# Uniqueness for Riccati equations with unbounded operator coefficients

Paolo Acquistapace  Università di Pisa, Dipartimento di Matematica, Largo
Bruno Pontecorvo 5, 56127 Pisa, ITALY  paolo.acquistapace(at)unipi.it and
Francesca Bucci  Francesca Bucci, Università degli Studi di Firenze,
Dipartimento di Matematica e Informatica, Via S. Marta 3, 50139 Firenze, ITALY
francesca.bucci(at)unifi.it

###### Abstract.

In this article we address the issue of uniqueness for differential and
algebraic operator Riccati equations, under a distinctive set of assumptions
on their unbounded coefficients. The class of boundary control systems
characterized by these assumptions encompasses diverse significant physical
interactions, all modeled by systems of coupled hyperbolic/parabolic partial
differential equations. The proofs of uniqueness provided tackle and overcome
the obstacles raised by the peculiar regularity properties of the composite
dynamics. These results supplement the theories of the finite and infinite
time horizon linear-quadratic problem devised by the authors jointly with
Lasiecka, as the unique solution to the Riccati equation enters the closed
loop form of the optimal control.

##  1\. Introduction

Well-posedness of Riccati equations is a fundamental question within control
theory of Partial Differential Equations (PDE). While the issues of existence
and uniqueness for the corresponding solutions are both natural to be
addressed and significant in themselves, when seen in the context of linear-
quadratic optimal control uniqueness proves particularly relevant, not
exclusively from a theoretical perspective. This is because it brings about in
a univocal manner the (optimal cost, or Riccati) operator which occurs in the
feedback representation of the optimal control, thereby allowing its
synthesis.

In the present work focus is on the differential and algebraic Riccati
equations arising from optimal control problems with quadratic functionals for
the class of infinite dimensional abstract control systems dealt with in our
earlier works [1] and [3], joint with Lasiecka. The basic characteristics of
these linear systems – which read as $y^{\prime}=Ay+Bu$, according to a
standard notation – are the following: the free dynamics operator $A$ is the
infinitesimal generator of a $C_{0}$-semigroup $\\{e^{At}\\}_{t\geq 0}$ on the
state space $Y$, while the control operator $B$ is unbounded, meaning that $B$
maps continuously the control space $U$ into a larger functional space than
$Y$, that is the extrapolation space $[{\mathcal{D}}(A^{*})]^{\prime}$; see
the Assumptions 2.1. It is well known that the latter is an intrinsic feature
of differential systems describing evolutionary PDE with boundary (and also
point) control, already recognized by the end of the sixties in the pioneering
work of Fattorini [18].

We require in addition and more specifically that several assumptions on the
operators $A,B$ are fulfilled, recorded in Section 2 as Assumptions 2.4. These
are regularity properties that pertain to the operator $B^{*}e^{A^{*}t}$, with
respective PDE counterparts. It is worth emphasizing here that not only the
aforesaid control-theoretic properties do not prescribe analyticity of the
semigroup $e^{At}$, in accordance with the fact that the class of systems
under consideration – introduced by these authors with Lasiecka in [1] – is
inspired by and tailored on systems of coupled hyperbolic-parabolic PDE,
subjected to boundary/interface control. They are also weaker (and somewhat
trickier) than the full singular estimates for $e^{At}B$ which are known to be
equally effective for the study of the quadratic optimal control problem on
both a finite and infinite time horizon, as proved in [21], [22], [25, 26]. In
this respect, we remark that it was first discovered in [10] that a very same
thermoelastic system may or may not yield a singular estimate for the
corresponding operator $e^{At}B$, depending on which combination of boundary
conditions/actions (e.g., clamped/Neumann vs clamped/Dirichlet) is taken into
consideration.

Another distinguishing feature of the coefficients of the Riccati equations
under study, which when the unknown $P$ is time-independent read as

$(Px,Az)_{Y}+(Ax,Pz)_{Y}-(B^{*}Px,B^{*}Pz)_{U}+(Rx,Rz)_{Z}=0\,,\quad
x,z\in{\mathcal{D}}(A)\,,$ (1.1)

is that $R$ – that is the observation operator in the optimization problem –
does not need to be smoothing; see (2.7), that is iiib) of the Assumptions
2.4.

Working in the framework described above, a theory for both the finite and
infinite time horizon LQ-problem has been devised in [1] and [3], the latter
under the Assumptions 2.9 (replacing Assumptions 2.4). The strenght of these
theories is confirmed by the trace regularity results that have been
established for the solutions to significant PDE systems comprising hyperbolic
and parabolic components, over the years. Indeed, the novel class introduced
in [1] has proven successful in describing a diverse range of physical
interactions such as mechanical-thermal, acoustic-structure, fluid-elasticity
ones. And above all, successful in order to attain solvability of the
associated optimization problems; see [2], [8], [11, 12], [3], and the recent
[9].

We recall that since in [1] and [3] we followed a variational approach, by
using the optimality conditions a bounded operator – to wit, $P(t)$ or $P$, in
accordance with either a finite or infinite time interval – is constructed in
terms of the optimal state and only subsequently shown to satisfy the
corresponding Riccati equations. For this reason the works [1] and [3] provide
existence for Riccati equations, but not uniqueness.

By contrast and as it is well-known, in a (sometimes called) ‘direct’
approach, the well-posedness of the nonlinear Riccati equation is studied in a
first step, independently from the minimization problem. For the Cauchy
problems associated with the differential Riccati equations, uniqueness is
established along with existence, as joint outcomes of a method of proof based
on fixed point theorems. Then, in order to achieve the actual feedback
representation of the optimal control, that is the ultimate goal from a
mathematical as well as a practical perspective, further steps are necessary.
(This is not the case here.)

Thus, our aim with the present work is to provide uniqueness results for both
the differential and algebraic Riccati equations, thereby completing the
complex of findings of [1] and [3], respectively. These results – our main
results – are stated as Theorems 2.7 and 2.11. Uniqueness holds in appropriate
(respective) classes of linear bounded operators, which are consistent with
the distinctive property of the gain operator emerged in [1, 3]; see the
statements S4. and A4. in Theorem 2.6 and Theorem 2.10, respectively.

We feel it is important to emphasize that although often following approaches
and arguments previously used in the past literature, our proofs of uniqueness
need to face novel challenges. This is due to the specific assumptions on the
operator coefficients of the Riccati equations, notably weaker than the ones
which characterize the parabolic class (and its generalization), somewhat a
little bit stronger than mere admissibility which is typical of the hyperbolic
class, yet not requiring smoothness of the observations.

We find it useful to devote a separate section to several comments and points
concerning our analysis.

###  1.1. An insight into the mathematical proofs

We provide two proofs of Theorem 2.7, that pertains to uniqueness for the
differential Riccati equations (2.13) (DRE, in short). This result is relevant
for the optimal control problem on a finite time horizon (i.e. Problem 2.3
with $T<+\infty$), under the Assumptions 2.4.

The first proof, given in Section 3, follows the method employed by Lasiecka
and Triggiani in [27, Vol. I, Theorem 1.5.3.3; Vol. II, Theorems 8.3.7.1], up
to a certain point. The basic rationale is standard: one proceeds by
contradiction, assuming there exists another solution $P_{1}(t)$ to the DRE,
besides the optimal cost operator $P(t)$. (In our case, that $P(t)$ solves the
DRE has been proved in [1]; see the statement S6. of Theorem 2.6.) On the
basis of the integral form of the DRE – in the present case two forms, derived
in Lemma 3.1 –, one finds that the difference $Q(t)=P_{1}(t)-P(t)$ solves a
suitable integral equation. It is in the estimates performed afterwards, that
the paths diverge, with iiic) of the Assumptions 2.4 playing a major role
here, together with the class of operators $P(t)$ and $P_{1}(t)$ belong to.
The proofs carried out in the aforementioned results instead take advantage of
either the enhanced regularity of the analytic semigroup $e^{At}$ that
describes the free dynamics, or the additional regularity assumed on the
operators $Re^{At}$ and $Re^{At}B$.

It is unlikely that the method of proof described above could be adjusted in
order to establish uniqueness for the algebraic Riccati equation, relevant for
Problem 2.3 with $T=+\infty$. This owing to the argument employed when
$T<+\infty$: $Q(t)$ is shown to be zero on some subinterval of $[0,T]$, with
the soughtafter goal attained in a finite number of steps. Other methods of
proof are certainly worth to be explored. One might attempt to proceed along
the lines of the proof of [27, Vol. I, Theorem 2.4.5], despite the absence of
analyticity of the semigroup $e^{At}$, as well as of the optimal state
semigroup $\Phi(t)$. If this were the case, a preliminary analysis which
appears unavoidable would pertain to issues connected to a given solution
$P_{1}$ to the ARE (a priori, distinct from the optimal cost operator $P$),
such as the possible generation of a $C_{0}$-semigroup on $Y$ by the operator
$A-BB^{*}P_{1}$, in turn to be suitably defined. It is unclear in this respect
whether a full description of the domain of the generator itself $A_{P}$ of
$\Phi(t)$, would be required. Improving the statement A5. of Theorem 2.10 is
an independent interesting subtlety; in any case, we leave this question open.

Thus, in order to prove Theorem 2.11 we choose to return to the dynamic
programming approach to the LQ-problem, and borrow from it a key element in
attaining that the optimal control admits a (pointwise in time) feedback
representation. This element is fulfilled by the so called fundamental
identity. In a direct approach, the fundamental identity builds a bridge
between the nonlinear Riccati equation – whose well-posedness is studied in a
first step, independently from the minimization problem, as recalled above –
and the actual closed loop form of the optimal control. The latter goal (i.e.
the feedback representation of the optimal control) was already attained in
[1] and [3]; see the statements S4. of in Theorem 2.6 and A4. of Theorem 2.10,
respectively. And yet, the identities we establish in Lemma 4.1 and Lemma 4.3
constitute a major (and technically nontrivial) step in our analysis, allowing
to achieve uniqueness for both differential and algebraic Riccati equations,
respectively. Theorem 2.7 and Theorem 2.11 are thus established, via methods
of proof which are akin to each other.

###  1.2. Riccati equations: a historical synopsis, and a few open questions

Historically, the appearance of matrix Riccati equations – named after Jacopo
Riccati (XVII century) – as a research subject is recognized to date back to
the sixties, with the independent contributions of Kalman [20] in the USA and
of Letov [30] in the former Soviet Union. Their study has already lasted for
more than half a century, owing to its connections to a wide variety of topics
such as stability and stabilization problems, the linear-quadratic (LQ)
optimal control problem, differential games, just to name a few. We refer the
reader to the introductory monograph [38], along with its references.

We begin by briefly mentioning a broader problem, and then we focus on well-
posedness of Riccati equations with unbounded operator coefficients. We make
an attempt to retrace the main stages of its evolution, which spans over
decades, thereby arriving at the problem at hand.

####  1.2.1. The non-standard LQR-problem, matrix/operator inequalities

A related significant problem which sees its development around the same years
is the one referred to in the literature as the non-standard (linear)
quadratic regulator problem. Associated with the Cauchy problems for the
linear control system $y^{\prime}=Ay+Bu$ is a quadratic functional, which may
not be coercive; it is indeed the integral of a general quadratic form

$F(y,u)=\langle Qy,y\rangle+2\text{Re}\langle Su,y\rangle+\langle
Ru,u\rangle\,,$

where for instance $Q$ is allowed to be negative definite. The goal, in short,
is then to characterize – given an arbitrary initial state $y_{0}$ – the
condition

$\inf_{u\in L^{2}(0,T;U)}\int_{0}^{T}F(y(t),u(t))\,dt>-\infty\,,$

where $y(\cdot)$ is the solution to the Cauchy problem corresponding to
$y_{0}$ and to the control function $u(\cdot)$. As it is well known, one is
led to solving matrix/operator inequalities, rather than (matrix/operator)
Riccati equations; in addition, when $T=+\infty$ the analysis moves from the
time domain to the frequency domain.

It is far beyond the scope of this article to go into any detail of this
problem, whose key result is the celebrated Kalman-Popov-Yakubovich Lemma.
However, we find it fitting to include in this synopsis a mention of the above
described scenario, owing to the variety of its motivations and applications –
some of them dating back to much earlier times than the sixties –, as well as
of the extensions of Yakubovich’s results to the PDE realm that have been
attained in the nineties. We refer the interested reader to [33] and [14],
which include historical comments and pertinent references.

####  1.2.2. Operator Riccati equations

During the seventies, the theories of the LQ-problem and of Riccati equations
is extended to the infinite dimensional setting. The abstract formulation of
initial-boundary value problems (IBVP) for PDE in bounded domains, in the
presence of distributed control, brings about control systems of the usual
form $y^{\prime}=Ay+Bu$, where $A$ is the infinitesimal generator of a
$C_{0}$-semigroup in a certain Banach space $Y$, while $B$ is a bounded
operator from a (control) space $U$ to $Y$; see [7]. (The same framework
encompasses also other kinds of differential problems such as, e.g., delay
systems.) The first contributions to the study of Riccati equations with
unbounded operator coefficients (still, with $B$ bounded) are due to the works
of Da Prato [16], Tartar, Curtain and Pritchard, Zabczyk; see [38, Part IV,
Ch. 4] and [7].

####  1.2.3. Riccati equations with unbounded $B$

The proof of well-posedness performed by Da Prato in [16] paves the way for
subsequent direct studies of Riccati equations with an unbounded (control)
operator $B$. These extensions were initially focused on mixed problems for
parabolic PDE. Hence, they could exploit the regularity properties brought
about by the analytic semigroup that describes the free dynamics. Although
well known to those acquainted with the subject, it is important to emphasize
that the transition to infinite-dimensional systems describing PDE with
boundary or point control (as opposed to PDE with distributed control), has
drastically impacted on the mathematical analysis of the corresponding Riccati
equations. Whatever approach and method of proof is chosen, the actual meaning
of the term $B^{*}P$ which appears in the quadratic term of the equation – or
lack thereof – is the central issue that must be tackled. It is here that the
parabolic and the hyperbolic analyses split apart.

Even more notably, when a Riccati theory in connection with optimal boundary
control of hyperbolic-like PDE is ultimately attained, greater mathematical
challenges come from the trace regularity results that are essential in order
to invoke it, with respective proofs on an ad hoc basis.

We refer the reader to [24] for a useful overview of the main results on the
subject until the early nineties, and to [7] for a broader picture. The in-
depth monograph [27] goes into much more detail; the notes at the end of any
chapter provide a roadmap as well as bibliographical information. Principal
contributions to the Riccati theory for parabolic (and parabolic-like) PDEs
are due to Balakrishnan (1977), Lasiecka and Triggiani (1983), Pritchard and
Salamon (1984), Flandoli (1984), Da Prato and Ichikawa (1985). The unifying
functional-analytic approach such as the one of Salamon [34] appears upweighed
by the Riccati theory developed specifically for hyperbolic (hyperbolic-like,
including Petrovskiĭ-type) PDE. We recall explicitly the direct study of
differential Riccati equations by Da Prato, Lasiecka and Triggiani [17]; the
most conclusive results on algebraic Riccati equations are due to Flandoli
(1988) and Barbu (2000), both joint with Lasiecka and Triggiani.

####  1.2.4. A glimpse of peculiarities and challenges relevant to hyperbolic-
like PDE

The hyperbolic character of the dynamics – in the present work, of one
component of the coupled PDE system –, combined with the unboundedness of the
control operator, can raise obstacles to the well-posedness of the Riccati
equations corresponding to the associated optimal control problems. It will
suffice to highlight that under the abstract trace regularity assumption or
admissibility condition

$\exists C_{T}>0\colon\quad\int_{0}^{T}\|B^{*}e^{A^{*}t}x\|_{U}^{2}\leq
C_{T}\|x\|_{Y}^{2}\qquad\forall x\in Y$

which is characteristic of hyperbolic-like dynamics,

  * •

well-posedness of differential Riccati equations holds true provided the
observation operator $R$ has appropriate smoothing properties;

  * •

given the optimal cost operator $P$ for the infinite time horizon problem, it
turns out that the gain operator $B^{*}P$ that occurrs in the algebraic
Riccati equation may be even not densely defined. (Then, appropriate
extensions of $B^{*}P$ are called for.)

A simple illustration of the latter weak point is given by Weiss and Zwart
[37]. This work exhibits a first-order hyperbolic PDE (in one space
dimension), with point control; given a certain quadratic functional, the
optimal cost operator $P$ is computed explicitly, and then it is shown that
$B^{*}$ is intrinsically not defined on $Py$, $y\in{\mathcal{D}}(A)$. In this
connection we remark that, to the best of our knowledge, the question as to
whether there actually exist examples of hyperbolic PDE with boundary (rather
than point) control which give rise to the same ‘failure’, is open.

We like to point out that in the case of systems of hyperbolic-parabolic PDEs
(with boundary control) which fulfil the requirements of our Assumptions 2.4
(or Assumptions 2.9, when time varies in the positive half line), the
highlighted difficulties are overcome, as the theory of the Riccati equations
devised in [1, 3] and in the present work shows.

The (full) Bolza problem – that is a finite time horizon problem where the
quadratic functional to be minimized includes a penalization of the state at
the final time $T$ – renders the study of well-posedness of Riccati equations
more intricate. A technical discussion of this subject is beyond the scope of
this synopsis. We refer the reader to [27] and its references.

We just remark that the case of functionals of Bolza type is not addressed in
[1]; the Bolza problem for the class of systems characterized by the
Assumptions 2.4 is indeed open.

####  1.2.5. The non-autonomous case

The case of time dependent (operator) coefficients in the Riccati equation is
considerably more difficult. The major contributions, that all pertain to the
parabolic case, are due to Acquistapace and Terreni, in part jointly with
Flandoli; see [4], [5].

####  1.2.6. Approaching the present scenario

We turn now our attention to PDE systems comprising two (or more) evolutionary
PDE of different type, which further display a strong coupling and are subject
to boundary/interface/point control actions. We owe to Avalos and Lasiecka the
understanding of the role played by certain (local in time) estimates for the
kernel $e^{At}B$ for the purpose of boundedness of the gain operator
$B^{*}P(t)$, even in the absence of analyticity of the semigroup $e^{At}$. It
was the theoretical study of the quadratic optimal control problem for a PDE
model of an acoustic-structure interaction, successfully carried out in [6],
to motivate the subsequent introduction of the class of control systems which
yield the said singular estimates. The theory relative to the corresponding
(differential as well as algebraic) Riccati equations is well recognized:
after the former works [21], [22], [25, 26], a great deal of attention has
been devoted to the Bolza problem by Lasiecka and Tuffaha (see [28], [29],
[36]). It should be pointed out that the question of uniqueness of Riccati
equations appear to be left out of the aforementioned works, with the
exception of [36].

The class of control systems at hand here – introduced by these authors,
jointly with Lasiecka, in [1] – is instead characterized by more elaborate
regularity properties of the adjoint operator $B^{*}e^{A^{*}t}$, recorded in
the next section as Assumptions 2.4 (these assume the slightly strenghtened
form of Assumptions 2.9 for the problem on the infinite time interval). These
assumptions have proved effective in describing significant systems of
strongly coupled hyperbolic/parabolic PDE which do not fall in the previous
class, as has been pointed out already. Within this framework, complete
theories of the quadratic optimal control problem on a finite and infinite
time horizon have been devised in [1] and [3], which bring about existence for
the corresponding differential and algebraic Riccati equations, with the
distinctive feature that $B^{*}P$ is defined on a dense set. An insightful
comparison among all the mentioned abstract classes, along with their
respective Riccati theories, is provided by [1, Section 2.1].

Thus, focus of the present work is on uniqueness of Riccati equations, an
issue which was not discussed and is absent in the aforesaid works [1] and
[3]. While benefiting from the already established existence, our proofs of
uniqueness tackle (and overcome) the challenges brought about by the peculiar
regularity properties of the composite dynamics, singled out by the
Assumptions 2.4 and 2.9.

###  1.3. Outline of the paper

The structure of the paper is outlined readily. In the next Subsection 2 we
give the statements of our uniqueness results, that are Theorem 2.7 and
Theorem 2.11, after having recalled the framework and the core statements of
the theories of the LQ-problem devised in [1] and [3].

In Section 3 we present a first proof of Theorem 2.7. This is preceded by
Lemma 3.1, which establishes two integral forms of the differential Riccati
equation. An integral form of the algebraic Riccati equation is derived as
well, in Lemma 3.2.

Section 4 provides a second proof of Theorem 2.7 and the proof of Theorem
2.11. Instrumental to the proofs are Lemma 4.1 and Lemma 4.2 for the former
result, and Lemma 4.3 along with Lemma 4.4 for the latter. These lemmas
establish the fundamental identities and discuss certain built closed loop
equations.

In Appendix A we gather several regularity results (some old, some new) which
are used throughout the paper.

##  2\. Abstract framework, main results

###  2.1. The LQ problem: abstract dynamics and setting

Let $Y$ and $U$ be two separable Hilbert spaces, the state and control space,
respectively. We consider the abstract (linear) control system
$y^{\prime}=Ay+Bu$ and the corresponding Cauchy problem

$\begin{cases}y^{\prime}(t)=Ay(t)+Bu(t)\,,&\quad 0\leq t<T\\\\[5.69054pt]
y(0)=y_{0}\in Y\,,&\end{cases}$ (2.1)

under the following basic Assumptions.

######  Assumptions 2.1 (Basic Assumptions).

Let $Y$, $U$ be separable complex Hilbert spaces.

  * •

The closed linear operator $A:{\mathcal{D}}(A)\subset Y\to Y$ is the
infinitesimal generator of a strongly continuous semigroup
$\\{e^{At}\\}_{t\geq 0}$ on $Y$;

  * •

$B\in{\mathcal{L}}(U,[{\mathcal{D}}(A^{*})]^{\prime})$.

######  Remarks 2.2.

The apparent weakness of the basic assumptions on the operators $A$ and $B$
which characterize the initial/boundary value problems described by the
abstract equation in (2.1) is a reflection of the modeling of boundary control
problems for systems of coupled hyperbolic/parabolic PDEs. It is worth
recalling its most prominent features: (i) first, the control operator $B$
will not be bounded from the control space $U$ into the state space $Y$; (ii)
secondly, the semigroup $e^{At}$ will not be analytic.
We remind the reader that (i) is intrinsic to the mathematical modeling of
control actions on the boundary of the domain (or on some part of it), as
first shown in [18]. We also note that the presence of control actions
concentrated on points (in $1$-D) or on curves (in $2$-D) in the interior of
domain results in unboundedness of the control operator as well; illustrations
of both situations are found in [24], [7] and [27]. As for the simple
requirement in (ii), it is a natural feature of composite dynamics comprising
solely a parabolic component: analiticity of the overall semigroup should not
be expected.

Thus, given $y_{0}\in Y$, the Cauchy problem (2.1) possesses a unique mild
solution given by

$y(t)=e^{At}y_{0}+\int_{0}^{t}e^{A(t-s)}Bu(s)\,ds\,,\qquad t\in[0,T)\,,$ (2.2)

where

$L\colon u(\cdot)\longrightarrow(Lu)(t):=\int_{0}^{t}e^{A(t-s)}Bu(s)\,ds$

is the input-to-state mapping, that is the operator which associates to any
control function $u(\cdot)$ the solution to the Cauchy problem (2.1) with
$y_{0}=0$, and (2.2) makes sense at least in the extrapolation space
$[{\mathcal{D}}(A^{*})]^{\prime}$; see [27, § 0.3, p. 6, and Remark 7.1.2, p.
646].
We will use the notation $L$ throughout the paper. We point out here the
definition (A.1) of the operator $L_{s}$, which will occur later; the symbol
$L_{0}$, in place of $L$, is avoided for the sake of simplicity.

To the state equation (2.1) we associate the quadratic functional

$J(u)=\int_{0}^{T}\left(\|Ry(t)\|_{Z}^{2}+\|u(t)\|_{U}^{2}\right)dt\,,$ (2.3)

where $Z$ is a third separable Hilbert space – the so called observation space
(possibly, $Z\equiv Y$) – and at the outset the observation operator $R$
simply satisfies

$R\in{\mathcal{L}}(Y,Z)\,.$ (2.4)

The formulation of the optimal control problem under study is classical. The
adjectives finite or infinite time horizon problem refer to the cases
$T<+\infty$ or $T=+\infty$, respectively.

######  Problem 2.3 (The optimal control problem).

Given $y_{0}\in Y$, seek a control function $u\in L^{2}(0,T;U)$ which
minimizes the cost functional (2.3), where $y(\cdot)=y(\cdot\,;y_{0},u)$ is
the solution to (2.1) corresponding to the control function $u(\cdot)$ (and
with initial state $y_{0}$) given by (2.2).

It is well known that aiming at solving Problem 2.3, certain principal facts
need to be ascertained, beside the existence of a unique optimal pair
$(\hat{u}(\cdot,s;y_{0}),\hat{y}(\cdot,s;y_{0}))$ (which is readily
established by using classical variational arguments); namely,

  * -

that the optimal control $\hat{u}(t)$ admits a (pointwise in time) feedback
representation, in terms of the optimal state $\hat{y}(t)$;

  * -

that the optimal cost operator $P(t)$ ($P$, when $T=+\infty$) solves the
corresponding Differential (Algebraic) Riccati equation; thus, the issue of
well-posedness of the DRE (ARE) arises, requiring

  * -

that a meaning is given to the gain operator $B^{*}P(t)$ ($B^{*}P$) on the
state space $Y$ (by means of extensions, or – and this will be the case here
–, as a bounded operator on a dense subset of $Y$).

###  2.2. Theoretical results: finite and infinite time horizon problems

We begin by recalling the theory of the LQ-problem on a finite time interval
developed in [1]. This theory pertains to the class of control systems –
introduced in the very same [1] – whose dynamics, control and observation
operators are subject to the following assumptions.

######  Assumptions 2.4 (Finite time horizon case).

Let $Y$, $U$ and $Z$ be separable complex Hilbert spaces, and let $T>0$ be
given. The pair $(A,B)$ (which describes the state equation (2.1)) fulfils
Assumptions 2.1, with the additional property $A^{-1}\in{\mathcal{L}}(Y)$,
while the observation operator $R$ (which occurs in the cost functional (2.3))
satisfies the basic condition (2.4).

The operator $B^{*}e^{A^{*}t}$ can be decomposed as

$B^{*}e^{A^{*}t}x=F(t)x+G(t)x\,,\qquad 0\leq t\leq
T\,,\;x\in{\mathcal{D}}(A^{*})\,,$ (2.5)

where $F(t)\colon Y\longrightarrow U$ and
$G(t)\colon{\mathcal{D}}(A^{*})\longrightarrow U$, $t>0$, are bounded linear
operators satisfying the following assumptions:

  1. i)

there exist constants $\gamma\in(0,1)$ and $N>0$ such that

$\|F(t)\|_{{\mathcal{L}}(Y,U)}\leq N\,t^{-\gamma}\,,\qquad 0<t\leq T\,;$ (2.6)

  2. ii)

the operator $G(\cdot)$ belongs to ${\mathcal{L}}(Y,L^{p}(0,T;U))$ for all
$p\in[1,\infty)$;

  3. iii)

there exists $\epsilon>0$ such that:

    1. a)

the operator $G(\cdot){A^{*}}^{-\epsilon}$ belongs to
${\mathcal{L}}(Y,C([0,T];U))$, with

$\sup_{t\in[0,T]}\|G(t){A^{*}}^{-\epsilon}\|_{{\mathcal{L}}(Y,U)}<\infty\,;$

    2. b)

the operator $R^{*}R$ belongs to
${\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),{\mathcal{D}}({A^{*}}^{\epsilon}))$,
i.e.

$\|{A^{*}}^{\epsilon}R^{*}RA^{-\epsilon}\|_{{\mathcal{L}}(Y)}\leq c<\infty\,;$
(2.7)

    3. c)

there exists $q\in(1,2)$ (depending, in general, on $\epsilon$) such that the
map $x\longmapsto B^{*}e^{A^{*}t}{A^{*}}^{\epsilon}x$ has an extension which
belongs to ${\mathcal{L}}(Y,L^{q}(0,T;U))$.

######  Remarks 2.5.

1\. We note that it is assumed at the very outset that $0\in\rho(A)$, i.e. the
dynamics operator $A$ is boundedly invertible on $Y$. It is important to
emphasize that this property happens to hold true for an ample variety of
composite systems of hyperbolic-parabolic PDE, such as e.g. thermoelastic
systems, structural-acoustics models, fluid-elasticity interactions; see [2],
[8], [11, 12], [9]. This allows in particular to define the fractional powers
$(-A)^{\alpha}$, $\alpha\in(0,1)$; see [35, § 1.15.1-2], [32], [31, § 2.2.2].
(In order to make the notation lighter, we wrote $A^{\alpha}$ instead of
$(-A)^{\alpha}$; the same will happen throughout the paper.)

On the other hand, when $\lambda=0$ is not in the resolvent set of $A$, one
can find $\omega_{0}>0$ – the type of the semigroup – such that the
translation $\hat{A}:=\omega-A$ is a positive operator for any
$\omega>\omega_{0}$; then $\hat{A}$ is boundedly invertible, and the
fractional powers $\hat{A}^{\theta}$ of $\hat{A}$ are well-defined. The
extension of the present theory to the case of unstable semigroups $e^{At}$ is
particularly relevant in the infinite time horizon case ($T=+\infty$). It
would certainly require a tedious series of technical changes and is so far
lacking.

2\. If the singular estimate (2.6) for the component $F$ (cf. i) of
Assumptions 2.4) is shown to hold true in an arbitrarily small right
neighbourhood of $t=0$, then it usually extends to all $t\in(0,T]$, by using
semigroup theory.

3\. We note that iiia) of the Assumptions 2.4 tells us that the ‘basic’ (time)
regularity of the $G$ component, that is $G(\cdot)y\in L^{p}(0,T;U)$ for $y\in
Y$ and all finite summability exponents $p\geq 1$, improves to $G(\cdot)y\in
C([0,T];U)$, when $y\in{\mathcal{D}}({A^{*}}^{\epsilon})$.

4\. The findings of the work [1], summarized in the next Theorem 2.6, were
actually established under the weaker regularity assumption

  1. iiic)’

there exists $q\in(1,2)$ such that the map $x\longmapsto
B^{*}e^{A^{*}t}R^{*}R{A}^{\epsilon}x$ has an extension which belongs to
${\mathcal{L}}(Y,L^{q}(0,T;U))$.

Indeed, iiic)’ of Assumptions 2.4, combined with iiib), implies readily iiic),
as already pointed out in [1, p. 1401] (with a reversed notation, though).

However, on one side the present iiic) – more precisely, the boundary
regularity result that (case by case) the control-theoretic condition iiic)
translates to – has been shown over the years to hold true in the case of
distinct PDE systems studied in the aforementioned references ([2], [8], [11,
12], [9]). On the other side, uniqueness of solutions to the Riccati equations
appears to be in need of it: both within the first proof of Theorem 2.6 given
in the next section (specifically to perform the estimates which bring about
(3.7)), and also to show Lemma 4.2, instrumental to the distinct proof of the
same result proposed in Section 4. Furthermore, the stronger (A.3) – which is
central to the proof of Lemma 4.4 relevant to the infinite time horizon case –
is based upon iiic) of Assumptions 2.4.

Under the listed Assumptions 2.4, a full solution to the optimal control
Problem 2.3, as detailed by the complex of statements S1.–S6. collected in
Theorem 2.6 below, was obtained in [1]. These include, in particular, two
specific novel features over the parabolic or hyperbolic cases ([24]):

  * •

the lack of continuity (in time) of the optimal control $\hat{u}(\cdot)$ (see
S1.), and

  * •

that the gain operator $B^{*}P(t)$ is bounded only on a certain dense subset
of $Y$, yet not preventing well-posedness of the Differential Riccati
Equations corresponding to the LQ problem.

######  Theorem 2.6 (Finite time horizon theory; cf. [1], Theorem 2.3).

With reference to the control problem (2.1)–(2.3), under the Assumptions 2.4,
the following statements are valid for each $s\in[0,T)$.

  1. S1.

For each $x\in Y$ the optimal pair $(\hat{u}(\cdot,s;x),\hat{y}(\cdot,s;x))$
satisfies

$\hat{y}(\cdot,s;x)\in C([s,T];Y),\quad\hat{u}(\cdot,s;x)\in\bigcap_{1\leq
p<\infty}L^{p}(s,T;U).$

  2. S2.

The linear bounded (on $Y$) operator $\Phi(t,s)$, defined by

$\Phi(t,s)x=\hat{y}(t,s;x)=e^{A(t-s)}x+[L_{s}\hat{u}(\cdot,s;x)](t)\,,\quad
s\leq t\leq T\,,\;x\in Y\,,$ (2.8)

is an evolution operator, i.e.

$\Phi(t,t)=I_{Y}\,,\qquad\Phi(t,s)=\Phi(t,\sigma)\Phi(\sigma,s)\quad\textrm{for
$s\leq\sigma\leq t\leq T$.}$

  3. S3.

For each $t\in[0,T]$ the operator $P(t)\in{\mathcal{L}}(Y)$, defined by

$P(t)x=\int_{t}^{T}e^{A^{*}(\tau-t)}R^{*}R\Phi(\tau,t)x\ d\tau\,,\qquad x\in
Y,$ (2.9)

is self-adjoint and positive; it belongs to ${\mathcal{L}}(Y,C([0,T];Y))$ and
is such that

$(P(s)x,x)_{Y}=J_{s}(\hat{u}(\cdot,s;x),\hat{y}(\cdot,s;x))\qquad\forall
s\in[0,T]\,.$

  4. S4.

The gain operator $B^{*}P(\cdot)$ belongs to
${\mathcal{L}}({\mathcal{D}}(A^{\varepsilon}),C([0,T];U))$ and the optimal
pair satisfies for $s\leq t\leq T$

$\hat{u}(t,s;x)=-B^{*}P(t)\hat{y}(t,s;x)\qquad\forall x\in Y.$ (2.10)

  5. S5.

The operator $\Phi(t,s)$ defined in (2.8) satisfies for $s<t\leq T$:

$\frac{\partial\Phi}{\partial s}(t,s)x=-\Phi(t,s)(A-BB^{*}P(s))x\in
L^{1/\gamma}(s,T;[{\mathcal{D}}({A^{*}}^{\varepsilon})]^{\prime})$ (2.11)

for all $x\in{\mathcal{D}}(A)$, and

$\frac{\partial\Phi}{\partial t}(t,s)x=(A-BB^{*}P(t))\Phi(t,s)x\in
C([s,T],[{\mathcal{D}}(A^{*})]^{\prime})$ (2.12)

for all $x\in{\mathcal{D}}(A^{\varepsilon})$.

  6. S6.

The operator $P(t)$ defined by (2.9) satisfies the following (differential)
Riccati equation in $[0,T)$:

$\begin{cases}\frac{d}{dt}\,(P(t)x,y)_{Y}+(P(t)x,Ay)_{Y}+(Ax,P(t)y)_{Y}+(Rx,Ry)_{Z}\\\\[2.84526pt]
\qquad\qquad\qquad\quad-(B^{*}P(t)x,B^{*}P(t)y)_{U}=0\qquad\qquad\forall\,x,y\in{\mathcal{D}}(A)\\\\[5.69054pt]
P(T)=0\,.\end{cases}$ (2.13)

Among the fundamental conclusions of Theorem 2.6 is assertion S6., namely the
property that the optimal cost operator $P(\cdot)$ defined in (2.9) does solve
the differential Riccati equation (DRE) corresponding to Problem (2.1)-(2.3).
That $P(\cdot)$ is actually the unique solution to the DRE (2.13), at least
within an appropriate class of operators, is an issue which was not explicitly
dealt with in the paper [1].

Thus, in order to render the finite time horizon theory devised in [1]
complete, we complement assertion S6. of Theorem 2.6 about existence of
solutions to the DRE (2.13) with the (novel) achievement of uniqueness,
thereby concluding the proof of well-posedness of the DRE. As we will see,
uniqueness is meant within a suitable class – that is class
${\mathcal{Q}}_{T}$ in (2.14) below – of linear, bounded, self-adjoint
operators also meeting an additional requirement, which is consistent with the
regularity property displayed by the gain operator in assertion S4. above.

######  Theorem 2.7 (Uniqueness for the DRE).

With reference to the control problem (2.1)–(2.3), let the Assumptions 2.4
hold. Then, the differential Riccati equation (2.13) has a unique solution
within the class

$\begin{split}{\mathcal{Q}}_{T}&=\big{\\{}Q\in
C([0,T];{\mathcal{L}}(Y))\colon\;Q(t)=Q(t)^{*}\geq
0\,,\;Q(T)=0\,,\\\\[2.84526pt] &\qquad\qquad\qquad\qquad\qquad\qquad
B^{*}Q(\cdot)\in{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([0,T];U))\big{\\}}\,.\end{split}$
(2.14)

The optimal cost operator $P(\cdot)$ defined by (2.9) is consequently that
solution.

######  Remark 2.8.

We give two distinct proofs of Theorem 2.7: a first one in Section 3 and a
second one in Section 4. Both proofs utilize the link between the differential
and the integral forms of the Riccati equation, as clarified in the beginning
of the next section. The standing Assumptions 2.4 play a key role in both
proofs, as expected, with the trickier iiic) influencing them in a decisive
and crucial way. A major technical step in the longer second proof of
uniqueness is the derivation of a fundamental identity which is classical in
control theory, stated as Lemma 4.1.

In the infinite time horizon case – i.e., when $T=+\infty$ in (2.3) –
appropriate requirements on the decay (as $t\to+\infty$) of the semigroup
$e^{At}$ as well as of the component $F(t)$ involved in the decomposition of
the operator $B^{*}e^{A^{*}t}$ are introduced, which both appear very natural;
see (2.15) and (2.16) (the latter being (i)’ of Assumptions 2.9) below,
respectively.

Interestingly, as a consequence of the aforesaid asymptotic behaviour, the
requirements on the $L^{p}$ (in time) regularity of the component $G(\cdot)$
as well as of the operator $B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}$ will need
to hold only on a certain bounded interval $[0,T]$, rather than on the entire
half-line $(0,\infty)$.
For the sake of completeness and the reader’s convenience, the hypotheses
pertaining to the infinite time horizon are wholly recorded below.

######  Assumptions 2.9 (Infinite time horizon case).

Let $Y$, $U$ and $Z$ be separable complex Hilbert spaces, and let the basic
Assumptions 2.1 be valid, with the additional property that the
$C_{0}$-semigroup $e^{At}$ is exponentially stable on $Y$, $t\geq 0$; namely,
there exist constants $M\geq 1$ and $\omega>0$ such that

$\|e^{At}\|_{{\mathcal{L}}(Y)}\leq M\,e^{-\omega t}\qquad\forall t\geq 0\,.$
(2.15)

Then in particular, $A^{-1}\in{\mathcal{L}}(Y)$.

The operator $B^{*}e^{A^{*}t}$ admits the decomposition (2.5), where
$F(t)\colon Y\longrightarrow U$, $t\geq 0$, is a bounded linear operator such
that

  1. i)’

there exist constants $\gamma\in(0,1)$ and $N,\eta>0$ such that

$\|F(t)\|_{{\mathcal{L}}(Y,U)}\leq N\,t^{-\gamma}\,e^{-\eta t}\qquad\forall
t>0\,,$ (2.16)

while ii)-iiia)-iiib)-iiic) of the Assumptions 2.4 on the (linear, bounded)
component $G(t)\colon{\mathcal{D}}(A^{*})\longrightarrow U$, $t\geq 0$, hold
true for some $T>0$.

We note that the functional (2.3) with $T=+\infty$ makes sense at least for
$u\equiv 0$. This again in view of the exponential stability of the semigroup
$e^{At}$ ((2.15) of Assumptions 2.9), which combined with (2.4) ensures
$Ry(\cdot,y_{0};0)\in L^{2}(0,\infty;Y)$.
(The analysis carried out in the present paper easily extends to more general
quadratic functionals, like

$J(u)=\int_{0}^{\infty}\big{(}\|Ry(t)\|_{Z}^{2}+\|\tilde{R}u(t)\|_{U}^{2}\big{)}dt\,,$

provided $\tilde{R}$ is a coercive operator in $U$. We take $\tilde{R}=I$ just
for the sake of simplicity and yet without loss of generality.)

######  Theorem 2.10 (Infinite time horizon theory; cf. [3], Theorem 1.5).

Under the Assumptions 2.9, the following statements are valid.

  1. A1.

For any $y_{0}\in Y$ there exists a unique optimal pair
$(\hat{u}(\cdot),\hat{y}(\cdot))$ for Problem (2.1)-(2.3), which satisfies the
following regularity properties

$\displaystyle\hat{u}\in\bigcap_{2\leq p<\infty}L^{p}(0,\infty;U)\,,$
$\displaystyle\hat{y}\in C_{b}([0,\infty);Y)\cap\big{[}\bigcap_{2\leq
p<\infty}L^{p}(0,\infty;Y)\big{]}\,.$

  2. A2.

The family of operators $\Phi(t)$, $t\geq 0$, defined by

$\Phi(t)y_{0}:=\hat{y}(t)=y(t,y_{0};\hat{u})$ (2.17)

is a $C_{0}$-semigroup on $Y$, $t\geq 0$, which is exponentially stable.

  3. A3.

The operator $P\in{\mathcal{L}}(Y)$ defined by

$Px:=\int_{0}^{\infty}e^{A^{*}t}R^{*}R\Phi(t)x\,dt,\qquad x\in Y,$ (2.18)

is the optimal cost operator; $P$ is (self-adjoint and) non-negative.

  4. A4.

The following (pointwise in time) feedback representation of the optimal
control is valid for any initial state $y_{0}\in Y$:

$\hat{u}(t)=-B^{*}P\hat{y}(t)\qquad\textrm{for a.e. $t\in(0,\infty)$},$

where the gain operator satisfies
$B^{*}P\in{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),U)$ (that is, it is just
densely defined on $Y$ and yet it is bounded on
${\mathcal{D}}(A^{\epsilon})$).

  5. A5.

The infinitesimal generator $A_{P}$ of the (optimal state) semigroup $\Phi(t)$
defined in (2.17) coincides with the operator $A(I-A^{-1}BB^{*}P)$; more
precisely,

$\displaystyle A_{P}\equiv A(I-A^{-1}BB^{*}P)\,,$
$\displaystyle{\mathcal{D}}(A_{P})\subset\big{\\{}x\in
Y:\;x-A^{-1}B\,B^{*}Px\in{\mathcal{D}}(A)\big{\\}}\,.$

  6. A6.

The operator $e^{At}B$, defined in $U$ and a priori with values in
$[{\mathcal{D}}(A^{*})]^{\prime}$, is such that

$e^{\delta\cdot}e^{A\cdot}B\in{\mathcal{L}}(U,L^{p}(0,\infty;[{\mathcal{D}}({A^{*}}^{\epsilon})]^{\prime})\quad\forall
p\in[1,1/{\gamma})$ (2.19)

for all $\delta\in[0,\omega\wedge\eta)$; almost the very same regularity is
inherited by the operator $\Phi(t)B$:

$e^{\delta\cdot}\Phi(\cdot)B\in{\mathcal{L}}(U,L^{p}(0,\infty;[{\mathcal{D}}({A^{*}}^{\epsilon})]^{\prime})\quad\forall
p\in[1,1/{\gamma})\,,$

with $\delta>0$ sufficiently small.

  7. A7.

The optimal cost operator $P$ defined in (2.18) is a solution to the algebraic
Riccati equation (ARE) corresponding to Problem (2.1)-(2.3), that is (1.1).
The ARE reads as

$(A^{*}Px,z)_{Y}+(x,A^{*}Pz)_{Y}-(B^{*}Px,B^{*}Pz)_{U}+(Rx,Rz)_{Z}=0$

when $x,z\in{\mathcal{D}}(A_{P})$.

In order to render the infinite time horizon theory devised in [3] complete,
we complement assertion A7. of Theorem 2.10 about existence of solutions to
the ARE (1.1) corresponding to Problem (2.1)-(2.3), with the achievement of
uniqueness, thereby concluding the proof of well-posedness of the ARE.

######  Theorem 2.11 (Uniqueness for the ARE).

Consider the optimal control problem (2.1)-(2.3), with $T=+\infty$, under the
Assumptions 2.9. Then, the algebraic Riccati equation (1.1) has a unique
solution $P$ within the class ${\mathcal{Q}}$ defined as follows:

${\mathcal{Q}}:=\big{\\{}Q\in{\mathcal{L}}(Y)\colon Q=Q^{*}\geq
0\,,\;B^{*}Q\in{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),U)\big{\\}}\,.$
(2.20)

The optimal cost operator $P$ defined by (2.18) is consequently that solution.

######  Remark 2.12.

As in the finite time horizon case, the (linear, bounded, self-adjoint)
operators that belong to the class ${\mathcal{Q}}$ in (2.20) are characterized
by a requirement that is consistent with the regularity property displayed by
the gain operator in assertion A4. of Theorem 2.10.

##  3\. A first proof of uniqueness for the DRE

In this Section we derive integral forms of both the differential and
algebraic Riccati equations, and present a first proof of Theorem 2.7. The
argument employed in this proof is pretty standard: the difference between
another possible solution $P_{1}(t)$ to the DRE and the Riccati operator
$P(t)$ is shown – in a series of steps – to be identically zero on the
interval $[0,T]$. The final goal is attained fully exploiting the Assumptions
2.4 and more specifically iiic), having taken as a starting point the integral
form of the DRE.

By contrast, in the next Section 4 a unified approach and method of proof will
prove effective in showing uniqueness for both cases.

###  3.1. Finite time interval, differential Riccati equations

In this subsection we make reference to the optimal control problem
(2.1)–(2.3), with $T<+\infty$. We address the issue of uniqueness of solutions
to the Cauchy problem (2.13) for the Riccati equation corresponding to problem
(2.1)–(2.3), under the Assumptions 2.4.

We begin by relating the differential form (2.13) of the Riccati equation to
an integral form of it, which in turn can be further interpreted.

######  Lemma 3.1 (Integral forms of the Riccati equation).

Let ${\mathcal{Q}}_{T}$ be the class defined in (2.14), and let
$Q(\cdot)\in{\mathcal{Q}}_{T}$ be a solution to the DRE (2.13). Then the
following assertions hold true.

1. $Q(\cdot)$ solves the integral Riccati equation (in short, IRE), that is

$\begin{split}&\big{(}Q(t)e^{A(t-s)}x,e^{A(t-s)}y\big{)}_{Y}=(Q(s)x,y)_{Y}-\int_{s}^{t}\big{(}Re^{A(r-s)}x,Re^{A(r-s)}y\big{)}_{Z}\,dr\\\\[2.84526pt]
&\qquad\qquad\qquad+\int_{s}^{t}\big{(}B^{*}Q(r)e^{A(r-s)}x,B^{*}Q(r)e^{A(r-s)}y\big{)}_{U}\,dr\,,\end{split}$
(3.1)

with $0\leq s\leq t\leq T$ and $x,y\in{\mathcal{D}}(A^{\epsilon})$.

2. $B^{*}Q(\cdot)e^{A(\cdot-s)}\in{\mathcal{L}}(Y,L^{2}(s,T;U))$.

3. The IRE (3.1) can be rewritten in the form

$\begin{split}&\big{(}e^{A^{*}(t-s)}Q(t)e^{A(t-s)}x,y\big{)}_{Y}=(Q(s)x,y)_{Y}-\int_{s}^{t}\big{(}e^{A^{*}(r-s)}R^{*}Re^{A(r-s)}x,y\big{)}_{Y}\,dr\\\\[2.84526pt]
&\qquad\qquad\qquad+\int_{s}^{t}\big{(}e^{A^{*}(r-s)}Q(r)BB^{*}Q(r)e^{A(r-s)}x,y\big{)}_{Y}\,dr\,,\end{split}$
(3.2)

valid for any $x,y\in Y$ and with $0\leq s\leq t\leq T$.

###### Proof.

1. Let $x,y\in{\mathcal{D}}(A)$: then $e^{A\cdot}x$, $e^{A\cdot}y$ are differentiable, and therefore, using (2.13), there exists

$\begin{split}&\frac{d}{dr}\big{(}Q(r)e^{A(r-s)}x,e^{A(r-s)}y\big{)}_{Y}\\\\[5.69054pt]
&\quad=-\big{(}Q(r)e^{A(r-s)}x,Ae^{A(r-s)}y\big{)}_{Y}-\big{(}Ae^{A(r-s)}x,Q(r)e^{A(r-s)}y\big{)}_{Y}\\\\[5.69054pt]
&\qquad-\big{(}Re^{A(r-s)}x,Re^{A(r-s)}y\big{)}_{Z}\,+\big{(}B^{*}Q(r)e^{A(r-s)}x,B^{*}Q(r)e^{A(r-s)}y\big{)}_{U}\\\\[5.69054pt]
&\qquad+\big{(}Q(r)Ae^{A(r-s)}x,e^{A(r-s)}y\big{)}_{Y}+\big{(}Q(r)e^{A(r-s)}x,Ae^{A(r-s)}y\big{)}_{Y}\\\\[5.69054pt]
&\quad=-\big{(}Re^{A(r-s)}x,Re^{A(r-s)}y\big{)}_{Z}\,+\big{(}B^{*}Q(r)e^{A(r-s)}x,B^{*}Q(r)e^{A(r-s)}y\big{)}_{U}\,.\end{split}$

Integrating the above identity in $r\in[s,t]$, one readily obtains the IRE
(3.1), valid for $x,y\in{\mathcal{D}}(A)$. In view of Lemma A.5, the validity
of the IRE is extended to all $x,y\in{\mathcal{D}}(A^{\epsilon})$ by density.

2. By taking now in (3.1) $t=T$, $x=y\in{\mathcal{D}}(A^{\epsilon})$, since $P(T)=0$ we establish

$\int_{s}^{T}\big{\|}B^{*}P(r)e^{A(r-s)}x\big{\|}_{U}^{2}\,dr\leq\int_{s}^{T}\big{\|}Re^{A(r-s)}x\big{\|}_{Z}^{2}\,dr\leq
C\|x\|_{Y}^{2}$

by density.

3. The equivalent form (3.2) of the IRE follows in view of 2. and by density.

∎

A first proof of Theorem 2.7. We follow the proof of Theorem 1.5.3.3 in [27],
up to a point. The subsequent arguments and estimates are driven by the
distinctive assumptions on the adjoint of the kernel $e^{At}B$, as well as by
the different class of regularity the solutions to the DRE are sought.

We know already that the optimal cost operator $P(\cdot)$ defined by (2.9)
solves (the Cauchy problem (2.13) for) the differential Riccati equation, as
well as that $P\in{\mathcal{Q}}_{T}$. Assume there exists another operator in
${\mathcal{Q}}_{T}$, say $P_{1}(\cdot)$, which solves (2.13), and set
$Q(t):=P_{1}(t)-P(t)$, $t\in[0,T]$; we aim to prove that $Q(t)\equiv 0$. By
construction $Q(\cdot)\in{\mathcal{Q}}_{T}$. By Lemma 3.1, both $P_{1}(\cdot)$
and $P(\cdot)$ satisfy the IRE (3.1). Then, taking in particular $t=T$, we
find that $Q(s)$ satisfies

$\begin{split}(Q(s)x,y)_{Y}&=-\int_{s}^{T}(B^{*}Q(r)e^{A(r-s)}x,B^{*}P_{1}(r)e^{A(r-s)}y)_{U}\,dr\\\\[2.84526pt]
&\qquad-\int_{s}^{T}(B^{*}P(r)e^{A(r-s)}x,B^{*}Q(r)e^{A(r-s)}y)_{U}\,dr\,,\end{split}$
(3.3)

for any $x,y\in D(A^{\epsilon})$. To render the computations cleaner, set
$V(r):=B^{*}Q(r)$ (that $r$ belongs to $[s,T]$ is omitted here and below, as
clear from the context). Because $Q(\cdot)\in{\mathcal{Q}}_{T}$, it holds
$V(r)^{*}\in{\mathcal{L}}(U,[{\mathcal{D}}(A^{\epsilon})]^{\prime})$, along
with

$\|V(r)^{*}\|_{{\mathcal{L}}(U,[{\mathcal{D}}(A^{\epsilon})]^{\prime})}=\|V(r)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),U)}\leq\|V(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s,T];U)}=:c\,.$

We see that

$\left|\langle
V(r)^{*}w,y\rangle_{[{\mathcal{D}}(A^{\epsilon})]^{\prime},{\mathcal{D}}(A^{\epsilon})}\right|\leq
c\,\|w\|_{U}\|y\|_{D(A^{\epsilon})}$

consequently, as well as that
${A^{*}}^{-\epsilon}V(r)^{*}\in{\mathcal{L}}(U,Y)$, with

$\big{|}\big{(}{A^{*}}^{-\epsilon}V(r)^{*}w,x\big{)}_{Y}\big{|}=\left|\langle
V(r)^{*}w,A^{-\epsilon}x\rangle_{[{\mathcal{D}}(A^{\epsilon})]^{\prime},{\mathcal{D}}(A^{\epsilon})}\right|\leq
c\,\|w\|_{U}\|x\|_{Y}\,.$

The same observations apply to $[B^{*}P_{1}(r)]^{*}$ and $[B^{*}P(r)]^{*}$,
bringing about analogous estimates.

We may now rewrite (3.3) as

$\begin{split}(Q(s)x,y)_{Y}&=-\int_{s}^{T}(e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}[B^{*}P_{1}(r)]^{*}V(r)e^{A(r-s)}x,A^{\epsilon}y)_{Y}\,dr\\\\[2.84526pt]
&\qquad-\int_{s}^{T}(e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}V(r)^{*}B^{*}P(r)e^{A(r-s)}x,A^{\epsilon}y)_{Y}\,dr\,,\end{split}$

which tells us that

$\begin{split}&{A^{*}}^{\epsilon}\int_{s}^{T}\Big{[}e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}[B^{*}P_{1}(r)]^{*}V(r)e^{A(r-s)}\\\\[2.84526pt]
&\qquad\qquad\qquad+e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}V(r)^{*}B^{*}P(r)e^{A(r-s)}\Big{]}x\,dr\,,\end{split}$

a priori an element of $[{\mathcal{D}}(A^{\epsilon})]^{\prime}$, in fact
coincides with $-Q(s)x\in Y$ by the very definition of adjoint operator. We
deduce

$\begin{split}Q(s)x&=-{A^{*}}^{\epsilon}\int_{s}^{T}\Big{[}e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}[B^{*}P_{1}(r)]^{*}V(r)e^{A(r-s)}\\\\[2.84526pt]
&\qquad\qquad\qquad+e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}V(r)^{*}B^{*}P(r)e^{A(r-s)}\Big{]}x\,dr\,,\end{split}$
(3.4)

valid for every $x\in D(A^{\epsilon})$, where, as pointed out above, the right
hand side is an element of $Y$. As $x\in{\mathcal{D}}(A^{\epsilon})$,
$B^{*}Q(s)x$ is meaningful, and we are allowed to apply $B^{*}$ to both sides
of (3.4), thus obtaining

$\begin{split}V(s)x&=-B^{*}{A^{*}}^{\epsilon}\int_{s}^{T}\Big{[}e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}[B^{*}P_{1}(r)]^{*}V(r)e^{A(r-s)}\\\\[2.84526pt]
&\qquad\qquad\qquad+e^{A^{*}(r-s)}{A^{*}}^{-\epsilon}V(r)^{*}B^{*}P(r)e^{A(r-s)}\Big{]}x\,dr\,.\end{split}$
(3.5)

It is here where iiic) of Assumptions 2.4, that is

$\exists\,q\in(1,2)\,,\;C=C(T)>0\colon\quad\|B^{*}e^{A^{*}(\cdot-s)}{A^{*}}^{\epsilon}x\|_{L^{q}(s,T;U)}\leq
C\,\|x\|_{Y}\qquad\forall x\in Y\,,$

becomes crucially important: indeed, it yields as well

$\|[B^{*}e^{A^{*}(\cdot-s)}{A^{*}}^{\epsilon}]^{*}g(\cdot)\|_{Y}\leq
C\,\|g\|_{L^{q^{\prime}}(s,T;U)}$

($q^{\prime}$ denotes the conjugate exponent of $q$), so that in particular

$\|[B^{*}e^{A^{*}(\cdot-s)}{A^{*}}^{\epsilon}]^{*}w\|_{Y}\leq
C\,(T-s)^{1/q^{\prime}}\|w\|_{U}\qquad\forall w\in U\,.$ (3.6)

We return to (3.5), and highlight a few blocks within its right hand side, as
follows:

$\begin{split}V(s)x&=-\int_{s}^{T}\big{[}B^{*}e^{A^{*}(r-s)}{A^{*}}^{\epsilon}\big{]}\,{A^{*}}^{-\epsilon}[B^{*}P_{1}(r)]^{*}V(r)e^{A(r-s)}x\,dr\\\\[2.84526pt]
&\qquad\qquad\qquad-\int_{s}^{T}\big{[}B^{*}e^{A^{*}(r-s)}{A^{*}}^{\epsilon}\big{]}{A^{*}}^{-\epsilon}V(r)^{*}B^{*}P(r)e^{A(r-s)}x\,dr\,;\end{split}$

multiply next both members by $w\in U$, to find

$\begin{split}(V(s)x,w)_{U}&=-\int_{s}^{T}\big{(}V(r)e^{A(r-s)}x,[B^{*}P_{1}(r)A^{-\epsilon}]\,\big{[}B^{*}e^{A^{*}(r-s)}{A^{*}}^{\epsilon}\big{]}^{*}w\big{)}_{U}\,dr\\\\[2.84526pt]
&\qquad-\int_{s}^{T}\big{(}B^{*}P(r)e^{A(r-s)}x,[V(r)A^{-\epsilon}]\,\big{[}B^{*}e^{A^{*}(r-s)}{A^{*}}^{\epsilon}\big{]}^{*}w\big{)}_{U}\,dr\,.\end{split}$

We now proceed to estimate either summand in the right hand side, making use
of (3.6); this leads to

$\begin{split}&|(V(s)x,w)_{U}|\\\ &\;\leq
M\,\|V(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s,T];U))}\|x\|_{{\mathcal{D}}(A^{\epsilon})}\|B^{*}P_{1}(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s,T];U))}\|w\|_{U}(T-s)^{1/q^{\prime}}\\\
&\quad+M\,\|B^{*}P(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s,T];U))}\|x\|_{{\mathcal{D}}(A^{\epsilon})}\|V(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s,T];U))}\|w\|_{U}(T-s)^{1/q^{\prime}}\,.\end{split}$

Therefore, there exists a positive constant $C$ (depending on $P$ and $P_{1}$)
such that

$\big{|}(V(s)x,w)_{U}\big{|}\leq
C\,\|V(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s,T];U))}(T-s)^{1/q^{\prime}}\|w\|_{U}\|x\|_{{\mathcal{D}}(A^{\epsilon})}\,,$

which establishes

$\|V(s)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),U)}\leq
C\,\|V(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s,T];U))}(T-s)^{1/q^{\prime}}\,,$
(3.7)

for any $s\in[0,T)$.

The argument is now pretty standard: set $s_{0}$ such that
$(T-s_{0})^{1/q^{\prime}}<1/C$; since the estimate (3.7) holds true in
particular for any $s\in[s_{0},T)$, we have

$\|V(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s_{0},T];U))}\leq
C\,(T-s_{0})^{1/q^{\prime}}\,\|V(\cdot)\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),C([s_{0},T];U))}$

which is impossible unless $V(\cdot)\equiv 0$ on $[s_{0},T]$. Iterating the
same argument, in a finite number of steps we obtain $V(s)\equiv 0$ on
$[0,T]$. This in turn implies, by (3.3),

$(Q(s)x,y)_{Y}=0\qquad\forall s\in[0,T]\,,\;\forall
x,y\in{\mathcal{D}}(A^{\epsilon})\,;$

by density we obtain ($Q(s)x=0$ for any $x\in Y$ first, and then)
$Q(\cdot)\equiv 0$, that is $P_{1}(\cdot)\equiv P(\cdot)$, as desired.

∎

###  3.2. Infinite time interval. Preparatory material

We turn now our attention to the optimal control problem (2.1)–(2.3), with
$T=+\infty$. In order to establish a uniqueness result for the corresponding
algebraic Riccati equation (1.1), we will employ a different method of proof
than the one utilized in the previous subsection for the DRE. Still, an
integral form of the ARE will prove more effective (than its algebraic form)
to accomplish this goal, just like the integral forms of the DRE in Lemma 3.1
provide fundamental tools for both proofs of Theorem 2.7. This is the reason
why we derive the said integral form of the ARE here. The following Lemma
contributes to the preparatory material for the forthcoming analysis in
Section 4. Its proof is not difficult, yet it is explicitly given for the
reader’s convenience.

######  Lemma 3.2 (Integral form of the ARE).

Let ${\mathcal{Q}}$ be the class defined in (2.20), and let
$P_{1}\in{\mathcal{Q}}$ be a solution to the algebraic Riccati equation (1.1).
Then, $P_{1}$ solves the following integral form of the ARE valid for all
$x,y\in{\mathcal{D}}(A^{\epsilon})$:

$\begin{split}&\big{(}P_{1}e^{A(t-s)}x,e^{A(t-s)}y\big{)}_{Y}=(P_{1}x,y)_{Y}+\int_{s}^{t}\big{(}B^{*}P_{1}e^{A(r-s)}x,B^{*}P_{1}e^{A(r-s)}y\big{)}_{U}\,dr\\\\[2.84526pt]
&\qquad\qquad\qquad-\int_{s}^{t}\big{(}Re^{A(r-s)}x,Re^{A(r-s)}y\big{)}_{Z}\,dr\,,\end{split}$
(3.8)

with $0\leq s\leq t$.

###### Proof.

Let $P_{1}\in{\mathcal{Q}}$ be a solution to the ARE (1.1), that we record for
the reader’s convenience:

$(P_{1}x,Ay)_{Y}+(Ax,P_{1}y)_{Y}-(B^{*}P_{1}x,B^{*}P_{1}y)_{U}+(Rx,Ry)_{Z}=0\,,\quad
x,y\in{\mathcal{D}}(A)\,.$

With $e^{A(t-s)}x$, $e^{A(t-s)}y\in{\mathcal{D}}(A)$ in place of $x,y$, and
with $0\leq s\leq t$, the equation becomes

$\begin{split}&(P_{1}e^{A(t-s)}x,Ae^{A(t-s)}y)_{Y}+(Ae^{A(t-s)}x,P_{1}e^{A(t-s)}y)_{Y}\\\\[2.84526pt]
&\qquad-(B^{*}P_{1}e^{A(t-s)}x,B^{*}P_{1}e^{A(t-s)}y)_{U}+(Re^{A(t-s)}x,Re^{A(t-s)}y)_{Z}=0\,,\end{split}$

that is nothing but

$\begin{split}&\frac{d}{dt}\big{(}P_{1}e^{A(t-s)}x,e^{A(t-s)}y\big{)}_{Y}=\big{(}B^{*}P_{1}e^{A(t-s)}x,B^{*}P_{1}e^{A(t-s)}y\big{)}_{U}\\\\[2.84526pt]
&\qquad\qquad\qquad\qquad\qquad\qquad\quad-\big{(}Re^{A(t-s)}x,Re^{A(t-s)}y\big{)}_{Z}\,,\quad
x,y\in{\mathcal{D}}(A)\,.\end{split}$ (3.9)

Integrating both sides of (3.9) between $s$ and $t$ we attain (3.8), initially
for any $x,y\in{\mathcal{D}}(A)$. Its validity is then extended to all
$x,y\in{\mathcal{D}}(A^{\epsilon})$ by density, since $P_{1}\in{\mathcal{Q}}$.
∎

While the integral form (3.8) of the ARE will constitute the starting point
for the proof of Theorem 2.11, it is important to emphasize the central role
of the distinguishing (and improved) regularity properties of the operator
$B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}$. We refer the reader to Appendix A,
where we collected and highlighted several instrumental results, with the aim
of displaying their statements in a clear sequence and framework. See, more
specifically, Proposition A.6 therein.

##  4\. A unified method of proof of uniqueness for both DRE and ARE

In this Section we provide a second proof of Theorem 2.7 and then show Theorem
2.11, thereby settling the question of uniqueness for the differential and
algebraic Riccati equations corresponding to the optimal control problem
(2.2)–(2.3). We recall from Section 1.1 that the a crucial intermediate step
to achieve either goal is an identity which is classical in control theory.

###  4.1. Finite time interval, differential Riccati equations

In this subsection we focus on the optimal control problem (2.1)–(2.3), with
$T<+\infty$, along with the corresponding Riccati equation. In approaching the
second proof of Theorem 2.7, we start by showing the above-mentioned
fundamental identity. Despite being a standard element in classical optimal
control theory, the identity should not be taken for granted in the absence of
evident beneficial regularity properties of the kernel $e^{At}B$ – such as
analiticity of the semigroup or more generally singular estimates. Achieving
the said equality requires that the Assumptions 2.4 are fully exploited. The
delicate, careful computations are carried out in the following Lemma.

######  Lemma 4.1 (Fundamental identity).

Let $Q\in{\mathcal{Q}}_{T}$ be a solution to the integral Riccati equation
(3.1). With $u\in L^{2}(s,T;U)$ and $x\in{\mathcal{D}}(A^{\epsilon})$, let
$y(\cdot)$ be the semigroup solution to the state equation in (2.1)
corresponding to $u(\cdot)$, with $y(s)=x$, that is

$y(t)=e^{A(t-s)}x+\int_{s}^{t}e^{A(t-r)}Bu(r)\,dr=e^{A(t-s)}x+L_{s}u(t)\,,\quad
t\in[s,T]\,.$

Then, the following identity is valid: for $t\in[s,T]$

$\begin{split}&(Q(t)y(t),y(t))_{Y}-(Q(s)x,x)_{Y}=-\int_{s}^{t}\left[\|Ry(r)\|_{Z}^{2}+\|u(r)\|_{U}^{2}\right]\,dr\\\\[2.84526pt]
&\qquad\qquad\qquad+\int_{s}^{t}\|u(r)+B^{*}Q(r)y(r)\|_{U}^{2}\,dr\,.\end{split}$
(4.1)

###### Proof.

Assume initially that $u\in L^{\infty}(s,T;U)$. We examine the right hand side
of the identity (4.1). For the first term we have

$\displaystyle-\int_{s}^{t}\|Ry(r)\|_{Z}^{2}\,dr$
$\displaystyle=-\int_{s}^{t}\big{\|}Re^{A(r-s)}x\big{\|}_{Z}^{2}\,dr-\int_{s}^{t}\big{\|}RL_{s}u(r)\big{\|}_{Z}^{2}\,dr$
$\displaystyle\quad-2\textrm{Re}\int_{s}^{t}\big{(}Re^{A(r-s)}x,RL_{s}u(r)\big{)}_{Z}\,dr=:\sum_{j=1}^{3}R_{j}\,.$

We note that each summand $R_{j}$ makes sense, just considering the space
regularity originally singled out in [1] and here recalled in Proposition A.1;
more specifically, $u\in L^{\infty}(s,T;U)$ implies $L_{s}u\in C([s,T];Y)$ by
its fourth assertion. We consider next the remainder

$-\int_{s}^{t}\|u(r)\|_{U}^{2}\,dr+\int_{s}^{t}\|u(r)+B^{*}Q(r)y(r)\|_{U}^{2}\,dr\,.$

Computing the square in the second integral, discarding additive inverses and
replacing again the expression of $y(r)$, we get

$\displaystyle-\int_{s}^{t}\|u(r)\|_{U}^{2}\,dr+\int_{s}^{t}\|u(r)+B^{*}Q(r)y(r)\|_{U}^{2}\,dr$
$\displaystyle\quad=2\textrm{Re}\int_{s}^{t}\big{(}B^{*}Q(r)e^{A(r-s)}x,u(r)\big{)}_{U}\,dr+2\textrm{Re}\int_{s}^{t}\big{(}B^{*}Q(r)L_{s}u(r),u(r)\big{)}_{U}\,dr$
$\displaystyle\qquad+\int_{s}^{t}\big{\|}B^{*}Q(r)e^{A(r-s)}x\big{\|}_{U}^{2}\,dr+2\textrm{Re}\int_{s}^{t}(B^{*}Q(r)e^{A(r-s)}x,B^{*}Q(r)L_{s}u(r)\big{)}_{U}\,dr$
$\displaystyle\qquad+\int_{s}^{t}\big{\|}B^{*}Q(r)L_{s}u(r)\big{\|}_{U}^{2}\,dr=:\sum_{j=1}^{5}C_{j}\,.$

That each summand $C_{j}$ makes sense as well is justified by the following
observations: $B^{*}Q(\cdot)e^{A(\cdot-s)}x\in L^{2}(s,T;U)$ because of item
2. of Lemma 3.1; in addition, since $L^{\infty}(s,T;U)\subset
L^{q^{\prime}}(s,T;U)$, Lemma A.2 yields the improved regularity $L_{s}u\in
C([s,T];{\mathcal{D}}(A^{\epsilon}))$, which in turn implies
$B^{*}Q(\cdot)L_{s}u(\cdot)\in C([s,T];U)$, as shown in Lemma A.5.

By using the original form (3.1) of the integral Riccati equation (IRE), with
$x=y$, we find that

$\begin{split}R_{1}+C_{3}&=-\int_{s}^{t}\|Re^{A(r-s)}x\|_{Z}^{2}\,dr+\int_{s}^{t}\big{\|}B^{*}Q(r)e^{A(r-s)}x\big{\|}_{U}^{2}\,dr\\\\[2.84526pt]
&=\big{(}Q(t)e^{A(t-s)}x,e^{A(t-s)}x\big{)}_{Y}-(Q(s)x,x)_{Y}\,.\end{split}$
(4.2)

Next,

$\begin{split}R_{3}+C_{4}+C_{1}&=-2\textrm{Re}\int_{s}^{t}\big{(}Re^{A(r-s)}x,RL_{s}u(r)\big{)}_{Z}\,dr\\\\[2.84526pt]
&\quad+2\textrm{Re}\int_{s}^{t}\big{(}B^{*}Q(r)e^{A(r-s)}x,B^{*}Q(r)L_{s}u(r)\big{)}_{U}\,dr\\\\[2.84526pt]
&\quad+2\textrm{Re}\int_{s}^{t}\big{(}B^{*}Q(r)e^{A(r-s)}x,u(r)\big{)}_{U}\,dr\\\\[2.84526pt]
&=-2\textrm{Re}\int_{s}^{t}\Big{(}R^{*}Re^{A(r-s)}x,\int_{s}^{r}e^{A(r-\sigma)}Bu(\sigma)\,d\sigma\Big{)}_{Y}\,dr\\\\[2.84526pt]
&\quad+2\textrm{Re}\int_{s}^{t}\Big{\langle}Q(r)BB^{*}Q(r)e^{A(r-s)}x,\int_{s}^{r}e^{A(r-\sigma)}Bu(\sigma)\,d\sigma\Big{\rangle}_{[{\mathcal{D}}(A^{\epsilon})]^{\prime},{\mathcal{D}}(A^{\epsilon})}\,dr\\\\[2.84526pt]
&\quad+2\textrm{Re}\int_{s}^{t}\big{(}B^{*}Q(r)e^{A(r-s)}x,u(r)\big{)}_{U}\,dr\,,\end{split}$

where the duality in the penultimate term is based on the membership
$Q(\cdot)\in{\mathcal{Q}}_{T}$ (along with the estimate (A.2)) which yields
$Q(r)B\in{\mathcal{L}}(U,[{\mathcal{D}}(A^{\epsilon})]^{\prime})$, combined as
before with $L_{s}u\in C([s,T];{\mathcal{D}}(A^{\epsilon}))$. The above leads
to

$\begin{split}R_{3}+C_{4}+C_{1}&=-2\textrm{Re}\int_{s}^{t}\int_{s}^{r}\big{(}B^{*}e^{A^{*}(r-\sigma)}R^{*}Re^{A(r-s)}x,u(\sigma)\big{)}_{U}\,d\sigma\,dr\\\\[2.84526pt]
&\quad+\,2\textrm{Re}\int_{s}^{t}\int_{s}^{r}\big{(}B^{*}e^{A^{*}(r-\sigma)}Q(r)BB^{*}Q(r)e^{A(r-s)}x,u(\sigma)\big{)}_{U}\,d\sigma
dr\\\\[2.84526pt]
&\quad+2\textrm{Re}\int_{s}^{t}(B^{*}Q(\sigma)e^{A(\sigma-s)}x,u(\sigma)\big{)}_{U}\,d\sigma\,,\end{split}$

which can be rewritten, exchanging the order of integration, as follows:

$\begin{split}R_{3}+C_{4}+C_{1}=&-2\textrm{Re}\int_{s}^{t}\Big{(}B^{*}\Big{\\{}\int_{\sigma}^{t}e^{A^{*}(r-\sigma)}R^{*}R\big{[}e^{A(r-s)}x\big{]}\,dr\\\\[2.84526pt]
&\qquad\quad-\int_{\sigma}^{t}e^{A^{*}(r-\sigma)}Q(r)BB^{*}Q(r)e^{A(r-\sigma)}\big{[}e^{A(\sigma-s)}x\big{]}\,dr\\\\[2.84526pt]
&\qquad\quad-Q(\sigma)\big{[}e^{A(\sigma-s)}x\big{]}\Big{\\}},u(\sigma)\Big{)}_{U}\,d\sigma\,.\end{split}$
(4.3)

Let us focus on the expression inside the curly bracket. Because $Q(\cdot)$
solves the IRE (3.1), as well as its second form (3.2) valid for any pair
$x,y\in Y$, then the following identity – a strong form of the IRE, when
$Q(\cdot)$ is unknown – holds true:

$\begin{split}&e^{A^{*}(t-\sigma)}Q(t)e^{A(t-\sigma)}z=Q(\sigma)z-\int_{\sigma}^{t}e^{A^{*}(r-\sigma)}R^{*}Rz\,dr\\\\[2.84526pt]
&\qquad\qquad\qquad+\int_{\sigma}^{t}e^{A^{*}(r-\sigma)}Q(r)BB^{*}Q(r)e^{A(r-\sigma)}z\,dr\,,\quad
0\leq\sigma\leq t\leq T\,,\;z\in Y\,.\end{split}$

Thus, returning to (4.3) with this information and setting in particular
$z=e^{A(\sigma-s)}x$, we find that $R_{3}+C_{4}+C_{1}$ simply reads as
follows:

$\begin{split}R_{3}+C_{4}+C_{1}&=-2\textrm{Re}\int_{s}^{t}\big{(}B^{*}e^{A^{*}(t-\sigma)}Q(t)e^{A(t-s)}x,u(\sigma)\big{)}_{U}\,d\sigma\\\\[2.84526pt]
&=-2\textrm{Re}\big{(}Q(t)e^{A(t-s)}x,L_{s}u(t)\big{)}_{Y}\,.\end{split}$
(4.4)

We examine next the sum

$R_{2}+C_{5}=-\int_{s}^{t}\|RL_{s}u(r)\|_{Z}^{2}\,dr+\int_{s}^{t}\big{\|}B^{*}Q(r)L_{s}u(r)\big{\|}_{U}^{2}\,dr\,,$

where, again, since $u\in L^{\infty}(s,T;U)\subset L^{q^{\prime}}(s,T;U)$, we
know from Lemma A.2 that $L_{s}u\in C([s,T];{\mathcal{D}}(A^{\epsilon}))$.
Consequently, one gets

$\begin{split}&R_{2}+C_{5}=-\int_{s}^{t}\big{\langle}\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}L_{s}u(r),L_{s}u(r)\big{\rangle}_{[{\mathcal{D}}(A^{\epsilon})]^{\prime},{\mathcal{D}}(A^{\epsilon})}\,dr\\\\[2.84526pt]
&\;=-\textrm{Re}\int_{s}^{t}\Big{(}{A^{*}}^{-\epsilon}\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}A^{-\epsilon}\,\int_{s}^{r}A^{\epsilon}e^{A(r-\lambda)}Bu(\lambda)\,d\lambda,\\\\[1.42262pt]
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_{s}^{r}A^{\epsilon}e^{A(r-\mu)}Bu(\mu)\,d\mu\Big{)}_{Y}\,dr\,.\end{split}$
(4.5)

It is important to emphasize that in going from the duality to the inner
product in (4.5), two facts have been crucially used, besides
$Q(\cdot)\in{\mathcal{Q}}_{T}$: the hypothesis (2.7) on the observation
operator $R$ (that is iiib) of the Assumptions 2.4), and once again, Lemma
A.2. Further handling of the right hand side of (4.5) leads to the triple
integral

$R_{2}+C_{5}=-\textrm{Re}\int_{s}^{t}I(r,s)\,dr\,,$

having set

$I(r,s)=\int_{s}^{r}\\!\\!\int_{s}^{r}\big{(}B^{*}e^{A^{*}(r-\mu)}\,\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}Bu(\lambda),u(\mu)\big{)}_{U}\,d\lambda\,d\mu\,.$

Let us focus on the inner double integral $I(r,s)$. We note that this integral
pertains to a symmetric function of $(\lambda,\mu)$, and hence the integral
over the square $[s,r]\times[s,r]$ can be replaced by twice the integral over
the triangle

$\\{(\lambda,\mu)\colon s\leq\mu\leq\lambda\leq r\\}\,.$

It follows that

$\begin{split}&I(r,s)=2\,\int_{s}^{r}\\!\\!d\lambda\int_{s}^{\lambda}\\!\\!d\mu\,\big{(}B^{*}e^{A^{*}(r-\mu)}\,\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}Bu(\lambda),u(\mu)\big{)}_{U}\\\\[2.84526pt]
&\,=2\,\int_{s}^{r}\\!\\!\Big{[}\int_{s}^{\lambda}\\!\\!\big{(}B^{*}e^{A^{*}(\lambda-\mu)}\,e^{A^{*}(r-\lambda)}\,\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}Bu(\lambda),u(\mu)\big{)}_{U}\,d\mu\,\Big{]}\,d\lambda\\\\[2.84526pt]
&\,=2\,\int_{s}^{r}\\!\\!\big{(}\,e^{A^{*}(r-\lambda)}\,\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}Bu(\lambda),\int_{s}^{\lambda}e^{A(\lambda-\mu)}Bu(\mu)\,d\mu\big{)}_{Y}\,d\lambda\\\\[2.84526pt]
&\,=2\,\int_{s}^{r}\\!\\!\big{(}\,e^{A^{*}(r-\lambda)}\,\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}Bu(\lambda),L_{s}u(\lambda)\big{)}_{Y}\,d\lambda\,.\end{split}$

Inserting the expression of $I(r,s)$ obtained above in the outer integral
yields

$\begin{split}&R_{2}+C_{5}\\\\[2.84526pt]
&\;=-2\textrm{Re}\int_{s}^{t}\\!\\!\int_{s}^{r}\big{(}\,e^{A^{*}(r-\lambda)}\,\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}Bu(\lambda),L_{s}u(\lambda)\big{)}_{Y}d\lambda\,dr\,;\end{split}$

next we exchange the order of integration and also move the first argument of
the inner product, to achieve

$\begin{split}&R_{2}+C_{5}\\\\[2.84526pt]
&\;=-2\textrm{Re}\int_{s}^{t}\\!\\!\int_{\lambda}^{t}\big{(}\,e^{A^{*}(r-\lambda)}\,\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}Bu(\lambda),L_{s}u(\lambda)\big{)}_{Y}dr\,d\lambda\\\\[2.84526pt]
&\;=-2\textrm{Re}\int_{s}^{t}\\!\\!\int_{\lambda}^{t}\big{(}u(\lambda),B^{*}e^{A^{*}(r-\lambda)}\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}L_{s}u(\lambda)\big{)}_{Y}dr\,d\lambda\\\\[2.84526pt]
&\;=-2\textrm{Re}\int_{s}^{t}\big{(}u(\lambda),B^{*}\int_{\lambda}^{t}e^{A^{*}(r-\lambda)}\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}\,L_{s}u(\lambda)\,dr\big{)}_{U}\,d\lambda\,.\end{split}$
(4.6)

It is apparent that the second form (3.2) of the IRE (with $\lambda$ in place
of $s$) – in fact, a strong form of it – provides once more the tool, just
like in deriving (4.4) from (4.3). With $z=L_{s}u(\lambda)$, replace the
integral

$\int_{\lambda}^{t}e^{A^{*}(r-\lambda)}\big{[}R^{*}R-Q(r)BB^{*}Q(r)\big{]}e^{A(r-\lambda)}z\,dr$

by $[Q(\lambda)-e^{A^{*}(t-\lambda)}Q(t)e^{A(t-\lambda)}]z$, to find

$\begin{split}&R_{2}+C_{5}=-2\textrm{Re}\int_{s}^{t}\big{(}u(\lambda),B^{*}\big{[}Q(\lambda)-e^{A^{*}(t-\lambda)}Q(t)e^{A(t-\lambda)}\big{]}L_{s}u(\lambda)\big{)}_{U}\,d\lambda\\\\[2.84526pt]
&\qquad=-2\textrm{Re}\int_{s}^{t}\big{(}B^{*}\big{[}Q(\lambda)-e^{A^{*}(t-\lambda)}Q(t)e^{A(t-\lambda)}\big{]}L_{s}u(\lambda),u(\lambda)\big{)}_{U}\,d\lambda\,.\end{split}$
(4.7)

Thus, adding $C_{2}$ to (4.7), we see that a useful simplification occurs, as
detailed below:

$\begin{split}R_{2}+C_{5}+C_{2}&=-2\textrm{Re}\int_{s}^{t}\big{(}B^{*}Q(\lambda)L_{s}u(\lambda),u(\lambda)\big{)}_{U}\,d\lambda\\\\[2.84526pt]
&\qquad+2\textrm{Re}\int_{s}^{t}\big{(}B^{*}e^{A^{*}(t-\lambda)}Q(t)e^{A(t-\lambda)}L_{s}u(\lambda),u(\lambda)\big{)}_{U}\,d\lambda\\\\[2.84526pt]
&\qquad+2\textrm{Re}\int_{s}^{t}\big{(}B^{*}Q(r)L_{s}u(r),u(r)\big{)}_{U}\,dr\\\\[2.84526pt]
&=2\textrm{Re}\int_{s}^{t}\\!\\!\int_{s}^{\lambda}\big{(}Q(t)e^{A(t-\sigma)}Bu(\sigma),e^{A(t-\lambda)}Bu(\lambda)\big{)}_{Y}\,d\sigma\,d\lambda\,.\end{split}$

Owing to the simmetry of the latter integrand in $(\sigma,\lambda)$, we may
replace twice the integral over the triangle $\\{(\lambda,\sigma)\colon
s\leq\lambda\leq\sigma\leq t\\}$ by the integral over the square
$[s,t]\times[s,t]$, and finally get

$\begin{split}R_{2}+C_{5}+C_{2}&=\textrm{Re}\int_{s}^{t}\\!\\!\int_{s}^{t}\big{(}Q(t)e^{A(t-\sigma)}Bu(\sigma),e^{A(t-\lambda)}Bu(\lambda)\big{)}_{Y}\,d\lambda\,d\sigma\\\\[2.84526pt]
&=\textrm{Re}\big{(}Q(t)L_{s}u(t),L_{s}u(t)\big{)}_{Y}=\big{(}Q(t)L_{s}u(t),L_{s}u(t)\big{)}_{Y}\,.\end{split}$
(4.8)

Combining (4.8) with (4.2) and (4.4), we finally obtain

$\begin{split}\sum_{i=1}^{3}R_{i}+\sum_{j=1}^{5}C_{j}&=\big{(}Q(t)e^{A(t-s)}x,e^{A(t-s)}x\big{)}_{Y}-(Q(s)x,x)_{Y}\\\\[-2.84526pt]
&\qquad+2\textrm{Re}(Q(t)e^{A(t-s)}x,L_{s}u(t))_{Y}+(Q(t)L_{s}u(t),L_{s}u(t))_{Y}\\\\[5.69054pt]
&=(Q(t)y(t),y(t))_{Y}-(Q(s)x,x)_{Y}\,,\end{split}$

which establishes the fundamental identity (4.1) in the case $u\in
L^{\infty}(s,T;U)$. Finally, the identity extends to $u\in L^{2}(s,T;U)$ by
density, which concludes the proof of Lemma 4.1.

∎

We next introduce an integral equation that involves a given operator solution
$Q(t)$ to the Riccati equation corresponding to optimal control problem
(2.1)–(2.3). Once uniqueness for the DRE (2.13) is established, so that $Q(t)$
must coincide with the Riccati operator $P(t)$, then it will be clear that the
said integral equation ((4.9) below) is nothing but the well known closed-loop
equation, of central importance for the synthesis of the optimal control.
(This justifies the use of the term “closed-loop equation” for (4.9)).

As we shall see, the following Lemma 4.2 and (the independent) Lemma 4.1
constitute the core elements for the proof of Theorem 2.7.

######  Lemma 4.2.

Let $\epsilon$ be as in iii) of Assumptions 2.4. Let $Q\in{\mathcal{Q}}_{T}$,
where ${\mathcal{Q}}_{T}$ is the class defined by (2.14). Then, for every
$x\in{\mathcal{D}}(A^{\epsilon})$, the closed loop equation

$y(t)=e^{At}x-\int_{0}^{t}e^{A(t-s)}BB^{*}Q(\sigma)y(\sigma)\,d\sigma\,,\qquad
t\in[0,T]\,,$ (4.9)

has a unique solution in the space

$X=\Big{\\{}y\in
C([0,T];{\mathcal{D}}(A^{\epsilon}))\colon\sup_{t\in[0,T]}\big{(}e^{-rt}\|y(t)\|_{{\mathcal{D}}(A^{\varepsilon})})<\infty\Big{\\}}$
(4.10)

endowed with the norm

$\|y\|_{X,r}=\sup_{t\in[0,T]}e^{-rt}\|y(t)\|_{{\mathcal{D}}(A^{\epsilon})}\,,\qquad
y\in X\,,$

provided $r>0$ is chosen sufficiently large.

###### Proof.

With $x\in{\mathcal{D}}(A^{\epsilon})$, we set $E(t)=e^{At}x$. By semigroup
theory we know that $E(\cdot)\in C([0,T];{\mathcal{D}}(A^{\epsilon}))$; even
more, since $e^{At}$ is exponentially stable, it holds $E(\cdot)\in X$
provided $r$ is sufficiently large. As the integral equation (4.9) has the
clear form

$y(t)+\big{[}LB^{*}Q(\cdot)y(\cdot)\big{]}(t)=E(t)\,,\quad t\in[0,T]\,,$

we appeal to a classical argument of functional analysis: we will prove that
$LB^{*}Q(\cdot)$ is a contraction mapping in $X$, having chosen $r$
sufficiently large. This will in turn imply that $I+LB^{*}Q(\cdot)$ is
invertible in $X$, thus providing the sought unique solution to (4.9).

For each $y\in X$, $z\in{\mathcal{D}}({A^{*}}^{\epsilon})$, $t\in[0,T]$, we
have by Lemma A.5

$\displaystyle\big{|}(e^{-rt}LB^{*}Q(\cdot)y(\cdot),{A^{*}}^{\epsilon}z)_{Y}\big{|}$
$\displaystyle\quad=\Big{|}\int_{0}^{t}e^{-r(t-s)}\,\big{(}B^{*}Q(s)e^{-rs}y(s),B^{*}e^{A^{*}(t-s)}{A^{*}}^{\epsilon}z\big{)}_{U}\,ds\Big{|}$
$\displaystyle\quad\leq\int_{0}^{t}e^{-r(t-s)}\|B^{*}Q(\cdot)e^{-r\cdot}y(\cdot)\|_{C([0,T];U)}\,\big{\|}B^{*}e^{(t-s)A^{*}}{A^{*}}^{\epsilon}z\big{\|}_{U}\,ds$
$\displaystyle\quad\leq\|B^{*}Q(\cdot)\|_{{\mathcal{L}}(C([0,T];{\mathcal{D}}(A^{\epsilon})),C([0,T];U))}\,\|y\|_{X,r}\int_{0}^{t}e^{-r\sigma}\|B^{*}e^{A^{*}\sigma}{A^{*}}^{\epsilon}z\|_{U}\,d\sigma$
$\displaystyle\quad\leq\|B^{*}Q(\cdot)\|_{{\mathcal{L}}(C([0,T];{\mathcal{D}}(A^{\epsilon})),C([0,T];U))}\,\|y\|_{X,r}\,\Big{[}\int_{0}^{t}e^{-r\sigma
q^{\prime}}\,d\sigma\Big{]}^{1/q^{\prime}}\|B^{*}e^{\cdot
A^{*}}{A^{*}}^{\epsilon}z\|_{L^{q}(0,T;U)}$
$\displaystyle\quad\leq\|B^{*}Q(\cdot)\|_{{\mathcal{L}}(C([0,T];{\mathcal{D}}(A^{\epsilon})),C([0,T];U))}\,\frac{1}{(rq^{\prime})^{1/q^{\prime}}}\,\|B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}\|_{{\mathcal{L}}(Y,L^{q}(0,T;U))}\,\|z\|_{Y}\,\|y\|_{X,r}\,.$

We note that in going from the antepenultimate to the penultimate estimate we
used iiic) of the Assumptions 2.4. Therefore, there exist positive constants
$c,c^{\prime}$ such that

$\big{\|}e^{-rt}\big{[}L\,B^{*}Q(\cdot)y(\cdot)\big{]}(t)\big{\|}_{{\mathcal{D}}(A^{\epsilon})}\leq\frac{c}{(rq^{\prime})^{1/q^{\prime}}}\,\|y\|_{X,r}\leq\frac{c^{\prime}}{r^{1/q^{\prime}}}\,\|y\|_{X,r}\,,$

so that by taking a sufficiently large $r$ we see that $LB^{*}Q(\cdot)$ is a
contraction mapping in $X$. The conclusion of the Lemma follows. ∎

Uniqueness for the DRE is now a consequence of Lemmas 4.1 and 4.2, its proof
following a somewhat familiar path.

Proof of Theorem 2.7. For the optimal pair $(\hat{y},\hat{u})$ corresponding
to the initial state $x\in Y$ it holds

$(P(s)x,x)_{Y}=J(\hat{u})=\int_{s}^{T}\Big{(}\|R\hat{y}(r)\|_{Z}^{2}+\|\hat{u}(r)\|_{U}^{2}\Big{)}\,dr\,,\quad
0\leq s\leq T\,,$

where $P(\cdot)$ is the Riccati operator defined in (2.9), i.e.

$P(t)x=\int_{t}^{T}e^{A^{*}(r-t)}R^{*}R\Phi(r,t)x\,dr\,,\qquad x\in Y\,,$

while $\Phi(r,t)$ denotes the evolution operator

$\Phi(r,t)x=e^{A(r-t)}x+L_{t}\hat{u}(r)\,,\qquad r\in[t,T]\,.$

Let $Q(\cdot)\in{\mathcal{Q}}_{T}$ be another solution to the DRE (2.13): by
Lemma 3.1 $Q(\cdot)$ solves the IRE (3.1) as well; then, with $u\in
L^{2}(s,T;U)$ and $x\in{\mathcal{D}}(A^{\epsilon})$, the identity (4.1) holds
true by Lemma 4.1. With $t=T$, since $Q(T)=0$, from (4.1) we see that

$\begin{split}(Q(s)x,x)_{Y}&=\int_{s}^{T}\left[\|Ry(r)\|_{Z}^{2}+\|u(r)\|_{U}^{2}\right]\,dr-\int_{s}^{T}\|u(r)+B^{*}Q(r)y(r)\|_{U}^{2}\,dr\\\\[2.84526pt]
&\leq\int_{s}^{T}\left[\|Ry(r)\|_{Z}^{2}+\|u(r)\|_{U}^{2}\right]\,dr=J(u)\,.\end{split}$

In particular, when $u=\hat{u}$, we establish

$(Q(s)x,x)_{Y}\leq J(\hat{u})=(P(s)x,x)_{Y}\qquad\forall s\in[0,T]\,,\;\forall
x\in{\mathcal{D}}(A^{\epsilon})\,.$ (4.11)

Conversely, let $y(\cdot)$ be the solution to the closed-loop equation (4.9)
corresponding to $x\in{\mathcal{D}}(A^{\epsilon})$, guaranteed by Lemma 4.2,
and let $u(\cdot)=-B^{*}Q(\cdot)y(\cdot)$. By construction $u\in
L^{2}(s,T;U)$, and the fundamental identity becomes

$(Q(s)x,x)_{Y}=\int_{s}^{t}\Big{(}\|Ry(r)\|_{Z}^{2}+\|u(r)\|_{U}^{2}\Big{)}\,dr+(Q(t)y(t),y(t))_{Y}\,,$

which in turn gives, for $t=T$,

$(Q(s)x,x)_{Y}=J(u)\geq J(\hat{u})=(P(s)x,x)_{Y}\qquad\forall
s\in[0,T]\,,\;\forall x\in{\mathcal{D}}(A^{\epsilon})\,.$ (4.12)

The inequality (4.12), combined with (4.11), establishes – via the usual
polarization (first) and density (next) arguments – $Q(s)\equiv P(s)$ on
$[0,T]$, as desired.

∎

###  4.2. Infinite time interval, algebraic Riccati equations

In this Section we prove our second main result, that is Theorem 2.11, which
pertains to uniqueness for the algebraic Riccati equation (1.1), under the
standing Assumptions 2.9. Instrumental results are the counterparts of Lemmas
4.1 and 4.2, along with the integral form (3.8) of the ARE, already obtained
in Section 3; see Lemma 3.2 therein.

The first Lemma is the infinite time horizon version of the fundamental
identity established in Lemma 4.1.

######  Lemma 4.3 (Fundamental identity ($T=+\infty$)).

Recall the class ${\mathcal{Q}}$ defined in (2.20). Let $Q\in{\mathcal{Q}}$ be
a solution to the integral Riccati equation (3.8). With $u\in
L^{2}_{\textrm{loc}}(0,\infty;U)$ and $x\in{\mathcal{D}}(A^{\epsilon})$, let
$y(\cdot)$ be the semigroup solution to the state equation (2.1) corresponding
to $u(\cdot)$, with initial state $x$, given by (2.2). Then, the following
identity holds true, for any $t\geq 0$:

$\begin{split}&(Qy(t),y(t))_{Y}-(Qx,x)_{Y}\\\\[5.69054pt]
&\qquad=-\int_{0}^{t}\Big{(}\|Ry(s)\|_{Z}^{2}+\|u(s)\|_{U}^{2}\Big{)}\,ds+\int_{0}^{t}\big{\|}u(s)+B^{*}Qy(s)\big{\|}_{U}^{2}\,ds\,.\end{split}$
(4.13)

###### Proof.

It suffices to proceed along the lines of the proof of Lemma 4.1, replacing
the interval $[s,t]$ by $[0,t]$ and assuming initially $u\in
L^{\infty}_{\textrm{loc}}(0,\infty;U)$; the proof is actually slightly
simpler, since here $Q$ is independent of $t$. The details are omitted for the
sake of conciseness. ∎

The next Lemma is the infinite time horizon version of Lemma 4.2, dealing with
an integral equation which – once uniquenss for the ARE is ascertained – will
turn out to be the closed-loop equation.

######  Lemma 4.4.

Let $\epsilon$ be as in the Assumptions 2.9. Recall the class ${\mathcal{Q}}$
defined by (2.20), and let $Q\in{\mathcal{Q}}$. For every
$x\in{\mathcal{D}}(A^{\epsilon})$ and a suitably large $r>0$ there exists a
unique solution $y(\cdot)$ to the closed loop equation

$y(t)=e^{At}x-\int_{0}^{t}e^{A(t-s)}BB^{*}Qy(s)\,ds\,,\qquad t>0\,,$ (4.14)

in the space

$X=\big{\\{}y\in
C([0,\infty);{\mathcal{D}}(A^{\epsilon}))\colon\quad\sup_{t\geq
0}e^{-rt}\|y(t)\|_{{\mathcal{D}}(A^{\epsilon})}<\infty\big{\\}}$ (4.15)

endowed with the norm

$\|y\|_{X,r}=\sup_{t>0}e^{-rt}\|y(t)\|_{D(A^{\epsilon})}\quad\forall y\in
X\,,\quad r>0\,.$

###### Proof.

The argument is pretty much the same employed in the proof of Lemma 4.2. A
technically decisive (distinct) element here comes from the extended (and
enhanced) regularity in time of the operator
$B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}$ over the half line $[0,\infty)$, which
is guaranteed by [3, Proposition 3.2], recalled here as Proposition A.6. The
computation is included for the reader’s convenience.

Let $x\in{\mathcal{D}}(A^{\epsilon})$ be given. By setting $E(t)=e^{At}x$, and
recalling the input-to-state map $L$, the integral equation (4.14) reads as
$\big{(}[I+LB^{*}Q]y(\cdot)\big{)}(t)=E(t)$, in short. For any function
$y(\cdot)\in X$ and any $z\in{\mathcal{D}}({A^{*}}^{\epsilon})$, we have

$\begin{split}&|(e^{-rt}LB^{*}Qy(t),A^{*\epsilon}z)_{Y}=\left|\int_{0}^{t}e^{-r(t-s)}(B^{*}Qy(s)e^{-rs},B^{*}e^{A^{*}(t-s)}{A^{*}}^{\epsilon}z)_{Y}\,ds\right|\\\\[2.84526pt]
&\qquad\leq\int_{0}^{t}e^{-r(t-s)}\|B^{*}Q\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),U)}\|y\|_{X,r}\,e^{-\delta(t-s)}\,\|e^{\delta(t-s)}B^{*}e^{A^{*}(t-s)}{A^{*}}^{\epsilon}z\|_{U}\,ds\\\\[2.84526pt]
&\qquad\leq\|B^{*}Q\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),U)}\|y\|_{X,r}\,\Big{(}\int_{0}^{t}e^{-(r+\delta)(t-s)q^{\prime}}\,ds\Big{)}^{1/q^{\prime}}\,\cdot\\\\[2.84526pt]
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\Big{(}\int_{0}^{t}\|e^{\delta(t-s)}B^{*}e^{A^{*}(t-s)}{A^{*}}^{\epsilon}z\|_{U}^{q}\,ds\Big{)}^{1/q}\\\\[2.84526pt]
&\qquad\leq\frac{1}{[(r+\delta)q^{\prime}]^{1/q^{\prime}}}\,\|B^{*}Q\|_{{\mathcal{L}}({\mathcal{D}}(A^{\epsilon}),U)}\,\|e^{\delta\cdot}B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}\|_{{\mathcal{L}}(Y,L^{q}(0,\infty;U))}\,\|y\|_{X,r}\,\|z\|_{Y}\,,\end{split}$

where $\delta$ belongs to the interval $(0,\omega\wedge\eta)$ ($\omega$ and
$\eta$ being like in the Assumptions 2.9). The above estimate implies readily
that there exists a constant $C>0$ such that

$\|LB^{*}Qy\|_{X,r}\leq\frac{C}{(r+\delta)^{1/q^{\prime}}}\,\|y\|_{X,r}\|e^{\delta\cdot}B^{*}e^{A^{*}\cdot}A^{*\epsilon}\|_{{\mathcal{L}}(Y,L^{q}(0,\infty;U))}$

so that

$\|LB^{*}Qy\|_{X,r}\leq\frac{1}{2}\|y\|_{X,r}\,,$

provided $r$ is sufficiently large. The conclusive argument is standard. ∎

Proof of Theorem 2.11. Let $y_{0}\in Y$, and let $(\hat{y},\hat{u})$ the
optimal pair of the optimal control problem (2.1)-(2.3) (with $T=+\infty$),
corresponding to the initial state $y_{0}$. Recall that

$(Py_{0},y_{0})_{Y}=J(\hat{u})=\int_{0}^{\infty}\|R\hat{y}(s)\|^{2}_{Z}\,ds+\int_{0}^{\infty}\|\hat{u}(s)\|_{U}^{2}\,ds\,,$

where the (optimal cost) operator $P$ is defined in terms of the optimal state
semigroup $\Phi(t)$ via (2.18). In addition, $P$ belongs to the class
${\mathcal{Q}}$ and solves the ARE (1.1); consequently, by Lemma 3.2 $P$
solves the integral form (3.8) of the ARE.

Let now $Q\in{\mathcal{Q}}$ be another solution to the ARE. By Lemma 4.3, we
know that for any given $y_{0}\in{\mathcal{D}}(A^{\epsilon})$, and any
admissible control $u(\cdot)$, the identity (4.13) holds true (with $x$
replaced by $y_{0}$), where $y(\cdot)$ is the solution to the state equation
corresponding to the control $u$ and the initial state $y_{0}$. Consequently,

$(Qy_{0},y_{0})_{Y}\leq(Qy(t),y(t))_{Y}+J(u)\qquad\forall u\in
L^{2}_{\textrm{loc}}(0,\infty;U)\,,\;\forall t>0\,;$

by choosing in particular the admissible pair $(y_{T},u_{T})$ defined as
follows,

$u_{T}=\hat{u}\cdot\chi_{[0,T]}\,,\qquad
y_{T}(t)=\begin{cases}\hat{y}(t)&\textrm{if $t\leq T$}\\\\[5.69054pt]
e^{At}y_{0}+e^{A(t-T)}L\hat{u}(T)&\textrm{if $t>T$,}\end{cases}$

we find $(Qy_{0},y_{0})_{Y}\leq(Qy_{T}(t),y_{T}(t))_{Y}+J(u_{T})$, valid for
arbitrary $t\geq T>0$. By letting $t\to+\infty$ in the previous inequality,
one obtains readily

$(Qy_{0},y_{0})_{Y}\leq J(u_{T})\qquad\forall
y_{0}\in{\mathcal{D}}(A^{\epsilon})\,\quad\forall T>0\,,$ (4.16)

in view of the fact that the semigroup $e^{At}$ decays exponentially; so
$\|y_{T}(t)\|_{Y}\longrightarrow 0$, as $t\to+\infty$.

Observe now that

$\begin{split}J(u_{T})&=\int_{0}^{\infty}\|Ry_{T}(s)\|^{2}_{Z}\,ds+\int_{0}^{\infty}\|u_{T}(s)\|_{U}^{2}\,ds\\\\[5.69054pt]
&=\int_{0}^{T}\|R\hat{y}(s)\|^{2}_{Z}\,ds\,+\int_{T}^{\infty}\big{\|}R\big{(}e^{sA}y_{0}+e^{(s-T)A}L\hat{u}(T)\big{)}\big{\|}^{2}\,ds+\int_{0}^{T}\|\hat{u}(s)\|_{U}^{2}\,ds\,,\end{split}$

so that $J(u_{T})\longrightarrow J(\hat{u})$, as $T\rightarrow+\infty$.
Keeping this in mind, return to (4.16) and let $T\to+\infty$ to find

$(Qy_{0},y_{0})_{Y}\leq J(\hat{u})=(Py_{0},y_{0})_{Y}\qquad\forall
y_{0}\in{\mathcal{D}}(A^{\epsilon})\,.$ (4.17)

On the other hand, given $y_{0}\in{\mathcal{D}}(A^{\epsilon})$ (and still with
$Q\in{\mathcal{Q}}$ another solution to the ARE), let $y(\cdot)$ be the
solution to the closed loop equation guaranteed by Lemma 4.4; by construction
$y\in L^{2}_{\textrm{loc}}(0,\infty;Y)$. Take now the control
$u(\cdot)=-B^{*}Qy(\cdot)$, which belongs to
$L^{2}_{\textrm{loc}}(0,\infty;U)$. Then, the identity (4.13) holds true for
any positive $t$, that is

$\begin{split}&(Qy_{0},y_{0})_{Y}=(Qy(t),y(t))_{Y}+\int_{0}^{t}\Big{(}\|Ry(s)\|_{Z}^{2}+\|u(s)\|_{U}^{2}\Big{)}\,ds\\\\[5.69054pt]
&\qquad-\cancel{\int_{0}^{t}\big{\|}u(s)+B^{*}Qy(s)\big{\|}_{U}^{2}\,ds}\geq\int_{0}^{t}\|Ry(s)\|_{Z}^{2}\,ds+\int_{0}^{t}\|u(s)\|_{U}^{2}\,ds\,.\end{split}$
(4.18)

As $t\to+\infty$, this shows that $Ry\in L^{2}(0,\infty;Z)$, $u\in
L^{2}(0,\infty;U)$, as well as $(Qy_{0},y_{0})_{Y}\geq J(u)$. By minimality we
find

$(Qy_{0},y_{0})_{Y}\geq J(u)\geq J(\hat{u})=(Py_{0},y_{0})_{Y}\qquad\forall
y_{0}\in{\mathcal{D}}(A^{\epsilon})\,.$ (4.19)

On the basis of (4.17) and (4.19), a standard polarization (first) and density
(next) argument confirms that $Q=P$, thereby concluding the proof of Theorem
2.11. ∎

##  Appendix A Instrumental results

In this appendix we gather several results (some old, some new) which single
out certain regularity properties – in time and space – of

  * •

the input-to-state map $L$,

  * •

the operator $B^{*}Q(\cdot)$, when $Q(t)\in{\mathcal{Q}}_{T}$,

  * •

the operator $B^{*}e^{A^{*}t}{A^{*}}^{\epsilon}$ and its adjoint.

All of them stem from the Assumptions 2.4 or 2.9 on the (dynamics and control)
operators $A$ and $B$. The role played by the assertions of the novel Lemma
A.2 and Lemma A.5 in the proofs of our uniqueness results is absolutely
critical.

Initially, it is useful to recall from [1] and [3] the basic regularity
properties of the input-to-state map $L$. The first result pertains to the
finite time horizon problem. The reader is referred to [1, Appendix B] for the
details of the computations leading to the various statements in the following
Proposition.

######  Proposition A.1 ([1], Proposition B.3).

Let $L_{s}$ be the operator defined by

$L_{s}\colon
u(\cdot)\longrightarrow(L_{s}u)(t):=\int_{s}^{t}e^{A(t-r}Bu(r)\,dr\,,\qquad
0\leq s\leq t\leq T\,.$ (A.1)

Under the Assumptions 2.4, the following regularity results hold true.

  1. (1)

If $p=1$, then
$L_{s}\in{\mathcal{L}}(L^{1}(s,T;U),L^{1/\gamma}(s,T;[{\mathcal{D}}({A^{*}}^{\epsilon})]^{\prime})$;

  2. (2)

if $1<p<\frac{1}{1-\gamma}$, then
$L_{s}\in{\mathcal{L}}(L^{p}(s,T;U),L^{r}(s,T;Y))$, with
$r=\frac{p}{1-(1-\gamma)p}$;

  3. (3)

if $p=\frac{1}{1-\gamma}$, then
$L_{s}\in{\mathcal{L}}(L^{p}(s,T;U),L^{r}(s,T;Y))$ for all $r\in[1,\infty)$;

  4. (4)

if $p>\frac{1}{1-\gamma}$, then
$L_{s}\in{\mathcal{L}}(L^{p}(s,T;U),C([s,T];Y))$.

Moreover, in all cases the norm of $L_{s}$ does not depend on $s$.

The space regularity in the last assertion can be actually enhanced. To be
more precise, $L_{s}$ maps control functions $u(\cdot)$ which belong to
$L^{q^{\prime}}(s,T;U)$ into functions which take values in
${\mathcal{D}}(A^{\epsilon})$ ($q^{\prime}$ being the conjugate exponent of
$q$ in the Assumptions 2.4). We highlight this property – appparently left out
of the work [1] – as a separate result, since it will be used throughout in
the paper. The proof is omitted, as it is akin to (and somewhat simpler than)
the one carried out to establish assertion (v) of the subsequent Proposition
A.3.

######  Lemma A.2.

Let $\epsilon$ and $q$ be as in (iii) of the Assumptions 2.4. Then, for the
operator $L_{s}$ defined in (A.1) we have

$L_{s}\in{\mathcal{L}}(L^{q^{\prime}}(s,T;U),C([s,T];{\mathcal{D}}(A^{\epsilon})))\,.$

A counterpart of Proposition A.1 specific for the infinite time horizon
problem was proved in [3, Proposition 3.6]. The collection of findings on the
regularity of the input-to-state map $L$ is recorded here for the reader’s
convenience.

######  Proposition A.3 ([3], Proposition 3.6).

Let $L$ be the operator defined by

$L\colon
u(\cdot)\longrightarrow(Lu)(t):=\int_{0}^{t}e^{A(t-r}Bu(r)\,dr\,,\qquad t\geq
0\,.$

Under the Assumptions 2.9, the following regularity results hold true.

  1. (i)

$L\in{\mathcal{L}}(L^{1}(0,\infty;U),L^{r}(0,\infty;[{\mathcal{D}}({A^{*}}^{\epsilon})]^{\prime})$,
for any $r\in[1,1/\gamma)$;

  2. (ii)

$L\in{\mathcal{L}}(L^{p}(0,\infty;U),L^{r}(0,\infty;Y))$, for any
$p\in(1,1/(1-\gamma))$ and any $r\in[p,p/(1-(1-\gamma)p)]$;

  3. (iii)

$L\in{\mathcal{L}}(L^{\frac{1}{1-\gamma}}(0,\infty;U),L^{r}(0,\infty;Y))$, for
any $r\in[1/(1-\gamma),\infty)$;

  4. (iv)

$L\in{\mathcal{L}}(L^{p}(0,\infty;U),L^{r}(0,\infty;Y)\cap
C_{b}([0,\infty);Y))$, for any $p\in(1/(1-\gamma),\infty)$ and any
$r\in[p,\infty)$;

  5. (v)

$L\in{\mathcal{L}}(L^{r}(0,\infty;U),C_{b}([0,\infty);{\mathcal{D}}(A^{\epsilon}))$,
for any $r\in[q^{\prime},\infty]$.

Because they occur in the present work, besides being central to the analysis
of [3], we need to recall the $L^{p}$-spaces with weights. Set

$L^{p}_{g}(0,\infty;X):=\big{\\{}f\colon(0,\infty)\longrightarrow
X\,,\;g(\cdot)f(\cdot)\in L^{p}(0,\infty;X)\big{\\}}\,,$

where $g\colon(0,\infty)\longrightarrow\mathbb{R}$ is a given (weight)
function. We will use more specifically the exponential weights
$g(t)=e^{\delta t}$, along with the following (simplified) notation:

$L^{p}_{\delta}(0,\infty;X):=\big{\\{}f\colon(0,\infty)\longrightarrow
X\,,\;e^{\delta\cdot}f(\cdot)\in L^{p}(0,\infty;X)\big{\\}}\,.$

######  Remark A.4.

As pointed out in [3, Remark 3.8], all the regularity results provided by the
statements contained in the Propositions A.1 and A.3 extend readily to natural
analogues involving $L^{p}_{\delta}$ spaces (rather than $L^{p}$ ones),
maintaining the respective summability exponents $p$.

We now move on to a result which clarifies the regularity of the operator
$B^{*}Q(\cdot)$, $Q(t)\in{\mathcal{Q}}_{T}$, when acting upon functions (with
values in ${\mathcal{D}}(A^{\epsilon})$) rather than on vectors – namely, on
elements of the space ${\mathcal{D}}(A^{\epsilon})$.

######  Lemma A.5.

Let $\epsilon$ be as in (iii) of the Assumptions 2.4. If
$Q(\cdot)\in{\mathcal{Q}}_{T}$ and $f\in
C([0,T];{\mathcal{D}}(A^{\epsilon}))$, then

$B^{*}Q(\cdot)f(\cdot)\in C([0,T];U)\,.$

###### Proof.

We proceed along the lines of the proof of [1, Lemma A.3]. Let
$Q(\cdot)\in{\mathcal{Q}}_{T}$ and let $t_{0}\in[0,T]$. By the definition of
${\mathcal{Q}}_{T}$, there exists $c_{1}>0$ such that

$\|B^{*}Q(t)z\|_{U}\leq c_{1}\|z\|_{{\mathcal{D}}(A^{\epsilon})}\qquad\forall
t\in[0,T]\,,\;\forall z\in{\mathcal{D}}(A^{\epsilon})\,.$ (A.2)

Since $f(t_{0})\in{\mathcal{D}}(A^{\epsilon})$, then $B^{*}Q(\cdot)f(t_{0})\in
C([0,T];U)$. Then

$\begin{split}&\|B^{*}Q(t)f(t)-B^{*}Q(t_{0})f(t_{0})\|_{U}\\\\[2.84526pt]
&\qquad\leq\|B^{*}Q(t)[f(t)-f(t_{0})]\|_{U}+\|B^{*}Q(t)f(t_{0})-B^{*}Q(t_{0})f(t_{0})\|_{U}\\\\[2.84526pt]
&\qquad\leq
c_{1}\|f(t)-f(t_{0})\|_{{\mathcal{D}}(A^{\epsilon})}+\|[B^{*}Q(t)-B^{*}Q(t_{0})]f(t_{0})\|_{U}=o(1)\,,\quad
t\longrightarrow t_{0}\,.\end{split}$

∎

Essential as well in this work, and more specifically in the proof of Theorem
2.11, is a stronger property of the operator
$B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}$, namely, (A.3) below, which holds true
for appropriate $\delta$, under the Assumptions 2.9. Originally devised in
[3], this result reveals that once the validity of iiic) of Assumptions 2.4 is
ascertained on some bounded interval $[0,T]$, then the very same regularity
estimate extends to the half line, along with an enhanced summability of the
function $B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}x$, $x\in Y$. The key to this
is the exponential stability of the semigroup, i.e. (2.15); see [3,
Proposition 3.2].

######  Proposition A.6 ([3], Proposition 3.2).

Let $\omega$, $\eta$ and $\epsilon$ like in the Assumptions 2.9. For each
$\delta\in(0,\omega\wedge\eta)$ the map

$t\longrightarrow e^{\delta t}B^{*}e^{A^{*}t}{A^{*}}^{\epsilon}$

has an extension which belongs to ${\mathcal{L}}(Y,L^{q}(0,\infty;U))$. In
short,

$B^{*}e^{A^{*}\cdot}{A^{*}}^{\epsilon}\in{\mathcal{L}}(Y,L^{q}_{\delta}(0,\infty;U))\,.$
(A.3)

We conclude providing a result that takes a more in-depth glance at the
regularity of the operator $B^{*}e^{A^{*}t}{A^{*}}^{\epsilon}$ and its
adjoint.

######  Lemma A.7.

Under the Assumptions 2.9, the following regularity results are valid, for any
$\delta\in(0,\omega\wedge\eta)$:

$\begin{split}&a)\quad
e^{\delta\cdot}A^{\epsilon}e^{A\cdot}B\in{\mathcal{L}}(L^{q^{\prime}}(0,\infty;U),Y)\,,\\\\[2.84526pt]
&b)\quad
e^{\delta\cdot}B^{*}e^{A^{*}\cdot}{A^{*}}^{-\epsilon}\in{\mathcal{L}}(L^{r}(0,\infty;Y),U)\qquad\forall
r>\frac{1}{1-\gamma}\,.\end{split}$ (A.4)

The respective actions of the operators in (A.4) are made explicit by (A.5)
and (A.6).

###### Proof.

The regularity results in (A.4) are, in essence, dual properties of the
regularity result in Proposition A.6 and of assertion A6. in Theorem 2.10,
respectively. To infer (a), we introduce the notation $S$ for the mapping from
$Y$ into $L^{q}(0,\infty;U)$ defined by

$Y\ni z\longrightarrow[Sz](t):=e^{\delta
t}B^{*}e^{A^{*}t}{A^{*}}^{\epsilon}z\,,\qquad\;t>0\,.$

For any $z\in Y$ and any $h\in L^{q^{\prime}}(0,\infty;U)$, it must be
$S^{*}\in{\mathcal{L}}(L^{q^{\prime}}(0,\infty;U),Y)$ and more precisely,

$\begin{split}\langle S^{*}h,z\rangle_{Y}&=\langle
h,Sz\rangle_{L^{q^{\prime}}(0,\infty;U),L^{q}(0,\infty;U)}=\int_{0}^{\infty}\big{\langle}h(t),e^{\delta
t}B^{*}e^{A^{*}t}{A^{*}}^{\epsilon}z\big{\rangle}_{U}\,dt\\\\[2.84526pt]
&=\Big{\langle}\int_{0}^{\infty}e^{\delta
t}A^{\epsilon}e^{At}Bh(t)\,dt,z\Big{\rangle}_{Y}\,.\end{split}$

Therefore,

$S^{*}h=\int_{0}^{\infty}e^{\delta t}A^{\epsilon}e^{At}Bh(t)\,dt\,,\qquad h\in
L^{q^{\prime}}(0,\infty;Y)\,.$ (A.5)

To achieve (b) of (A.4), we recall instead the assertion A6. in Theorem 2.10,
which further tells us that

$e^{\delta\cdot}A^{-\epsilon}e^{A\cdot}B\in{\mathcal{L}}(U,L^{p}(0,\infty;Y))\,,\quad\text{for
any $p$ such that $1\leq p<\frac{1}{\gamma}$.}$

Similarly as above, we introduce the notation $T$ for the mapping from $U$
into $L^{p}(0,\infty;Y)$ defined by

$U\ni w\longrightarrow[Tw](t):=e^{\delta t}A^{-\epsilon}e^{At}Bw\,,\qquad
t>0\,;$

by construction, $T^{*}\in{\mathcal{L}}(L^{p^{\prime}}(0,\infty;Y),U)$ for all
$p^{\prime}>1/(1-\gamma)$. More precisely, for any $w\in U$ and any $g\in
L^{p^{\prime}}(0,\infty;Y)$ we have

$\begin{split}\langle T^{*}g,w\rangle_{U}&=\langle
g,Tw\rangle_{L^{p^{\prime}}(0,\infty;Y),L^{p}(0,\infty;Y)}=\int_{0}^{\infty}\big{\langle}g(t),e^{\delta
t}A^{-\epsilon}e^{At}Bw\big{\rangle}_{Y}\,dt\\\\[2.84526pt]
&=\Big{\langle}\int_{0}^{\infty}e^{\delta
t}B^{*}e^{A^{*}t}{A^{*}}^{-\epsilon}g(t)\,dt,w\Big{\rangle}_{U}\,,\end{split}$

which establishes

$T^{*}g=\int_{0}^{\infty}e^{\delta
t}B^{*}e^{A^{*}t}{A^{*}}^{-\epsilon}g(t)\,dt\,.\qquad\forall g\in
L^{p^{\prime}}(0,\infty;Y)\,.$ (A.6)

The integrals in (A.5) and (A.6) are the sought respective representations of
the adjoint operators in (A.4).

∎

## Acknowledgements

The research of F. Bucci was partially supported by the Università degli Studi
di Firenze under the 2019 Project Metodi ed Applicazioni per Equazioni
Differenziali Ordinarie e a Derivate Parziali. Bucci is a member of the Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), and participant
to the GNAMPA Projects Controllabilità di PDE in modelli fisici e in scienze
della vita (2019) and Problemi inversi e di controllo per equazioni di
evoluzione e loro applicazioni (2020). She is also a member of the French-
German-Italian Laboratoire International Associé (LIA) COPDESC in Applied
Analysis.
The research of P. Acquistapace was partially supported by the PRIN-MIUR
Project 2017FKHBA8 of the Italian Education, University and Research Ministry.

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