# Equivalent Conditions on Periodic Feedback Stabilization for Linear Periodic
Evolution Equations

Gengsheng Wang Yashan Xu  School of Mathematics and Statistics, Wuhan
University, Wuhan, 430072, China<EMAIL_ADDRESS>The author was partially
supported by the National Natural Science Foundation of China under grant
11161130003.School of Mathematical Sciences, Fudan University, KLMNS, Shanghai
200433, China<EMAIL_ADDRESS>
###### Abstract

This paper studies the periodic feedback stabilization for a class of linear
$T$-periodic evolution equations. Several equivalent conditions on the linear
periodic feedback stabilization are obtained. These conditions are related
with the following subjects: the attainable subspace of the controlled
evolution equation under consideration; the unstable subspace (of the
evolution equation with the null control) provided by the Kato projection; the
Poincar$\acute{e}$ map associated with the evolution equation with the null
control; and two unique continuation properties for the dual equations on
different time horizons $[0,T]$ and $[0,n_{0}T]$ (where $n_{0}$ is the sum of
algebraic multiplicities of distinct unstable eigenvalues of the
Poincar$\acute{e}$ map). It is also proved that a $T$-periodic controlled
evolution equation is linear $T$-periodic feedback sabilizable if and only if
it is linear $T$-periodic feedback sabilizable with respect to a finite
dimensional subspace. Some applications to heat equations with time-periodic
potentials are presented.

Keywords. periodic evolution equations, periodic feedback stabilization,
equivalent conditions, attainable subspaces, unique continuation properties,
the Poincar$\acute{e}$ map, the Kato projection

2010 MSC. 34H15 49N20

##  1 Introduction

###  1.1 The problem and the motivation

Consider the following controlled evolution equation:

$y^{\prime}(t)+Ay(t)+B(t)y(t)=D(t)u(t)\;\;\mbox{in}\;\;\mathbb{R}^{+}\triangleq[0,\infty).$
(1.1)

Here and throughout this paper, we make the following assumptions.

$({\cal H}_{1})$ The operator $(-A)$, with its domain ${\cal D}(-A)$,
generates a $C_{0}$ compact semigroup $\\{S(t)\\}_{t\geq 0}$ in a real Hilbert
space $H$ (identified with its dual) with its norm and inner product denoted
by $\|\cdot\|$ and $\mathop{\langle}\cdot,\cdot\mathop{\rangle}$,
respectively.

$({\cal H}_{2})$ The operator-valued function $B(\cdot)\in
L^{1}_{loc}(\mathbb{R}^{+};{\cal L}(H))$ is $T$-periodic, i.e., $B(t+T)=B(t)$
for a.e. $t\in\mathbb{R}^{+}$, where $T>0$ and ${\cal L}(H)$ denotes the space
of all linear bounded operators on $H$.

$({\cal H}_{3})$ The operator-valued function $D(\cdot)\in
L^{\infty}(\mathbb{R}^{+};{\cal L}(U,H))$ is $T$-periodic. Here $U$ is also a
real Hilbert space (identified with its dual) with its norm and inner product
denoted by $\|\cdot\|_{U}$ and
$\mathop{\langle}\cdot,\cdot\mathop{\rangle}_{U}$, respectively; and ${\cal
L}(U,H)$ stands for the space of all linear bounded operators from $U$ to $H$.
Controls $u(\cdot)$ are taken from the space $L^{2}(\mathbb{R}^{+};U)$.

For each $h\in H$, $s\geq 0$ and $u(\cdot)\in L^{2}(\mathbb{R}^{+};U)$,
Equation (1.1) (over $[s,\infty)$) with the initial condition that $y(s)=h$
has a unique mild solution $y(\cdot;s,h,u)\in C([s,\infty);H)$. (See, for
instance, Proposition 5.3 on Page 66 in [14].) The following definitions about
the periodic feedback stabilization will be used throughout this paper:

  * •

Equation (1.1) is said to be linear periodic feedback stablizable (LPFS, for
short) if there is a $T$-periodic $K(\cdot)\in
L^{\infty}\left(\mathbb{R}^{+};{\cal L}(H,U)\right)$ such that the feedback
equation

$y^{\prime}(t)+Ay(t)+B(t)y(t)=D(t)K(t)y(t)\quad\;\;\mbox{in}\;\;\mathbb{R}^{+}$
(1.2)

is exponentially stable, i.e., there are two positive constants $M$ and
$\delta$ such that for each $h\in H$, the solution $y_{K}(\cdot;0,h)$ to the
equation (1.2) with the initial condition that $y(0)=h$ satisfies that
$\|y_{K}(t;0,h)\|\leq Me^{-\delta t}\|h\|$ for all $t\geq 0$. Any such a
$K(\cdot)$ is called an LPFS law for Equation (1.1).

  * •

Equation (1.1) is said to be LPFS with respect to a subspace $Z$ of $U$ if
there is a $T$-periodic $K(\cdot)\in L^{\infty}\left(\mathbb{R}^{+};{\cal
L}(H,Z)\right)$ such that the equation (1.2) is exponentially stable. Any such
a $K(\cdot)$ is called an LPFS law for Equation (1.1) with respect to $Z$.

Let

${\cal U}^{FS}\triangleq\big{\\{}Z\;\big{|}\;Z\;\;\mbox{is a subspace
of}\;\;U\;\;\mbox{s.t. Equation (\ref{state}) is LPFS w.r.t.}\;Z\big{\\}}.$
(1.3)

In this paper, we provide three criteria for judging whether a subspace $Z$
belongs to ${\cal U}^{FS}$. We also show that if $U\in{\cal U}^{FS}$, then
there is a finite dimensional subspace $Z$ in ${\cal U}^{FS}$. The
aforementioned three criteria are related with the following subjects: the
attainable subspace of (1.1); the unstable subspace (of (1.1) with the null
control) provided by the Kato projection; the Poincar$\acute{e}$ map
associated to (1.1) with the null control; and two unique continuation
properties for the dual equations of (1.1) (with the null control) on
different time horizons $[0,T]$ and $[0,n_{0}T]$ (where $n_{0}$ is the sum of
algebraic multiplicities of distinct unstable eigenvalues of the
Poincar$\acute{e}$ map). Among three criteria, the most important one is a
geometric condition connecting the attainable set with the unstable subspace
of the system (1.1); while the other two are analytic conditions related with
the unique continuation of the dual equations of (1.1) over different time
horizons and with initial data in different finite dimensional subspaces of
$H$.

The motivation for this work is as follows. First, the equation (1.1) with the
null control is exponentially stable if and only if the spectrum of the
Poincar$\acute{e}$ map associated with the system is contained in the unit
open ball of the complex plane. (This can be proved by the exactly same way to
show Corollary 7.2.4 on page 200, [11].) Thus, it is a natural problem to
explore equivalent conditions on the periodic stabilization for a linear
periodic controlled evolution system. Second, there are two important kinds of
solutions for evolution equations: equilibrium and periodic solutions. The
stabilization for equilibrium solutions of time-invariant systems has been
extensively studied (see for instance [4], [8], [22], [23] and the references
therein). However, the understanding on the periodic stabilization of periodic
solutions for time-varying evolution systems is quite limited. (See [5], [16],
[17] and [21]. Here, we would like to mention [6] which establishes a feedback
law stabilizing a smooth non-stationary solutions, for instance, around a
periodic trajectory, for Navier-Stokes equations.) Finally, when the system
(1.1) is LPFS, it should be important and interesting to answer if there is a
finite dimensional subspace $Z$ of $U$ such that (1.1) is LPFS w.r.t. $Z$,
from perspectives of both mathematics and applied sciences.

###  1.2 Main results

Before stating our main results, we give some preliminaries in order:

$(I)$ Notations We will use $\|\cdot\|$ to denote the usual norm of ${\cal
L}(H)$ when there is no risk of causing any confusion. Given $L\in{\cal
L}(X,Y)$ (where $X$ and $Y$ are two Hilbert spaces), we write $L^{*}$ for its
adjoint operator. For $L\in{\cal L}(X)\triangleq{\cal L}(X,X)$, we denote by
$\sigma(L)$ the spectrum of $L$. When $X$ is a real Hilbert space and
$L\in{\cal L}(H)$, we denote by $X^{C}$ and $L^{C}$ their complexification,
respectively, i.e., $X^{C}=X+iX$ and $L^{C}(\alpha+i\beta)=L\alpha+iL\beta$
for any $\alpha,\beta\in X$, where $i$ is the imaginary unit. We write
$\mathbb{B}$ for the open unit ball in $\mathbb{C}^{1}$ and
$\mathbb{B}(0,\delta)$ for the open ball in $\mathbb{C}^{1}$, centered at the
origin and of radius $\delta>0$. Denote by $\partial\mathbb{B}(0,\delta)$ the
boundary of $\mathbb{B}(0,\delta)$.

(II) The Poincar$\acute{e}$ map Let $\bigr{\\{}\Phi(t,s)\bigl{\\}}_{0\leq
s\leq t<+\infty}$ be the evolution system generated by $(-A-B(\cdot))$. It
follows from Lemma 5.6 in [14] (see Page 68, [14]) that $\Phi(t,s)$ is
strongly continuous over
$\\{(t,s)\in\mathbb{R}^{+}\times\mathbb{R}^{+}\big{|}0\leq s\leq t<\infty\\}$,
and that

$\Phi(t,s)h=S(t-s)h+\displaystyle\int_{s}^{t}S(t-r)B(r)\Phi(r,s)hdr,\;\;\mbox{when}\;\;0\leq
s\leq t<\infty\;\;\mbox{and}\;\;h\in H.$ (1.4)

By Proposition 5.7 on Page 69 in [14], for each $s\geq 0$, $h\in H$ and
$u(\cdot)\in L^{2}(\mathbb{R}^{+};U)$, it holds that

$y(t;s,h,u)=\Phi(t,s)h+\int^{t}_{s}\Phi(t,r)D(r)u(r)dr,\;\;s\in[t,\infty).$
(1.5)

By the $T$-periodicity of $B(\cdot)$ and (1.4), one can easily check that
$\Phi(\cdot,\cdot)$ is $T$-periodic, i.e.,

$\Phi(t+T,s+T)=\Phi(t,s)\;\;\mbox{ for all }\;0\leq s\leq t<\infty.$ (1.6)

Now, we introduce the following Poincar$\acute{e}$ map (see Page 197, [11]):

${\cal P}(t)\triangleq\Phi(t+T,t),\;\;t\in\mathbb{R}^{+}.$ (1.7)

It is proved that (see Lemma 2.1)

$\sigma({\cal
P}(t)^{C})\setminus\\{0\\}=\\{\lambda_{j}\\}^{\infty}_{j=1}\;\;\mbox{for
each}\;\;t\geq 0,$ (1.8)

where $\lambda_{j}$, $j=1,2,\dots$, are all distinct non-zero eigenvalues of
the compact operator ${\cal P}(0)^{C}$ such that
$\lim_{j\rightarrow\infty}|\lambda_{j}|=0$. Thus, there is a unique
$n\in\mathbb{N}$ such that

$|\lambda_{j}|\geq 1,\quad
j\in\\{1,2,\cdots,n\\}\quad~{}{\mbox{and}}~{}\quad|\lambda_{j}|<1,\quad
j\in\\{n+1,n+2,\cdots\\}.$ (1.9)

Set

$\bar{\delta}\triangleq\max\\{|\lambda_{j}|,\,j>n\\}<1.$ (1.10)

Let $l_{j}$ be the algebraic multiplicity of $\lambda_{j}$ for each
$j\in\mathbb{N}$, and write

$n_{0}\triangleq l_{1}+\cdots+l_{n}.$ (1.11)

$(III)$ The Kato projection Arbitrarily fix a $\delta\in(\bar{\delta},1)$,
where $\bar{\delta}$ is given by (1.10). Let $\Gamma$ be the circle
$\partial\mathbb{B}\left(0,\delta\right)$ with the clockwise direction in
$\mathbb{C}^{1}$. We introduce the Kato projections (see [12]):

$\hat{P}(t)=\displaystyle\frac{1}{2\pi i}\int_{\Gamma}(\lambda I-{\cal
P}(t)^{C})^{-1}d\lambda,\;\;t\geq 0.$ (1.12)

It is proved that (see Lemma 2.2) for each $t\geq 0$, the operator $P(t)$,
defined by

$P(t)\triangleq\hat{P}(t)\bigm{|}_{H}\;\;(\mbox{the restriction
of}\;\hat{P}(t)\;\mbox{on}\;H),$ (1.13)

is a projection on $H$; $H=H_{1}(t)\bigoplus H_{2}(t)\triangleq
P(t)H\bigoplus(I-P(t))H$ for each $t\geq 0$; both $H_{1}(t)$ and $H_{2}(t)$
are invariant subspaces of ${\cal P}(t)$; $\sigma({\cal
P}(t)^{C}|_{H_{1}(t)^{C}})=\\{\lambda_{j}\\}^{n}_{j=1}$, $\sigma({\cal
P}(t)^{C}|_{H_{2}(t)^{C}})\setminus\\{0\\}=\\{\lambda_{j}\\}^{\infty}_{j=n+1}$;
and $\mbox{dim}H_{1}(t)=n_{0}$. It is also shown that (see Lemma 2.2)
$P(\cdot)$ is $T$-periodic. We simply write

$H_{1}\triangleq H_{1}(0),\;H_{2}\triangleq H_{2}(0),\;P\triangleq
P(0)\;\;\mbox{and}\;\;{\cal P}\triangleq{\cal P}(0).$ (1.14)

The subspaces $H_{1}$ and $H_{2}$ are respectively called the unstable
subspace and the stable subspace of Equation (1.1) with the null control. Each
eigenvalue in $\\{\lambda_{j}\\}_{j=1}^{n}$ (or in
$\\{\lambda_{j}\\}_{j=n+1}^{\infty}$) is called an unstable (or stable)
eigenvalue of ${\cal P}^{C}$. Each eigenfunction of ${\cal P}^{C}$
corresponding to an unstable (or stable) eigenvalue is called an unstable (or
stable) eigenfunction of ${\cal P}^{C}$.

$(IV)$ Attainable subspaces For each subspace $Z\subset U$, we let

$V^{Z}_{k}\mathop{\buildrel\Delta\over{=}}\Big{\\{}\displaystyle\int^{kT}_{0}\Phi(kT,s)D(s)u(s)ds~{}\Bigm{|}u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\Big{\\}}\;\;{\hbox{ for all }}\;\;k\in\mathbb{N}.$
(1.15)

The space $V^{Z}_{k}$ is called the attainable subspace of Equation (1.1)
(over $(0,kT)$) w.r.t. $Z$. Let

$\hat{V}^{Z}_{k}=PV^{Z}_{k},\qquad k\in\mathbb{N},$ (1.16)

where $P$ is given by (1.14).

Now the main results of this paper are presented by the following two
theorems:

###### Theorem 1.1

Let $P$, ${\cal P}$ and $H_{j}$ with $j=1,2$ be given by (1.14). Let $n_{0}$
be given by (1.11). Then, for each subspace $Z\subseteq U$, the following
statements are equivalent:

$(a)$ Equation (1.1) is LPFS with respect to $Z$, i.e., $Z\in{\cal U}^{FS}$.

$(b)$ The subspace $Z$ satisfies

$\hat{V}^{Z}_{n_{0}}=H_{1},\;\;\mbox{where}\;\;\hat{V}^{Z}_{n_{0}}\;\;\mbox{is
given by}\;\;(\ref{2-14}).$ (1.17)

$(c)$ The subspace $Z$ satisfies

$\xi\in
P^{*}H_{1}\;\;\mbox{and}\;\;\left(D(\cdot)\bigl{|}_{Z}\right)^{*}\Phi(n_{0}T,\cdot)^{*}\xi=0\;\mbox{over}\;(0,n_{0}T)\Rightarrow\xi=0.$
(1.18)

$(d)$ The subspace $Z$ satisfies

$\mu\notin\mathbb{B},\;\xi\in H^{C},\;\big{(}\mu I-{\cal
P}^{*C}\big{)}\xi=0,\;\mbox{ and
}\;\big{(}D(\cdot)\bigl{|}_{Z}\big{)}^{*C}\Phi(T,\cdot)^{*C}\xi=0\;\mbox{over}\;(0,T)\;\Rightarrow\xi=0.$
(1.19)

###### Theorem 1.2

Equation (1.1) is LPFS if and only if it is LPFS with respect to a finite
dimensional subspace $Z$ of $U$.

It is worthwhile to make the following remarks:

  * •

The key to show the above two theorems is to build up the equivalence
$(a)\Leftrightarrow(b)$ in Theorem 1.1.

  * •

The functions $\Phi(n_{0}T,\cdot)^{*}\xi$ with $\xi\in H$ and
$\Phi(T,\cdot)^{*C}\xi$ with $\xi\in H^{C}$ are respectively the solutions to
the following dual equations:

$\psi_{t}(t)-A^{*}\psi(t)-B(t)^{*}\psi(t)=0\;\;\mbox{for
a.e.}\;t\in(0,n_{0}T),\;\;\psi(n_{0}T)=\xi$

and

$\psi_{t}(t)-A^{*C}\psi(t)-B(t)^{*C}\psi(t)=0\;\;\mbox{for
a.e.}\;t\in(0,T),\;\;\psi(T)=\xi.$

Thus, the condition $(c)$ in Theorem 1.1 presents a unique continuation
property for solutions of the first dual equation with initial data in
$P^{*}H_{1}$; while the condition $(d)$ in Theorem 1.1 presents a unique
continuation property for solutions of the second equation where initial data
are unstable eigenfunctions of ${\cal P}^{*C}$.

  * •

There have been studies, in the past, on equivalence conditions of periodic
feedback stabilization for linear periodic evolution systems. In [16] and
[21], the authors established an equivalent condition on stabilizability for
linear time-periodic parabolic equations with open-loop controls. Their
equivalence (see Theorem 3.1 in [16] and Proposition 3.1 in [21]) can be
stated, under our framework, as follows: the condition $(d)$ (in our Theorem
1.1 where $Z=U$) is equivalent to the statement that for any $h\in H$, there
is a control $u^{h}(\cdot)\in C(\mathbb{R}^{+};U)$, with
$\sup\limits_{t\in\mathbb{R}^{+}}\|e^{\bar{\delta}t}u^{h}(t)\|_{U}$ bounded
(where $\bar{\delta}$ is given by (1.10)), such that the solution
$y(\cdot;0,h,u^{h})$ is stable. Meanwhile, it was pointed out in [16] (see the
paragraph before the last one in Section 1 in [16]) that when open-looped
stabilization controls exist, one can construct a periodic feedback
stabilization law through using a method provided in [20]. From this point of
view, the equivalence $(a)\Leftrightarrow(d)$ in Theorem 1.1 is not new,
though our way to approach the equivalence differs from those in [16] and [21]
and our method to construct the stabilization feedback law is different from
that in [20].

  * •

To the best of our knowledge, both Theorem 1.2 and the equivalences:
$(a)\Leftrightarrow(b)$ and $(a)\Leftrightarrow(c)$ in Theorem 1.1 appear to
be new. It is worth mentioning that the equivalence $(a)\Leftrightarrow(b)$ in
Theorem 1.1 is an extension of a result in our previous paper [25] which
studies the stabilization of finite-dimensional periodic systems.

  * •

A byproduct of this study (see Proposition 3.3, and Remark 3.1) shows that
when both $B(\cdot)$ and $D(\cdot)$ are time-invariant, linear time-period
functions $K(\cdot)$ will not aid the linear stabilization of Equation (1.1),
i.e., Equation (1.1) is linear $\hat{T}$-periodic feedback stabilizable for
some $\hat{T}>0$ if and only if Equation (1.1) is linear time-invariant
feedback stabilizable. On the other hand, when Equation (1.1) is periodic
time-varying, linear time-periodic $K(\cdot)$ do aid in the linear
stabilization of this equation.

The rest of this paper is organized as follows. Section 2 provides some
properties on Poincar$\acute{e}$ map, Kato projection and attainable
subspaces. Section 3 studies the multi-periodic feedback stabilization.
Section 4 proves Theorem 1.1 and 1.2. Section 5 presents some applications of
the main theorems to internally controlled heat equations with time-periodic
potentials.

##  2  Poincar$\acute{e}$ map, Kato projection and attainable subspaces

In this section, we will first present three lemmas (Lemma 2.2, 2.3, 2.4)
which show certain properties on the Poincar$\acute{e}$ map and the Kato
projection. (These lemmas are slightly different versions of the existing
results. For the sake of the completeness of the paper, we provide their
detailed proof in Appendix.) Then, we show certain properties on attainable
subspaces $V^{Z}_{k}$ defined by (1.15).

The first one is another version of Lemma 7.2.2 in [11] where $B(\cdot)$ is
assumed to be $T$-periodic and H$\ddot{o}$lder continuous (see Page 197,
[11]); while in our case, $B(\cdot)\in L^{1}(\mathbb{R}^{+};{\cal L}(H))$ is
$T$-periodic.

###### Lemma 2.1

Let ${\cal P}(\cdot)$ be defined by (1.7). Then $\sigma({\cal
P}(t)^{C})\setminus\\{0\\}$ is independent of $t\in\mathbb{R}^{+}$. Moreover,
$\sigma({\cal P}(t)^{C})\setminus\\{0\\}$ consists entirely of distinct
eigenvalues $\\{\lambda_{j}\\}_{j=1}^{\infty}$ (of ${\cal P}(0)$) with
$\lim_{j\rightarrow\infty}|\lambda_{j}|=0$.

The second one is another version of Theorem 7.2.3 in [11] where complex case
is studied (see Page 198, [11]); while in following lemma, we consider the
real case.

###### Lemma 2.2

Let ${\cal P}(\cdot)$ and $P(\cdot)$ be defined by (1.7) and (1.13)
respectively. Then each $P(t)$ (with $t\geq 0$) is a projection on $H$ such
that

$H=H_{1}(t)\bigoplus H_{2}(t),$ (2.1)

where

$H_{1}(t)\mathop{\buildrel\Delta\over{=}}P(t)H\;\;\mbox{and}\;\;H_{2}(t)\mathop{\buildrel\Delta\over{=}}(I-P(t))H.$
(2.2)

Moreover, $P(\cdot)$, $H_{1}(\cdot)$ and $H_{2}(\cdot)$ have the following
properties:

$(a)$ $P(\cdot)$, $H_{1}(\cdot)$ and $H_{2}(\cdot)$ are $T$-periodic.

$(b)$ For each $t\geq 0$, both $H_{1}(t)$ and $H_{2}(t)$ are invariant
subspaces of ${\cal P}(t)$.

$(c)$ If $\\{\lambda_{j}\\}_{j=1}^{\infty}$, $n$ and $n_{0}$ are given by
(1.8), (1.9) and (1.11), then

$\sigma({\cal
P}(t)^{C}|_{H_{1}(t)^{C}})=\\{\lambda_{j}\\}^{n}_{j=1},\qquad\sigma({\cal
P}(t)^{C}|_{H_{2}(t)^{C}})\setminus\\{0\\}=\\{\lambda_{j}\\}^{\infty}_{j=n+1};$
(2.3) $\mbox{dim}H_{1}(t)=n_{0}.$ (2.4)

$(d)$ When $0\leq s\leq t<\infty$, $\Phi(t,s)\in{\cal L}(H_{j}(s),H_{j}(t))$
with $j=1,2$.

$(e)$ It holds that

$\Phi(t,s)P(s)=P(t)\Phi(t,s),\;\;\mbox{when}\;\;0\leq s\leq t.$ (2.5)

$(f)$ Let $\bar{\rho}\triangleq(-{\ln\bar{\delta}})/{T}>0$ with $\bar{\delta}$
given by (1.10). For any $\rho\in(0,\bar{\rho})$, there is a positive constant
$C_{\rho}$ such that

$\|\Phi(t,s)h_{2}\|\leq C_{\rho}e^{-\rho(t-s)}\|h_{2}\|,\mbox{ when}\;\;0\leq
s\leq t<\infty\;\;\mbox{and}\;\;h_{2}\in H_{2}(s).$ (2.6)

The third one is essentially another version of Theorem 6.22 in [12] (see Page
184, [12]). To state it, we recall that ${\cal P}^{*}$ and $P^{*}$ are the
adjoint operators of ${\cal P}$ and $P$. It is clear that

$\sigma\big{(}{\cal P}^{*C}\big{)}=\overline{\sigma\big{(}{\cal
P}^{C}\big{)}}.$ (2.7)

Since $\sigma\big{(}{\cal
P}^{C}\big{)}\setminus\\{0\\}=\\{\lambda_{j}\\}^{\infty}_{j=1}$ (see (1.8)),
it holds that $\sigma\big{(}{\cal
P}^{*C}\big{)}\setminus\\{0\\}=\\{\bar{\lambda}_{j}\\}^{\infty}_{j=1}$. Write
$\bar{l}_{j}$ for the algebraic multiplicity of $\bar{\lambda}_{j}$ w.r.t.
${\cal P}^{*C}$. It is clear that

$\bar{l}_{j}=l_{j}\;\;\mbox{for
all}\;\;j,\;\;\mbox{and}\;\;\bar{l}_{1}+\cdots+\bar{l}_{n}=n_{0},$ (2.8)

where $l_{j}$ is the algebraic multiplicity of $\lambda_{j}$ w.r.t. ${\cal
P}^{C}$; $n$ and $n_{0}$ are given by (1.9) and (1.11) respectively. Let
$\Gamma$ be the circle used to defined $\hat{P}\triangleq\hat{P}(0)$ (see
(1.12)). Define the Kato projection with respect to ${\cal P}^{*C}$ as
follows:

$\hat{\tilde{P}}=\displaystyle\frac{1}{2\pi i}\int_{\Gamma}(\lambda I-{\cal
P}^{*C})^{-1}d\lambda.$ (2.9)

From Theorem 6.17 on Page 178 in [12], it follows that

$H^{C}=\hat{\tilde{H}}_{1}\bigoplus\hat{\tilde{H}}_{2}\triangleq\hat{\tilde{P}}H^{C}\bigoplus(I-\hat{\tilde{P}})H^{C}$
(2.10)

and

$\mbox{both}\;\;\hat{\tilde{H}}_{1}\;\mbox{and}\;\hat{\tilde{H}}_{2}\;\;\mbox{are
invariant w.r.t.}\;\;{\cal P}^{*C}.$ (2.11)

###### Lemma 2.3

Let $\hat{\tilde{P}}$ be defined by (2.9). Then,
$\tilde{P}\triangleq\hat{\tilde{P}}\big{|}_{H}$ is a projection on $H$;
$H=\tilde{H}_{1}\bigoplus\tilde{H}_{2}$, where
$\tilde{H}_{1}\triangleq\tilde{P}H$ and
$\tilde{H}_{2}\triangleq(I-\tilde{P})H$; ${\cal
P}^{*}\tilde{H}_{1}\subseteq\tilde{H}_{1}$; $\sigma\big{(}{\cal
P}^{*C}|_{\tilde{H}_{1}^{C}}\big{)}=\\{\bar{\lambda}_{j}\\}^{n}_{j=1}$ and
$\sigma\big{(}{\cal P}^{*C}|_{\tilde{H}_{2}^{C}}\big{)}\subseteq\mathbb{B}$;
and $\mbox{dim}\tilde{H}_{1}=n_{0}$. It further holds that

$\tilde{P}=P^{*};$ (2.12)
$\tilde{H}_{1}=P^{*}H=P^{*}H_{1},\;\;\mbox{where}\;\;H_{1}\;\;\mbox{is given
by}\;\;(\ref{wgs1.15});$ (2.13)
$\xi\in\tilde{H}_{1}^{C},\;\;\mbox{when}\;\;\mu\in\sigma\big{(}{\cal
P}^{*C}\big{)}\setminus\mathbb{B}\;\;\mbox{and}\;\;(\mu I-{\cal
P}^{*C})\xi=0.$ (2.14)

Next, we will introduce certain properties on attainable subspaces
$V^{Z}_{k}$, $k\in\mathbb{N}$. They will play important roles in the proof of
our main theorems. We start with recalling (2.1) and (1.15). Since $H_{1}$ is
invariant w.r.t. ${\cal P}$ (see Part $(b)$ of Lemma 2.2), we can define
${\cal P}_{1}:H_{1}\rightarrow H_{1}$ by setting

${\cal P}_{1}\triangleq{\large{\mbox{${\cal P}$}}}\bigm{|}_{H_{1}}.$ (2.15)

By (2.3), it holds that

$\sigma({\cal P}_{1})\bigcap\mathbb{B}=\emptyset.$ (2.16)

###### Lemma 2.4

Let ${\cal P}_{1}$ and $n_{0}$ be given by (2.15) and (1.11), respectively.
Suppose that $Z\subseteq U$ is a subspace with $V^{Z}_{k}$ and
$\hat{V}^{Z}_{k}$ given by (1.15) and (1.16), respectively. Then for each
$k\in\mathbb{N}$,

$V^{Z}_{k}=V^{Z}_{1}+{\cal P}V^{Z}_{1}+\cdots+{\cal
P}^{k-1}V^{Z}_{1};\qquad\hat{V}^{Z}_{k}=\hat{V}^{Z}_{1}+{\cal
P}_{1}\hat{V}^{Z}_{1}+\cdots+{\cal P}_{1}^{k-1}\hat{V}^{Z}_{1}.$ (2.17)

Furthermore, ${\cal P}_{1}$ is invertible and it holds that

$\hat{V}^{Z}=\hat{V}^{Z}_{n_{0}};\;\;\;\;{\cal
P}_{1}\hat{V}^{Z}=\hat{V}^{Z}={\cal P}_{1}^{-1}\hat{V}^{Z},$ (2.18)

where

$\hat{V}^{Z}\triangleq\bigcup_{k=1}^{\infty}\hat{V}^{Z}_{k}.$ (2.19)

Proof. We begin with proving the first equality in (2.17) by the mathematical
induction. Clearly, it stands when $k=1$. Assume that it holds in the case
when $k=k_{0}$ for some $k_{0}\geq 1$, i.e.,

$V^{Z}_{k_{0}}=V^{Z}_{1}+{\cal P}V^{Z}_{1}+\cdots+{\cal
P}^{k_{0}-1}V^{Z}_{1}.$ (2.20)

Because of (1.6) and (1.14), we have that
$\Phi((k_{0}+1)T,T)=\Phi(T,0)^{k_{0}}={\cal P}^{k_{0}}$. This, along with
(1.15), the $T$-periodicity of $D(\cdot)$ and (2.20), indicates that

$\begin{array}[]{rl}V^{Z}_{k_{0}+1}&=\Big{\\{}\displaystyle\int^{(k_{0}+1)T}_{0}\Phi((k_{0}+1)T,s)D(s)u(s)ds\bigm{|}u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\Big{\\}}\\\
&=\Big{\\{}\Phi((k_{0}+1)T,T)\displaystyle\int^{T}_{0}\Phi(T,s)D(s)u(s)ds\bigm{|}u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\Big{\\}}\\\
&\quad+\Big{\\{}\displaystyle\int^{k_{0}T}_{0}\Phi((k_{0}+1)T,s+T)D(s+T)u(s+T)ds\bigm{|}u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\Big{\\}}\\\ &={\cal
P}^{k_{0}}V^{Z}_{1}+\Big{\\{}\displaystyle\int_{0}^{k_{0}T}\Phi(k_{0}T,s)D(s)u(s+T)ds\bigm{|}u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\Big{\\}}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr&={\cal P}^{k_{0}}V^{Z}_{1}+V^{Z}_{k_{0}}=V^{Z}_{1}+{\cal
P}V^{Z}_{1}+\cdots+{\cal P}^{k_{0}}V^{Z}_{1}.\end{array}$

which leads to the first equality in (2.17).

We next show the second equality in (2.17). By (1.14) and (2.5) with $t=T$ and
$s=0$, we have

${\cal P}P=P{\cal P}.$ (2.21)

Since $P$ is a projection from $H$ onto $H_{1}$ (see Lemma 2.2), the second
equality in (2.17) follows from the first one in (2.17), (1.16) and (2.21).

Then we show the first equality in (2.18). It follows respectively from (2.19)
and (2.17) that

$\hat{V}^{Z}_{n_{0}}\subseteq\hat{V}^{Z}\;\;\mbox{and}\;\;\hat{V}^{Z}_{k}\subseteq\hat{V}^{Z}_{n_{0}},\;\;\mbox{when}\;\;k\leq
n_{0}.$ (2.22)

Since $\mbox{dim}H_{1}=n_{0}$ (see (2.4)) and ${\cal P}_{1}:H_{1}\rightarrow
H_{1}$ (see (2.15)), according to the Hamilton-Cayley theorem, each ${\cal
P}_{1}^{j}$ with $j\geq n_{0}$ is a linear combination of
$\big{\\{}~{}I,~{}{\cal P}_{1}^{1},~{}{\cal P}_{1}^{2},\cdots,~{}{\cal
P}_{1}^{(n_{0}-1)}\big{\\}}$. This, along with the second equality in (2.17),
indicates that

$\hat{V}^{Z}_{k}=\sum_{j=0}^{k-1}{\cal
P}_{1}^{j}(\hat{V}^{Z}_{1})\subseteq\sum_{j=0}^{n_{0}-1}{\cal
P}_{1}^{j}(\hat{V}^{Z}_{1})=\hat{V}^{Z}_{n_{0}},\;\;\mbox{when}\;\;k\geq
n_{0}.$ (2.23)

Now the first equality in (2.18) follows from (2.22) and (2.23).

Finally, we show the non-singularity of ${\cal P}_{1}$ and the second equality
in (2.18). By the first equality in (2.18) and the Hamilton-Cayley theorem, we
see that

${\cal P}_{1}\hat{V}^{Z}={\cal P}_{1}\hat{V}^{Z}_{n_{0}}={\cal
P}_{1}\sum_{j=0}^{n_{0}-1}{\cal
P}_{1}^{j}(\hat{V}^{Z}_{1})=\sum_{j=1}^{n_{0}}{\cal
P}_{1}^{j}(\hat{V}^{Z}_{1})\subseteq\sum_{j=0}^{n_{0}-1}{\cal
P}_{1}^{j}(\hat{V}^{Z}_{1})=\hat{V}^{Z}_{n_{0}},$

from which, it follows that

${\cal P}_{1}\hat{V}^{Z}\subseteq\hat{V}^{Z}.$ (2.24)

Because $0\notin\sigma({\cal P}_{1}^{C})$ (see (2.3) as well as (1.9)), and
since the domain of ${\cal P}_{1}^{C}$, $H_{1}^{C}$ is a finite dimensional
subspace, the operator ${\cal P}_{1}^{C}$ is invertible. Hence, the operator
${\cal P}_{1}$ is also invertible. This implies that $\mbox{dim}({\cal
P}_{1}\hat{V}^{Z})=\mbox{dim}\hat{V}^{Z}$, which, together with (2.24), yields
that ${\cal P}_{1}\hat{V}^{Z}=\hat{V}^{Z}$. This completes the proof.

###### Lemma 2.5

Let $n_{0}$ be given by (1.11). Then, for each subspace $Z$ of $U$, there is a
finite dimension subspace of $\hat{Z}$ of $Z$ such that

$\hat{V}^{Z}_{n_{0}}=\hat{V}^{\hat{Z}}_{n_{0}},$ (2.25)

where $\hat{V}^{Z}_{n_{0}}$ and $\hat{V}^{\hat{Z}}_{n_{0}}$ are defined by
(1.16).

Proof. Let $Z$ be a subspace of $U$. Since $\hat{V}^{Z}_{n_{0}}$ is a subspace
of $H_{1}$ and $\mbox{dim}H_{1}=n_{0}<\infty$ (see (2.4), we can assume that
$\mbox{dim}\hat{V}^{Z}_{n_{0}}\triangleq m\leq n_{0}$. Write
$\\{\xi_{1},\dots,\xi_{m}\\}$ for an orthonormal basis of
$\hat{V}^{Z}_{n_{0}}$. By the definition of $\hat{V}^{Z}_{n_{0}}$ (see (1.16),
as well as (1.15)), there are $u_{j}(\cdot)\in L^{2}(\mathbb{R}^{+};Z)$,
$j=1,\dots,m$, such that

$\displaystyle\displaystyle\int^{n_{0}T}_{0}P\Phi(n_{0}T,s)D(s)u_{j}(s)ds=\xi_{j}\;\;\mbox{
for all }\;\;j=1,\dots,m.$ (2.26)

By the boundedness of $\Phi(n_{0}T,\cdot)$ and $D(\cdot)$ over $[0,n_{0}T]$,
there is a constant $C>0$ such that

$\|P\Phi(n_{0}T,\cdot)D(\cdot)\|_{L^{\infty}(0,n_{0}T;\mathcal{L}(U;H))}\leq
C.$ (2.27)

Let $\hat{\varepsilon}>0$ small enough to satisfy

$(1-\hat{\varepsilon})^{2}-(2\hat{\varepsilon}+\hat{\varepsilon}^{2})(m-1)>0.$
(2.28)

By the definition of the Bochner integration (see [7]), there are simple
functions

$v_{j}(\cdot)=\sum\limits_{l=1}^{k_{j}}{\mbox{\Large$\chi$}}_{E_{j\hskip
1.42262ptl}}(\cdot)z_{j\hskip
1.42262ptl}\;\;\mbox{over}\;\;(0,n_{0}T),\;\;j=1,\dots,m,$ (2.29)

with $z_{jl}\in Z$, $E_{jl}$ measurable sets in $(0,n_{0}T)$ and
$\chi_{E_{jl}}$ the characteristic function of ${E_{jl}}$, such that

$\displaystyle\int_{0}^{n_{0}T}\|u_{j}(s)-v_{j}(s)\|_{U}ds\leq\hat{\varepsilon}/C.$

This, along with (2.27), yields that for each $j\in\\{1,\dots,m\\}$,

$\displaystyle\Big{\|}\int^{n_{0}T}_{0}P\Phi(n_{0}T,s)D(s)u_{j}(s)ds-\displaystyle\int^{n_{0}T}_{0}P\Phi(n_{0}T,s)D(s)v_{j}(s)ds\Big{\|}\leq\hat{\varepsilon}.$
(2.30)

Let

$\eta_{j}=\displaystyle\int^{n_{0}T}_{0}P\Phi(n_{0}T,s)D(s)v_{j}(s)ds,\;\;j=1,\dots,m.$
(2.31)

By (1.16), (1.15), (2.29) and (2.31), we see that
$\eta_{j}\in\hat{V}^{Z}_{n_{0}}$ for all $j=1,\dots,m$. Meanwhile, it follows
from (2.30) that

$\|\eta_{j}-\xi_{j}\|\leq\hat{\varepsilon}\;\;\mbox{ for all
}\;\;j\in\\{1,\dots,m\\}.$ (2.32)

Now we claim that $\\{\eta_{1},\dots,\eta_{m}\\}$ is a basis of
$\hat{V}^{Z}_{n_{0}}$. In fact, since $\\{\xi_{1},\dots,\xi_{m}\\}$ is
orthonormal, it follows from (2.32) that

$|\mathop{\langle}\eta_{j},\eta_{l}\mathop{\rangle}|=|\mathop{\langle}\xi_{j}+(\eta_{j}-\xi_{j}),\xi_{l}+(\eta_{l}-\xi_{l})\mathop{\rangle}|\leq
2\hat{\varepsilon}+\hat{\varepsilon}^{2}\;\;\mbox{for
all}\;\;j,l\in\\{1,\dots,m\\}\;\;\mbox{with}\;\;j\neq l.$

From this and (2.32), one can directly check that

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\mathop{\langle}\sum^{m}_{j=1}\alpha_{j}\eta_{j},\,\sum^{m}_{l=1}\alpha_{l}\eta_{l}\mathop{\rangle}=\displaystyle\sum^{m}_{j=1}\alpha_{j}^{2}\|\eta_{j}\|^{2}+\sum^{m}_{j=1}\sum_{l\neq
j}\alpha_{l}\alpha_{j}\mathop{\langle}\eta_{j},\,\eta_{l}\mathop{\rangle}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\geq&\displaystyle\sum^{m}_{j=1}\alpha_{j}^{2}\left(\|\xi_{j}\|-\|\xi_{j}-\eta_{j}\|\right)^{2}-\sum^{m}_{j=1}\sum_{l\neq
j}|\alpha_{l}\alpha_{j}|\cdot|\mathop{\langle}\eta_{j},\,\eta_{l}\mathop{\rangle}|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\geq&\displaystyle\left((1-\hat{\varepsilon})^{2}-(2\hat{\varepsilon}+\hat{\varepsilon}^{2})(m-1)\right)\sum^{m}_{j=1}\alpha_{j}^{2},\;\;\mbox{when}\;\;\alpha_{1},\dots,\alpha_{m}\in\mathbb{R}.\end{array}$

This, along with (2.28), indicates that $\alpha_{1}=\cdots=\alpha_{m}=0$
whenever $\sum^{m}_{j=1}\alpha_{j}\eta_{j}=0$. Namely,
$\\{\eta_{1},\dots,\eta_{m}\\}$ is linearly independent group in the subspace
$\hat{V}^{Z}_{n_{0}}$ which has the dimension $m$. Hence,
$\\{\eta_{1},\dots,\eta_{m}\\}$ is a basis of $\hat{V}^{Z}_{n_{0}}$.

Let

$\hat{Z}=\hbox{\rm
span$\,$}\left\\{z_{11},\dots,z_{1k_{1}},z_{21},\dots,z_{2k_{2}},\dots,z_{m1},\dots,z_{mk_{m}}\right\\},$

where $z_{jl}$, $j=1,\dots,m$, $l=1,\dots,k_{j}$, are given by (2.29).
Clearly, $\hat{Z}$ is a finite-dimensional subspace of $Z$ and all
$v_{j}(\cdot)$, $j=1,\dots,m$, (given by (2.29)) belong to
$L^{2}(\mathbb{R}^{+};\hat{Z})$. These, along with (2.31), yield that all
$\eta_{j}$, $j=1,\dots,m$, are in $\hat{V}^{\hat{Z}}_{n_{0}}$. Therefore, it
holds that

$\hat{V}^{Z}_{n_{0}}\supseteq\hat{V}^{\hat{Z}}_{n_{0}}\supseteq\hbox{\rm
span$\,$}\\{\eta_{1},\dots,\eta_{m}\\}=\hat{V}^{Z}_{n_{0}}.$

This leads to (2.25) and completes the proof.

##  3 The multi-periodic feedback stabilization

In this section, we will introduce three propositions. The first two
propositions will be used in the proof of our main theorems. The last one is
independent interesting. We begin with the following definitions:

  * •

Equation (1.1) is said to be linear multi-periodic feedback stabilizable
(LMPFS, for short) if there is a $kT$-periodic $K(\cdot)\in
L^{\infty}\left(\mathbb{R}^{+};{\cal L}(H,U)\right)$ for some $k\in\mathbb{N}$
such that the equation (1.2) is exponentially stable. Any such a $K(\cdot)$ is
called an LMPFS law for Equation (1.1).

  * •

Equation (1.1) is said to be LMPFS with respect to a subspace $Z$ of $U$ if
there is a $kT$-periodic $K(\cdot)\in L^{\infty}\left(\mathbb{R}^{+};{\cal
L}(H,Z)\right)$ for some $k\in\mathbb{N}$ such that the equation (1.2) is
exponentially stable. Any such a $K(\cdot)$ is called an LMPFS law for
Equation (1.1) with respect to $Z$.

###### Proposition 3.1

Let $n_{0}$ and $H_{1}$ be given by (1.11) and (1.14) respectively. Suppose
that $Z\subseteq U$ is a finite dimensional subspace satisfying (1.17). Then,
Equation (1.1) is LMPFS with respect to $Z$.

Proof. Let $Z\subseteq U$ satisfy (1.17). We organize the proof by several
steps as follows.

Step 1. For any $h_{1}\in H_{1}$, to construct control $u^{h_{1}}(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)$ such that $Py(n_{0}T;0,h_{1},u^{h_{1}})=0$

Because $\mbox{dim}H_{1}=n_{0}$ (see (2.4)), we can set
$\\{\eta_{1},\cdots,\eta_{n}\\}$ to be an orthonormal basis of $H_{1}$. Define
a linear map $\mathcal{F}:\mathbb{R}^{n_{0}}\rightarrow H_{1}$ by setting

$\mathcal{F}(\textbf{a})\mathop{\buildrel\Delta\over{=}}\sum_{j=1}^{n_{0}}a_{j}\eta_{j}\triangleq(\eta_{1},\dots,\eta_{n_{0}})(a_{1},\dots,a_{n_{0}})^{*}\;\;\mbox{for
each}\;\;\textbf{a}=(a_{1},\dots,a_{n_{0}})^{*}\in\mathbb{R}^{n_{0}}.$ (3.1)

Clearly, $\mathcal{F}$ is invertible and

$\mathcal{F}^{-1}(h_{1})=(\mathop{\langle}h_{1},\eta_{1}\mathop{\rangle},\dots,\mathop{\langle}h_{1},\eta_{n_{0}}\mathop{\rangle})^{*}.$
(3.2)

Since $Z$ is finite dimensional, we can assume that
$\mbox{dim}Z=m_{0}<\infty$. Write $\\{z_{1},\dots,z_{m_{0}}\\}$ for an
orthonormal basis of $Z$. Define a linear map
$\mathcal{G}:L^{2}(\mathbb{R}^{+};\mathbb{R}^{m_{0}})\rightarrow
L^{2}(\mathbb{R}^{+};Z)$ by setting

$\mathcal{G}(\beta)(t)\mathop{\buildrel\Delta\over{=}}(z_{1},\dots,z_{m_{0}})\beta(t)\triangleq\sum_{j=1}^{m_{0}}\beta_{j}(t)z_{j},\;\mbox{a.e.}\;t\geq
0,$ (3.3)

for each
$\beta(\cdot)\triangleq(\beta_{1}(\cdot),\dots,\beta_{m_{0}}(\cdot))^{*}\in
L^{2}(\mathbb{R}^{+};\mathbb{R}^{m_{0}})$ with $\beta_{j}(\cdot)\in
L^{2}(\mathbb{R}^{+};\mathbb{R})$. Clearly, $\mathcal{G}$ is invertible and it
holds that

$u(t)=\mathcal{G}\mathcal{G}^{-1}(u)(t)=(z_{1},\dots,z_{m_{0}})\mathcal{G}^{-1}(u)(t)\;\;\mbox{for
a.e.}\;\;t\geq 0,\;\;\mbox{when}\;\;u(\cdot)\in L^{2}(\mathbb{R}^{+};Z).$
(3.4)

By the facts that $P{\cal P}={\cal P}P$, $H_{1}$ is invariant with respect to
${\cal P}$ and $P:H\rightarrow H_{1}$ is a projection (see Lemma 2.2), and by
(2.15), we obtain that

$P{\cal P}^{n_{0}}h_{1}={\cal P}^{n_{0}}h_{1}={\cal
P}^{n_{0}}_{1}h_{1}\;\;\mbox{ for each}\;\;h_{1}\in H_{1}.$

From this, we see that

$\begin{array}[]{rl}&Py(n_{0}T;0,h_{1},u)=P{\cal
P}^{n_{0}}h_{1}+\displaystyle\int^{n_{0}T}_{0}P\Phi(n_{0}T,s)D(s)u(s)ds\\\
&={\cal
P}^{n_{0}}_{1}h_{1}+\displaystyle\int^{n_{0}T}_{0}P\Phi(n_{0}T,s)D(s)u(s)ds\;\;\mbox{for
all}\;\;h_{1}\in H_{1},u(\cdot)\in L^{2}(\mathbb{R}^{+};Z).\end{array}$ (3.5)

Let $\hat{A}\in\mathbb{R}^{n_{0}\times n_{0}}$ be the matrix of ${\cal P}_{1}$
under $\\{\eta_{1},\dots,\eta_{n_{0}}\\}$, i.e., ${\cal
P}_{1}(\eta_{1},\dots,\eta_{n_{0}})=(\eta_{1},\dots,\eta_{n_{0}})\hat{A}$.
Then, it follows from (3.1) and (3.2) that

${\cal P}_{1}^{n_{0}}h_{1}={\cal
P}_{1}^{n_{0}}(\eta_{1},\dots,\eta_{n_{0}})\mathcal{F}^{-1}(h_{1})=(\eta_{1},\dots,\eta_{n_{0}})\hat{A}^{n_{0}}\mathcal{F}^{-1}(h_{1}),\;\;\mbox{when}\;\;h_{1}\in
H_{1}.$ (3.6)

Since $Py(n_{0}T;0,h_{1},u)\in H_{1}$, it follows by (3.1) and (3.2) that

$Py(n_{0}T;0,h_{1},u)=(\eta_{1},\dots,\eta_{n_{0}})\mathcal{F}^{-1}(Py(n_{0}T;0,h_{1},u))\;\;\mbox{for
all}\;\;h_{1}\in H_{1},u(\cdot)\in L^{2}(\mathbb{R}^{+};Z).$ (3.7)

Since $D(\cdot)\in L^{\infty}(\mathbb{R}^{+};\mathcal{L}(U;H))$ and
$P\Phi(n_{0}T,s)D(s)(z_{1},\dots,z_{m_{0}})\in(H_{1})^{m_{0}}$ for a.e. $s\geq
0$, there is a unique $\hat{B}(\cdot)\in
L^{\infty}(\mathbb{R}^{+};\mathbb{R}^{n_{0}\times m_{0}})$ such that

$P\Phi(n_{0}T,s)D(s)(z_{1},\dots,z_{m_{0}})=(\eta_{1},\dots,\eta_{n_{0}})\hat{B}(s)\;\;\mbox{for
a.e.}\;\;s\in\mathbb{R}^{+}.$ (3.8)

Now, from (3.5), (3.6), (3.7), (3.4) and (3.8), we see that for each $h_{1}\in
H_{1}$ and $u(\cdot)\in L^{2}(\mathbb{R}^{+};Z)$,

$\mathcal{F}^{-1}(Py(n_{0}T;0,h_{1},u))=\hat{A}^{n_{0}}\mathcal{F}^{-1}(h_{1})+\displaystyle\int^{n_{0}T}_{0}\hat{B}(s)\mathcal{G}^{-1}(u)(s)ds.$
(3.9)

Meanwhile, it follows from (1.16), (1.15), (3.4), (3.8) and (3.3) that

$\hat{V}_{n_{0}}^{Z}=\Big{\\{}(\eta_{1},\dots,\eta_{n_{0}})\displaystyle\int_{0}^{n_{0}T}\hat{B}(s)\beta(s)ds\;\big{|}\;\beta(\cdot)\in
L^{2}(\mathbb{R}^{+};\mathbb{R}^{m_{0}})\Big{\\}}.$

Since $Z$ satisfies (1.17), the above equality yields

$\mbox{span}\\{\eta_{1},\dots,\eta_{n_{0}}\\}=\Big{\\{}(\eta_{1},\dots,\eta_{n_{0}})\displaystyle\int_{0}^{n_{0}T}\hat{B}(s)\beta(s)ds\;\big{|}\;\beta(\cdot)\in
L^{2}(\mathbb{R}^{+};\mathbb{R}^{m_{0}})\Big{\\}},$

which is equivalent to

$\Big{\\{}\displaystyle\int_{0}^{n_{0}T}\hat{B}(s)\beta(s)ds\;\big{|}\;\beta(\cdot)\in
L^{2}(\mathbb{R}^{+};\mathbb{R}^{m_{0}})\Big{\\}}=\mathbb{R}^{n_{0}}.$ (3.10)

From (3.10), we see that the finite-dimensional controlled system
$x^{\prime}(t)=\hat{B}(t)\beta(t),\;t\geq 0$, (where
$x(\cdot)=(x_{1}(\cdot),\dots,x_{n_{0}}(\cdot))^{*}$ is treated as a state and
$\beta(\cdot)$ is treated as a control) is exactly controllable. Hence, the
matrix $\displaystyle\int^{n_{0}T}_{0}\hat{B}(s)\hat{B}(s)^{*}ds$ is positive
definite (see, for instance, [24]).

Now for each $h_{1}\in H_{1}$, we define $\beta^{h_{1}}(\cdot)\in
L^{2}(\mathbb{R}^{+};\mathbb{R}^{m_{0}})$ by setting

$\beta^{h_{1}}(t)=\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr-\hat{B}(t)^{*}\displaystyle\Big{(}\int^{n_{0}T}_{0}\hat{B}(s)\hat{B}(s)^{*}ds\Big{)}^{-1}\hat{A}^{n_{0}}\mathcal{F}^{-1}(h_{1})&\mbox{for
a.e.}\;\;t\in[0,n_{0}T);\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr 0&\mbox{for
a.e.}\;\;t\in[n_{0}T,+\infty).\end{array}\right.$ (3.11)

It is clear that

$\hat{A}^{n_{0}}\mathcal{F}^{-1}(h_{1})+\displaystyle\int_{0}^{n_{0}T}\hat{B}(s)\beta^{h_{1}}(s)ds=0.$
(3.12)

Then, for each $h_{1}\in H_{1}$, we construct a control

$u^{h_{1}}(\cdot)=\mathcal{G}(\beta^{h_{1}})(\cdot)\;\;\mbox{over}\;\;\mathbb{R}^{+}.$
(3.13)

By (3.3), $u^{h_{1}}(\cdot)\in L^{2}(\mathbb{R}^{+};Z)$. Meanwhile, it follows
from (3.12) and (3.9) that

$\mathcal{F}^{-1}(Py(n_{0}T;0,h_{1},u^{h_{1}}))=0,\;\;\mbox{when}\;\;h_{1}\in
H_{1}.$

This, implies that

$Py(n_{0}T;0,h_{1},u^{h_{1}})=0,\;\;\mbox{when}\;\;h_{1}\in H_{1}.$ (3.14)

Step 2. To show the existence of an $N_{0}\in\mathbb{N}$ such that

$\|y(NT;0,h,{\cal L}(Ph))\|\leq\delta_{0}\|h\|\;\;\;\mbox{for all}\;\;h\in
H\;\;\mbox{and}\;\;N\geq N_{0},$ (3.15)

where $\delta_{0}\triangleq(1+\bar{\delta})/2$ with $\bar{\delta}$ given by
(1.10).

Define an operator ${\cal L}:H_{1}\rightarrow L^{2}(\mathbb{R}^{+};Z)$ by
setting

${\cal L}h_{1}(\cdot)=u^{h_{1}}(\cdot)\;\;\mbox{for all}\;\;h_{1}\in H_{1},$
(3.16)

where $u^{h_{1}}(\cdot)$ is given by (3.13). Several observations on ${\cal
L}$ are given in order. First, it is clear that $\mathcal{L}$ is linear.
Second, from (3.16), (3.13), (3.11) and the fact that $\hat{B}(\cdot)\in
L^{\infty}(\mathbb{R}^{+};\mathbb{R}^{n_{0}\times m_{0}})$, we have

$\|{\cal L}\|\triangleq\|{\cal L}\|_{{\cal
L}(H_{1},L^{2}(\mathbb{R}^{+};Z))}<+\infty.$ (3.17)

Next, by (3.16), (3.13), (3.3) and (3.11), we see that

${\cal
L}h_{1}(\cdot)=0\;\;\mbox{over}\;\;[n_{0}T,+\infty),\;\;\mbox{when}\;\;h_{1}\in
H_{1}.$ (3.18)

Finally, it follows from (3.14) that

$Py(n_{0}T;0,h_{1},{\cal L}h_{1})=0,\;\;\mbox{when}\;\;h_{1}\in H_{1}.$ (3.19)

Let $\rho_{0}=-\ln\delta_{0}/{T}$. Since
$\delta_{0}\triangleq(1+\bar{\delta})/2$, we see that
$0<\rho_{0}<-\ln\bar{\delta}/{T}\triangleq\bar{\rho}$. Then, by Part $(f)$ of
Lemma 2.2, there is a constant $C_{\rho_{0}}>0$ such that

$\|y(kT;0,h_{2},0)\|=\|\Phi(kT,0)h_{2}\|\leq
C_{\rho_{0}}e^{-\rho_{0}kT}\|h_{2}\|=C_{\rho_{0}}\delta_{0}^{k}\|h_{2}\|,\;\;\mbox{when}\;\;k\in\mathbb{N},h_{2}\in
H_{2}.$ (3.20)

We claim that there is a constant $C>0$ such that

$\|y(NT;0,h_{1},{\cal L}h_{1})\|\leq
CC_{\rho_{0}}\delta_{0}^{N-n_{0}}\|h_{1}\|,\;\;\mbox{when}\;\;h_{1}\in
H_{1},N\geq n_{0},$ (3.21)

where ${\cal L}$ is given by (3.16). In fact, because

$\big{\|}y(n_{0}T;0,h_{1},{\cal
L}h_{1})\big{\|}\leq\displaystyle\big{\|}\Phi(n_{0}T,0)h_{1}\big{\|}+\left\|\int^{n_{0}T}_{0}\Phi(n_{0}T,s)D(s){\cal
L}h_{1}(s)ds\right\|\;\;\mbox{for each}\;\;h_{1}\in H_{1},$

there is a constant $C>0$ such that

$\|y(n_{0}T;0,h_{1},{\cal L}h_{1})\|\leq C\|h_{1}\|\;\;\mbox{for
all}\;\;h_{1}\in H_{1}.$ (3.22)

Here, we used facts that $D(\cdot)\in
L^{\infty}(\mathbb{R}^{+};\mathcal{L}(U;H))$ (see the assumption $({\cal
H}_{3})$) and ${\cal L}$ is linear and bounded (see (3.17)). Meanwhile, it
follows from (3.18), (1.6) and (1.5) with $u(\cdot)\equiv 0$ that when $N\geq
n_{0}$ and $h_{1}\in H_{1}$,

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&y(NT;0,h_{1},{\cal
L}h_{1})=y(NT;n_{0}T,y(n_{0}T;0,h_{1},{\cal L}h_{1}),{\cal L}h_{1})\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr=&y(NT;n_{0}T,y(n_{0}T;0,h_{1},{\cal
L}h_{1}),0)=\Phi(NT,n_{0}T)y(n_{0}T;0,h_{1},{\cal L}h_{1})\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr=&\Phi((N-n_{0})T,0)y(n_{0}T;0,h_{1},{\cal
L}h_{1})=y((N-n_{0})T;0,y(n_{0}T;0,h_{1},{\cal L}h_{1}),0).\end{array}$ (3.23)

Because of (3.19) and (2.1)-(2.2) with $t=0$, it holds that
$y(n_{0}T;0,h_{1},{\cal L}h_{1})\in H_{2}$, when $h_{1}\in H_{1}$. This along
with (3.23), (3.20) and (3.22), leads to (3.21).

Let

$N_{0}=\max\Big{\\{}\displaystyle\frac{\ln
C_{\rho_{0}}+\ln\left(C\delta_{0}^{-n_{0}}\|P\|+\|I-P\|\right)}{\ln(1/\delta_{0})}+2,~{}n_{0}\Big{\\}}.$
(3.24)

(Here, $[r]$ with $r\in\mathbb{R}$ denotes the integer such that $r-1<[r]\leq
r$.) Then, it follows from (3.21), (3.20) and (3.24) that

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&\|y(NT;0,h,{\cal
L}(Ph))\|\leq\|y(NT;0,Ph,{\cal L}(Ph))\|+\|y(NT;0,(I-P)h,0)\|\\\ \vskip 3.0pt
plus 1.0pt minus
1.0pt\cr\leq&C_{\rho_{0}}\delta_{0}^{N}(C\delta_{0}^{-n_{0}}\|P\|+\|I-P\|)\|h\|\leq\delta_{0}\|h\|,\qquad\mbox{when}\;\;N\geq
N_{0}\;\;\mbox{and}\;\;h\in H.\end{array}$

This leads to (3.15).

Step 3. To study a value function associated with a class of optimal control
problems

Given $N\in\mathbb{N}$ and $t\in[0,NT)$, $h\in H$ and $u(\cdot)\in
L^{2}(0,NT;Z)$, consider the equation:

$\left\\{\begin{array}[]{ll}y^{\prime}(s)+Ay(s)+B(s)y(s)=D(s)\bigl{|}_{Z}u(s)\;\;\mbox{in}\;\;(t,NT),\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(t)=h,\end{array}\right.$ (3.25)

where $D(t)\bigl{|}_{Z}$ is the restriction of $D(t)$ on the subspace $Z$.
Because of assumptions $({\cal H}_{1})$-$({\cal H}_{3})$, Equation (3.25) has
a unique mild solution $y^{Z}_{N}(\cdot;t,h,u)\in C([0,NT];H)$ (see
Proposition 5.3 on Page 66 in [14]). Clearly,
$y^{Z}_{N}(\cdot;t,h,u)=y(\cdot;t,h,u)\big{|}_{[t,NT]}$. For each
$\varepsilon>0$, we define a cost functional
$J^{Z}_{\varepsilon,N,t,h}(\cdot):L^{2}(0,NT;Z)\rightarrow\mathbb{R}^{+}$ by
setting

$J^{Z}_{\varepsilon,N,t,h}(u)=\displaystyle\int^{NT}_{t}\varepsilon\|u(s)\|_{U}^{2}ds+\|y^{Z}_{N}(NT;t,h,u)\|^{2},\;\;u\in
L^{2}(0,NT;Z).$ (3.26)

Then, for each $N\in\mathbb{N}$, $\varepsilon>0$, $t\in[0,NT]$ and $h\in H$,
we define the optimal control problem

$(P)^{Z}_{\varepsilon,N,t,h}:\;\;\;\;\;\;\;\displaystyle\inf_{u\in
L^{2}(0,NT;\,Z)}J^{Z}_{\varepsilon,N,t,h}(u).$

This a classical linear quadratic optimal control problem (see Page 370 in
[14]). For each $\varepsilon>0$ and $N\in\mathbb{N}$, the value function
associated with the above optimal control problems is

$W^{Z}_{\varepsilon,N}(t,h)\mathop{\buildrel\Delta\over{=}}\displaystyle\inf_{u\in
L^{2}(0,NT;\,Z)}J^{Z}_{\varepsilon,N,t,h}(u),\;\;t\in[0,NT]\;\;\mbox{and}\;\;h\in
H.$ (3.27)

Let

$\varepsilon_{0}\mathop{\buildrel\Delta\over{=}}(\delta_{0}-\delta_{0}^{2})/{(\|{\cal
L}\|\|P\|+1)^{2}},$ (3.28)

where $\delta_{0}\in(0,1)$ and ${\cal L}$ are given by (3.15) and (3.16)
respectively. Because of (3.17), it holds that $0<\varepsilon_{0}<+\infty$. We
claim that

$W^{Z}_{\varepsilon,N}(0,h)\leq\delta_{0}\|h\|^{2}\;\mbox{ for all }\;h\in
H,\;\;\mbox{when}\;\;N\geq
N_{0}\;\;\mbox{and}\;\;\varepsilon\in(0,\varepsilon_{0}],$ (3.29)

where $N_{0}$ is given by (3.24). In fact, it follows from (3.28) that

$\varepsilon\|{\cal
L}(Ph)(\cdot)\|_{L^{2}(\mathbb{R}^{+};Z)}^{2}\leq\varepsilon_{0}\|{\cal
L}\|^{2}\|P\|^{2}\|h\|^{2}\leq(\delta_{0}-\delta_{0}^{2})\|h\|^{2},\;\;\mbox{when}\;\;h\in
H\;\;\mbox{and}\;\;\varepsilon\in(0,\varepsilon_{0}].$ (3.30)

By (3.16) and the fact that $P$ is a projection from $H$ to $H_{1}$ (see Lemma
2.2), we find

${\cal L}(Ph)\in L^{2}(\mathbb{R}^{+};Z)\;\;\mbox{for all}\;\;h\in H.$ (3.31)

Since $y^{Z}_{N}(\cdot;t,h,u\bigl{|}_{(0,NT)})=y(\cdot;t,h,u)\big{|}_{[t,NT]}$
for any $u\in L^{2}(\mathbb{R}^{+};Z)$, it holds that

$y^{Z}_{N}(NT;0,h,{\cal L}(Ph)\bigl{|}_{(0,NT)})=y(NT;0,h,{\cal L}(Ph)).$

This, together with (3.27), (3.26), (3.31), (3.30) and (3.15), indicates that

$W^{Z}_{\varepsilon,N}(0,h)\leq J^{Z}_{\varepsilon,N,0,h}({\cal
L}(Ph)\bigl{|}_{(0,NT)})=\varepsilon\displaystyle\int_{0}^{NT}\|{\cal
L}(Ph)(s)\|^{2}ds+\|y^{Z}_{N}(NT;0,h,{\cal
L}(Ph))\|^{2}\leq\delta_{0}\|h\|^{2},$

when $N\geq N_{0}$, $\varepsilon\in(0,\varepsilon_{0}]$, $h\in H$, i.e.,
(3.29) stands.

Step 4. To construct an $NT$-periodic $K^{Z}_{\varepsilon,N}(\cdot)\in
L^{\infty}(\mathbb{R}^{+};{\cal L}(H,Z))$

Arbitrarily fix an $\varepsilon\in(0,\varepsilon_{0}]$ and an $N\geq N_{0}$,
where $N_{0}$ and $\varepsilon_{0}$ are given by (3.24) and (3.28)
respectively. By the exactly same way to show Corollary 2.10 on Page 379 and
Theorem 4.3 on Page 397 in [14], we can verify that

$W^{Z}_{\varepsilon,N}(t,h)=\mathop{\langle}Q^{Z}_{\varepsilon,N}(t)h,h\mathop{\rangle},\;\;\mbox{when}\;\;h\in
H,$

where $Q^{Z}_{\varepsilon,N}(\cdot)\in C([0,NT);{\cal L}(H))$ has the
following properties: $(i)$ for each $t\geq 0$, $Q^{Z}_{\varepsilon,N}(t)$ is
self-adjoint; $(ii)$ it solves the Riccati integral equation:

$\begin{array}[]{r}{Q}(t)h=\Phi(NT,t)^{*}\Phi(NT,t)h-\displaystyle\frac{1}{\varepsilon}\int^{NT}_{t}\Phi(s,t)^{*}Q(s)^{*}D(s)\bigl{|}_{Z}\left(D(s)\bigl{|}_{Z}\right)^{*}Q(s)\Phi(s,t)hds,\\\
{\mbox{ for all }}h\in H{\mbox{ and }}t\in[0,NT).\end{array}$ (3.32)

Besides, it follows form (3.26) that

$0\leq\left\langle h,Q^{Z}_{\varepsilon,N}(t)h\right\rangle\leq
J^{Z}_{\varepsilon,N,t,h}(0)\leq\|\Phi(NT,t)\|^{2}\|h\|^{2}\;\mbox{ for all
}\;h\in H.$

Define $K^{Z}_{\varepsilon,N}(\cdot):[0,NT)\rightarrow{\cal L}(H;Z)$ by

$K^{Z}_{\varepsilon,N}(t)=-\displaystyle\frac{1}{\varepsilon}\left(D(s)\bigl{|}_{Z}\right)^{*}Q^{Z}_{\varepsilon,N}(t)\;\;\mbox{for
a.e.}\;\;t\in[0,NT).$ (3.33)

One can easily check that $K^{Z}_{\varepsilon,N}(\cdot)\in
L^{\infty}(0,NT;{\cal L}(H;Z))$. From this and assumptions $({\cal
H}_{1})$-$({\cal H}_{3})$, we see that (see Proposition 5.3 on Page 66 in
[14]) the feedback equation:

$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
y^{\prime}(s)+Ay(s)+B(s)y(s)=D(s)\bigl{|}_{Z}K^{Z}_{\varepsilon,N}(s)y(s)&\mbox{in}\;\;(0,NT),\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(0)=h\in H\end{array}\right.$ (3.34)

has a unique mild solution $y^{Z}_{\varepsilon,N}(\cdot;0,h)\in C([0,NT];H)$.
Let

$u^{Z}_{\varepsilon,N,0,h}(s)\mathop{\buildrel\Delta\over{=}}K^{Z}_{\varepsilon,N}(s)y^{Z}_{\varepsilon,N}(s;0,h)\qquad\mbox{for
a.e.}\;\;s\in(0,NT).$ (3.35)

By the state feedback representation of optimal controls for linear quadratic
control problems (see Section 3.4 in Chapter 9 in [14], in particular, (3.71)
on Page 392 and (4.12) on Page 397 in [14]), we see that
$u^{Z}_{\varepsilon,N,0,h}(\cdot)$ defined by (3.35) is the optimal control to
$(P)^{Z}_{\varepsilon,N,0,h}$. This, along with (3.27), yields that

$W^{Z}_{\varepsilon,N}(0,h)=J^{Z}_{\varepsilon,N,0,h}\big{(}u^{Z}_{\varepsilon,N,0,h}\big{)},\;\;\mbox{when}\;\;h\in
H.$ (3.36)

By (3.25) with $t=0$, (3.34) and (3.35), we see that
$y^{Z}_{\varepsilon,N}(NT;0,h)=y^{Z}_{N}(NT;0,h,u^{Z}_{\varepsilon,N,0,h})$.
From this, (3.26), (3.36) and (3.29), it follows that

$\|y^{Z}_{\varepsilon,N}(NT;0,h)\|^{2}\leq
J^{Z}_{\varepsilon,N,0,h}(u^{Z}_{\varepsilon,N,0,h})=W^{Z}_{\varepsilon,N}(0,h)\leq\delta_{0}\|h\|^{2},\;\;\mbox{when}\;\;h\in
H.$ (3.37)

Now, we extend $NT$-periodically $K^{Z}_{\varepsilon,N}(\cdot)$ over
$\mathbb{R}^{+}$ by setting

$K^{Z}_{\varepsilon,N}(t+kNT)=K^{Z}_{\varepsilon,N}(t)\;\;\mbox{for
all}\;\;t\in[0,NT),~{}k\in\mathbb{N}.$ (3.38)

Clearly, $K^{Z}_{\varepsilon,N}(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal
L}(H;Z))$ is $NT$-periodic.

Step 5. To prove that when $\varepsilon\in(0,\varepsilon_{0}]$ and $N\geq
N_{0}$ (where $N_{0}$ and $\varepsilon_{0}$ are given by (3.24) and (3.28),
respectively), $K^{Z}_{\varepsilon,N}(\cdot)$ defined by (3.33) and (3.38) is
an LMPFS law for Equation (1.1) with respect to $Z$

Consider the feedback equation:

$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
y^{\prime}(s)+Ay(s)+B(s)y(s)=D(s)\bigl{|}_{Z}K^{Z}_{\varepsilon,N}(s)y(s)&\mbox{in}\;\;\mathbb{R}^{+},\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(0)=h\in H.\end{array}\right.$ (3.39)

By assumptions $({\cal H}_{2})$ and $({\cal H}_{3})$, and by the fact that
$K^{Z}_{\varepsilon,N}(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal L}(H;Z))$, we
have

$B(\cdot)-D(\cdot)\bigl{|}_{Z}K^{Z}_{\varepsilon,N}(\cdot)\in
L^{1}_{loc}(\mathbb{R}^{+};{\cal L}(H)).$

Thus, for each $h\in H$, Equation (3.39) has a unique mild solution
$y^{Z}_{\varepsilon}(\cdot;0,h)\in C(\mathbb{R}^{+};H)$ (see Proposition 5.3
on Page 66 in [14]). Clearly,

$y^{Z}_{\varepsilon}(t;0,h)=y^{Z}_{\varepsilon,N}(t;0,h)\;\;\mbox{for
each}\;\;t\in[0,NT].$

Write $\\{\Phi^{Z}_{\varepsilon,N}(t,s)\bigl{\\}}_{0\leq s\leq t<+\infty}$ for
the evolution system generated by
$-A-B(\cdot)+D(\cdot)\bigl{|}_{Z}K^{Z}_{\varepsilon,N}(\cdot)$. Then it holds
that (see Proposition 5.7, Page 69, [14])

$y^{Z}_{\varepsilon,N}(NT;0,h)=y^{Z}_{\varepsilon}(NT;0,h)=\Phi^{Z}_{\varepsilon,N}(NT,0)h\;\;\mbox{
for all }\;\;h\in H.$

This, along with (3.37) and the fact that $\delta_{0}<1$ (see (3.15)), yields

$\|\Phi^{Z}_{\varepsilon,N}(NT,0)\|\leq\sqrt{\delta_{0}}<1.$ (3.40)

Since $B(\cdot)$ and $D(\cdot)$ are $T$-periodic and
$K^{Z}_{\varepsilon,N}(\cdot)$ is $NT$-periodic, it follows that

$\Phi^{Z}_{\varepsilon,N}(t+NT,s+NT)=\Phi^{Z}_{\varepsilon,N}(t,s),\;\;\mbox{when}\;\;0\leq
s\leq t<+\infty.$ (3.41)

By (3.41) and (3.40), one can easily shows that Equation (3.39) is
exponentially stable. Hence, $K^{Z}_{\varepsilon,N}(\cdot)$, with
$\varepsilon\in(0,\varepsilon_{0}]$ and $N\geq N_{0}$, is an LMPFS law for
Equation (3.39). This completes the proof.

###### Proposition 3.2

Let $Z$ be a subspace of $U$. Then, Equation (1.1) is LPFS with respect to $Z$
if and only if it is LMPFS with respect to $Z$.

Proof. It is clear that $\mbox{Equation}\;(\ref{state})\;\mbox{is LPFS
w.r.t.}\;Z\Rightarrow\mbox{Equation}\;(\ref{state})\;\mbox{is LMPFS
w.r.t.}\;Z$. We will show the reverse. Suppose that Equation (1.1) is LMPFS
with respect to $Z$. Then there are an $N\in\mathbb{N}$ and an $NT$-periodic
$\hat{K}^{Z}_{N}(\cdot)\in L^{\infty}(\mathbb{R}^{+};\mathcal{L}(H;Z))$ such
that the feedback equation

$y^{\prime}(s)+Ay(s)+B(s)y(s)=D(s)\bigl{|}_{Z}\hat{K}^{Z}_{N}(s)y(s),\;\;s\geq
0$ (3.42)

is exponentially stable. From this, assumptions
$(\mathcal{H}_{1})$-$(\mathcal{H}_{3})$ and the fact that
$\hat{K}^{Z}_{N}(\cdot)\in L^{\infty}(\mathbb{R}^{+};\mathcal{L}(H;Z))$ is
$NT$-periodic, one can easily verify that for each $t\geq 0$ and $h\in H$, the
solution $\hat{y}^{Z}_{N}(\cdot;t,h)$ to the equation

$\left\\{\begin{array}[]{l}y^{\prime}(s)+Ay(s)+B(s)y(s)=D(s)\bigl{|}_{Z}\hat{K}^{Z}_{N}(s)y(s)\;\;\mbox{in}\;\;[t,\infty),\\\
y(t)=h\end{array}\right.$ (3.43)

satisfies

$\|\hat{y}^{Z}_{N}(s;t,h)\|\leq
M_{1}e^{-\delta_{1}(s-t)}\|h\|,\;\;\mbox{when}\;\;s\geq
t\;\;\mbox{and}\;\;h\in H,$ (3.44)

where $M_{1}$ and $\delta_{1}$ are two positive constants independent of $h$,
$t$ and $s$. Write

$\hat{C}_{0}\triangleq\left\|\hat{K}^{Z}_{N}(\cdot)\right\|_{L^{\infty}(\mathbb{R}^{+};{\cal
L}(H;Z))}.$ (3.45)

The rest of the proof is organized by two steps as follows.

Step 1. To study a value function associated with a class of optimal control
problems

For each $t\geq 0$ and $h\in H$, we define an optimal control problem:

$(P)^{Z}_{t,h}:\;\;\;\;\;\displaystyle\inf\limits_{u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)}\int^{\infty}_{t}\left(\|y(s;t,h,u)\|^{2}+\|u(s)\|_{U}^{2}\right)ds.$
(3.46)

Associated with $\big{\\{}(P)^{Z}_{t,h}\big{\\}}_{t\geq 0,h\in H}$, we define
a value function $W^{Z}(\cdot,\cdot):\mathbb{R}^{+}\times
H\rightarrow\mathbb{R}$ by

$W^{Z}(t,h)=\displaystyle\inf\limits_{u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)}\int^{\infty}_{t}\left(\|y(s;t,h,u)\|^{2}+\|u(s)\|_{U}^{2}\right)ds,\;\;(t,h)\in\mathbb{R}^{+}\times
H.$ (3.47)

It is well defined. In fact, define a control $u^{Z}_{N,t,h}(\cdot)$ by
setting

$u^{Z}_{N,t,h}(s)\triangleq\left\\{\begin{array}[]{ll}K^{Z}_{N}(s)\hat{y}^{Z}_{N}(s;t,h)&\mbox{for
a.e.}\;\;s\in[t,\infty),\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr 0&\mbox{for
a.e.}\;\;s\in[0,t).\end{array}\right.$

By (3.44), we see that $u^{Z}_{N,t,h}(\cdot)\in L^{2}(\mathbb{R}^{+};Z)$.
Meanwhile, it is clear that
$\hat{y}^{Z}_{N}(\cdot;t,h)=y(\cdot;t,h,u^{Z}_{N,t,h})$. These, along with
(3.44) and (3.45), yields that for each $t\geq 0$ and $h\in H$,

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&0\leq
W^{Z}(t,h)\leq\displaystyle\int^{\infty}_{t}\left(\big{\|}\hat{y}^{Z}_{N}(s;t,h)\big{\|}^{2}+\big{\|}u^{Z}_{N,t,h}(s))\big{\|}_{U}^{2}\right)ds\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\leq\displaystyle\int^{\infty}_{t}\big{(}1+\big{\|}\hat{K}^{Z}_{N}(s)\big{\|}_{{\cal
L}(H;Z)}^{2}\big{)}\big{\|}\hat{y}^{Z}_{N}(s;t,h)\big{\|}^{2}ds\leq\displaystyle\frac{(1+\hat{C}_{0}^{2})M^{2}_{1}}{2\delta_{1}}\big{\|}h\big{\|}^{2}.\end{array}$
(3.48)

Thus $W^{Z}(\cdot,\cdot)$ is well-defined. By (3.47), one can directly check
that when $h,g\in H,t\geq 0$, $c\in\mathbb{R}$,

$W^{Z}(t,ch)=c^{2}W^{Z}(t,h)\;\;\mbox{and}\;\;\;W^{Z}(t,h+g)+W^{Z}(t,h-g)=2W^{Z}(t,h)+2W^{Z}(t,g).$

These, together with (3.48), imply that (see [13]) there is a unique
$Q^{Z}(\cdot):\mathbb{R}^{+}\rightarrow\mathcal{L}(H)$, with $Q^{Z}(t)$ self-
adjoint for each $t\geq 0$, such that

$W^{Z}(t,h)=\mathop{\langle}Q^{Z}(t)h,h\mathop{\rangle}\;\;\mbox{for
all}\;\;t\in\mathbb{R}^{+}\;\;\mbox{and}\;\;h\in H.$ (3.49)

This, together with (3.48), implies that

$0\leq
Q^{Z}(t)\leq\displaystyle\frac{(1+\hat{C}_{0}^{2})M^{2}_{1}}{2\delta_{1}}I\;\;\mbox{for
all}\;\;t\in\mathbb{R}^{+}.$

Meanwhile, from the $T$-periodicity of $B(\cdot)$ and $D(\cdot)$, one can
easily derive the $T$-periodicity of $W^{Z}(\cdot,h)$ for each $h\in H$. Thus,
by (3.49), $Q^{Z}(\cdot)$ is $T$-periodic, i.e.,

$Q^{Z}(t)=Q^{Z}(T+t)\;\;\mbox{for all}\;\;t\in\mathbb{R}^{+}.$ (3.50)

Now we present other properties of $Q^{Z}(\cdot)$. By the Bellman optimality
principle (see Section 1, Chapter 6 in [14]), it holds that for any
$t\in[0,T]$ and $h\in H$,

$W^{Z}(t,h)=\displaystyle\inf\limits_{u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)}\Big{\\{}\int^{T}_{t}\left(\|y(s;t,h,u)\|^{2}+\|u(s)\|_{U}^{2}\right)ds+W^{Z}(T,y(T;t,h,u))\Big{\\}}.$
(3.51)

This, along with (3.49) and (3.50), yields that for any $t\in[0,T]$ and $h\in
H$,

$W^{Z}(t,h)=\displaystyle\inf\limits_{u\in
L^{2}(\mathbb{R}^{+};Z)}\Big{\\{}\int^{T}_{t}\left(\|y(s;t,h,u)\|^{2}+\|u(s)\|_{U}^{2}\right)ds+\|\big{(}{Q^{Z}(0)}\big{)}^{1/2}y(T;t,h,u)\|\Big{\\}},$
(3.52)

i.e., $W^{Z}(\cdot,\cdot)\big{|}_{[0,T]\times H}$ is the value function
associated with the LQ problems $(P)^{Z,T}_{t,h}$, $t\in[0,T]$, $h\in H$:

$\displaystyle\inf\limits_{v\in
L^{2}(0,T;Z)}\Big{\\{}\int^{T}_{t}\left(\|y^{T}(s;t,h,v)\|^{2}+\|v(s)\|_{U}^{2}\right)ds+\mathop{\langle}y^{T}(T;t,h,v),Q^{Z}(0)y^{T}(T;t,h,v)\mathop{\rangle}\Big{\\}},$
(3.53)

where $y^{T}(\cdot;t,h,v)$ is the solution of Equation (1.1) (over $[t,T]$),
with the initial condition that $y(t)=h$ and with the control $v(\cdot)\in
L^{2}(0,T;Z)$. Furthermore, by the exactly same way to show Corollary 2.10 on
Page 379 and Theorem 4.3 on Page 397 in [14], we can obtain that

$W^{Z}(t,h)=\mathop{\langle}h,\hat{Q}(t)h\mathop{\rangle},\;\;\mbox{when}\;\;t\in[0,T]\;\;\mbox{and}\;\;h\in
H,$ (3.54)

where $\hat{Q}(\cdot)\in C([0,T);{\cal L}(H))$, with $\hat{Q}(t)$ self-adjoint
and non-negative for each $t\in[0,T]$, satisfies the following Riccati
integral equation:

$\begin{array}[]{rl}\hat{Q}(t)h=&\Phi(T,t)^{*}Q(0)\Phi(T,t)h+\displaystyle\int^{T}_{t}\Phi(s,t)^{*}\Phi(s,t)hds\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&-\displaystyle\int^{T}_{t}\Phi(s,t)^{*}\hat{Q}(s)^{*}D(s)\bigl{|}_{Z}\left(D(s)\bigl{|}_{Z}\right)^{*}\hat{Q}(s)\Phi(s,t)hds\;\;\mbox{for
all}\;\;h\in H,t\in[0,T).\end{array}$ (3.55)

Because $Q(t)$ and $\hat{Q}(t)$ are selfadjoint, it follows from (3.49) and
(3.54) that

$Q^{Z}(\cdot)\bigr{|}_{[0,T]}=\hat{Q}(\cdot).$ (3.56)

Step 2. To construct a $T$-periodic $\widetilde{K}(\cdot)\in
L^{\infty}(\mathbb{R}^{+};{\cal L}(H;Z))$

Let $Q^{Z}(\cdot)$ be given by (3.49). We define $\widetilde{K}(\cdot)\in
L^{\infty}(\mathbb{R}^{+};{\cal L}(H;Z))$ by setting

$\widetilde{K}(s)=-\displaystyle\left(D(s)\bigl{|}_{Z}\right)^{*}Q^{Z}(s)\;\;\mbox{for
a.e.}\;\;s\in\mathbb{R}^{+}.$ (3.57)

Since both $Q^{Z}(\cdot)$ and $D(\cdot)$ are $T$-periodic, so is
$\widetilde{K}(\cdot)$. Let $\widetilde{y}_{T}(\cdot;0,g)\in C([0,T];H)$ be
the mild solution to the feedback equation:

$\left\\{\begin{array}[]{l}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
y^{\prime}(s)+Ay(s)+B(s)y(s)=D(s)\bigl{|}_{Z}\widetilde{K}(s)y(s)\;\;\mbox{for
a.e.}\;\;s\in(0,T);\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr y(0)=g\in
H.\end{array}\right.$

The existence and uniqueness of the solution to the above equation is ensured
by of assumptions $({\cal H}_{1})$-$({\cal H}_{3})$ and the fact that
$\widetilde{K}(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal L}(H;Z))$ (see
Proposition 5.3 on Page 66 in [14]). For each $g\in H$, we define

$\widetilde{u}_{g}(t)\triangleq\widetilde{K}(t)\widetilde{y}_{T}(t;0,g)\;\;\mbox{for
a.e.}\;\;t\in(0,T).$

By the state feedback representation of optimal controls for linear quadratic
control problems (see Section 3.4 in Chapter 9, in particular the formula
(3.71) on Page 392 in [14]), it follows from (3.57) and (3.56) that
$\widetilde{u}_{g}(\cdot)$ is the optimal control to $(P)^{Z,T}_{0,g}$ (which
is defined by (3.53)). By (3.52), (3.54) and (3.56), we see that for each
$g\in H$,

$\mathop{\langle}g,Q^{Z}(0)g\mathop{\rangle}=\displaystyle\int^{T}_{0}\|\widetilde{y}_{T}(t;0,g)\|^{2}+\|\widetilde{u}_{g}(t)\|_{U}^{2}dt+\mathop{\langle}\widetilde{y}_{T}(T;0,g),Q^{Z}(0)\widetilde{y}_{T}(T;0,g)\mathop{\rangle}.$
(3.58)

Associated with each $h\in H$, we define three sequences
$\\{h_{k}\\}_{k=0}^{\infty}\subseteq H$,
$\\{y^{h}_{k}(\cdot)\\}_{k=1}^{\infty}\subseteq C([0,T];H)$ and
$\\{u^{h}_{k}(\cdot)\\}_{k=1}^{\infty}\subseteq L^{2}(0,T;Z)$ by setting

$h_{0}\triangleq
h,\;h_{k}\triangleq\widetilde{y}_{T}(T;0,h_{k-1}),\;y^{h}_{k}(\cdot)\triangleq\widetilde{y}_{T}(\cdot;0,h_{k-1}),\;u^{h}_{k}(\cdot)\triangleq\widetilde{u}_{h_{k-1}}(\cdot)\;\;\mbox{for
all}\;\;k\in\mathbb{N},$ (3.59)

Taking $g=h_{k-1}$ in (3.58), we find that

$\mathop{\langle}h_{k-1},Q^{Z}(0)h_{k-1}\mathop{\rangle}=\displaystyle\int^{T}_{0}\|y^{h}_{k}(t)\|^{2}+\|u^{h}_{k}(t)\|_{U}^{2}dt+\mathop{\langle}h_{k},Q^{Z}(0)h_{k}\mathop{\rangle}\;\;\mbox{for
each}\;\;k\in\mathbb{N}.$ (3.60)

Step 3. To prove that $\widetilde{K}$ given by (3.57) is an LPFS law for
Equation (1.1) with respect to $Z$

Consider the feedback equation:

$y^{\prime}(t)+Ay(t)+B(t)y(t)=D(t)\bigl{|}_{Z}\widetilde{K}(t)y(t)\;\;\mbox{in}\;\;\mathbb{R}^{+}.$
(3.61)

Because of assumptions $({\cal H}_{1})$-$({\cal H}_{3})$ and the fact that
$\widetilde{K}(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal L}(H;Z))$,
corresponding to each initial condition that $y(0)=h\in H$, Equation (3.61)
has a unique mild solution $y_{\widetilde{K}}(\cdot;0,h)\in
C(\mathbb{R}^{+};H)$ (see Proposition 5.3 on Page 66 in [14]). Let
$\\{\Phi_{\widetilde{K}}(t,s)\\}_{t\geq s\geq 0}$ for the evolution system
generated by $-A-B(\cdot)+D(\cdot)\bigl{|}_{Z}\widetilde{K}(\cdot)$. Then
$y_{\widetilde{K}}(t;0,h)=\Phi_{\widetilde{K}}(t,0)h$ for each $t\geq 0$ and
$h\in H$ (see Proposition 5.7 on Page 69 in [14]). By the $T$-periodicity of
$B(\cdot)$, $D(\cdot)$ and $\widetilde{K}(\cdot)$, and by (3.59), one can
easily check that

$\Phi_{\widetilde{K}}(t+T,s+T)=\Phi_{\widetilde{K}}(t,s),\;\;\mbox{when}\;\;t\geq
s\geq 0.$ (3.62)

Meanwhile, it follows from the definition of $y^{h}_{k}$ (see (3.59)) that
given $h\in H$,

$y_{\widetilde{K}}(t;0,h)=y^{h}_{[t/T]+1}(t-[t/T]T)\;\;{\mbox{ for all
}}\;\;t\in\mathbb{R}^{+}.$ (3.63)

We first claim that

$\int^{\infty}_{0}\|\Phi_{\widetilde{K}}(t,0)h\|^{2}dt\leq\|Q^{Z}(0)\|\|h\|^{2},\;\;\mbox{when}\;\;h\in
H.$ (3.64)

Indeed, it follows from (3.63) that

$\int^{\infty}_{0}\|\Phi_{\widetilde{K}}(t,0)h\|^{2}dt=\sum\limits^{\infty}_{k=1}\int^{kT}_{(k-1)T}\|y_{\widetilde{K}}(t,0,h)\|^{2}dt=\sum\limits^{\infty}_{k=1}\int^{T}_{0}\|y^{h}_{k}(t)\|^{2}dt.$
(3.65)

Meanwhile, by (3.60), we find that

$\sum\limits^{N}_{k=1}\mathop{\langle}h_{k-1},Q^{Z}(0)h_{k-1}\mathop{\rangle}=\sum\limits^{N}_{k=1}\big{[}\displaystyle\int^{T}_{0}\|y^{h}_{k}(t)\|^{2}+\|u^{h}_{k}(t)\|_{U}^{2}dt+\mathop{\langle}h_{k},Q^{Z}(0)h_{k}\mathop{\rangle}\big{]},\qquad
N\in\mathbb{N}.$

Thus

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\sum\limits^{\infty}_{k=1}\int^{T}_{0}\|y^{h}_{k}(t)\|^{2}dt=\lim\limits_{N\rightarrow\infty}\sum\limits^{N}_{k=1}\displaystyle\int^{T}_{0}\|y^{h}_{k}(t)\|^{2}dt\\\
=&\lim\limits_{N\rightarrow\infty}\big{[}\displaystyle\mathop{\langle}h,Q^{Z}(0)h\mathop{\rangle}-\sum\limits^{N}_{k=1}\displaystyle\int^{T}_{0}\|u^{h}_{k}(t)\|_{U}^{2}dt-\mathop{\langle}h_{N},Q^{Z}(0)h_{N}\mathop{\rangle}\big{]}\\\
\leq&\mathop{\langle}h,Q^{Z}(0)h\mathop{\rangle}\leq\|Q^{Z}(0)\|\|h\|^{2}.\end{array}$

This, along with (3.65), leads to (3.64).

Since $\widetilde{K}(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal L}(H;U))$ is
T-periodic, in order to show that $\widetilde{K}(\cdot)$ is an LPFS law for
Equation (1.1) with respect to $Z$, it suffices to prove that there are
positive $M$ and $\delta$ such that $\|y_{\tilde{K}(t;0,h)}\|\leq Me^{-\delta
t}\|h\|$ for all $h\in H$. This will be done if one can show that
$\hat{\delta}<1$, where

$\hat{\delta}\triangleq\lim\limits_{k\rightarrow\infty}\|\Phi_{\widetilde{K}}(T,0)^{k}\|^{1/k}.$
(3.66)

The reason is that $\Phi_{\tilde{K}}(\cdot,\cdot)$ is $T$-periodic and
$y_{\widetilde{K}}(t;0,h)=\Phi_{\widetilde{K}}(t,0)h$.

The rest is to prove that $\hat{\delta}<1$. First we can assume that

$\hat{\rho}\triangleq{\ln(\hat{\delta}+1)}/{T}>0,$ (3.67)

for otherwise $\hat{\delta}=0<1$. By (3.66) and (3.67), one can easily check
that there is a constant $\hat{C}_{1}>0$ such that

$\left\|\Phi_{\widetilde{K}}(kT,0)\right\|\leq\hat{C}_{1}e^{\hat{\rho}kT}\;\;\mbox{for
all}\;\;k\in\mathbb{N}.$ (3.68)

Because of $(\mathcal{H}_{1})$, $(\mathcal{H}_{2})$, $(\mathcal{H}_{3})$ and
the fact that $\widetilde{K}(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal
L}(H;U))$, one can easily check that
$\\{\|\Phi_{\widetilde{K}}(t,s)\|_{\mathcal{L}(H)}\\}_{0\leq s\leq t\leq T}$
is bounded. Thus, we can write

$\hat{C}_{2}\triangleq\max\limits_{0\leq t_{1}\leq t_{2}\leq
T}\left\|\Phi_{\widetilde{K}}(t_{2},t_{1})\right\|\in\mathbb{R}^{+}\;\;\mbox{and}\;\;\hat{C}_{3}\triangleq\max\\{\hat{C}_{2},~{}\hat{C}_{1}\hat{C}_{2}^{2}\\}\in\mathbb{R}^{+}.$
(3.69)

We claim that

$\|\Phi_{\widetilde{K}}(t,s)\|\leq\hat{C}_{3}e^{\hat{\rho}(t-s)},\;\;\mbox{when}\;\;t\geq
s\geq 0.$ (3.70)

In fact, given $0\leq s\leq t$, there are only two cases: $(i)$
$\left[{t}/{T}\right]=\left[{s}/{T}\right]$ and $(ii)$
$\left[{t}/{T}\right]\neq\left[{s}/{T}\right]$. In the first case, we have

$0\leq t-\left[{s}/{T}\right]\leq T\;\;\mbox{and}\;\;0\leq
s-\left[{s}/{T}\right]\leq T.$

These, along with (3.62) and (3.69), yields

$\|\Phi_{\widetilde{K}}(t,s)\|=\|\Phi_{\widetilde{K}}(t-\left[{s}/{T}\right]T,s-\left[{s}/{T}\right]T)\leq\hat{C}_{2}\leq\hat{C}_{3}e^{\hat{\rho}(t-s)},$

i.e., (3.70) holds for the first case. In the second case, we have

$\left[t/T\right]T\geq\left[s/T\right]T+T\;\;\mbox{and}\;\;(\left[t/T\right]-\left[s/T\right]-1)T\leq
t-s.$

These, together with (3.62), (3.68) and (3.69), yields

$\begin{array}[]{rl}&\big{\|}\Phi_{\widetilde{K}}(t,s)\big{\|}\leq\big{\|}\Phi_{\widetilde{K}}\big{(}t,\,\big{[}{t}/{T}\big{]}T\big{)}\big{\|}\cdot\big{\|}\Phi_{\widetilde{K}}\big{(}\big{[}{t}/{T}\big{]}T,\,\big{[}{s}/{T}\big{]}T+T\big{)}\big{\|}\cdot\big{\|}\Phi_{\widetilde{K}}\big{(}\big{[}{s}/{T}\big{]}T+T,\,s\big{)}\big{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\leq\big{\|}\Phi_{\widetilde{K}}\big{(}t-\big{[}{t}/{T}\big{]}T,\,0\big{)}\big{\|}\cdot\big{\|}\Phi_{\widetilde{K}}\big{(}\big{(}\big{[}{t}/{T}\big{]}-\big{[}{s}/{T}\big{]}-1\big{)}T,~{}0\big{)}\big{\|}\cdot\big{\|}\Phi_{\widetilde{K}}\big{(}T,\,s-\big{[}{s}/{T}\big{]}T\big{)}\big{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\leq\hat{C}_{1}\hat{C}_{2}^{2}e^{\hat{\rho}(t-s)}\leq\hat{C}_{3}e^{\hat{\rho}(t-s)},\end{array}$

i.e., (3.70) holds for the second case. In summary, we conclude that (3.70)
stands.

Let

$\hat{C}_{4}\triangleq\max\Big{\\{}\max_{t\in[0,T]}\|\Phi_{\widetilde{K}}(t,0)\|,\;\hat{C}_{3}\sqrt{\|Q^{Z}(0)\|{2\hat{\rho}}/(1-e^{-2\hat{\rho}T}})\Big{\\}}\;\;\mbox{and}\;\;\hat{C}_{5}\triangleq\max\\{\hat{C}_{2},\hat{C}_{4}\hat{C}_{2}\\},$
(3.71)

where $\hat{C}_{2}$ and $\hat{C}_{3}$ are given by (3.69), $\hat{\rho}$ is
given by (3.67). Then, we claim

$\|\Phi_{\widetilde{K}}(t,s)\|\leq\hat{C}_{5},\;\;\mbox{when}\;\;t\geq s\geq
0\;(\mbox{where}\;\hat{C}_{5}\;\mbox{is given by}\;(\ref{WANGwgs3.76})).$
(3.72)

For this purpose, we first show

$\|\Phi_{\widetilde{K}}(t,0)\|\leq\hat{C}_{4}\;\;\mbox{for all}\;\;t\geq
0\;(\mbox{where}\;\hat{C}_{4}\;\mbox{is given by}\;(\ref{WANGwgs3.76})).$
(3.73)

In fact, by (3.70) and (3.64), we have

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\frac{1-e^{-2\hat{\rho}t}}{2\hat{\rho}}\|\Phi_{\widetilde{K}}(t,0)h\|^{2}=\displaystyle\int^{t}_{0}e^{-2\hat{\rho}(t-r)}\|\Phi_{\widetilde{K}}(t,0)h\|^{2}dr\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\leq\displaystyle\int^{t}_{0}e^{-2\hat{\rho}(t-r)}\|\Phi_{\widetilde{K}}(t,r)\|^{2}\|\Phi_{\widetilde{K}}(r,0)h\|^{2}dr\leq\displaystyle\int^{t}_{0}e^{-2\hat{\rho}(t-r)}\hat{C}_{3}^{2}e^{2\hat{\rho}(t-r)}\|\Phi_{\widetilde{K}}(r,0)h\|^{2}dr\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&<\hat{C}_{3}^{2}\displaystyle\int^{\infty}_{0}\|\Phi_{\widetilde{K}}(r,0)h\|^{2}dr\leq\hat{C}_{3}^{2}\|Q^{Z}(0)\|\|h\|^{2},\;\;\mbox{when}\;\;h\in
H\;\;\mbox{and}\;\;t\geq 0,\end{array}$

from which, it follows that

$\displaystyle\|\Phi_{\widetilde{K}}(t,0)\|\leq\hat{C}_{3}\sqrt{\|Q^{Z}(0)\|{2\hat{\rho}}/(1-e^{-2\hat{\rho}t})}\;\;\mbox{for
all}\;\;t>0.$

This, along with (3.71), yields

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\sup\limits_{t\in\mathbb{R}^{+}}\|\Phi_{\widetilde{K}}(t,0)\|=\max\big{\\{}\max\limits_{t\in[0,T]}\|\Phi_{\widetilde{K}}(t,0)\|,\sup\limits_{t\in[T,\infty)}\|\Phi_{\widetilde{K}}(t,0)\|\big{\\}}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\leq\max\big{\\{}\max\limits_{t\in[0,T]}\|\Phi_{\widetilde{K}}(t,0)\|\,,~{}\hat{C}_{3}\sqrt{\|Q^{Z}(0)\|{2\hat{\rho}}/(1-e^{-2\hat{\rho}T})}\big{\\}}=\hat{C}_{4},\end{array}$

which leads to (3.73). Now, given $0\leq s\leq t$, there are only two
possibilities: $(i)$ $[t/T]=[s/T]$; and $(ii)$ $[t/T]>[s/T]$. In the first
case, (3.72) follows from (3.69) and (3.71). In the second case, we have that
$t\geq([s/T]+1)T\geq s$. This, along with (3.62), (3.73), (3.69) and (3.71),
indicates that

$\begin{array}[]{rl}\big{\|}\Phi_{\widetilde{K}}(t,s)\big{\|}&\leq\big{\|}\Phi_{\widetilde{K}}\big{(}t,\big{[}{s}/{T}\big{]}T+T\big{)}\|\cdot\|\Phi_{\widetilde{K}}\big{(}\big{[}{s}/{T}\big{]}T+T,s\big{)}\big{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&=\big{\|}\Phi_{\widetilde{K}}\big{(}t-\big{[}{s}/{T}\big{]}T-T,0\big{)}\big{\|}\cdot\big{\|}\Phi_{\widetilde{K}}\big{(}T,s-\big{[}{s}/{T}\big{]}T\big{)}\big{\|}\leq\hat{C}_{4}\hat{C}_{2}\leq\hat{C}_{5}.\end{array}$

In summary, we conclude that (3.72) holds.

Finally, it follows from (3.72) and (3.64) that when $t\geq 0$ and $h\in H$,

$t\big{\|}\Phi_{\widetilde{K}}(t,0)h\big{\|}^{2}=\displaystyle\int^{t}_{0}\big{\|}\Phi_{\widetilde{K}}(t,0)h\big{\|}^{2}ds\leq\displaystyle\int^{t}_{0}\big{\|}\Phi_{\widetilde{K}}(t,s)\big{\|}^{2}\big{\|}\Phi_{\widetilde{K}}(s)h\big{\|}^{2}ds\leq\hat{C}_{5}^{2}\big{\|}Q^{Z}(0)\big{\|}\big{\|}h\big{\|}^{2}.$

This implies that

$\big{\|}\Phi_{\widetilde{K}}(t,0)\big{\|}\leq\hat{C}_{5}\sqrt{{\|Q^{Z}(0)\|}/{t}},\;\;\mbox{when}\;\;t>0.$

Therefore, there is an $N_{0}\in\mathbb{N}$ such that
$\|\Phi_{\widetilde{K}}(N_{0}T,0)\|<1$. This, together with (3.66) and (3.62),
yields that

$\hat{\delta}=\lim\limits_{k\rightarrow\infty}\big{\|}\Phi_{\widetilde{K}}(T,0)^{N_{0}k}\big{\|}^{\frac{1}{N_{0}k}}=\lim\limits_{k\rightarrow\infty}\big{\|}\Phi_{\widetilde{K}}(N_{0}T,0)^{k}\big{\|}^{\frac{1}{N_{0}k}}\leq\big{\|}\Phi_{\widetilde{K}}(N_{0}T,0)\big{\|}^{\frac{1}{N_{0}}}<1.$

This completes the proof.

###### Proposition 3.3

When both $B(\cdot)$ and $D(\cdot)$ are time-invariant, i.e., $B(t)\equiv B$
and $D(t)\equiv D$ for all $t\geq 0$, the following statements are equivalent:

$(i)$ Equation (1.1) is linear $\hat{T}$-periodic feedback stabilizable for
some $\hat{T}>0$.

$(ii)$ Equation (1.1) is linear $\hat{T}$-periodic feedback stabilizable for
any $\hat{T}>0$.

$(iii)$ Equation (1.1) is linear time-invariant feedback stabilizable.

Proof. It suffices to show that $(i)\Rightarrow(iii)$. For this purpose, we
suppose that $(i)$ holds. Let $N\in\mathbb{N}$ with $N\geq 2$ and let
$T=\hat{T}/N$. Since $B(\cdot)$ and $D(\cdot)$ are time-invariant, Equation
(1.1) is $T$-periodic. Because of $(i)$, there is an $NT$-periodic
$\hat{K}^{U}_{N}(\cdot)\in L^{\infty}(\mathbb{R}^{+};\mathcal{L}(H;U))$ such
that the feedback equation (3.42), where $Z=U$, is exponentially stable. Now,
by the same way to show that Equation (1.1) is LMPFS $\Rightarrow$ Equation
(1.1) is LPFS in the proof of Proposition 3.2 (where $Z=U$), we see that
$\widetilde{K}(\cdot)$ given by (3.57), where $Z=U$, is a LPFS law for
Equation (1.1). We claim that this $\widetilde{K}(\cdot)$ is time-invariant in
the case that $B(\cdot)$ and $D(\cdot)$ are time-invariant. When this is done,
$\widetilde{K}(\cdot)\equiv\widetilde{K}\in{\cal L}(H;U)$ is a feedback law
for Equation (1.1), which leads to $(iii)$.

The rest is to show that $\widetilde{K}(\cdot)$ is time-invariant. By the
time-invariance of $D(\cdot)$, and by (3.49) and (3.57), where $Z=U$, it
suffices to show the valued function $W^{U}(t,h)$, given by (3.47) with $Z=U$
is time-invariant. The later will be proved as follows. Since Equation (1.1)
is time-invariant, we have that for each $t\in\mathbb{R}^{+}$, $h\in H$ and
$u(\cdot)\in L^{2}(\mathbb{R}^{+};U)$,

$y(s;t,h,u)=y(s-t;0,h,v)\;\;\mbox{for all}\;\;s\geq t,$

where $v(\cdot)$ is defined by $v(s)=u(s+t)$ for all $s\geq 0$. Hence, given
$t\in\mathbb{R}^{+}$ and $h\in H$,

$\displaystyle\int^{\infty}_{t}\left(\|y(s;t,h,u(s))\|^{2}+\|u(s)\|_{U}^{2}\right)ds=\displaystyle\int^{\infty}_{0}\left(\|y(r;0,h,u(r+t))\|^{2}+\|u(r+t)\|_{U}^{2}\right)dr,$

for all $u(\cdot)\in L^{2}(\mathbb{R}^{+};U)$. Taking the infimum on the both
sides of the above equation with respect to $u(\cdot)\in
L^{2}(\mathbb{R}^{+};U)$, we get that $W^{U}(t,h)=W^{U}(0,h)$, i.e., the value
function $W^{U}(t,h)$ is time-invariant. This completes the proof.

###### Remark 3.1

By Proposition 3.3, we see that linear time-periodic functions $K(\cdot)$ will
not aid in the linear stabilization of Equation (1.1) when both $B(\cdot)$ and
$D(\cdot)$ are time-invariant. On the other hand, when Equation (1.1) is
$T$-periodically time-varying, linear time-periodic functions $K(\cdot)$ do
aid in the linear stabilization of Equation (1.1). This can be seen from the
following $2$-periodic ordinary differential equation:

$y^{\prime}(t)=\sum\limits_{j=1}^{\infty}\left[\chi_{[2j,2j+1)}(t)-\chi_{[2j+1,2j+2)}(t)\right]u(t).$

For each $k\in\mathbb{R}$, consider the feedback equation

$y^{\prime}(t)=\sum\limits_{j=1}^{\infty}\left[\chi_{[2j,2j+1)}(t)-\chi_{[2j+1,2j+2)}(t)\right]ky(t).$

Clearly, the corresponding Poincar$\acute{e}$ map ${\cal P}_{k}\equiv 1$. Thus
any linear time-invariant feedback equation is not exponentially stable. On
the other hand, by a direct computation, one can easily check that the
2-periodic time-varying feedback law given by

$k(t)=\sum\limits_{j=1}^{\infty}\left[\chi_{[2j,2j+1)}(t)+2\chi_{[2j+1,2j+2)}(t)\right].$

is an LPFS law.

##  4 The proof of Theorem 1.1 and Theorem 1.2

Proof of Theorem 1.1. $(a)\Leftrightarrow(b)$: We first show that
$(b)\Rightarrow(a)$. Suppose that a subspace $Z$ of $U$ satisfies (1.17). Then
by Lemma 2.5, there is a finite dimension subspace $\hat{Z}$ of $Z$ such that
$\hat{V}^{\hat{Z}}_{n_{0}}=H_{1}.$ From this, we can apply Proposition 3.1 to
get that Equation (1.1) is LMPFS with respect to $\hat{Z}$. This, along with
Proposition 3.2, yields that Equation (1.1) is LPFS with respect to $\hat{Z}$.
Because $\hat{Z}$ is a subspace of $Z$, Equation (1.1) is also LPFS with
respect to $Z$, i.e., $(a)$ stands.

We next show that $(a)\Rightarrow(b)$. Seeking for a contradiction, we suppose
that $Z\in\mathcal{U}^{FS}$, but (1.17) does not hold. Then
$\hat{V}^{Z}_{n_{0}}$ would be a proper subspace of $H_{1}$. This, along with
(2.18), yields that $\hat{V}^{Z}$ is a proper subspace of $H_{1}$. Write
$(\hat{V}^{Z})^{\bot}$ for the orthogonal complement subspace of $\hat{V}^{Z}$
in $H$. Then, one can directly check that $H_{1}\bigcap(\hat{V}^{Z})^{\bot}$
is the orthogonal complement subspace of $\hat{V}^{Z}$ in $H_{1}$, i.e.,

$\big{(}H_{1}\bigcap(\hat{V}^{Z})^{\bot}\big{)}\bot\hat{V}^{Z};\;\;H_{1}=\hat{V}^{Z}\bigoplus\big{(}H_{1}\bigcap(\hat{V}^{Z})^{\bot}\big{)}.$
(4.1)

Since $\hat{V}^{Z}$ is a proper subspace of $H_{1}$ and
$\mbox{dim}H_{1}=n_{0}$ (see (2.4)), we have

$n_{0}\geq l\triangleq{\rm
dim}\left(H_{1}\bigcap(\hat{V}^{Z})^{\bot}\right)\geq 1.$ (4.2)

By (4.1) and (4.2), we can let $\\{\eta_{1},\dots,\eta_{n_{0}}\\}$ be a basis
of $H_{1}$ such that $\\{\eta_{1},\cdots,\eta_{l}\\}$ and
$\\{\eta_{l+1},\cdots,\eta_{n_{0}}\\}$ are bases of
$H_{1}\bigcap(\hat{V}^{Z})^{\bot}$ and $\hat{V}^{Z}$, respectively. By (2.18)
in Lemma 2.4, $\hat{V}^{Z}$ is an invariant subspace under ${\cal P}_{1}$ .
Thus there are matrices $A_{1}\in\mathbb{R}^{l\times l}$,
$A_{2}\in\mathbb{R}^{(n_{0}-l)\times l}$,
$A_{3}\in\mathbb{R}^{(n_{0}-l)\times(n_{0}-l)}$ such that

${\cal
P}_{1}\left(\begin{array}[]{cc}\eta_{1},\cdots,\eta_{l},\bigm{|}\eta_{l+1},\cdots,\eta_{n_{0}}\end{array}\right)=\left(\begin{array}[]{cc}\eta_{1},\cdots,\eta_{l},\bigm{|}\eta_{l+1},\cdots,\eta_{n_{0}}\end{array}\right)\left(\begin{array}[]{cc}A_{1}&0_{l\times(n_{0}-l)}\\\
A_{2}&A_{3}\end{array}\right).$ (4.3)

Let $P_{11}$ be the orthogonal projection from $H_{1}$ onto
$H_{1}\bigcap(\hat{V}^{Z})^{\bot}$. Define a linear bijection
$\mathcal{J}:\mathbb{R}^{l}\rightarrow\big{(}H_{1}\bigcap(\hat{V}^{Z})^{\bot}\big{)}$
by setting

$\mathcal{J}(\alpha)\mathop{\buildrel\Delta\over{=}}(\eta_{1},\dots,\eta_{l})\alpha,\;\;\alpha\in\mathbb{R}^{l},$
(4.4)

where $\alpha$ denotes the column vectors. By (4.3) and (4.4), we see that

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&P_{11}{\cal
P}_{1}^{k}\mathcal{J}(\alpha)=P_{11}{\cal
P}_{1}^{k}(\eta_{1},\cdots,\eta_{l})\alpha\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr=&P_{11}(\eta_{1},\cdots,\eta_{l},\bigm{|}\eta_{l+1},\cdots,\eta_{n_{0}})\left(\begin{array}[]{cc}A_{1}&0_{l\times(n_{0}-l)}\\\
A_{2}&A_{3}\end{array}\right)^{k}\left(\begin{array}[]{l}\alpha\\\
0_{(n_{0}-l)\times 1}\end{array}\right)\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr=&(\eta_{1},\cdots,\eta_{l})A_{1}^{k}\alpha,\;\;\mbox{when}\;\;\alpha\in\mathbb{R}^{l}\;\;\mbox{and}\;\;k\in\mathbb{N}.\end{array}$
(4.5)

On the other hand, since $Z\in\mathcal{U}^{FS}$, there is a $T$-periodic
$K(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal L}(H;Z))$ such that Equation
(1.2) is exponentially stable, which implies that

$\lim\limits_{t\rightarrow+\infty}y_{K}\left(t;0,h)\right)=0,\;\;\mbox{when}\;\;h\in
H,$ (4.6)

where $y_{K}\left(\cdot;0,h)\right)$ denotes the solution of Equation (1.2)
with the initial condition that $y(0)=h$. Let $u_{K}^{h}(t)=K(t)y_{K}(t;0,h)$
for a.e. $t\geq 0$. The by Proposition 5.7 on Page 69 in [14], we have

$y_{K}(t;0,h)=\Phi(t,0)h+\displaystyle\int^{t}_{0}\Phi(t,s)D(s)u^{h}_{K}(s)ds,\;\;\mbox{when}\;\;t\in\mathbb{R}^{+}\;\;\mbox{and}\;\;h\in
H.$ (4.7)

From (4.7) and (1.15), it follows that

$P_{11}Py_{K}\left(kT;0,h)\right)\in P_{11}P({\cal
P}^{k}h+V^{Z}_{k})\;\;\mbox{for all}\;\;h\in
H\;\;\mbox{and}\;\;k\in\mathbb{N}.$ (4.8)

Since $PV^{Z}_{k}\triangleq\hat{V}^{Z}_{k}\subseteq\hat{V}^{Z}$ for all
$k\in\mathbb{N}$ (see (1.16) and (2.19)) and $P_{11}$ is the orthogonal
projection from $H_{1}$ onto $H_{1}\bigcap(\hat{V}^{Z})^{\bot}$, and because
of (4.1), we have

$P_{11}PV^{Z}_{k}\subset P_{11}\hat{V}^{Z}=\\{0\\}.$

This, along with (4.8) and the fact that $P{\cal P}^{k}={\cal P}^{k}P$ for all
$k\in\mathbb{N}$ (see Parts $(a)$ and $(e)$ in Lemma 2.2), indicates that

$P_{11}Py_{K}(kT;0,h)=P_{11}P{\cal P}^{k}h=P_{11}{\cal P}^{k}Ph\;\;\mbox{for
all}\;\;h\in H\;\;\mbox{and}\;\;k\in\mathbb{N}.$ (4.9)

Since $P:H\rightarrow H_{1}$ is a projection (see Lemma 2.2), it follows from
(4.6) and (4.9) that

$\lim\limits_{k\rightarrow+\infty}P_{11}{\cal
P}^{k}h_{1}=0,\;\;\mbox{when}\;\;h\in H_{1}.$ (4.10)

Now by (4.10), (4.5) and (2.15), we have that
$\lim\limits_{k\rightarrow\infty}A_{1}^{k}\alpha=0$, when
$\alpha\in\mathbb{R}^{l}$, from which, it follows that

$\sigma(A_{1})\in\mathbb{B}\;\;(\mbox{the open unit ball
in}\;\;\mathbb{C}^{1}).$ (4.11)

By (4.3), it holds that $\sigma(A_{1})\subset\sigma({\cal P}_{1})$. This,
together with (4.11) and (2.16), leads to a contradiction. Hence,
$(a)\Rightarrow(b)$. This completes the proof of $(a)\Leftrightarrow(b)$.

$(b)\Leftrightarrow(c)$: First of all, we introduce two complex adjoint
equations as follows:

$\psi^{\prime}(t)-A^{*C}\psi(t)-B(t)^{*C}\psi(t)=0\;\;\mbox{in}\;\;(0,n_{0}T),\;\;\psi(n_{0}T)\in
H^{C};$ (4.12)
$\psi^{\prime}(t)-A^{*C}\psi(t)-B(t)^{*C}\psi(t)=0\;\;\mbox{in}\;\;(0,T),\;\;\psi(T)\in
H^{C}.$ (4.13)

For each $\xi\in H^{C}$, Equation (4.12) (or (4.13)) with the initial
condition that $\psi^{\xi}_{n_{0}}(n_{0}T)={\xi}$ (or $\psi^{\xi}(T)={\xi}$)
has a unique solution in $C[0,n_{0}T];H^{C})$ (or $C([0,T];H^{C})$). We denote
this solution by $\psi^{\xi}_{n_{0}}(\cdot)$ (or $\psi^{\xi}(\cdot)$).
Clearly, when $\xi\in H$, $\psi^{\xi}_{n_{0}}(\cdot)\in C[0,n_{0}T];H)$ and
$\psi^{\xi}(\cdot)\in C([0,T];H)$ are accordingly the solutions of (4.12) and
(4.13) where $A^{C}$ and $B(t)^{C}$ are replaced by $A$ and $B(t)$
respectively. One can easily check that

$\psi^{\xi}(0)={\cal
P}^{*C}\xi\;\;\mbox{and}\;\;\psi^{\xi}_{n_{0}}(0)=\left({\cal
P}^{*C}\right)^{n_{0}}\xi\;\;\mbox{for all}\;\;\xi\in H^{C}.$ (4.14)

By the $T$-periodicity of $B^{*}(\cdot)$, we see that for each $\xi\in H^{C}$,

$\psi^{\xi}_{n_{0}}((k-1)T+t)=\psi^{{}^{\xi_{k}}}(t),\;t\in[0,T],\,k\in\\{1,\dots,n_{0}\\},\;\;\mbox{where}\;\;\xi_{k}\mathop{\buildrel\Delta\over{=}}\left({\cal
P}^{*C}\right)^{n_{0}-k}\xi.$ (4.15)

Now we carry out the proof of $(b)\Leftrightarrow(c)$ by several steps as
follows.

Step 1. To prove that (1.17) is equivalent to the following condition:

$\forall\;h\in H,\;\exists\;u^{h}(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\;\mbox{s.t.}\;Py(n_{0}T;0,h,u^{h})=0,\;\;\mbox{where}\;\;P\;\;\mbox{is
given by}\;\;(\ref{wgs1.15}).$ (4.16)

Suppose that (1.17) holds. Then by (1.15), we have

$P\Big{\\{}\displaystyle\int^{n_{0}T}_{0}\Phi(n_{0}T,t)D(t)u(t)dt\bigm{|}u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\Big{\\}}=H_{1}.$ (4.17)

Given $h\in H$, it holds that $P\Phi(n_{0}T,0)h\in H_{1}$ (see (2.2)). This,
along with (4.17), yields that there is a $u^{h}(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)$ such that

$Py(n_{0}T;0,h,u^{h})=P\Phi(n_{0}T,0)h+P\displaystyle\int^{n_{0}T}_{0}\Phi(n_{0}T,t)D(t)u^{h}(t)dt=0,$

which leads to (4.16).

Assume that (4.16) holds. Then for any $h\in H$, there exists $u^{h}(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)$ such that $Py(n_{0}T;0,h,u^{h})=0$, i.e.,

$-P\Phi(n_{0}T,0)h=P\displaystyle\int_{0}^{n_{0}T}\Phi(n_{0}T,t)D(t)u^{h}(t)dt.$

Thus, we have

$\begin{array}[]{rl}H_{1}&\supseteq\hat{V}_{n_{0}}^{Z}\mathop{\buildrel\Delta\over{=}}P\Big{\\{}\displaystyle\int^{n_{0}T}_{0}\Phi(n_{0}T,t)D(t)u(t)dt\bigm{|}u(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)\Big{\\}}\\\ &\supseteq
P\Big{\\{}\displaystyle\int^{n_{0}T}_{0}\Phi(n_{0}T,t)D(t)u^{h}(t)dt\bigm{|}h\in
H\Big{\\}}\\\ &=-P\left\\{\Phi(n_{0}T,0)h\bigm{|}h\in H\right\\}=P{\cal
P}^{n_{0}}H.\end{array}$ (4.18)

By the facts that $P{\cal P}={\cal P}P$ (see (2.5)), $PH=H_{1}$ and ${\cal
P}H_{1}={\cal P}_{1}H_{1}=H_{1}$ (see (2.15) and lemma 2.4), we see that
$P{\cal P}^{n_{0}}H=H_{1}$. This, together with (4.18), leads to (1.17).

Step 2. To show that $\xi\in P^{*}H_{1}$ and $\psi_{n_{0}}^{\xi}(0)=0$
$\Rightarrow$ $\xi=0$

Recall Lemma 2.3. Because $\tilde{H}_{1}$ is a invariant subspace of ${\cal
P}^{*}$, it follows from (4.14) that

$\psi_{n_{0}}^{\xi}(0)=({\cal P}^{*})^{n_{0}}\xi=\big{(}{\cal
P}^{*}\bigl{|}_{\tilde{H}_{1}}\big{)}^{n_{0}}\xi\in\tilde{H}_{1},\;\mbox{when}\;\xi\in\tilde{H}_{1}.$
(4.19)

By Lemma 2.3, we have

$\sigma\big{(}{\cal P}^{*C}|_{\tilde{H}_{1}^{C}}\big{)}\bigcap{\cal
B}=\varnothing\;\;\mbox{and}\;\;\mbox{dim}\tilde{H}_{1}=n_{0}<\infty.$

Thus, the map $\big{(}{\cal P}^{*}\bigl{|}_{\tilde{H}_{1}}\big{)}^{n_{0}}$ is
invertible from $\tilde{H}_{1}$ onto $\tilde{H}_{1}$. Then by (4.19), we see
that $\xi=0$ when $\xi\in\tilde{H}_{1}$ and $\psi_{n_{0}}^{\xi}(0)=0$. This,
together with (2.13), implies that $\xi=0$ when $\xi\in P^{*}H_{1}$ and
$\psi_{n_{0}}^{\xi}(0)=0$.

Step 3. To show that (4.16) $\Rightarrow$ (1.18)

Clearly, when $\eta,\,h\in H$ and $u(\cdot)\in L^{2}(\mathbb{R}^{+};Z)$,

$\big{\langle}\psi^{\eta}_{n_{0}}(0),\,h\big{\rangle}=\bigr{\langle}\eta,\,y(n_{0}T;0,h,u)\bigl{\rangle}-\displaystyle\int^{n_{0}T}_{0}\big{\langle}\bigr{(}D(t)\bigl{|}_{Z}\bigl{)}^{*}\psi^{\eta}_{n_{0}}(t),\,u(t)\big{\rangle}dt.$
(4.20)

Let $\xi\in P^{*}H_{1}$ satisfy the conditions on the left side of (1.18).
Then by (4.20) where $\eta=\xi$ and
$\psi^{\xi}_{n_{0}}(t)=\Phi(n_{0}T,t)^{*}\xi$, we find

$\big{\langle}\psi^{\xi}_{n_{0}}(0),\,h\big{\rangle}=\bigr{\langle}\xi,\,y(n_{0}T;0,h,u)\bigl{\rangle},{\mbox{
when }}h\in H{\mbox{ and }}u(\cdot)\in L^{2}(\mathbb{R}^{+};Z).$ (4.21)

By (4.16), given $h\in H$, there is a $u^{h}(\cdot)\in
L^{2}(\mathbb{R}^{+};Z)$ such that

$Py(n_{0}T;0,h,u^{h})=0.$ (4.22)

Since $\xi\in P^{*}H_{1}$, there is $g\in H_{1}$ such that $\xi=P^{*}g$. This,
along with (4.21) and (4.22), indicates

$\big{\langle}\psi^{\xi}_{n_{0}}(0),\,h\big{\rangle}=\bigr{\langle}\xi,\,y(n_{0}T;0,h,u^{h})\bigl{\rangle}=\bigr{\langle}P^{*}g,\,y(n_{0}T;0,h,u^{h})\bigl{\rangle}=\bigr{\langle}g,\,Py(n_{0}T;0,h,u^{h})\bigl{\rangle}=0,\;\forall\;h\in
H.$

Hence, $\psi^{\xi}_{n_{0}}(0)=0$. Then by the result in Step 2, we have
$\xi=0$, i.e., (1.18) holds.

Step 4. To show that (1.18) $\Rightarrow$ (4.16)
Assume that (1.18) holds. Define two subspaces

$\Gamma\mathop{\buildrel\Delta\over{=}}\left\\{\left(D(\cdot)\bigl{|}_{Z}\right)^{*}\psi^{\xi}_{n_{0}}(\cdot)\bigm{|}\xi\in
P^{*}H_{1}\right\\}\subseteq
L^{2}(0,n_{0}T;Z)\;\mbox{and}\;\Gamma_{0}\mathop{\buildrel\Delta\over{=}}\left\\{\psi^{\xi}_{n_{0}}(0)\bigm{|}\xi\in
P^{*}H_{1}\right\\}\subseteq H.$ (4.23)

Let

${\cal
L}_{1}\left(\left(D(\cdot)\bigl{|}_{Z}\right)^{*}\psi^{\xi}_{n_{0}}(\cdot)\right)=\psi^{\xi}_{n_{0}}(0)\;\;{\mbox{
for all }}\;\;\xi\in P^{*}H_{1}.$ (4.24)

By (1.18) and the result in Step 2, we see that ${\cal
L}_{1}:\Gamma\mapsto\Gamma_{0}$ is well defined. Clearly, it is linear. Given
$h\in H$, define a linear functional ${\cal F}^{h}$ on $\Gamma$ by

${\cal F}^{h}(\gamma)=\mathop{\langle}{\cal
L}_{1}(\gamma),h\mathop{\rangle}\;\;{\mbox{ for all }}\;\;\gamma\in\Gamma.$
(4.25)

Since $\mbox{dim}(P^{*}H_{1})=\mbox{dim}\tilde{H}_{1}=n_{0}<\infty$, it holds
that $\mbox{dim}\Gamma<\infty$. Thus, ${\cal F}^{h}\in{\cal
L}(\Gamma;\mathbb{R})$. By the Hahn-Banach theorem, there is a $\bar{\cal
F}^{h}\in{\cal L}(L^{2}(0,n_{0}T;Z);\mathbb{R})$ such that

$\bar{\cal F}^{h}(\gamma)={\cal F}^{h}(\gamma)\;\;\mbox{for
all}\;\;\gamma\in\Gamma;\;\;\mbox{and}\;\;\|\bar{\cal F}^{h}\|=\|{\cal
F}^{h}\|.$ (4.26)

By making use of the Riesz representation theorem (see Page 59 in [9]), there
exists a function $u^{h}(\cdot)\in L^{2}(0,n_{0}T;Z)$ such that

$\bar{\cal
F}^{h}(\gamma)=-\displaystyle\int^{n_{0}T}_{0}\mathop{\langle}u^{h}(t),\gamma(t)\mathop{\rangle}_{U}dt\;\;\mbox{for
all}\;\;\gamma\in L^{2}(0,n_{0}T;Z).$ (4.27)

Now, since $P^{*}H_{1}=P^{*}H$ (see (2.13)), it follows from (4.24), (4.25),
(4.26) and (4.27) that

$-\displaystyle\int^{n_{0}T}_{0}\big{\langle}\bigr{(}D(t)\bigl{|}_{Z}\bigl{)}^{*}\psi^{P^{*}\eta}_{n_{0}}(t),\,u^{h}(t)\big{\rangle}dt=\big{\langle}\psi^{P^{*}\eta}_{n_{0}}(0),\,h\big{\rangle}\;\;\mbox{
for all }\;\;\eta\in H.$

Meanwhile, it follows by (4.20) that for each $\eta\in H$,

$\big{\langle}\psi^{P^{*}\eta}_{n_{0}}(0),\,h\big{\rangle}=\bigr{\langle}{P^{*}\eta},\,y(n_{0}T;0,h,u^{h})\bigl{\rangle}-\displaystyle\int^{n_{0}T}_{0}\big{\langle}\bigr{(}D(t)\bigl{|}_{Z}\bigl{)}^{*}\psi^{P^{*}\eta}_{n_{0}}(t),\,u^{h}(t)\big{\rangle}dt$

The above two equalities imply that

$\bigr{\langle}\eta,\,Py(n_{0}T;0,h,u^{h})\bigl{\rangle}=\bigr{\langle}{P^{*}\eta},\,y(n_{0}T;0,h,u^{h})\bigl{\rangle}=0\;\;{\mbox{
for all }}\;\;\eta\in H,$

i.e., $Py(n_{0}T;0,h,u^{h})=0$, which leads to (4.16).

From Step 1-Step 4, one can easily check that $(b)\Leftrightarrow(c)$.

$(c)\Leftrightarrow(d)$: We first show that $(c)\Rightarrow(d)$. Suppose that
a subspace $Z$ of $U$ satisfies (1.18). Let $\mu$ and $\xi$ satisfy the
conditions on the left side of (1.19) with the aforementioned $Z$. Then by
(2.14) (see Lemma 2.3), it holds that $\xi\in{\tilde{H}_{1}^{C}}$. Hence, we
can write $\xi\mathop{\buildrel\Delta\over{=}}\xi_{1}+i\xi_{2}$ with
$\xi_{1},\,\xi_{2}\in\tilde{H}_{1}$. By (2.13), we have $\xi_{1},\xi_{2}\in
P^{*}H_{1}$. By the last condition on the left side of (1.19), we have

$\left(D(t)\bigl{|}_{Z}\right)^{*C}\psi^{\xi}(t)=0\;\;\mbox{for
a.e.}\;\;t\in(0,T).$

By (4.15) and the third condition on the left side in (1.19), it holds that

$\psi^{\xi}_{n_{0}}((k-1)T+t)=\psi^{\mu^{n_{0}-k}\xi}(t)=\mu^{n_{0}-k}\psi^{\xi}(t)\;\;\mbox{for
all}\;\;t\in[0,T],k=1,\dots,n_{0}.$

Notice that
$\psi^{\xi}_{n_{0}}(\cdot)=\psi^{\xi_{1}}_{n_{0}}(\cdot)+i\psi^{\xi_{2}}_{n_{0}}(\cdot)$.
This, along with the above two equalities, yields that

$\left(D(\cdot)\bigl{|}_{Z}\right)^{*}\psi^{\xi_{1}}_{n_{0}}(\cdot)+i\left(D(\cdot)\bigl{|}_{Z}\right)^{*}\psi^{\xi_{2}}_{n_{0}}(\cdot)=\left(D(\cdot)\bigl{|}_{Z}\right)^{*}\psi^{\xi}_{n_{0}}(\cdot)=0\;\;\mbox{over}\;\;(0,n_{0}T).$

Since $\xi_{1},\xi_{2}\in P^{*}H_{1}$, the above-mentioned equality, along
with (1.18), leads to $\xi_{1}=\xi_{2}=0$, i.e., $\xi=0$. Hence, $Z$ satisfies
(1.19). Thus, $(c)\Rightarrow(d)$.

We next show that $(d)\Rightarrow(c)$. Suppose that a subspace $Z$ satisfies
(1.19). In order to show that $Z$ satisfies (1.18), it suffices to prove

$\hat{\xi}\in(P^{*}H_{1})^{C}\;\;\mbox{and}\;\;\left(D(\cdot)\bigl{|}_{Z}\right)^{*C}\psi_{n_{0}}^{\hat{\xi}}(\cdot)=0\;\;\mbox{over}\;\;(0,n_{0}T)\Rightarrow\hat{\xi}=0.$
(4.28)

First of all, we notice that $(P^{*}H_{1})^{C}=\tilde{H}_{1}^{C}$ and
$\mbox{dim}\tilde{H}_{1}^{C}=n_{0}$ (see Lemma 2.3). In this step, we simply
write

${\cal Q}\mathop{\buildrel\Delta\over{=}}{\cal
P}^{*C}\bigl{|}_{\tilde{H}_{1}^{C}}\in{\cal
L}(\tilde{H}_{1}^{C})\;\;\mbox{and}\;\;D_{1}(\cdot)\mathop{\buildrel\Delta\over{=}}\big{(}\big{(}D(\cdot)\bigl{|}_{Z}\big{)}^{*C}\big{)}\bigl{|}_{(0,T)}\in
L^{2}(0,T;{\cal L}(H,Z)).$

By Lemma 2.3 and (2.8), we have that $\sigma({\cal
Q})=\\{\bar{\lambda}_{j}\\}_{j=1}^{n}$; $l_{j}$ is the algebraic multiplicity
of $\bar{\lambda}_{j}$. Hence,
$\mathbb{P}_{0}(\lambda)\triangleq\prod\limits^{n}_{j=1}\left(\lambda-\bar{\lambda}_{j}\right)^{l_{j}}$
is the characteristic polynomial of ${\cal Q}$. Write $\hat{l}_{j}$ for the
geometric multiplicity of $\bar{\lambda}_{j}$. Clearly, $\hat{l}_{j}\leq
l_{j}$ for all $j$. Let $\beta\triangleq\\{\beta_{1},\dots,\beta_{n_{0}}\\}$
be a basis of $(P^{*}H)^{C}=\tilde{H}_{1}^{C}$ such that

${\cal Q}(\beta_{1},\dots,\beta_{n_{0}})=J(\beta_{1},\dots,\beta_{n_{0}}).$
(4.29)

Here $J$ is the Jordan matrix: ${\rm
diag}\big{\\{}J_{11},\dots,J_{1\hat{l}_{1}},J_{21},~{}\dots,~{}J_{2\hat{l}_{2}},\dots,J_{n1},~{}\dots,~{}J_{n\hat{l}_{n}}\big{\\}}$
with

$J_{jk}=\left(\begin{array}[]{cccc}\bar{\lambda}_{j}&1\\\ &\ddots&\ddots\\\
&&\ddots&1\\\
&&&\bar{\lambda}_{j}\end{array}\right)\;\;\mbox{a}\;\;d_{jk}\times
d_{jk}\;\;\mbox{matrix},$

where $j=1,\dots,n$, $k=1,\dots,\hat{l}_{j}$, and for each $j$,
$\\{d_{jk}\\}_{k=1}^{\hat{l}_{j}}$ is decreasing. It is clear that
$\sum\limits^{\hat{l}_{j}}_{k=1}d_{jk}=l_{j}$ for each $j=1,\dots,n$, and
$\sum\limits^{n}_{j=1}\sum\limits^{\hat{l}_{j}}_{k=1}d_{jk}=n_{0}$. We rewrite
the basis $\beta$ as

$\beta=\big{\\{}\xi_{111},\dots,\xi_{11d_{11}},\xi_{1\hat{l}_{1}1},\dots,\xi_{1\hat{l}_{1}d_{1\hat{l}_{1}}},\dots,\xi_{n11},\dots,\xi_{n1d_{n1}},\xi_{n\hat{l}_{n}1},\dots,\xi_{n\hat{l}_{n}d_{n\hat{l}_{n}}}\big{\\}}.$

Then by (4.29), one can easily check that for each $j\in\\{1,\dots,n\\}$ and
$k\in\\{1,\dots,\hat{l}_{j}\\}$,

$\begin{array}[]{r}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\big{(}\bar{\lambda}_{j}I-{\cal
Q}\big{)}^{q}\xi_{jkr}=\left\\{\begin{array}[]{cl}\xi_{jk(r-q)}&{\rm
when}~{}r>q,\\\ 0&{\rm when}~{}r\leq q.\end{array}\right.\end{array}$ (4.30)

Now we assume $\hat{\xi}$ satisfies the conditions on the left side of (4.28).
Since $\hat{\xi}\in(P^{*}H_{1})^{C}=\tilde{H}_{1}^{C}$, there is a vector

$\big{(}C_{111},\dots,C_{11d_{11}},C_{1\hat{l}_{1}1},\dots,C_{1\hat{l}_{1}d_{1\hat{l}_{1}}},\dots,C_{n11},\dots,C_{n1d_{n1}},C_{n\hat{l}_{n}1},\dots,C_{n\hat{l}_{n}d_{n\hat{l}_{n}}}\big{)}^{*}\in\mathbb{C}^{n_{0}},$

such that

$\hat{\xi}=\sum\limits^{n}_{j=1}\sum\limits^{\hat{l}_{j}}_{k=1}\sum\limits^{d_{jk}}_{r=1}C_{jkr}\xi_{jkr}.$
(4.31)

From (4.14) and the second condition on the left side of (4.28), it follows
that for each $m\in\\{0,\dots,n_{0}-1\\}$,

$D_{1}(\cdot)\psi_{n_{0}}^{\hat{\xi}}(\cdot)\bigl{|}_{((n_{0}-m-1)T,(n_{0}-m)T)}=0\;\;\mbox{i.\,e.}\;\;\sum\limits^{n}_{j=1}\sum\limits^{\hat{l}_{j}}_{k=1}\sum\limits^{d_{jk}}_{r=1}C_{jkr}D_{1}(t)\psi^{{\cal
Q}^{m}\xi_{jkr}}(t)=0\;\;\mbox{for a.e.}\;\;t\in(0,T),$

from which, we see that

$\begin{array}[]{c}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\sum\limits^{n}_{j=1}\sum\limits^{\hat{l}_{j}}_{k=1}\sum\limits^{d_{jk}}_{r=1}C_{jkr}D_{1}(\cdot)\psi^{g({\cal
Q})\xi_{jkr}}=0\;\;\mbox{over}\;\;(0,T)\end{array}$ (4.32)

for any polynomial $g$ with $\mbox{degree}(g)\leq n_{0}-1$. Arbitrarily fix a
$\widetilde{j}\in\\{1,\dots,n\\}$. Let

$\mathbb{P}_{\widetilde{j}}(\lambda)=\prod\limits_{j=1,j\neq\widetilde{j}}^{n}\left(\lambda-\bar{\lambda}_{j}\right)^{l_{j}}.$

By taking $g(\lambda)=\lambda^{m}\mathbb{P}_{\widetilde{j}}(\lambda)$, with
$m=0,1,\dots,l_{\tilde{j}}-1$, in (4.32), we have

$\sum\limits^{n}_{j=1}\sum\limits^{\hat{l}_{j}}_{k=1}\sum\limits^{d_{jk}}_{r=1}C_{jkr}D_{1}(\cdot)\psi^{{\cal
Q}^{m}\mathbb{P}_{\widetilde{j}}({\cal
Q})\xi_{jkr}}(\cdot)=0\;\;\mbox{over}\;\;(0,T),\;\;\mbox{when}\;\;m\in\\{0,1,\dots,l_{\widetilde{j}}-1\\}.$

By (4.30), we see that

$\mathbb{P}_{\widetilde{j}}({\cal Q})\xi_{jkr}=0,\;\;\mbox{ when
}\;\;j\in\\{1,\dots,,n\\},j\neq\widetilde{j},\,k\in\\{1,\dots,\hat{l}_{j}\\},\,r\in\\{1,\dots,d_{jk}\\}.$

The above two equalities imply that for each
$m\in\\{0,1,\dots,l_{\widetilde{j}}-1\\}$,

$\sum\limits^{\hat{l}_{\widetilde{j}}}_{k=1}\sum\limits^{d_{{\widetilde{j}}k}}_{r=1}C_{{\widetilde{j}}kr}D_{1}(\cdot)\psi^{{\cal
Q}^{m}\mathbb{P}_{\widetilde{j}}({\cal
Q})\xi_{{\widetilde{j}}kr}}(\cdot)=0\;\;\mbox{over}\;\;(0,T),$

from which, it follows that

$\begin{array}[]{c}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\sum\limits^{\hat{l}_{\widetilde{j}}}_{k=1}\sum\limits^{d_{{\widetilde{j}}k}}_{r=1}C_{{\widetilde{j}}kr}D_{1}(\cdot)\psi^{f({\cal
Q})\mathbb{P}_{\widetilde{j}}({\cal
Q})\xi_{{\widetilde{j}}kr}}(\cdot)=0\;\;\mbox{over}\;\;(0,T),\end{array}$
(4.33)

for any polynomial $f$ with $\mbox{degree}(f)\leq l_{\tilde{j}}-1$. Given
$m\in\\{0,1,2,\cdots,l_{\widetilde{j}}-1\\}$, since
$\mathbb{P}_{\widetilde{j}}(\lambda)$ and
$(\lambda-\bar{\lambda}_{\widetilde{j}})^{m+1}$ are coprime, there are
polynomials $g^{1}_{m}(\lambda)$ and $g^{2}_{m}(\lambda)$ with
$\mbox{degree}(g^{1}_{m})\leq m$ and
$\mbox{degree}(g^{2}_{m})\leq\mbox{degree}(\mathbb{P}_{\widetilde{j}})-1$,
respectively, such that

$g^{1}_{m}(\lambda)\mathbb{P}_{\widetilde{j}}(\lambda)+g^{2}_{m}(\lambda)(\lambda-\bar{\lambda}_{\widetilde{j}})^{m+1}\equiv
1,$

from which, we see that

$({\cal
Q}-\bar{\lambda}_{\widetilde{j}}I)^{l_{\widetilde{j}}-m-1}g^{1}_{m}({\cal
Q})\mathbb{P}_{\widetilde{j}}({\cal Q})\xi_{{\widetilde{j}}kr}\\\
+g^{2}_{m}({\cal Q})({\cal
Q}-\bar{\lambda}_{\widetilde{j}}I)^{l_{\widetilde{j}}}\xi_{{\widetilde{j}}kr}\equiv({\cal
Q}-\bar{\lambda}_{\widetilde{j}}I)^{l_{\widetilde{j}}-m-1}\xi_{{\widetilde{j}}kr},$
(4.34)

for all $m\in\\{0,1,\cdots,l_{\widetilde{j}}-1\\}$,
$k\in\\{1,2,\cdots,\hat{l}_{\widetilde{j}}\\}$, and
$r\in\\{1,2,\cdots,d_{\widetilde{j}k}\\}$. By (4.30), we have

$(\hat{\cal
Q}-\bar{\lambda}_{\widetilde{j}}I)^{l_{\widetilde{j}}}\xi_{{\widetilde{j}}kr}=0\;\;\mbox{for
all}\;\;k\in\\{1,2,\cdots,\hat{l}_{\widetilde{j}}\\},\,r\in\\{1,2,\cdots,d_{{\widetilde{j}}k}\\}.$
(4.35)

Taking
$f(\lambda)=(\lambda-\bar{\lambda}_{\widetilde{j}})^{l_{\widetilde{j}}-m-1}g^{1}_{m}(\lambda)$,
with $m=0,\dots,l_{\widetilde{j}}-1$, in (4.33), using (4.34) and (4.35), we
find

$\sum\limits^{\hat{l}_{\widetilde{j}}}_{k=1}\sum\limits^{d_{{\widetilde{j}}k}}_{r=1}C_{{\widetilde{j}}kr}D_{1}(\cdot)\psi^{({\cal
Q}-\bar{\lambda}_{\widetilde{j}}I)^{m}\xi_{{\widetilde{j}}kr}}(\cdot)=0\;\;\mbox{over}\;\;(0,T),\;\;\mbox{for
each}\;\;m\in\\{0,1,\dots,l_{\widetilde{j}}-1\\}.$ (4.36)

Now we are on the position to show

$C_{\widetilde{j}kr}=0\;\;\mbox{for
all}\;\;k\in\\{1,2,\cdots,\hat{l}_{\widetilde{j}}\\},\,r\in\\{1,\dots,d_{{\widetilde{j}}k}\\},$
(4.37)

which leads to $\hat{\xi}=0$ because of (4.31). For this purpose, we write

$K_{\widetilde{j}}^{m}=\left\\{k\in\\{1,2,\dots,\hat{l}_{\widetilde{j}}\\}\bigm{|}d_{{\widetilde{j}}k}>m\right\\},\;\;m=0,1,\dots,l_{\widetilde{j}}-1.$

One can easily check that (4.37) is equivalent to

$\mathcal{C}_{\hat{m}}\mathop{\buildrel\Delta\over{=}}\left\\{C_{\widetilde{j}k\hat{m}},~{}~{}k\in
K_{\widetilde{j}}^{\hat{m}-1}\right\\}=\\{0\\}\;\;\mbox{for
all}\;\;\hat{m}\in\\{1,\dots,d_{\widetilde{j}1}\\}$ (4.38)

We will use the mathematical induction method with respect to $\hat{m}$ to
prove (4.38). (Notice that $d_{\widetilde{j}k}$ is decreasing with respect to
$k$.) First of all, we let

$\mathbb{Q}_{\widetilde{j}}^{m}(\lambda)=\left(\bar{\lambda}_{\widetilde{j}}-\lambda\right)^{m},\;\;m=0,1,\dots,l_{\widetilde{j}}-1,$
(4.39)

In the case that $\hat{m}=d_{\widetilde{j}1}$, it follows from (4.39) and
(4.30) that

$\mathbb{Q}^{\hat{m}-1}_{\widetilde{j}}({\cal
Q})\xi_{\widetilde{j}k\hat{m}}=\big{(}\bar{\lambda}_{\widetilde{j}}I-{\cal
Q}\big{)}^{\hat{m}-1}\xi_{\widetilde{j}k\hat{m}}=\xi_{\widetilde{j}k1},\;\;\mbox{when}\;\;k\in
K^{\hat{m}-1}_{\widetilde{j}},\emph{}$

and

$\mathbb{Q}^{\hat{m}-1}_{\widetilde{j}}({\cal
Q})\xi_{\widetilde{j}kr}=0,\quad{\mbox{ when }}k\in
K^{\hat{m}-1}_{\widetilde{j}},\,r<\hat{m};{\mbox{ or }}k\notin
K^{\hat{m}-1}_{\widetilde{j}},\,r\in\\{1,\dots,d_{\widetilde{j}k}\\}.$

These, alone with (4.36) (where $m=\hat{m}-1$), imply that

$\sum\limits_{k\in
K_{\widetilde{j}}^{\hat{m}-1}}C_{\widetilde{j}k\hat{m}}D_{1}(\cdot)\psi^{\xi_{{\widetilde{j}}k1}}(\cdot)=0\;\;\mbox{over}\;\;(0,T).$

Let

$\bar{\xi}_{\hat{m}}\mathop{\buildrel\Delta\over{=}}\sum_{k\in
K_{\widetilde{j}}^{\hat{m}-1}}C_{{\widetilde{j}}k\hat{m}}\xi_{{\widetilde{j}}k1},\;\hat{m}=1,\dots,d_{\widetilde{j}1}.$

Then, it holds that

$D_{1}(\cdot)\displaystyle\psi^{\bar{\xi}_{\hat{m}}}(\cdot)=0\;\;\mbox{over}\;\;(0,T).$
(4.40)

Since for each $k\in\\{1,\dots,\hat{l}_{\tilde{j}}\\}$,
$\xi_{\widetilde{j}k1}$ is an eigenfunction of ${\cal Q}$ with respect to the
eigenvalue $\bar{\lambda}_{\widetilde{j}}$, it follows from the definition of
$\bar{\xi}_{\hat{m}}$ that $(\bar{\lambda}_{\widetilde{j}}I-{\cal
Q})\bar{\xi}_{\hat{m}}=0$. This, along with (4.40) and (1.19), yields that
$\bar{\xi}_{\hat{m}}=0$, i.e., $\bar{\xi}_{d_{\widetilde{j},1}}=0$, which
leads to $\mathcal{C}_{d_{\tilde{j}1}}=0$ because of the linear independence
of the group $\\{\xi_{{\widetilde{j}}k1},~{}k\in
K_{\widetilde{j}}^{\hat{m}-1}\\}$. Hence, (4.38) holds when
$\hat{m}=d_{\widetilde{j}1}$.

Suppose inductively that (4.38) holds when $\tilde{m}+1\leq\hat{m}\leq
d_{\widetilde{j}1}$ for some $\tilde{m}\in\\{1,\dots,d_{\widetilde{j}1}-1\\}$,
i.e.,

$\mathcal{C}_{\hat{m}}=\\{0\\},\;\;\mbox{when}\;\;\tilde{m}+1\leq\hat{m}\leq
d_{\widetilde{j}1}.$ (4.41)

We will show that (4.38) holds when $\hat{m}=\tilde{m}$, i.e.,
$\mathcal{C}_{\tilde{m}}=\\{0\\}$. In fact, it follows from (4.30) that

$\mathbb{Q}^{\tilde{m}-1}_{\widetilde{j}}({\cal
Q})\xi_{\widetilde{j}kr}=\left\\{\begin{array}[]{cl}\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\xi_{\widetilde{j}k(r-\tilde{m}+1)},&{\rm when}~{}k\in
K^{\tilde{m}-1}_{\widetilde{j}},\,r\geq\tilde{m},\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 0,&{\rm when}~{}k\in
K^{\tilde{m}-1}_{\widetilde{j}},\,r<\tilde{m},\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 0,&{\rm when}~{}k\notin
K^{\tilde{m}-1}_{\widetilde{j}},\,r\in\\{1,\dots,d_{\widetilde{j}k}\\}.\end{array}\right.$

This, alone with (4.36) (where $m=\tilde{m}-1$), indicates that

$\sum\limits^{\hat{l}_{\widetilde{j}}}_{k=1}\sum\limits^{d_{{\widetilde{j}}k}}_{r=1}C_{{\widetilde{j}}kr}D_{1}(\cdot)\psi^{\mathbb{Q}^{\tilde{m}-1}_{\widetilde{j}}({\cal
Q})\xi_{{\widetilde{j}}kr}}(\cdot)=\sum\limits_{k\in
K_{\widetilde{j}}^{\tilde{m}-1}}\sum\limits^{d_{{\widetilde{j}}k}}_{r=\tilde{m}}C_{{\widetilde{j}}kr}D_{1}(\cdot)\psi^{\xi_{{\widetilde{j}}k(r-\tilde{m}+1)}}(\cdot)=0\;\;\mbox{over}\;\;(0,T).$

Then, by (4.41), we have

$\sum\limits_{k\in
K_{\widetilde{j}}^{\tilde{m}-1}}C_{\widetilde{j}k\tilde{m}}D_{1}(\cdot)\psi^{\xi_{{\widetilde{j}}k1}}(\cdot)=0\;\;\mbox{over}\;\;(0,T).$
(4.42)

Let

$\bar{\xi}_{\tilde{m}}\mathop{\buildrel\Delta\over{=}}\sum_{k\in
K_{\widetilde{j}}^{\tilde{m}-1}}C_{{\widetilde{j}}k\tilde{m}}\xi_{{\widetilde{j}}k1}.$

Then, it follows from (4.42) that

$D_{1}(\cdot)\displaystyle\psi^{\bar{\xi}_{\tilde{m}}}(\cdot)=0\;\;\mbox{over}\;\;(0,T).$
(4.43)

Since for each $k\in\\{1,\dots,\hat{l}_{\widetilde{j}}\\}$,
$\xi_{\widetilde{j},k,1}$ is an eigenfunction of ${\cal Q}$ with respect to
the eigenvalue $\bar{\lambda}_{\widetilde{j}}$, it holds that
$(\bar{\lambda}_{\widetilde{j}}I-{\cal Q})\bar{\xi}_{\tilde{m}}=0$ . This,
along with (4.43) and (1.19), yields that $\bar{\xi}_{\hat{m}}=0$. Hence,
$\mathcal{C}_{\tilde{m}}=\\{0\\}$ because of the linear independence of the
group $\\{\xi_{{\widetilde{j}}k1},~{}k\in K_{\widetilde{j}}^{\tilde{m}-1}\\}$.
In summary, we conclude that $(d)\Rightarrow(c)$. This completes the proof of
Theorem 1.1.

Proof of Theorem 1.2. Clearly, it suffice to show the only if part. Assume
that Equation (1.1) is LPFS. By the equivalence of $(a)$ and $(b)$ in Theorem
1.1, it holds that

$\hat{V}^{U}_{n_{0}}=H_{1}.$ (4.44)

Meanwhile, according to Lemma 2.5, there is a finite dimensional subspace of
$\hat{Z}$ of $U$, such that

$\hat{V}^{U}_{n_{0}}=\hat{V}^{\hat{Z}}_{n_{0}}.$ (4.45)

From (4.44) and (4.45), it follows that $\hat{V}^{\hat{Z}}_{n_{0}}=H_{1}$.
This, along with the equivalence of $(a)$ and $(b)$ in Theorem 1.1, indicates
that Equation (1.1) is LPFS with respect to $\hat{Z}$. Hence, we complete the
proof of Theorem 1.2.

##  5 Applications

In this section, we will present some applications of Theorem 1.1 to the
internally controlled heat equations with time-periodic potential.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ ($d\geq 1$) with a
$C^{2}$-smooth boundary $\partial\Omega$. Write
$Q\triangleq\Omega\times\mathbb{R}^{+}$ and
$\sum\triangleq\partial\Omega\times\mathbb{R}^{+}$. Let
$\omega\subseteq\Omega$ be a non-empty open subset with its characteristic
function $\chi_{\omega}$. Let $T>0$ and $a\in L^{\infty}(Q)$ be $T$-periodic
(with respect to the time variable $t$), i.e., for a.e. $t\in\mathbb{R}^{+}$,
$a(\cdot,t)=a(\cdot,t+T)$ over $\Omega$. One can easily check that the
function $a$ can be treated as a $T$-periodic function in
$L^{1}_{loc}(\mathbb{R}^{+};L^{2}(\Omega)).$ Consider the following controlled
heat equation:

$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\partial_{t}y(x,t)-\triangle
y(x,t)+a(x,t)y(x,t)=\chi_{\omega}(x)u(x,t)&\mbox{in}&Q,\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr y(x,t)=0&\mbox{on}&\sum,\end{array}\right.$ (5.1)

where $u\in L^{2}(\mathbb{R}^{+};L^{2}(\Omega))$. Given $y_{0}\in
L^{2}(\Omega)$ and $u\in L^{2}(\mathbb{R}^{+};L^{2}(\Omega))$, Equation (5.1)
with the initial condition that $y(x,0)=y_{0}(x)$ has a unique solution
$y(\cdot;0,y_{0},u)\in C(\mathbb{R}^{+};L^{2}(\Omega))$.

Let $H=U=L^{2}(\Omega)$ and $A=-\triangle$ with
$\mathcal{D}(A)=H_{0}^{1}(\Omega)\bigcap H_{2}(\Omega)$. Define, for a.e.
$t\in\mathbb{R}^{+}$, $B(t):H\rightarrow H$ by $B(t)z(x)=a(x,t)z(x)$,
$x\in\Omega$, and $D(t):U\rightarrow H$ by $D(t)v(x)=\chi_{\omega}(x)v(x)$,
$x\in\Omega$. Clearly, $(-A)$ generates a compact semigroup on $L^{2}(\Omega)$
and both $B(\cdot)\in L^{1}_{loc}(\mathbb{R}^{+};L^{2}(\Omega))$ and
$D(\cdot)\in L^{\infty}(\mathbb{R}^{+};{\cal L}(U;H))$ are $T$-periodic. Thus,
we can study Equation (5.1) under the framework (1.1). Write
$\\{\Psi_{a}(t,s)\\}_{0\leq s\leq t}$ for the evolution system generated by
$-A-B(\cdot)$. We use notations $n_{0}$, $P$, $H_{j}$ (with $j=1,2$),
$V^{Z}_{k}$ and $\hat{V}^{Z}_{k}$ (with $k\in\mathbb{N}$) to denote the same
subjects as those introduced in section 1. We will use the different
equivalent conditions in Theorem 1.1 to show that Equation (5.1) is LPFS.

###### Corollary 5.1

Equation (5.1) is LPFS. Consequently, it is LPFS with respect to a finite
dimensional subspace of $L^{2}(\Omega)$.

Proof. We will provide two ways to show that Equation (5.1) is LPFS. We first
use the equivalence $(a)\Leftrightarrow(c)$ in Theorem 1.1. In fact,
$\psi(\cdot)\triangleq\Psi_{a}(n_{0}T,\cdot)^{*}\xi$ with $\xi\in H$ is the
solution to the equation:

$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\partial_{t}\psi(x,t)+\triangle\psi(x,t)-a(x,t)\psi(x,t)=0&\mbox{in}&\Omega\times(0,n_{0}T),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\psi(x,t)=0&\mbox{on}&\partial\Omega\times(0,n_{0}T),\\\ \vskip 3.0pt
plus 1.0pt minus
1.0pt\cr\psi(x,n_{0}T)=\xi(x)&\mbox{in}&\Omega,\end{array}\right.$ (5.2)

and it holds that

$D(t)\eta=\chi_{\omega}\eta\;\;\mbox{for any}\;\;\eta\in
H\;\;\mbox{and}\;\;t\in[0,T].$ (5.3)

These, along with the unique continuation property of parabolic equations
established in [15] (see also [18] and [19]), leads to the condition $(c)$ in
Theorem 1.1 for the current case. Then, according to the equivalence
$(a)\Leftrightarrow(c)$ in Theorem 1.1, Equation (5.1) is LPFS.

We next use the equivalence $(a)\Leftrightarrow(b)$ in Theorem 1.1. Without
loss of generality, we can assume that $n_{0}\geq 1$, for otherwise Equation
(5.1), with the null control $u=0$, is stable. When $n_{0}\geq 1$, we have
$H_{1}\neq\\{0\\}$ and $\|P\|>0$. Write
$\left\\{\xi_{1},\dots,\xi_{n_{0}}\right\\}$ for an orthonormal basis of
$H_{1}$. By the approximate controllability of the heat equation (see [10]),
$V^{U}_{1}$ is dense in $H$. Thus there are $\eta_{j}$, $j=1\dots,n_{0}$, in
$V^{U}_{1}$ such that such that

$\|\eta_{j}-\xi_{j}\|\leq\displaystyle\frac{1}{16n_{0}\|P\|}\;\;\mbox{for
all}\;\;j=1,\dots,n_{0}.$ (5.4)

Since $P$ is a projection from $H$ onto $H_{1}$, we have $P\xi_{j}=\xi_{j}$
for all $j=1,\dots,n_{0}$. This, along with (5.4), yields that for each
$j\in\\{1,\dots,n_{0}\\}$,

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\|P\eta_{j}\|\leq&\|\xi_{j}\|+\|P\eta_{j}-\xi_{j}\|=\|\xi_{j}\|+\|P(\eta_{j}-\xi_{j})\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\|\xi_{j}\|+\|P\|\|\eta_{j}-\xi_{j}\|\leq
1+\displaystyle{1\over{16n_{0}}};\end{array}$ (5.5)

and

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\mathop{\langle}P\eta_{j},\xi_{j}\mathop{\rangle}=&\mathop{\langle}\xi_{j}+(P\eta_{j}-\xi_{j}),\xi_{j}\mathop{\rangle}=1+\mathop{\langle}P\eta_{j}-\xi_{j},\xi_{j}\mathop{\rangle}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\geq&1-\|P\eta_{j}-\xi_{j}\|\geq
1-\|P\|\|\eta_{j}-\xi_{j}\|\geq 1-\displaystyle\frac{1}{16n_{0}}.\\\
\end{array}$ (5.6)

Since $P\eta_{j}\in H_{1}$ and $\\{\xi_{k}\\}_{k=1}^{n_{0}}$ is an orthonormal
basis of $H_{1}$, it holds that

$\|P\eta_{j}\|^{2}=\sum^{n_{0}}_{k=1}|\mathop{\langle}P\eta_{j},\xi_{k}\mathop{\rangle}|^{2},\;\;\mbox{when}\;\;j=1,\dots,n_{0}.$
(5.7)

From (5.7), (5.5) and (5.6), it follows that for each
$j\in\\{1,\dots,n_{0}\\}$,

$\begin{array}[]{rl}\sum\limits_{k\neq
j}\left|\mathop{\langle}P\eta_{j},\xi_{k}\mathop{\rangle}\right|&\leq(n_{0}-1)^{1/2}\left(\sum\limits_{k\neq
j}\left|\mathop{\langle}P\eta_{j},\xi_{k}\mathop{\rangle}\right|^{2}\right)^{1/2}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&=(n_{0}-1)^{1/2}\big{(}\|P\eta_{j}\|^{2}-|\mathop{\langle}P\eta_{j},\xi_{j}\mathop{\rangle}|^{2}\big{)}^{1/2}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&\leq
n_{0}^{1/2}\big{(}\left(1+{1}/(16n_{0})\right)^{2}-\left(1-{1}/(16n_{0})\right)^{2}\big{)}^{1/2}={1}/{2}.\end{array}$

This, together with (5.6), indicates that

$\mathop{\langle}P\eta_{j},\xi_{j}\mathop{\rangle}\geq
1-{1}/(16n_{0})>{1}/{2}\geq\sum_{k\neq
j}\left|\mathop{\langle}P\eta_{j},\xi_{k}\mathop{\rangle}\right|,\;j=1,2,\cdots,n_{0}.$
(5.8)

We claim that $\\{P\eta_{1},\dots,P\eta_{n_{0}}\\}$ is a linearly independent
group. In fact, suppose that

$\sum\limits_{j=1}^{n}c_{j}P\eta_{j}=0\;\;\mbox{for
some}\;\;c_{1},\dots,c_{n_{0}}\in\mathbb{R}.$ (5.9)

Write
$\hat{A}\triangleq(\mathop{\langle}P\eta_{j},\xi_{k}\mathop{\rangle})_{j,k}\in\mathbb{R}^{n_{0}\times
n_{0}}$ and
$\hat{c}\triangleq(c_{1},\dots,c_{n_{0}})^{*}\in\mathbb{R}^{n_{0}}$. By (5.8),
the matrix $\hat{A}$ is diagonally dominant, hence it is invertible. Then,
from (5.9), it follows that $\hat{A}^{*}\hat{c}=0,$ which implies $\hat{c}=0$.
Hence, $P\eta_{1},\dots,P\eta_{n_{0}}$ are linearly independent.

Since $\mbox{dim}H_{1}=n_{0}$, it follows that
$\mbox{span}\\{P\eta_{1},\dots,P\eta_{n_{0}}\\}=H_{1}$. Therefore

$H_{1}\supseteq\hat{V}^{U}_{n_{0}}\supseteq\hat{V}^{U}_{1}=PV^{U}_{1}\supseteq\mbox{span}\\{P\eta_{1},\cdots,P\eta_{n_{0}}\\}=H_{1}.$

from which, it follows that $H_{1}=\hat{V}^{U}_{n_{0}}$. This, along with the
equivalence of $(a)$ and $(b)$ in Theorem 1.1, indicates that Equation (5.1)
is LPFS.

Finally, according to Theorem 1.2, there is a finite-dimensional subspace $Z$
of $U$ such that Equation (5.1) is LPFS with respect to $Z$. This completes
the proof.

###### Corollary 5.2

Equation (5.1) is LPFS with respect to the subspace $P^{*}H$.

Proof. Let $Z=P^{*}H$. By the equivalence between $(a)$ and $(d)$ in Theorem
1.1, it suffices to show that $Z$ satisfies (1.19), i.e.,

$\mu\notin\mathbb{B},\;\xi\in H^{C},\;\big{(}\mu I-{\cal
P}^{*C}\big{)}\xi=0,\;\big{(}D(\cdot)\bigl{|}_{Z}\big{)}^{*C}\Psi_{a}(T,\cdot)^{*C}\xi=0\;\;\mbox{over}\;\;(0,T)\Rightarrow\xi=0.$
(5.10)

Suppose that $\mu$ and $\xi$ satisfy the conditions on the left side of
(5.10). Write $\xi=\xi_{1}+i\xi_{2}$ where $\xi_{1},\xi_{2}\in H$. Then, we
have

$\big{(}D(\cdot)\bigl{|}_{Z}\big{)}^{*}\Psi_{a}(T,\cdot)^{*}\xi_{j}=0,\;j=1,2.$
(5.11)

Clearly, $\psi_{j}(\cdot)\triangleq\Psi_{a}(T,\cdot)^{*}\xi_{j}$ (with
$j=1,2$) is the solution to the equation (5.2) where $n_{0}T$ and $\xi$ are
replaced by $T$ and $\xi_{j}$ respectively. Since
$\Psi_{a}(T,\cdot)^{*}\xi_{j}$ is continuous on $[0,T]$ and $D(t)$ is
independent of $t$, it follows from (5.11) that

$\big{(}D(0)\bigl{|}_{Z}\big{)}^{*}\Psi_{a}(T,0)^{*}\xi_{j}=0,\;j=1,2.$ (5.12)

For each $\eta\in H$, we have $P^{*}\eta\in P^{*}H$. This, along with (5.3),
yields

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\left\langle\big{(}D(0)\bigl{|}_{Z}\big{)}^{*}\Psi_{a}(T,0)^{*}\xi_{j},~{}P^{*}\eta\right\rangle=\left\langle\Psi_{a}(T,0)^{*}\xi_{j},~{}\big{(}D(0)\bigl{|}_{Z}\big{)}P^{*}\eta\right\rangle\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&=\left\langle\Psi_{a}(T,0)^{*}\xi_{j},~{}\chi_{\omega}P^{*}\eta\right\rangle=\left\langle
P\chi_{\omega}\Psi_{a}(T,0)^{*}\xi_{j},~{}\eta\right\rangle,\;j=1,2.\end{array}$

This, alone with (5.12), implies that
$P\chi_{\omega}\Psi_{a}(T,0)^{*}\xi_{j}=0,\;j=1,2$, from which, we have

$\mathop{\langle}P^{*}\Psi_{a}(T,0)^{*}\xi_{j},\,\chi_{\omega}\Psi_{a}(T,0)^{*}\xi_{j}\mathop{\rangle}=\mathop{\langle}\Psi_{a}(T,0)^{*}\xi_{j},\,P\chi_{\omega}\Psi_{a}(T,0)^{*}\xi_{j}\mathop{\rangle}=0,\;j=1,2.$
(5.13)

Tow facts are as follows. First, it follows from (2.5) that

$P^{*}\Psi_{a}(T,0)^{*}\xi_{j}=\Psi_{a}(T,0)^{*}P^{*}\xi_{j},\;j=1,2.$ (5.14)

Second, by (2.14), (2.13), and the first three conditions on the left side of
(5.10), we have $\xi\in\tilde{H}_{1}^{C}$. Since $P^{*}=\tilde{P}$ and
$\tilde{P}$ is a projection from $H$ to $\tilde{H}_{1}$ (see Lemma 2.3), we
see that $P^{*}:H\rightarrow\tilde{H}_{1}$ is a projection. Hence,
$P^{*C}:H^{C}\rightarrow\tilde{H}_{1}^{C}$ is a projection. These two facts
yields that $P^{*C}\xi=\xi$, from which, it follows that
$P^{*}\xi_{j}=\xi_{j}$, $j=1,2$. This along with (5.13) and (5.14), indicates
that $\left\|\chi_{\omega}\Psi_{a}(T,0)^{*C}\xi\right\|=0$, i.e.,
$\chi_{\omega}\psi_{j}(T)=0$. By the unique continuation property of parabolic
equations established in [15] (see also [18] and [19]), we find that
$\xi_{j}=0$, $j=1,2$, which leads to $\xi=0$. This completes the proof.

Finally, we will present a controlled heat equation which is not LPFS. Write
$\lambda_{1}$ and $\lambda_{2}$ for the first and the second eigenvalues of
the operator $-\triangle$ with ${\cal D}(-\triangle)=H_{0}^{1}(\Omega)\cap
H^{2}(\Omega)$, respectively. Let $\xi_{j}$, $j=1,2$, be an eigenfunction
corresponding to $\lambda_{j}$. Consider the following heat equation:

$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\partial_{t}y(x,t)-\triangle
y(x,t)-\lambda_{2}y(x,t)=\mathop{\langle}u(t),\xi_{1}\mathop{\rangle}\xi_{1}(x)&\mbox{in}&Q,\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
y(x,t)=0&\mbox{on}&\sum.\end{array}\right.$ (5.15)

where $u(\cdot)\in L^{2}(\mathbb{R}^{+};L^{2}(\Omega))$. By a direct
calculation, one has that

$V_{n_{0}}=\hbox{\rm
span$\,$}\\{\xi_{1}\\}\;\;\mbox{and}\;\;H_{1}\supseteq\mbox{span}\\{\xi_{1},\xi_{2}\\}.$

These, along with the equivalence $(a)\Leftrightarrow(b)$ in Theorem 1.1,
indicates that (5.15) is not LPFS.

## Appendix

The proof of Lemma 2.1. By the compactness of $\\{S(t)\\}_{t>0}$, the
assumption $({\cal H}_{2})$ and (1.4), one can easily check that each ${\cal
P}(t)$, with $t\geq 0$, is compact. Hence, each ${\cal
P}(t)^{C}:H^{C}\rightarrow H^{C}$, with $t\geq 0$, is also compact. Thus, for
each $t\geq 0$, $\sigma({\cal P}(t)^{C})\setminus\\{0\\}$ consists of all
nonzero eigenvalues $\\{\lambda_{j}(t)\\}^{\infty}_{j=1}$ (in $\mathbb{C}$) of
${\cal P}(t)^{C}$ such that
$\lim\limits_{j\rightarrow\infty}|\lambda_{j}(t)|=0$.

We next show that $\\{\lambda_{j}(t)\\}^{\infty}_{j=1}$ is independent of $t$.
For this purpose, we arbitrarily fix $s_{1}$ and $s_{2}$ with $0\leq s_{1}\leq
s_{2}+T$. Let $\lambda\in\mathbb{C}^{1}$ be a non-zero eigenfunction of ${\cal
P}(s_{1})^{C}$ and $\eta\in H^{C}$ be a corresponding eigenfunction, i.e.,

${\cal P}(s_{1})^{C}\eta=\lambda\eta.$ ( a.1)

Write $\lambda=\alpha_{1}+i\alpha_{2}$ with $\alpha_{1},a_{2}\in\mathbb{R}$
and $\eta=\eta_{1}+i\eta_{2}$ with $\eta_{1},\eta_{2}\in H$. By (a.1), we have

${\cal P}(s_{1})\eta_{1}=\alpha_{1}\eta_{1}-\alpha_{2}\eta_{2},\qquad{\cal
P}(s_{1})\eta_{2}=\alpha_{2}\eta_{1}+\alpha_{1}\eta_{2}.$ ( a.2)

From (a.2) and (1.6), one can easily check that ${\cal
P}(s_{2})^{C}(\Phi(s_{2},s_{1})^{C}\eta)=\lambda\Phi(s_{2},s_{1})^{C}\eta$.
This implies that $\lambda$ is an eigenvalue of ${\cal P}(s_{2})^{C}$ and
$\Phi(s_{2},s_{1})^{C}\eta$ is a corresponding eigenfunction. Hence,

$\sigma({\cal P}(s_{1})^{C})\setminus\\{0\\}\subseteq\sigma({\cal
P}(s_{2})^{C})\setminus\\{0\\}.$

Similarly, we can show

$\sigma({\cal P}(s_{2})^{C})\setminus\\{0\\}\subseteq\sigma({\cal
P}(s_{1}+T)^{C})\setminus\\{0\\}.$

Then by the $T$-periodicity of ${\cal P}(\cdot)$, $\sigma({\cal
P}(t)^{C})\setminus\\{0\\}$ is independent of $t$. This completes the proof.

The proof of Lemma 2.2. First of all, we let

$\hat{H}_{1}(t)\triangleq\hat{P}(t)H^{C}\;\;\mbox{and}\;\;\hat{H}_{2}(t)\triangleq(I-\hat{P}(t))H^{C},\;\;t\geq
0.$ ( a.3)

From Theorem 6.17 on Page 178 in [12], it follows that when $t\geq 0$, both
$\hat{H}_{1}(t)$ and $\hat{H}_{2}(t)$ are invariant w.r.t. ${\cal P}(t)^{C}$;

$\hat{P}(t):H^{C}(t)\rightarrow\hat{H}_{1}(t)\;\;\mbox{is a projection};$ (
a.4) $H^{C}=\hat{H}_{1}(t)\bigoplus\hat{H}_{2}(t);$ ( a.5)

and

$\sigma\big{(}{\cal
P}(t)^{C}|_{\hat{H}_{1}(t)}\big{)}=\\{\lambda_{j}\\}^{n}_{j=1}\;\;\mbox{and}\;\;\sigma({\cal
P}(t)^{C}|_{\hat{H}_{2}(t)})\setminus\\{0\\}=\\{\lambda_{j}\\}^{\infty}_{j=n+1},$
( a.6)

where $\\{\lambda_{j}\\}_{j=1}^{\infty}$ and $n$ are given by (1.8) and (1.9)
respectively.

Then we prove that the operator $P(t)$, with $t\geq 0$, is a linear operator
from $H$ to $H$. For this purpose, it suffices to show that

$\hat{P}(t)h\in H,\;\;\mbox{when}\;\;h\in H\;\;\mbox{and}\;\;t\geq 0.$ ( a.7)

The proof of (a.7) is as follows. By (1.12), it holds that

$\hat{P}(t)h=\displaystyle\frac{-\delta}{2\pi}\int^{2\pi}_{0}\big{(}\delta
e^{i\theta}I-{\cal P}(t)^{C}\big{)}^{-1}e^{i\theta}d\theta\
h,\,\;\;\mbox{when}\;\;h\in H\;\;\mbox{and}\;\;t\geq 0.$ ( a.8)

Write $F$ for the conjugate map from $H^{C}$ to $H^{C}$, i.e., $F(h+ig)=h-ig$
for any $h,g\in H$. We claim

$F\big{(}\big{(}\delta e^{i\theta}I-{\cal
P}(t)^{C}\big{)}^{-1}e^{i\theta}h\big{)}=(\delta e^{-i\theta}I-{\cal
P}(t)^{C})^{-1}e^{-i\theta}h\;\;\mbox{for all}\;\;\theta\in[0,2\pi],h\in
H\;\;\mbox{and}\;\;t\geq 0,$ ( a.9)

When (a.9) is proved, it follows from (a.8) and (a.9) that

$\begin{array}[]{rl}F(\hat{P}(t))h&=\displaystyle\frac{-\delta}{2\pi}\int^{2\pi}_{0}\left(\delta
e^{-i\theta}I-{\cal P}(t)^{C}\right)^{-1}e^{-i\theta}d\theta\,h\\\
&=\displaystyle\frac{-\delta}{2\pi}\int^{2\pi}_{0}\left(\delta
e^{i\theta}I-{\cal
P}(t)^{C}\right)^{-1}e^{i\theta}d\theta\,h=\hat{P}(t)h\;\;\mbox{for
each}\;\;t\geq 0,h\in H,\end{array}$

Which leads to (a.7). Now we are on the position to show (a.9). Arbitrarily
fix $\theta\in[0,\pi]$, $t\geq 0$ and $h\in H$. Write

$(\delta e^{i\theta}I-{\cal
P}(t)^{C})^{-1}e^{i\theta}h=g_{1}+ig_{2},\;\;g_{1},g_{2}\in H.$ ( a.10)

It is clear that $(\delta e^{i\theta}I-{\cal
P}(t)^{C})(g_{1}+ig_{2})=e^{i\theta}h$, from which, one can directly check
that

$(\delta e^{-i\theta}I-{\cal P}(t)^{C})(g_{1}-ig_{2})=e^{-i\theta}h.$

Hence, $(\delta e^{-i\theta}I-{\cal
P}(t)^{C})^{-1}(e^{-i\theta}h)=g_{1}-ig_{2}=F(g_{1}+ig_{2})$. This, along with
(a.10), leads to (a.9).

Next we prove that $P(t)$, with $t\geq 0$, is a projection on $H$. Let
$H_{1}(t)$ and $H_{2}(t)$, with $t\geq 0$, be defined by (2.2). Two
observations are given in order:

$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hat{H}_{1}(t)&\triangleq\hat{P}(t)H^{C}=\big{\\{}\hat{P}(t)(h_{1}+ih_{2})\bigm{|}h_{1},h_{2}\in
H\big{\\}}=\big{\\{}P(t)h_{1}+iP(t)h_{2}\bigm{|}h_{1},h_{2}\in H\big{\\}}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&=P(t)H+iP(t)H\triangleq
H_{1}(t)+iH_{1}(t)\triangleq H_{1}^{C}(t);\end{array}$ ( a.11)
$\hat{H}_{2}(t)=H_{2}^{C}(t).$ ( a.12)

By (a.4) and (a.11), we see that $\hat{P}(t)$ (with $t\geq 0$) is a projection
from $H^{C}$ onto $H_{1}(t)^{C}$. Thus, for each $t\geq 0$,

$P(t)(h_{1}+h_{2})=\hat{P}(t)(h_{1}+h_{2})=h_{1},\;\;\mbox{when}\;\;h_{1}\in
H_{1}(t),h_{2}\in H_{2}(t),$

i.e., $P(t)$ is a projection from $H$ onto $H_{1}(t)$. Besides, (2.1) follows
from (a.5), (a.11) and (a.12).

Finally, we will show properties $(a)$-$(f)$ one by one.

The proof of $(a)$: Since ${\cal P}(\cdot)$ is $T$-periodic, so is
$\hat{P}(\cdot)$ (see (1.12)). This, along with (1.13), indicates the
$T$-periodicity of $P(\cdot)$. Then by (2.2), both $H_{1}(\cdot)$ and
$H_{2}(\cdot)$ are $T$-periodic.

The proof of $(b)$: Let $t\geq 0$. Since $\hat{H}_{1}(t)$ and $\hat{H}_{2}(t)$
are invariant w.r.t. ${\cal P}(t)^{C}$, so are $H_{1}(t)^{C}$ and
$H_{2}(t)^{C}$ (see (a.11) and (a.12)). Hence, $H_{1}(t)$ and $H_{2}(t)$ are
invariant w.r.t. ${\cal P}(t)$.

The proof of $(c)$: (2.3) follows from (a.6), (a.11) and (a.12). Meanwhile, by
(1.11) and (2.3), we see that $\mbox{dim}H_{1}(t)^{C}=n_{0}$, which leads to
(2.4).

The proof of $(d)$ and $(e)$: Let $0\leq s\leq t<\infty$. By (1.6), we have
that $\Phi(t,s){\cal P}(s)={\cal P}(t)\Phi(t,s)$. From this, one can directly
verify that $\Phi(t,s)^{C}\hat{P}(s)=\hat{P}(t)\Phi(t,s)^{C}$. This, along
with (2.2), (1.13) and (a.7), indicates that

$\Phi(t,s)H_{1}(s)\subseteq P(t)H\triangleq H_{1}(t),$ ( a.13)

which leads to $(e)$. Meanwhile, it follows from (a.13) that
$\Phi(t,s)\in\mathcal{L}(H_{1}(s),H_{1}(t))$. Similarly, one can show that
$\Phi(t,s)\in\mathcal{L}(H_{2}(s),H_{2}(t))$. Hence, $(d)$ stands.

The proof of $(f)$: Let $\bar{\rho}\triangleq(-{\ln\bar{\delta}})/{T}>0$ with
$\bar{\delta}$ given by (1.10). Because of (2.3), it follows from Theorem 4 on
Page 212 in [26] that the spectral radius of ${\cal
P}^{C}(0)\bigl{|}_{H_{2}(0)^{C}}$ equals to $\bar{\delta}$. Thus, we have

$\bar{\delta}=\lim\limits_{k\rightarrow\infty}\big{\|}\big{(}{\cal
P}(0)^{C}\bigl{|}_{H_{2}(0)^{C}}\big{)}^{k}\big{\|}^{\frac{1}{k}}.$

Now we arbitrarily fix a $\rho\in(0,\bar{\rho})$ where $\bar{\rho}$ is given
by (1.10). Then it holds that $\bar{\delta}\triangleq
e^{-\bar{\rho}T}<e^{-\rho T}$. Thus there is positive integer $\hat{N}$ such
that $\big{\|}\big{(}{\cal
P}(0)^{C}\bigl{|}_{H_{2}(0)^{C}}\big{)}^{k}\big{\|}<e^{-\rho kT}$ for all
$k\geq\hat{N}$, which implies

$\big{\|}\big{(}{\cal P}(0)\bigl{|}_{H_{2}(0)}\big{)}^{k}\big{\|}<e^{-\rho
kT}\;\;\mbox{ for all }\;\;k\geq\hat{N}.$ ( a.14)

Notice that $\Phi(\cdot,\cdot)$ is continuous from $[0,T]\times[0,T]$ to
${\cal L}(H)$ (see Lemma 5.6 on Page 68 in [14]). Thus, we can write

$C_{1}\triangleq\max\limits_{0\leq t_{1}\leq t_{2}\leq
T}\left\|\Phi(t_{2},t_{1})\right\|\in\mathbb{R}^{+};\;\;C_{\rho}\triangleq(C_{1}+1)^{2}e^{3\rho
T}\in\mathbb{R}^{+}.$ ( a.15)

We are going to show that the above $C_{\rho}$ satisfies (2.6). For this
purpose, we let $0\leq s\leq t<\infty$ and $h_{2}\in H_{2}(s)$. For each
$r\in\mathbb{R}^{+}$, we denote by $[r]$ the integer such that $r-1<[r]\leq
r$. There are only two possibilities: $(i)$ $[t/T]=[s/T]$ and $(ii)$
$[t/T]\neq[s/T]$. In the first case, it follows from (1.6) and (a.15) that

$\begin{array}[]{rl}&\left\|\Phi(t,s)h_{2}\right\|=\left\|\Phi(t-\left[{s}/{T}\right]T,s-\left[{s}/{T}\right]T)h_{2}\right\|\leq\left\|\Phi(t-\left[{s}/{T}\right]T,s-\left[{s}/{T}\right]T)\right\|\|h_{2}\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&\leq C_{1}\|h_{2}\|\leq C_{1}e^{\rho
T}e^{-\rho(t-s)}\|h_{2}\|<(C_{1}+1)^{2}e^{3\rho
T}e^{-\rho(t-s)}\|h_{2}\|=C_{\rho}e^{-\rho(t-s)}\|h_{2}\|,\end{array}$

i.e., $C_{\rho}$ satisfies (2.6) in the first case. In the second case, we
have that $[t/T]T\geq[s/T]T+T$; and it follows from $(d)$ and $(a)$ that
$\Phi([s/T]T+T,s)h_{2}\in H_{2}(0)\triangleq H_{2}$. These, along with (1.6)
and (a.14), indicate that

$\begin{array}[]{rl}&\left\|\Phi(t,s)h_{2}\right\|\leq\big{\|}\Phi\big{(}t,\,\big{[}{t}/{T}\big{]}T\big{)}\big{\|}\cdot\big{\|}{\cal
P}(0)^{([{t}/{T}]-[{s}/{T}]-1)}\Phi\big{(}\big{[}{s}/{T}\big{]}T+T,\,s\big{)}h_{2}\big{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\leq\big{\|}\Phi\big{(}t-\big{[}{t}/{T}\big{]}T,\,0\big{)}\big{\|}\cdot
e^{-\rho
T([{t}/{T}]-[{s}/{T}]-1)}\cdot\big{\|}\Phi\big{(}T,\,s-\big{[}{s}/{T}\big{]}T\big{)}\big{\|}\|h_{2}\|.\end{array}$

By this and (a.15), one can directly check that
$\big{\|}\Phi(t,s)h_{2}\big{\|}\leq C_{\rho}e^{-\rho(t-s)}\|h_{2}\|$, i.e.,
$C_{\rho}$ satisfies (2.6) in the second case. This shows (2.6) and completes
the proof.

The proof of Lemma 2.3. By (2.7), (2.9), (2.8), (2.10) and (2.11), one can
make use of the exactly same way utilized in the proof of Lemma 2.2 to verify
all properties in Lemma 2.3, except for (2.12)-(2.14). Since $(\lambda I-{\cal
P}^{*C})^{-1}=\big{(}(\lambda I-{\cal P}^{C})^{-1}\big{)}^{*}$, (2.12) follows
from (2.9), (1.12) and (1.13). Now, we prove (2.13). The first equation of
(2.13) follows from the definition of $\tilde{H}_{1}$ and (2.12). It is clear
that $P^{*}H\supseteq P^{*}H_{1}$. On the other hand, since
$P^{*}Ph=0\Rightarrow\mathop{\langle}h,P^{*}Ph\mathop{\rangle}=0\Rightarrow
Ph=0$, it holds that ${\cal N}(P^{*}P)\subseteq{\cal N}(P)$. This, together
with the fact that $H_{1}=PH$ (see (2.1) and (1.14)), yields

$P^{*}H_{1}=P^{*}PH={\cal R}(P^{*}P)={\cal N}(P^{*}P)^{\bot}\supseteq{\cal
N}(P)^{\bot}={\cal R}(P^{*})=P^{*}H.$

Therefore, (2.13) holds.

The proof of (2.14) is as follows. Since ${\cal P}^{*C}\xi=\mu\xi$, we derive
from (2.9) that

$\hat{\tilde{P}}\xi=\displaystyle\frac{1}{2\pi i}\int_{\Gamma}\left(\lambda
I-{\cal P}^{*C}\right)^{-1}d\lambda\,\xi=\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\left(\lambda-\mu\right)^{-1}d\lambda\,\xi=\xi.$

Hence, $\xi\in\hat{\tilde{P}}H^{C}$. Meanwhile, by the definitions of
$\tilde{P}$ and $\tilde{H}_{1}$, we find that

$\tilde{H}_{1}^{C}=\big{(}\hat{\tilde{P}}\big{|}_{H}\big{)}^{C}H^{C}=\hat{\tilde{P}}H^{C}.$

Thus, it holds that $\xi\in\tilde{H}_{1}^{C}$. This completes the proof.

Acknowledgement The authors would like to thank Professor Xu Zhang for his
valuable suggestions on this paper.

## References

  * [1]
  * [2]
  * [3]
  * [4] V. Barbu, Stabilization of Navier-Stokes Flows, Springer-Verlag, London, 2011.
  * [5] V. Barbu and G. Wang, Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems, Indiana Univ. Math. J. 54 (2005), pp. 1521-1546.
  * [6] V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier-Stoke equations, SIAM J. Control Optim. 49, 4 (2011) 1454-1478.
  * [7] S. Bochner, Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind, Fundamenta Mathematicae 20 (1933) pp. 262-276.
  * [8] J. M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.
  * [9] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, AMS, Providence, RI, 1977.
  * [10] C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125A (1995), pp. 31-61.
  * [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981.
  * [12] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag, New York, 1966.
  * [13] S. Kurepa, On the quadratic functional, Acad. Serbe Sci. Publ. Inst. Math., 13 (1961), pp. 57-72.
  * [14] X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1995.
  * [15] F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 43 (1990), pp. 127-136.
  * [16] A. Lunardi, Stabilizability of time-periodic parabolic equations, SIAM J. Control and Optim., 29 (1991), pp. 810-828.
  * [17] K. D. Phung and G. Wang, Quantitative uniqueness for time-periodic heat equation with potential and its applications. Differential Integral Equations, 19, 6 (2006), 627-668.
  * [18] K. D. Phung and G. Wang, Quantitative unique continuation for the semiliear heat equation in a convex domain. J. Funct. Anal, 259 (2010) pp. 1230-1247.
  * [19] K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications. J. Eur. Math. Soc., 15, 2 (2013) pp. 681-703.
  * [20] G. Da Prato and A. Ichikawa, Quadratic control for linear time-varying systems. SIAM J. Control and Optim., 28 (1990) pp. 359-381.
  * [21] G. Da Prato and A. Lunaridi, Floquent exponents and Stabilitizability in Time-periodic parobolic systems, Appl. Math. Optim., 22 (1990) pp. 91-113.
  * [22] A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite dimensional systems, SIAM Rev., 23 (1981), pp. 25-52.
  * [23] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open problems, SIAM Rev., 20 (1978), pp. 639-739.
  * [24] E. D. Sontag, Mathematical Control Theory: Deterministic Finite-Dimensional Systems, Second edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.
  * [25] G. Wang and Y. Xu, Periodic Stabilization for Linear Time-Periodic Ordinary Differential Equations, submitted.
  * [26] K. Yosida, Functional analysis, Reprint of the sixth edition (1980), Classics in Mathematics, Springer-Verlag, Berlin, 1995.