,
we obtain the desired result.
#
(iii) Fix r ∈ (1, ∞). For each v ∈ Lr (s, T ; U ), since
r#
Ls v ∈ L 1−(1−γ)r# (s, T ; Y ) and (
we have similarly:
/
/" T
/
/
(L∗s y(t), v(t))U dt/ ≤ %y%
/
r!
r
)! =
,
!
1 − (1 − γ)r
1 + (1 − γ)r
r
L 1+(1−γ)r (s,T ;Y )
s
≤ cr,T %y%
%Ls v%
r
L 1+(1−γ)r (s,T ;Y )
r#
#
L 1−(1−γ)r (s,T ;Y )
%v%Lr# (s,T ;U ) .
Hence, L∗s v ∈ Lr (s, T ; U ).
1
. Let p ≥ p0 and y ∈ Lp (s, T ; D(A∗ ε )): noting that
(iv) Set p0 = 1−γ
1
p!0 = γ1 < 1−γ
= p0 (since γ > 12 ), as before, we get, by Corollary B.3(i),
/"
/
/
s
T
/
/
(L∗s y(t), v(t))U dt/
/"
/
=/
s
T
/
/
(y(t), Ls v(t))Y dt/
≤ %y%Lp0 (s,T ;D(A∗ ε )) %Ls v%Lp#0 (s,T ;D(A∗ ε )# )
≤ cp0 ,T %y%Lp0 (s,T ;D(A∗ ε )) %v%L1 (s,T ;Y ) ;
i.e., L∗s y ∈ L∞ (s, T ; Y ) and
%L∗s y%L∞ (s,T ;Y ) ≤ cT %y%Lp0 (s,T ;D(A∗ ε )) ≤ cT %y%Lp (s,T ;D(A∗ ε )) .
It remains to show that L∗s y is continuous when y ∈ Lp0 (s, T ; D(A∗ ε )). This
will be a consequence of the following properties:
(a) L∗s y ∈ C([s, T ], Y ) when y is a Y -valued step function;
(b) the Y -valued step functions are dense in Lp0 (s, T ; Y ).
Optimal boundary control and Riccati theory
1433
Property (b) is well known; we just have to prove (a). Let s = t0 < t1 <
. . . < tm = T , set Ik = [tk−1 , tk [ and define
y(t) =
m
;
vk χIk (t),
v1 , . . . , vm ∈ Y.
k=1
Then for t ∈ [s, T ] we have
L∗s y(t)
=
=
=
m "
;
T
∗
B ∗ e(τ −t)A vk χIk (τ ) dτ
k=1 t
m " T −t
;
(F (σ) + G(σ))vk χIk (σ + t) dσ
k=1 0
m " 0∨(tk −t)
;
0∨(tk−1 −t)
k=1
(F (σ) + G(σ))vk dσ.
By Hypothesis 2.2(i)-(ii), σ 3→ G(σ)vk and σ 3→ F (σ)vk are integrable functions in [0, T ]; hence L∗s y is a finite sum of continuous functions. This proves
(a) and completes the proof.
!
Proposition B.5. Under Hypotheses 2.1 and 2.2, let Ls be the operator
#
defined by (2.11). Then we have R∗ RLs ∈ L(Lq (s, T ; U ), C([s, T ], D(A∗ ε ))).
#
Proof. Let u ∈ Lq (s, T ; U ): we first show that R∗ RLs u ∈ L∞ (s, T ; D(A∗ ε )).
Indeed, for all z ∈ D(Aε ) we have by Hypothesis 2.2(iii)(c)
/" t
/
∗
/
/
|(R∗ RLs u(t), Aε z)Y | = / (u(σ), B ∗ e(t−σ)A R∗ RAε z)U dσ /
s
≤ %u%L
q#
(s,T ;U ) %B
∗ ·A∗
e
∗
R RAε %L(Y,Lq (0,T ;U )) %z%Y ≤ c%u%Lq# (s,T ;U ) %z%Y ,
so that A∗ ε R∗ RLs u ∈ L∞ (s, T ; U ) and
%R∗ RLs u%L∞ (s,T ;D(A∗ ε )) ≤ c%u%Lq# (s,T ;U ) .
#
In order to prove that R∗ R Ls u ∈ C([s, T ], D(A∗ ε )) for all u ∈ Lq (s, T ; U ),
it is enough to show that this result holds when u belongs to the space
#
1
, which is dense in Lq (s, T ; U ).
Z = {u ∈ H 1,p (s, T ; U ) : u(s) = 0}, p > 1−γ
1
Thus, fix p > 1−γ
and let u ∈ H 1,p (s, T ; U ) be such that u(s) = 0: then we
can write, integrating by parts,
" t
e(t−σ)A Bu(σ) dσ
R∗ RLs u(t) = R∗ R
s
1434
Paolo Acquistapace, Francesca Bucci, and Irena Lasiecka
" t7
d (t−σ)A −1 8
A
e
Bu(σ) dσ
dσ
s
= −R∗ RA−1 Bu(t) + R∗ RA−1 Ls u! (t).
∗
=R R
Now we have
R∗ RA−1 Ls u! ∈ C([s, T ], D(A∗ ε )),
since Ls ∈ C([s, T ], Y ) by Corollary B.3(iv) and
hence we have only to verify that
u!
(B.3)
−
(B.4)
R∗ RA−1
∈ L(Y, D(A∗ ε ));
R∗ RA−1 Bu ∈ C([s, T ], D(A∗ ε )).
(B.5)
Indeed, we have for s ≤ τ < t ≤ T and for all z ∈ D(Aε ):
(A∗ ε R∗ RA−1 Bu(t) − A∗ ε R∗ RA−1 Bu(τ ), z)Y
= (u(t) − u(τ ), B ∗ A∗ −1 R∗ RAε z)U
"
7
∗ (t−τ )A∗ ∗ −1 ∗ ε
= u(t) − u(τ ), B e
A R A z−
t−τ
∗
B ∗ eσA R∗ RAε z dσ
0
hence, using Hypothesis 2.2(i)-(iii), and the fact that u is
uous,
/
/
/
/ ∗ε ∗
ε
/(A R RA−1 Bu(t) − A∗ R∗ RA−1 Bu(τ ), z)Y /
1
p# -Hölder
8
U
;
contin-
≤ %u(t) − u(τ )%U %F (t − τ )A∗ −1 R∗ RAε z%U
+%u(t) − u(τ )%U %G(t − τ )A∗ −1 R∗ RAε z%U
" t−τ
∗
+%u(t) − u(τ )%U
%B ∗ eσA R∗ RAε z%U dσ
0
≤ c(t − τ )
1
−γ
p#
1
1
%z%Y + c(t − τ ) p# %z%Y + c(t − τ ) p#
+ q1#
%z%Y .
This proves (B.5). By (B.3), (B.4) and the above remarks, the result follows.
!
Remark B.6. The same proof shows that, for fixed s, t with 0 ≤ s < t ≤ T ,
one has
%R∗ Re·A Ls u(t)%C([0,T −t+s],D(A∗ ε )) ≤ c%u%Lq# (s,t;U )
#
∀u ∈ Lq (s, t; U ).
Corollary B.7. Under Hypotheses 2.1 and 2.2, let Ls be the operator defined
by (2.11). Then for its adjoint L∗s , defined by (2.12), we have
L∗s R∗ R ∈ L(L1 (s, T ; [D(A∗ ε )]! ), Lq (s, T ; U )) .
Optimal boundary control and Riccati theory
1435
Proof. It follows easily, by duality, from Proposition B.5.
!
Acknowledgements. The research of P. Acquistapace was supported by
the Italian Ministero dell’Istruzione, dell’Università e della Ricerca
(M.I.U.R.), under the project “Kolmogorov equations”; the research of F.
Bucci was partially supported by the Italian M.I.U.R., under the project
“Feedback controls and optimal controls”. The research of I. Lasiecka was
partially supported by the National Science Foundation under grant DMS
0104305 and by the Army Research Office under grant DAAD 19-02-1-0179.
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