remarkRemark defnDefinition lemLemma corCorollary propProposition exmpExample
hypothesisHypothesis claimClaim FINITE-TIME LINEAR-QUADRATIC OPTIMAL CONTROL
OF INDEX-1 PDAEsA. Alalabi and K. Morris

# Finite-time Linear-Quadratic optimal control of partial differential-
algebraic equations ††thanks: This work was supported by NSERC (Canada) and
the Faculty of Mathematics, University of Waterloo.

Ala’ Alalabi Authors are with the Department of Applied Mathematics,
University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L
3G1
<EMAIL_ADDRESS><EMAIL_ADDRESS>Kirsten Morris

###### Abstract

We consider finite-time linear-quadratic control of partial differential-
algebraic equations (PDAEs) . The discussion is restricted to those that are
radial with index $0$; this corresponds to a nilpotency degree of 1. We
establish the existence of a unique minimizing optimal control. A projection
is used to derive a system of differential Riccati-like equation coupled with
an algebraic equation, yielding the solution of the optimization problem in a
feedback form. This generalizes the well-known result for PDEs to this class
of PDAEs. These equations and hence the optimal control can be calculated
without construction of the projected PDAE. Finally, we provide numerical
simulations to illustrate application of the theoretical results.

###### keywords:

Partial differential-algebraic equations, linear-quadratic optimal control,
differential Riccati equation, calculus of variation.

##  1 Introduction

Partial differential-algebraic equations (PDAEs), also known as infinite-
dimensional descriptors [34], singular distributed parameter systems [14, 12],
and abstract DAEs [22] arise in many applications. Examples include fluid
dynamics specifically divergence-free fluid dynamics [24], electrical networks
[17], nano-electronics [4], biology [1] and chemical engineering [23]. The
well-posedness of linear PDAEs has received attention, for example, [37, 19,
11, 26] and the references therein.

Various definitions for the index of a PDAE exist in the literature[8, 25, 38,
33]. Unlike (finite-dimensional) DAEs [21], the different definitions of
indices for PDAEs are not equivalent [11]. The concept of the index in PDAEs
is crucial, serving as a key indicator of the numerical challenges expected
when solving such equations. Moreover, it plays a role in the assignment of
initial and boundary conditions to ensure the consistency relation that
results from the algebraic component of the system[25]. Failure to meet this
relation leads to distributions in the solution of the PDAE [13].

Controller synthesis methods for PDAEs are needed in order stabilize their
dynamics and/or to achieve desired performance. One valuable technique in
controller design is linear quadratic (LQ) control. Researchers have made
progress in establishing LQ controller design for DAEs in finite-dimensional
spaces, addressing index-1 [5, 27] as well as general higher-index [35, 32].
Using a singular value decomposition and calculus of variations, Bender and
Laub [6] studied the optimization problem for index-1 DAEs on a finite and
infinite horizons. Moreover, to solve for the optimal control, Bender and Laub
[6] have derived several differential Riccati equations. Mehrmann [27] studied
the finite-time LQ control of the same problem and showed that the existence
of a unique continuous optimal control depends on the solvability of a two-
point boundary value problem. This optimal control was assumed to satisfy the
consistency relation on the initial condition. On the other hand, Reis and
Voigt [35] used a behavior-based approach to study optimal control for DAEs
with arbitrary index. Petreczky and Zhuk [31] also used behaviors to study
optimal control for linear DAEs that are not regular.

Optimal control for DAEs on infinite-dimensional spaces, that is, PDAEs,
contains many open problems. In this context, we mention the work done by
Grenkin et al. [16] where they tackled the boundary optimal control problem
for a heat transfer model consisting of a coupled transient and steady-state
heat equations. Grenkin et al. [16] showed the existence of weak solutions of
this optimization problem under certain assumptions but without taking into
consideration the initial condition’s consistency. More recently, Gernandt and
Reis [15] studied LQ optimal control for a class of PDAEs with resolvent
index-one in the pseudo-resolvent sense by considering the mild solutions of
the system. Under specific conditions on the Popov operator, and provided that
the initial algebraic sub-state of the system adheres to a certain algebraic
constraint, Gernandt and Reis [15] showed the existence of unique minimizing
control in the space of square integrable functions. The optimal cost was
shown to be determined by a bounded Riccati operator.

This paper studies the LQ control problem over a finite-horizon for a class of
linear PDAEs, those with radiality-index 0. (The radiality index will be
defined in the next section.) Many equations arising in applications, such a
parabolic-elliptic systems and also the equation used to model the free
surface of seepage liquid have radiality-index  0 [19] have index zero. With
proofs different from those for finite-dimensional systems [27, 35], a fixed-
point argument is used to show that there is a unique continuous optimal
control, and this control can written in feedback form. We do not impose
assumption on the initial values such as been done in [15]. Instead, by
defining a set of admissible controls for the optimization problem, we demand
that the control signal maintain the consistency of the initial conditions,
thereby preventing distributions in the solutions [13]. Next, we derive a
coupled system consisting of a differential Riccati-like equation and an
algebraic equation that leads to the optimal control. It is important to note
that specific projections are involved in the derivation’s proof process.
However, after the state weight’s decomposition through these projections is
established, there is no need to use these projections in computing the
optimal control. This approach draws inspiration from the works of
Heinkenschloss [18] and Stykel [36], where they developed a Lyapunov equation
within the study of balanced truncation model reduction for specific finite-
dimensional systems. Similarly, Duan [10] derived a Lyapunov equation for a
particular class of index-1 DAEs. Breiten et al. [7] studied optimal control
of the Navier-Stokes equations and derived a certain Riccati equation. In
fact, our work marks a first effort towards deriving differential Riccati-like
equations for a general class of differential-algebraic equations, without
needing to know the projection explicitly. We demonstrate that although
projections can be a valuable tool in deriving the desired equations, they may
not be needed for computation of the control. Finally, we illustrate the
approach by design of an LQ-optimal control for an unstable coupled parabolic-
elliptic system [30, 2].

The paper is structured as follows: In Section 2, we formulate the problem,
and define the class of PDAEs. Section 3 establishes the existence of an
optimal control for the finite-horizon optimization problem. The derivation of
a differential Riccati equation, essential for determining this optimal
control, is elaborated in Section 4. Finally, in Section 5, numerical
simulations are given to illustrate the theoretical results.

##  2 Problem statement

Let $\mathcal{X},\;\mathcal{Z}$ and $\mathcal{U}$ be Hilbert spaces. Consider
a system modelled by

(1a) $\displaystyle\frac{d}{dt}Ex(t)=$ $\displaystyle Ax(t)+Bu(t),$ (1b)
$\displaystyle Ex(0)=$ $\displaystyle Ex_{i},$

where
$E\in\mathcal{L}(\mathcal{X},\mathcal{Z}),\;A:D(A)\subset\mathcal{X}\rightarrow\mathcal{Z}$
is closed and densely defined on $\mathcal{X},$
$B\in\mathcal{L}(\mathcal{U},\mathcal{Z})$. The initial condition
$x(0)=x_{i}\in\mathcal{X}$. Systems of the form (1) reduce to classical
infinite-dimensional systems [9] when $E$ is invertible. The situation where
$E$ is non-invertible, and /or unbounded is of particular interest in this
study.

It will be assumed throughout this paper that the PDAE (1b) is radial with
degree 0 [19, 37]. This implies that exist projections $P^{\mathcal{X}_{1}}$
and $P^{\mathcal{Z}_{1}}$ with ranges $\mathcal{X}_{1}$ and $\mathcal{Z}_{1}$
respectively, satisfying the two assumptions below. Let $\mathcal{X}_{0}$
indicate the complement of $\mathcal{X}_{1}$ and define similarly
$\mathcal{Z}_{0},\mathcal{Z}_{1}:$

(2)
$\displaystyle\mathcal{X}=\mathcal{X}_{1}\oplus\mathcal{X}_{0},\quad\mathcal{Z}=\mathcal{Z}_{1}\oplus\mathcal{Z}_{0}.$

Use the projections to define

(3a) $\displaystyle E_{0}=E|_{\mathcal{X}_{0}},\;\;$ $\displaystyle
E_{1}=E|_{\mathcal{X}_{1}},$ $\displaystyle A_{0}=A|_{\mathcal{X}_{0}\cap
D(A)},$ $\displaystyle A_{1}=A|_{\mathcal{X}_{1}\cap D(A)},$ (3b)
$\displaystyle B_{0}=P^{\mathcal{Z}_{0}}B,$ $\displaystyle
B_{1}=P^{\mathcal{Z}_{1}}B.$

Assumption 1.

(4a) $\displaystyle\text{For all
}x\in\mathcal{X},\;P^{\mathcal{Z}_{1}}Ex=EP^{\mathcal{X}_{1}}x,$ (4b)
$\displaystyle\text{For all }x\in D(A),\;P^{\mathcal{X}_{1}}x\in
D(A)\;\text{and }P^{\mathcal{Z}_{1}}Ax=AP^{\mathcal{X}_{1}}x.$

Assumption 2. The operator $A_{0}:D(A_{0})\rightarrow\mathcal{Z}_{0}$ has a
bounded inverse: $A_{0}^{-1}\in\mathcal{L}(\mathcal{Z}_{0},\mathcal{X}_{0}),$
and the operator $E_{0}$ is the zero operator. Also,
$E_{1}\in\mathcal{L}(\mathcal{X}_{1},\mathcal{Z}_{1})$ maps into
$\mathcal{Z}_{1}$, and has a bounded inverse:
$E_{1}^{-1}\in\mathcal{L}(\mathcal{Z}_{1},\mathcal{X}_{1}).$ The operator
$A_{1}:D(A)\cap\mathcal{X}_{1}\to\mathcal{Z}_{1}$ is closed and densely
defined, and $E_{1}^{-1}A_{1}$ generates a $C_{0}-$semigroup operator $T(t)$
on $\mathcal{X}_{1}$.

Letting $I^{\mathcal{X}}$ and $I^{\mathcal{Z}}$ indicate the identity operator
on the space $\mathcal{X}$ and $\mathcal{Z}$ , respectively, we define

$\displaystyle P^{\mathcal{X}_{0}}=I^{\mathcal{X}}-P^{\mathcal{X}_{1}},\quad
P^{\mathcal{Z}_{0}}=I^{\mathcal{Z}}-P^{\mathcal{Z}_{1}}.$

We also define

(5)
$\displaystyle\tilde{P}^{\mathcal{X}}=\begin{bmatrix}P^{\mathcal{X}_{1}}\\\
P^{\mathcal{X}_{0}}\end{bmatrix}\in\mathcal{L}(\mathcal{X},\mathcal{X}_{1}\times\mathcal{X}_{0}),\quad\tilde{P}^{\mathcal{Z}}=\begin{bmatrix}P^{\mathcal{Z}_{1}}\\\
P^{\mathcal{Z}_{0}}\end{bmatrix}\in\mathcal{L}(\mathcal{Z},\mathcal{Z}_{1}\times\mathcal{Z}_{0}),$
(6)
$\displaystyle(\tilde{P}^{\mathcal{X}})^{-1}=\begin{bmatrix}I^{\mathcal{X}_{1}}&I^{\mathcal{X}_{0}}\end{bmatrix}\in\mathcal{L}(\mathcal{X}_{1}\times\mathcal{X}_{0},\mathcal{X}),$
(7)
$\displaystyle(\tilde{P}^{\mathcal{Z}})^{-1}=\begin{bmatrix}I^{\mathcal{Z}_{1}}&I^{\mathcal{Z}_{0}}\end{bmatrix}\in\mathcal{L}(\mathcal{Z}_{1}\times\mathcal{Z}_{0},\mathcal{Z}).$

For convenience of notation, define

(8)
$\displaystyle\tilde{A}_{1}=E_{1}^{-1}A_{1},\;\;\;\tilde{B}_{1}=E_{1}^{-1}B_{1},\;\;\;\tilde{B}_{0}=A_{0}^{-1}B_{0},$

and

$\displaystyle P^{\mathcal{X}_{1}}x_{i}=(x_{i})_{1},\quad
P^{\mathcal{X}_{0}}x_{i}=(x_{i})_{0}.$

Pre-multiply system (1) with the operator $\tilde{P}^{\mathcal{Z}}$ and use
(4a)-(4b),

(9a) $\displaystyle\frac{d}{dt}x_{1}(t)=$
$\displaystyle\tilde{A}_{1}x_{1}(t)+\tilde{B}_{1}u(t),\;\;x_{1}(0)=(x_{i})_{1},$
(9b) $\displaystyle 0=$ $\displaystyle
x_{0}(t)+\tilde{B}_{0}u(t),\;\;\;\;x_{0}(0)=(x_{i})_{0},$

In common with finite-dimensions, this is known as the Weierstra$\beta$ form;
the systems considered here have degree of nilpotency 1 [11].

For continuous control $u(t)\in C(0,t_{f};\mathcal{U})$, the mild solution of
(9a) is [9, Chapter 5]

(10) $\displaystyle
x_{1}(t)=T(t)(x_{i})_{1}+\int_{0}^{t}T(t-s)\tilde{B}_{1}u(s)ds.$

For system (9b) (see [13])

(11) $\displaystyle
x_{0}(t)=-\tilde{B}_{0}u(t)+\delta(t)\Big{(}(x_{i})_{0}+\tilde{B}_{0}u(0)\Big{)},\quad
t\geq 0,$

where $\delta(t)$ is the Dirac delta function. If
$(x_{i})_{0}=-\tilde{B}_{0}u(0)$, then the distributional component of the
solution (11) is eliminated. This equation is known as the consistency
condition on the initial condition in the DAE literature [21, Theorem 2.12].
With a consistent initial condition

(12) $\displaystyle x_{0}(t)=-\tilde{B}_{0}u(t),\quad t\geq 0,$

and

(13) $\displaystyle x(t)$
$\displaystyle=-\tilde{B}_{0}u(t)+T(t)(x_{i})_{1}+\int_{0}^{t}T(t-s)\tilde{B}_{1}u(s)ds.$

The optimal control problem is to minimize, for arbitrary initial condition
$x_{i}$, the quadratic performance criterion

$\displaystyle J(x_{i},u;t_{f})=$ $\displaystyle\langle
x(t_{f}),Gx(t_{f})\rangle_{\mathcal{X}}+\int_{0}^{t_{f}}\langle
x(s),Qx(s)\rangle_{\mathcal{X}}$ (14) $\displaystyle+\langle
u(s),Ru(s)\rangle_{\mathcal{U}}ds.$

As usual, $R$ is assumed to be coercive, that is, $R$ is self-adjoint and
$R\geq\epsilon I$ for some $\epsilon>0$. Also, $G$ and $Q$ are self-adjoint
non-negative operators. The notation $\langle\cdot,\cdot\rangle_{\mathcal{U}}$
stands for the inner product on some Hilbert space $\mathcal{U}$.

In order to avoid distributions in the solution, the set of admissible
controls for minimization of the cost $J(x_{i},u;t_{f})$ is

(15) $\displaystyle\mathcal{U}_{a}=\\{u(t)\in
C([0,t_{f}];\mathcal{U}):\;(x_{i})_{0}=-\tilde{B}_{0}u(0)\\}.$

This leads to a formal definition of the optimal control problem as

(16) $\inf_{u\in\mathcal{U}_{a}}J(x_{i},u;t_{f}),$

subject to $x(t)$ that solves (1).

The notation $P^{\mathcal{Z}_{1}*}$ and $P^{\mathcal{Z}_{0}*}$ shall denote
the adjoint operators of $P^{\mathcal{Z}_{1}}$ and $P^{\mathcal{Z}_{0}}$,
respectively, and so

$\displaystyle P^{\mathcal{Z}_{1}*}:\mathcal{Z}\to\mathcal{Z}_{1},\quad
P^{\mathcal{Z}_{0}*}:\mathcal{Z}\to\mathcal{Z}_{0},$

In a similar way, we define $P^{\mathcal{X}_{1}*}$ and $P^{\mathcal{X}_{0}*}$.
From (4), we also have that

(17a) $\displaystyle\text{For all
}z\in\mathcal{Z},\;P^{\mathcal{X}_{1}*}E^{*}z=EP^{\mathcal{Z}_{1}*}z,$ (17b)
$\displaystyle\text{For all }z\in D(A^{*}),\;P^{\mathcal{Z}_{1}*}z\in
D(A^{*})\;\text{and
}P^{\mathcal{X}^{*}_{1}}A^{*}z=A^{*}P^{\mathcal{Z}_{1}*}z.$

Throughout the paper, it is assumed that the final cost vanishes on the
subspace $\mathcal{X}_{0}$

(18) $\displaystyle
P^{{\mathcal{X}_{1}*}}GP^{{\mathcal{X}}_{0}}=P^{{\mathcal{X}_{0}*}}GP^{{\mathcal{X}}_{1}}=P^{{\mathcal{X}_{0}*}}GP^{{\mathcal{X}}_{0}}=0.$

Since in many applications $G=0$, this assumption is in practice not difficult
to meet. It is also assumed

(19) $\displaystyle
P^{{\mathcal{X}_{0}*}}QP^{{\mathcal{X}_{1}}}=P^{{\mathcal{X}_{1}*}}QP^{{\mathcal{X}_{0}}}=0.$

We also define the following restrictions on the subspaces $\mathcal{X}_{1}$
and $\mathcal{X}_{0}$,

$\displaystyle G_{1}=G|_{\mathcal{X}_{1}},\quad
Q_{1}=Q|_{\mathcal{X}_{1}},\quad Q_{0}=Q|_{\mathcal{X}_{0}}.$

Note that the operator $G_{1}$ maps to $X_{1}$ due to assumption (18) and so
$G_{1}\in\mathcal{L}(\mathcal{X}_{1})$. Similarly, (19) implies that
$Q_{1}\in\mathcal{L}(\mathcal{X}_{1})$ and
$Q_{0}\in\mathcal{L}(\mathcal{X}_{0}).$

##  3 Existence of the optimal control

Given the control problem defined in the previous section, and in particular
the definition of the set of admissible controls in (15), the set of
admissible variations $\mathcal{V}_{a}$, that is those for which if
$u\in\mathcal{U}_{a},$ $v\in\mathcal{V}_{a}$ then $u+v\in\mathcal{U}_{a}$ is

(20) $\displaystyle\mathcal{V}_{a}=\\{h(t)\in
C([0,t_{f}];\mathcal{U}):\;0=-\tilde{B}_{0}h(0)\\}.$

The next proposition shows that an arbitrary variation in the control leads to
a change in cost that is a sum of quadratic and linear terms.

######  Proposition 1.

Consider any $u(t)\in\mathcal{U}_{a}$, $x_{i}\in\mathcal{X}$ and
$h(t)\in\mathcal{V}_{a}$. Define, letting $T^{*}(t)$ indicate the
$C_{0}$-semigroup on $\mathcal{Z}_{1}$ generated by $\tilde{A}_{1}^{*},$

$\displaystyle z_{u}(t)=$ $\displaystyle
2\Big{[}\tilde{B}_{1}^{*}T^{*}(t_{f}-t)G_{1}x_{1}(t_{f})+\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}x_{1}(r)dr$
(21) $\displaystyle-\tilde{B}_{0}^{*}Q_{0}x_{0}(t)+Ru(t)\Big{]},$ (22)
$\displaystyle\phi(t)h(t)=\int_{0}^{t}T(t-s)\tilde{B}_{1}h(s)ds-\tilde{B}_{0}h(t).$

Then variation

$\displaystyle J(x_{i},u+h;t_{f})-$ $\displaystyle
J(x_{i},u;t_{f})=\int_{0}^{t_{f}}\langle z_{u}(s),h(s)\rangle_{\mathcal{U}}ds$
$\displaystyle+\langle\phi(t_{f})h(t_{f}),G\phi(t_{f})h(t_{f})\rangle_{\mathcal{X}}+\int_{0}^{t_{f}}\langle\phi(s)h(s),Q\phi(s)h(s)\rangle_{\mathcal{X}}$
(23) $\displaystyle+\langle h(s),Rh(s)\rangle_{\mathcal{U}}ds.$

######  Proof 3.1.

The proof is similar to the classical argument in $LQ-$optimal control for
ODEs e.g. [28]. Let $u(t)\in\mathcal{U}_{a}$ and let $h(t)\in\mathcal{V}_{a}$.
Then, $(u+h)(t)\in\mathcal{U}_{a}$ since we have that $(u+h)(t)\in
C(0,t_{f};\mathcal{U})$, and

(24) $\displaystyle-\tilde{B}_{0}(u+h)(0)=(x_{i})_{0}.$

Let $x(t)$ indicate the trajectory corresponding to $u(t)$, then from (13)

(25) $\displaystyle
x(t)=-\tilde{B}_{0}u(t)+T(t)(x_{i})_{1}+\int_{0}^{t}T(t-s)\tilde{B}_{1}u(s)ds.$

With the definition of $\phi(t)h(t)$ in (22), the trajectory corresponding to
$(u+h)(t)$ is

(26) $\displaystyle x_{u+h}(t)=x(t)+\phi(t)h(t).$

Thus,

$\displaystyle J(x_{i},u+h;t_{f})$ $\displaystyle-J(x_{i},u;t_{f})=2\langle
x(t_{f}),G\phi(t_{f})h(t_{f})\rangle_{\mathcal{X}}+2\Big{(}\int_{0}^{t_{f}}\langle
u(s),Rh(s)\rangle_{\mathcal{U}}$ $\displaystyle+\langle
x(s),Q\phi(s)h(s)\rangle_{\mathcal{X}}ds\Big{)}+\langle\phi(t_{f})h(t_{f}),G\phi(t_{f})h(t_{f})\rangle_{\mathcal{X}}$
(27)
$\displaystyle+\int_{0}^{t_{f}}\langle\phi(s)h(s),Q\phi(s)h(s)\rangle_{\mathcal{X}}+\langle
h(s),Rh(s)\rangle_{\mathcal{U}}ds.$

Setting

$\displaystyle\nabla J(u,h,t_{f})$ $\displaystyle=2\langle
x(t_{f}),G\phi(t_{f})h(t_{f})\rangle_{\mathcal{X}}+2\Big{(}\int_{0}^{t_{f}}\langle
x(s),Q\phi(s)h(s)\rangle_{\mathcal{X}}$ (28) $\displaystyle+\langle
u(s),Rh(s)\rangle_{\mathcal{U}}ds\Big{)},$

we rewrite (27) as

$\displaystyle J(x_{i},u+h;t_{f})-J(x_{i},u;t_{f})$ $\displaystyle=\nabla
J(u,h,t_{f})+\langle\phi(t_{f})h(t_{f}),G\phi(t_{f})h(t_{f})\rangle_{\mathcal{X}}$
(29)
$\displaystyle+\int_{0}^{t_{f}}\langle\phi(s)h(s),Q\phi(s)h(s)\rangle_{\mathcal{X}}+\langle
h(s),Rh(s)\rangle_{\mathcal{U}}ds.$

In the following steps, we compute a $z_{u}\in L_{2}(0,t_{f};\mathcal{U})$ in
a way that

(30) $\displaystyle\nabla J(u,h,t_{f})=\int_{0}^{t_{f}}\langle
z_{u}(s),h(s)\rangle_{\mathcal{U}}ds.$

Using the definition of $\phi(t)h(t)$ and the assumptions (18) and (19) on the
operators $G$ and $Q$, we write

$\displaystyle\nabla J(u,h,t_{f})=2\Big{[}\langle
x_{1}(t_{f}),G_{1}\int_{0}^{t_{f}}T(t_{f}-s)\tilde{B}_{1}h(s)ds\rangle_{\mathcal{X}}+\int_{0}^{t_{f}}\langle
u(s),Rh(s)\rangle_{\mathcal{U}}$ $\displaystyle+\langle
x_{1}(s),Q_{1}\int_{0}^{s}T(s-r)\tilde{B}_{1}h(r)dr\rangle_{\mathcal{X}}-\langle
x_{0}(s),Q_{0}\tilde{B}_{0}h(s)\rangle_{\mathcal{X}}ds\Big{]}.$

To simplify the right-hand-side of (LABEL:eq25), we note that after
straightforward calculations

$\displaystyle\langle
x_{1}(t_{f}),G_{1}\int_{0}^{t_{f}}T(t_{f}-s)\tilde{B}_{1}h(s)ds\rangle_{\mathcal{X}}+\int_{0}^{t_{f}}\langle
x_{1}(s),Q_{1}$
$\displaystyle\int_{0}^{s}T(s-r)\tilde{B}_{1}h(r)dr\rangle_{\mathcal{X}}ds=\int_{0}^{t_{f}}\langle\tilde{B}_{1}^{*}T^{*}(t_{f}-s)G_{1}x_{1}(t_{f})$
(32)
$\displaystyle+\tilde{B}_{1}^{*}\int_{s}^{t_{f}}T^{*}(r-s)Q_{1}x_{1}(r)dr,h(s)\rangle_{\mathcal{U}}ds,$

and

(33) $\displaystyle-2\int_{0}^{t_{f}}\langle
x_{0}(s),Q_{0}\tilde{B}_{0}h(s)\rangle_{\mathcal{X}}ds=-2\int_{0}^{t_{f}}\langle\tilde{B}_{0}^{*}Q_{0}x_{0}(s),h(s)\rangle_{\mathcal{U}}ds.$

Subbing (32) and (33) in (LABEL:eq25), we arrive at

$\displaystyle\nabla J(u,h,t_{f})$
$\displaystyle=\int_{0}^{t_{f}}\Big{[}\langle
2\tilde{B}_{1}^{*}T^{*}(t_{f}-s)G_{1}x_{1}(t_{f})$ (34)
$\displaystyle+2\tilde{B}_{1}^{*}\int_{s}^{t_{f}}T^{*}(r-s)Q_{1}x_{1}(r)dr-2\tilde{B}_{0}^{*}Q_{0}x_{0}(s)+2Ru(s),h(s)\rangle\Big{]}ds.$

Defining $z_{u}(t)$ as given in (21), the conclusion follows by combining (29)
and (34).

The next theorem shows the optimal control problem (16) has a solution.

######  Theorem 3.2.

The control input

$\displaystyle u^{opt}(t)=$
$\displaystyle-R^{-1}\Big{[}\tilde{B}_{1}^{*}T^{*}(t_{f}-t)G_{1}x^{opt}_{1}(t_{f})+\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}x^{opt}_{1}(r)dr$
(35) $\displaystyle-\tilde{B}^{*}_{0}Q_{0}x^{opt}_{0}(t)\Big{]},$

is the unique solution of the optimization problem (16). Here $x^{opt}(t)$ is
the system state corresponding to control $u^{opt}(t)$ , and
$x^{opt}(t)=x_{1}^{opt}(t)+x_{0}^{opt}(t)$.

######  Proof 3.3.

The proof takes two main steps. First, we show that $u^{opt}(t)$ is the unique
solution in $C(0,t_{f};\mathcal{U})$ solving equation

(36) $\displaystyle z_{u}(t)=0.$

We rewrite (36) in terms of a fixed-point operator. Define the operator

(37)
$\displaystyle(V_{1}u)(t)=T(t)(x_{i})_{1}+\int_{0}^{t}T(t-s)\tilde{B}_{1}u(s)ds.$

It is straightforward to show that $V_{1}u(t):C([0,t_{f}];\mathcal{U})\to
C([0,t_{f}];\mathcal{U}),$ by using the fact that $T(t)$ is a
$C_{0}$-semigroup, and following a similar argument as the one given in [9,
Lemma 5.1.5]. Next, referring to (21), we rewrite equation (36) by using (10)
and (12)

$\displaystyle Ru(t)$
$\displaystyle=-\tilde{B}_{1}^{*}T^{*}(t_{f}-t)G_{1}x_{1}(t_{f})-\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}\Big{(}T(r)(x_{i})_{1}$
$\displaystyle+\int_{0}^{r}T(r-s)\tilde{B}_{1}u(s)ds\Big{)}dr-\tilde{B}^{*}_{0}Q_{0}A_{0}^{-1}B_{0}u(t).$

The previous equation can be now written with the help of operator $V_{1}$ as

$\displaystyle Ru(t)$
$\displaystyle=-\tilde{B}_{1}^{*}T^{*}(t_{f}-t)G_{1}x_{1}(t_{f})-\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}(V_{1}u)(r)dr$
$\displaystyle-\tilde{B}^{*}_{0}Q_{0}A_{0}^{-1}B_{0}u(t),$

that is

$\displaystyle(R+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0})u(t)$
$\displaystyle=-\tilde{B}_{1}^{*}T^{*}(t_{f}-t)G_{1}x_{1}(t_{f})$ (38)
$\displaystyle-\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}(V_{1}u)(r)dr.$

Since the operator $\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0}$ is positive semi-
definite and $R$ is coercive, it follows that
$R+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0}$ is also coercive, and so has a bounded
inverse. For convenience of notation, we set

(39) $\displaystyle\tilde{R}=R+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0}.$

For $u(t)\in C([0,t_{f}];\mathcal{U})$, define $F:C([0,t_{f}];\mathcal{U})\to
C([0,t_{f}];\mathcal{U})$ as

$\displaystyle(Fu)(t)$
$\displaystyle=-(\tilde{R})^{-1}\tilde{B}_{1}^{*}T^{*}(t_{f}-t)G_{1}x_{1}(t_{f})$
(40)
$\displaystyle-(\tilde{R})^{-1}\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}(V_{1}u)(r)dr.$

With this definition, equation (38) can be written

(41) $\displaystyle u(t)=(Fu)(t).$

It will now be shown that $F^{n}$ is a contraction for large enough $n$. For
$u_{1},\;u_{2}\in C(0,t_{f};\mathcal{U})$

$\displaystyle|(Fu_{1})(t)-(Fu_{2})(t)|$
$\displaystyle=|(\tilde{R})^{-1}\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}\Big{[}(V_{1}u_{1})(r)-(V_{1}u_{2})(r)\Big{]}\,dr|$
$\displaystyle\leq\|(\tilde{R})^{-1}\tilde{B}_{1}^{*}\|\int_{t}^{t_{f}}\|T^{*}(r-t)\|\|Q_{1}\||(V_{1}u_{1})(r)-(V_{1}u_{2})(r)|dr$
$\displaystyle\leq\|(\tilde{R})^{-1}\tilde{B}_{1}^{*}\|\int_{t}^{t_{f}}\|T^{*}(r-t)\|\|Q_{1}\|\int_{0}^{r}\|T(r-s)\|\|\tilde{B}_{1}\||u_{1}(s)-u_{2}(s)|dsdr.$

Since $T(t)$ is bounded on every finite subinterval of $[0,\infty)$; see [9,
Theorem 2.1.7 a], and so is $T^{*}(t)$, the operators
$(\tilde{R})^{-1}\tilde{B}_{1}^{*}=(R+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0})^{-1}\tilde{B}_{1}^{*}\in\mathcal{L}(\mathcal{X}_{1},\mathcal{U})$,
and $Q_{1}$, $\tilde{B}_{1}$ are also bounded linear operators, there is a
constant $M>0$ such that

$|(Fu_{1})(t)-(Fu_{2})(t)|\leq
M\int_{t}^{t_{f}}\int_{0}^{r}|u_{1}(s)-u_{2}(s)|dsdr.$

Thus,

$\displaystyle|(Fu_{1})(t)-(Fu_{2})(t)|$ $\displaystyle\leq
M\|u_{1}-u_{2}\|_{\infty}\int_{t}^{t_{f}}\int_{0}^{r}dsdr$ $\displaystyle\leq
t_{f}(t_{f}-t)M\|u_{1}-u_{2}\|_{\infty}.$

Replace $u_{1}$ and $u_{2}$ in the inequalities above with $Fu_{1}$ and
$Fu_{2}$, respectively,

$\displaystyle|(F^{2}u_{1})(t)-(F^{2}u_{2})(t)|$ $\displaystyle\leq
M\int_{t}^{t_{f}}\int_{0}^{r}|(Fu_{1})(s)-(Fu_{2})(s)|dsdr$ $\displaystyle\leq
M^{2}t_{f}\|u_{1}-u_{2}\|_{\infty}\int_{t}^{t_{f}}\int_{0}^{r}(t_{f}-s)dsdr$
$\displaystyle=M^{2}t_{f}\|u_{1}-u_{2}\|_{\infty}\int_{t}^{t_{f}}\frac{(t_{f}-r)^{2}}{2}dr$
(42) $\displaystyle\leq
M^{2}t_{f}^{2}\|u_{1}-u_{2}\|_{\infty}\frac{(t_{f}-t)^{2}}{2}.$

An induction argument shows that

(43) $\displaystyle|(F^{n}u_{1})(t)-(F^{n}u_{2})(t)|$ $\displaystyle\leq
M^{n}t_{f}^{n}\frac{(t_{f}-t)^{n}}{n!}\|u_{1}-u_{2}\|_{\infty}.$

Defining $a=(Mt_{f}^{2}),$ it follows that

(44) $\displaystyle\|(F^{n}u_{1})(t)-(F^{n}u_{2})(t)\|_{\infty}$
$\displaystyle\leq\frac{a^{n}}{n!}\|u_{1}-u_{2}\|_{\infty}.$

For sufficiently large $n$, $\frac{a^{n}}{n!}<1$ and hence $F^{n}$ is a
contraction for large enough $n.$ It follows from [20, Lemma 5.4-3] that $F$
has a unique fixed point. Hence, there exists a unique control in
$C(0,t_{f};\mathcal{U})$ that solves equation (41), and is also the unique
solution of equation (36).

The control $u^{opt}(t)\in C(0,t_{f};\mathcal{U})$, and is the unique control
leading to $z_{u}=0$. Since equation (23) was derived by allowing for the
variation $h(t)\in\mathcal{V}_{a}$, ensuring that $(u+h)(t)$ satisfies the
consistency condition, it follows that $u^{opt}(t)$ ensures the consistency of
the initial condition, that is, $u^{opt}(t)\in\mathcal{U}_{a}$.

Referring to (23), since $R>0,$ $Q\geq 0,$ $G\geq 0,$ for any admissible
variation $h,$

(45) $\displaystyle J(x_{i},u^{opt}+h;t_{f})-J(x_{i},u^{opt};t_{f})>0,$

and so $u^{opt}(t)$ is the unique control minimizing $J.$

##  4  Derivation of differential Riccati equations

Consider the cost functional (14) with a variable initial time $t_{0}$, $0\leq
t_{0}\leq t_{f}$,

$\displaystyle J(x_{i},u;t_{0},t_{f})=$ $\displaystyle\langle
x(t_{f}),Gx(t_{f})\rangle_{\mathcal{X}}+\int_{t_{0}}^{t_{f}}\langle
x(s),Qx(s)\rangle_{\mathcal{X}}$ (46) $\displaystyle+\langle
u(s),Ru(s)\rangle_{\mathcal{U}}ds.$

Since the governing equations are time-invariant, assuming $u(t_{0})$ is
consistent with $x(t_{0})$, the previous section’s results apply. Thus, there
is unique input that minimizes the cost functional (46) for trajectories of
(1) with initial condition $x(t_{0})=x_{i}$. The control that minimizes cost
functional (46) is denoted $u^{opt}(\cdot;x_{i},t_{0},t_{f})$, and its
corresponding optimal state trajectory is $x^{opt}(\cdot;x_{i},t_{0},t_{f}).$
The control that minimizes (14) shall be denoted by $u^{opt}(t)$ or
$u^{opt}(\cdot;x_{i},0,t_{f})$, and its corresponding state trajectory is
$x^{opt}(t)$ or $x^{opt}(\cdot;x_{i},0,t_{f})$.

The following result holds due to the uniqueness of the optimal control. It is
an extension of the principle of optimality from linear PDEs to linear PDAEs,
and for each of the sub-states.

######  Lemma 2.

Let $0\leq t_{0}\leq t_{f}$. For all $s\in[t_{0},t_{f}]$,

(47) $\displaystyle
x^{opt}(s;x_{i},0,t_{f})=x^{opt}(s;x^{opt}(t_{0},x_{i},0,t_{f}),t_{0},t_{f}).$

In addition, each of the dynamical and the algebraic sub-states satisfy the
principle of optimality, that is,

$\displaystyle
x_{0}^{opt}(s;(x_{i})_{0},0,t_{f})=x_{0}^{opt}(s;x^{opt}_{0}(t_{0},(x_{i})_{0},0,t_{f}),t_{0},t_{f}),$

and

(48) $\displaystyle
x_{1}^{opt}(s;(x_{i})_{1},0,t_{f})=x_{1}^{opt}(s;x^{opt}_{1}(t_{0},(x_{i})_{1},0,t_{f}),t_{0},t_{f}).$

######  Proof 4.1.

Since the optimal control $u^{opt}(t)$ is unique, we can use a similar line of
reasoning as the one presented in [9, Lemma 9.1.7] to establish equation (47)
for all $s\in[t_{0},t_{f}]$. Using once more the uniqueness of the optimal
control and also equation (9b), we have for any $s\in[t_{0},t_{f}]$

$\displaystyle x_{0}^{opt}(s;(x_{i})_{0},0,t_{f})$
$\displaystyle=\tilde{B}_{0}u^{opt}(s;x_{i},0,t_{f})$
$\displaystyle=\tilde{B}_{0}u^{opt}(s;x^{opt}(t_{0},x_{i},0,t_{f}),t_{0},t_{f})$
(49)
$\displaystyle=x_{0}^{opt}(s;x^{opt}_{0}(t_{0},(x_{i})_{0},0,t_{f}),t_{0},t_{f}).$

Thus, the algebraic sub-state of the PDAE conforms to the principle of
optimality. The trajectory of system (1) on $[0,t_{f}]$ with initial condition
$x_{i}$ is

(50) $\displaystyle x^{opt}(s;x_{i},0,t_{f})$
$\displaystyle=x_{1}^{opt}(s;(x_{i})_{1},0,t_{f})+x_{0}^{opt}(s;(x_{i})_{0},0,t_{f}),$

and the trajectory on $[t_{0},t_{f}]$ with initial condition
$x^{opt}(t_{0},x_{i},0,t_{f})$ is

$\displaystyle x^{opt}(s;x^{opt}(t_{0},x_{i},0,t_{f}),t_{0},t_{f})$
$\displaystyle=x_{1}^{opt}(s;x_{1}^{opt}(t_{0},(x_{i})_{1},0,t_{f}),t_{0},t_{f})$
(51)
$\displaystyle+x_{0}^{opt}(s;x^{opt}_{0}(t_{0},(x_{i})_{0},0,t_{f}),t_{0},t_{f}).$

Using (47), we deduce that the right-hand-side of equations (50) and (51) are
equal. It then follows from (49) that the dynamical sub-state of the PDAE,
i.e. $x^{opt}_{1}$, also satisfies the principle of optimality (48).

The next proposition demonstrates that at any given time $t\in[0,t_{f}]$, the
optimal control (35) can be written as a feedback of the dynamical state
$x^{opt}_{1}$.

######  Proposition 3.

The optimization problem (16) reduces to minimizing the cost functional

$\displaystyle J((x_{i})_{1},u;t_{f})=$ $\displaystyle\langle
x_{1}(t_{f}),G_{1}x_{1}(t_{f})\rangle_{\mathcal{X}}+\int_{0}^{t_{f}}\langle
x_{1}(s),Q_{1}x_{1}(s)\rangle_{\mathcal{X}}$ (52) $\displaystyle+\langle
u(s),\tilde{R}u(s)\rangle_{\mathcal{U}}ds,$

over the set of admissible control $\mathcal{U}_{a}$, where $x_{1}(t)$ solves
system (9a).

The minimizing optimal control $u^{opt}(t)$ can be rewritten as

(53) $\displaystyle u^{opt}(t)=$
$\displaystyle-\tilde{R}^{-1}\tilde{B}_{1}^{*}\big{[}T^{*}(t_{f}-t)G_{1}x^{opt}_{1}(t_{f})+\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}x^{opt}_{1}(r)dr\big{]}.$

######  Proof 4.2.

Rewrite the cost functional (14) as

$\displaystyle J(x_{i},u;t_{f})=$ $\displaystyle\langle
x_{1}(t_{f}),G_{1}x_{1}(t_{f})\rangle_{\mathcal{X}}+\int_{0}^{t_{f}}\langle
x_{1}(s),Q_{1}x_{1}(s)\rangle_{\mathcal{X}}+\langle
x_{0}(s),Q_{0}x_{0}(s)\rangle_{\mathcal{X}}$ (54) $\displaystyle+\langle
u(s),Ru(s)\rangle_{\mathcal{U}}ds.$

Since the optimization is over $\mathcal{U}_{a}$, use that
$x_{0}^{opt}(t)=-\tilde{B}_{0}u^{opt}(t)$ to obtain the cost (52). Note that
the existence of unique optimizing control for cost functional (52) over the
set $\mathcal{U}_{a}$ follows from the results in section3 concerning the
equivalent optimization problem (35). To prove statement (53), we first recall
that $\tilde{R}=R+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0}$. From (35), it follows
that

$\displaystyle u^{opt}(t)=$
$\displaystyle-R^{-1}\Big{[}\tilde{B}_{1}^{*}T^{*}(t_{f}-t)G_{1}x^{opt}_{1}(t_{f})+\tilde{B}_{1}^{*}\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}x^{opt}_{1}(r)dr$
$\displaystyle+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0}u^{opt}(t)\Big{]}.$

Now the previous equation can be rewritten as

$\displaystyle(R+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0})u^{opt}(t)=$
$\displaystyle-\tilde{B}_{1}^{*}\Big{[}T^{*}(t_{f}-t)G_{1}x^{opt}_{1}(t_{f})+\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}x^{opt}_{1}(r)dr\Big{]}.$

Since the operator $\tilde{R}=R+\tilde{B}^{*}_{0}Q_{0}\tilde{B}_{0}$ is
coercive, we arrive at equation (53).

The optimization problem (16) simplifies to a standard LQ problem on
$\mathcal{X}_{1}$. This simplification allows us to apply well-known results
on the finite-time LQ problem; see [9, Chapter 9].

######  Lemma 4.

[9, Lemma 9.1.9] For any $t\in[0,t_{f}]$ and any $x_{1}\in\mathcal{X}_{1}$,
define the operator $\Pi_{1}(t)$ on $\mathcal{X}_{1}$

$\displaystyle\Pi_{1}(t)x_{1}$
$\displaystyle=T^{*}(t_{f}-t)G_{1}x_{1}^{opt}(t_{f};x_{1},t,t_{f})$ (55)
$\displaystyle+\int_{t}^{t_{f}}T^{*}(r-t)Q_{1}x_{1}^{opt}(r;x_{1},t,t_{f})dr;$

$\Pi_{1}(t)\in\mathcal{L}(\mathcal{X}_{1})$ for all $t\in[0,t_{f}]$. The
optimal control (53) can be written as

(56) $\displaystyle u^{opt}(t;x_{i},0,t_{f})=$
$\displaystyle-\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t;(x_{i})_{1},0,t_{f}),$

and the minimum cost is

(57) $\displaystyle
J(x_{i},u^{opt};t_{f})=\langle(x_{i})_{1},\Pi_{1}(0)(x_{i})_{1}\rangle_{\mathcal{X}}.$

The optimal dynamical state $x^{opt}_{1}(t)$ is the mild solution to an
abstract evolution equation. This is stated in the corollary below.

######  Corollary 5.

The optimal sub-state $x^{opt}_{1}(t)$ is the mild solution of the abstract
evolution equation

(58) $\displaystyle\frac{d}{dt}x^{opt}_{1}(t)$
$\displaystyle=(E_{1}^{-1}A_{1}-\tilde{B}_{1}\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t))x^{opt}_{1}(t),\;$
$\displaystyle x_{1}^{opt}(0)$ $\displaystyle=(x_{i})_{1}.$

Also, the operator
$E_{1}^{-1}A_{1}-\tilde{B}_{1}\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t)$
generates the mild evolution operator $U(t,s)$ on the set $\\{(t,s);0\leq
s\leq t\leq t_{f}\\}$, so

(59) $\displaystyle
x^{opt}_{1}(t;(x_{i})_{1},t_{0},t_{f})=U(t,t_{0})(x_{i})_{1}.$

######  Proof 4.3.

This result follows directly by using the expression of the optimal control in
(53), and applying the results in [9, Corollary 9.1.10].

As a matter of fact, the operator-valued function $\Pi_{1}(t)$ is the unique
solution of a standard differential Riccati equation.

######  Lemma 6.

[9, Lemma 4.3.2,Theorem 9.1.11] The operator-valued function $\Pi_{1}(t)$
solves the following differential Riccati equation

$\displaystyle\frac{d}{dt}\Pi_{1}(t)x_{1}+\Pi_{1}(t)E_{1}^{-1}A_{1}x_{1}+A_{1}^{*}E_{1}^{-*}\Pi_{1}(t)x_{1}$
(60)
$\displaystyle-\Pi_{1}(t)\tilde{B}_{1}\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}+Q_{1}x_{1}$
$\displaystyle=0,\qquad\forall x_{1}\in D(A_{1}),$ (61)
$\displaystyle\tilde{\Pi}_{1}(t_{f})x_{1}$ $\displaystyle=G_{1}x_{1}.$

The operator-valued function $\Pi_{1}(t)$ is the unique solution of this
equation in the class of strongly continuous, self-adjoint operators in
$\mathcal{L}(\mathcal{X}_{1})$ such that $\langle
x_{1}^{a},\Pi_{1}(t)x^{b}_{1}\rangle_{\mathcal{X}}$ is differentiable for
$t\in(0,t_{f})$ and $x_{1}^{a},\;x_{1}^{b}\in D(A_{1})$.

This leads to the main result of this section: a characterization of the
optimal control (35) that does not require calculating the restrictions of
operators $A,E$ on the subspace $\mathcal{X}_{1}$.

######  Theorem 4.4.

Define the operator

(62) $\displaystyle\Pi_{0}$ $\displaystyle=-A_{0}^{-*}Q_{0};$

$\Pi_{0}\in\mathcal{L}(\mathcal{X}_{0},\mathcal{Z}_{0})$. Also, recalling the
operator-valued function (55), define

(63a)
$\displaystyle\tilde{\Pi}_{1}(t)=P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}P^{\mathcal{Z}_{1}},$
(63b)
$\displaystyle\tilde{\Pi}_{0}=P^{\mathcal{Z}_{0}*}\Pi_{0}P^{\mathcal{X}_{0}}.$

  1. 1.

The solution of the optimization problem (16) is

(64) $\displaystyle u^{opt}(t)=$
$\displaystyle-R^{-1}B^{*}(\tilde{\Pi}_{0}+\tilde{\Pi}_{1}(t)E)x^{opt}(t).$

  2. 2.

The operator $\tilde{\Pi}_{0}$ solves the algebraic equation

(65) $\displaystyle A^{*}\tilde{\Pi}_{0}x$
$\displaystyle=-QP^{\mathcal{X}_{0}}x,\quad\forall x\in\mathcal{X},$

and is uniquely defined on $\mathcal{X}_{0}.$

  3. 3.

In addition, $\tilde{\Pi}_{1}(t)\in C([0,t_{f}];\mathcal{L}(\mathcal{Z}))$
solves

$\displaystyle\frac{d}{dt}E^{*}\tilde{\Pi}_{1}(t)Ex+E^{*}\tilde{\Pi}_{1}(t)Ax+A^{*}\tilde{\Pi}_{1}(t)Ex$
$\displaystyle-E^{*}\tilde{\Pi}_{1}(t)BR^{-1}B^{*}\tilde{\Pi}_{1}(t)Ex$ (66a)
$\displaystyle-E^{*}\tilde{\Pi}_{1}(t)BR^{-1}B^{*}\tilde{\Pi}_{0}x+QP^{\mathcal{X}_{1}}x$
$\displaystyle=0,\qquad\forall x\in D(A),$ (66b) $\displaystyle
E^{*}\tilde{\Pi}_{1}(t_{f})Ex$ $\displaystyle=Gx.$

such that $\langle Ex^{a},\tilde{\Pi}_{1}(t)Ex^{b}\rangle_{\mathcal{Z}}$ is
differentiable for $t\in(0,t_{f})$ and $x^{a},x^{b}\in D(A).$

  4. 4.

The minimum cost is

(67) $\displaystyle J(x_{i},u^{opt};t_{f})=\langle
Ex_{i},\tilde{\Pi}_{1}(0)Ex_{i}\rangle_{\mathcal{Z}}.$

######  Proof 4.5.

1. Using the definitions of the operators $\tilde{B}_{0}$ and $\tilde{R}$ , write the optimal control (56) as follows:

$\displaystyle(R+B^{*}_{0}A_{0}^{-*}Q_{0}\tilde{B}_{0})u^{opt}(t)=$
$\displaystyle-\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t),$

and so

$\displaystyle u^{opt}(t)$
$\displaystyle=-R^{-1}\Big{(}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)+B^{*}_{0}A_{0}^{-*}Q_{0}\tilde{B}_{0}u^{opt}(t)\Big{)}.$

Since $u^{opt}(t)\in\mathcal{U}_{a}$, we use that
$x_{0}^{opt}(t)=-\tilde{B}_{0}u^{opt}(t)$ to rewrite the previous equation

$\displaystyle u^{opt}(t)$
$\displaystyle=-R^{-1}\Big{(}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)-B^{*}_{0}A_{0}^{-*}Q_{0}x_{0}^{opt}(t)\Big{)}$
(68)
$\displaystyle=-R^{-1}\Big{(}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)+B^{*}_{0}\Pi_{0}x_{0}^{opt}(t)\Big{)},$

where we also used to the definition of operator $\Pi_{0}$ in (62). From (4)a,
and (3), recall that for any $x(t)\in\mathcal{X}$

(69a) $\displaystyle
E_{1}x_{1}(t)=EP^{\mathcal{X}_{1}}x(t)=P^{\mathcal{Z}_{1}}Ex(t),$ (69b)
$\displaystyle B_{1}=P^{\mathcal{Z}_{1}}B,\quad B_{0}=P^{\mathcal{Z}_{0}}B.$

Using the statements above and that $\tilde{B}_{1}=E_{1}^{-1}B_{1}$, rewrite
(68) as

$\displaystyle u^{opt}(t)$
$\displaystyle=-R^{-1}B^{*}\Big{(}P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}EP^{\mathcal{X}_{1}}x^{opt}(t)+P^{\mathcal{Z}_{0}*}\Pi_{0}P^{\mathcal{X}_{0}}x^{opt}(t)\Big{)}.$

Equation (64) now follows from the definition of $\tilde{\Pi}_{1}(t)$ and
$\tilde{\Pi}_{0}$ in (63).

2. From (62), for any $x_{0}\in\mathcal{X}_{0}$,

(70) $\displaystyle A_{0}^{*}\Pi_{0}x_{0}$ $\displaystyle=-Q_{0}x_{0}.$

Writing $P^{\mathcal{X}_{0}}x=x_{0}$, we use (4b) and (3) to write the
previous equation as

(71) $\displaystyle
A^{*}P^{\mathcal{Z}_{0}*}\Pi_{0}P^{\mathcal{X}_{0}}x+Q_{0}P^{\mathcal{X}_{0}}x=0.$

Referring to (63b), we find that the operator $\tilde{\Pi}_{0}$ solves
equation (65). Deducing that $\tilde{\Pi}_{0}$ is the unique solution of (65)
on $\mathcal{X}_{0}$ is straightforward. If $x_{i}\in D(A)$, then
$(x_{i})_{1}\in D(A_{1})$. It follows from equation (59) that
$x_{1}^{opt}(t)\in D(A)$, and equation (60) implies

$\displaystyle\frac{d}{dt}\Pi_{1}(t)x^{opt}_{1}(t)+\Pi_{1}(t)E_{1}^{-1}A_{1}x_{1}^{opt}(t)+A_{1}^{*}E_{1}^{-*}\Pi_{1}(t)x_{1}^{opt}(t)$
(72)
$\displaystyle-\Pi_{1}(t)\tilde{B}_{1}\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)+Q_{1}x_{1}^{opt}(t)$
$\displaystyle=0.$

Since the right-hand-sides of equations (56) and (68) are equal,

$-\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)=-R^{-1}\Big{(}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)+B^{*}_{0}\Pi_{0}x_{0}^{opt}(t)\Big{)},$

and so

(73)
$\displaystyle-\Pi_{1}(t)\tilde{B}_{1}\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)=-\Pi_{1}(t)\tilde{B}_{1}R^{-1}\Big{(}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)+B^{*}_{0}\Pi_{0}x_{0}^{opt}(t)\Big{)}.$

Using (73) to replace the term
$-\Pi_{1}(t)\tilde{B}_{1}\tilde{R}^{-1}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)$
in equation (72),

$\displaystyle\frac{d}{dt}\Pi_{1}(t)x_{1}^{opt}(t)+\Pi_{1}(t)E_{1}^{-1}A_{1}x_{1}^{opt}(t)+A_{1}^{*}E_{1}^{-*}\Pi_{1}(t)x_{1}^{opt}(t)$
(74)
$\displaystyle-\Pi_{1}(t)\tilde{B}_{1}R^{-1}\Big{(}\tilde{B}_{1}^{*}\Pi_{1}(t)x_{1}^{opt}(t)+B^{*}_{0}\Pi_{0}x_{0}^{opt}(t)\Big{)}+Q_{1}x_{1}^{opt}(t)$
$\displaystyle=0$

From (4b), recall that for any $x_{1}(t)\in D(A)$

(75) $\displaystyle
A_{1}x_{1}(t)=AP^{\mathcal{X}_{1}}x(t)=P^{\mathcal{Z}_{1}}Ax(t).$

Hence, using the statement above along with (69), we rewrite equation (74) as

$\displaystyle\frac{d}{dt}E^{*}P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}P^{\mathcal{Z}_{1}}Ex^{opt}(t)+E^{*}P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}P^{\mathcal{Z}_{1}}Ax^{opt}(t)$
$\displaystyle+A^{*}P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}P^{\mathcal{Z}_{1}}Ex^{opt}(t)$
$\displaystyle-E^{*}P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}P^{\mathcal{Z}_{1}}BR^{-1}B^{*}P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}P^{\mathcal{Z}_{1}}Ex^{opt}(t)$
$\displaystyle-E^{*}P^{\mathcal{Z}_{1}*}E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}P^{\mathcal{Z}_{1}}BR^{-1}B^{*}P^{\mathcal{Z}_{0}*}\Pi_{0}P^{\mathcal{X}_{0}}x^{opt}(t)+Q_{1}P^{\mathcal{X}_{1}}x^{opt}(t)=0.$

Using the definitions of $\tilde{\Pi}_{1}(t)$ and $\tilde{\Pi}_{0}$ in (63),
the previous equation yields

$\displaystyle\Big{(}\frac{d}{dt}$ $\displaystyle
E^{*}\tilde{\Pi}_{1}(t)E+E^{*}\tilde{\Pi}_{1}(t)A+A^{*}\tilde{\Pi}_{1}(t)E-E^{*}\tilde{\Pi}_{1}(t)BR^{-1}B^{*}\tilde{\Pi}_{1}(t)E$
(76)
$\displaystyle-E^{*}\tilde{\Pi}_{1}(t)BR^{-1}B^{*}\tilde{\Pi}_{0}+Q_{1}P^{\mathcal{X}_{1}}\Big{)}x_{opt}(t)=0.$

Since the initial condition $x_{i}$ was arbitrary in $D(A)$, and equation (76)
holds for all $t\geq 0,$ it follows that $\tilde{\Pi}_{1}(t)$ solves (66a),
where $\tilde{\Pi}_{0}$ solves (65).

The final condition in (66b) is obtained from (61) with the help of (4) and
(18).

3. To obtain the minimum cost (67), we refer to statement (57). For any $x_{0}\in\mathcal{X}$,

(77) $\displaystyle J(x_{i},u^{opt};t_{f})=\langle
E_{1}(x_{i})_{1},E_{1}^{-*}\Pi_{1}(0)E_{1}^{-1}E_{1}(x_{i})_{1}\rangle_{\mathcal{X}}.$

Using (63a), the previous equation yields

(78) $\displaystyle J(x_{i},u^{opt};t_{f})=\langle
Ex_{i},\tilde{\Pi}_{1}(0)Ex_{i}\rangle_{\mathcal{Z}}.$

The natural question we now address is whether the differential equation (66)
has a unique solution. The following theorem shows that the solution to this
equation is unique on the range of $E$.

######  Theorem 4.6.

Recall the operator $\tilde{\Pi}_{0}$ in (63b) and the differential equation
(66a)

$\displaystyle\frac{d}{dt}E^{*}Z(t)Ex+E^{*}Z(t)Ax+A^{*}Z(t)Ex$
$\displaystyle-E^{*}Z(t)BR^{-1}B^{*}Z(t)Ex$ (79a)
$\displaystyle-E^{*}Z(t)BR^{-1}B^{*}\tilde{\Pi}_{0}x+QP^{\mathcal{X}_{1}}x$
$\displaystyle=0,\qquad\forall x\in D(A),$ (79b) $\displaystyle
E^{*}Z(t_{f})Ex$ $\displaystyle=Gx.$

  1. 1.

Let $\tilde{\Pi}_{1}(t)\in C([0,t_{f}];\mathcal{L}(\mathcal{Z}))$ be as
defined by (63a) where $\langle
Ex^{a},\tilde{\Pi}_{1}(t)Ex^{b}\rangle_{\mathcal{X}}$ is differentiable for
all $x^{a},\;x^{b}\in D(A)$ and $t\in(0,t_{f})$. Also, let

$Z_{2}(t)\in C([0,t_{f}];\mathcal{L}(\mathcal{Z}_{0},\mathcal{Z}_{1})),\quad
Z_{4}(t)\in C([0,t_{f}];\mathcal{L}(\mathcal{Z}_{0})),$

where $Z_{4}(t)$ is arbitrary and $Z_{2}(t)$ solves

(80)
$Z_{2}(t)(A_{0}-B_{0}R^{-1}B_{0}^{*}\tilde{\Pi}_{0})=Z_{1}(t)B_{1}R^{-1}B_{0}^{*}\tilde{\Pi}_{0}.$

The general solution to equation (79) is

(81) $\displaystyle
Z(t)=\tilde{\Pi}_{1}(t)+P^{\mathcal{Z}_{1}*}Z_{2}(t)P^{\mathcal{Z}_{0}}+P^{\mathcal{Z}_{0}*}Z_{4}(t)P^{\mathcal{Z}_{0}}.$

  2. 2.

For any solution $Z(t)$ of the equation (79), the operator-valued function
$Z(t)E\in\mathcal{L}(\mathcal{X},\mathcal{Z})$ is unique and leads to the
optimal control (64)

(82) $\displaystyle u^{opt}(t)=$
$\displaystyle-R^{-1}B^{*}(\tilde{\Pi}_{0}+Z(t)E)x^{opt}(t).$

The minimum cost is

(83) $\displaystyle J(x_{i},u^{opt};t_{f})=\langle
Ex_{i},Z(0)Ex_{i}\rangle_{\mathcal{Z}}$

######  Proof 4.7.

Decompose equation (79a) with the projections $\tilde{P}^{\mathcal{Z}}$ and
$\tilde{P}^{\mathcal{X}}$ in (5) and write an arbitrary solution $Z(t)$

$Z(t)=\begin{bmatrix}Z_{1}(t)&Z_{2}(t)\\\ Z_{3}(t)&Z_{4}(t)\end{bmatrix}$

where

$\displaystyle Z_{1}(t)\in\mathcal{L}(\mathcal{Z}_{1}),$ $\displaystyle\quad
Z_{2}(t)\in\mathcal{L}(\mathcal{Z}_{1},\mathcal{Z}_{0}),$ $\displaystyle
Z_{3}(t)\in\mathcal{L}(\mathcal{Z}_{0},\mathcal{Z}_{1}),$ $\displaystyle\quad
Z_{4}(t)\in\mathcal{L}(\mathcal{Z}_{0}).$

Use the expression of $\tilde{\Pi}_{0}$ in (63b) and that $Q$ satisfies (19) ,
to obtain

$\displaystyle\frac{d}{dt}\begin{bmatrix}E_{1}^{*}&0\\\
0&0\end{bmatrix}\begin{bmatrix}Z_{1}(t)&Z_{2}(t)\\\ Z_{3}(t)&Z_{4}(t)\\\
\end{bmatrix}\begin{bmatrix}E_{1}&0\\\
0&0\end{bmatrix}+\begin{bmatrix}E_{1}^{*}&0\\\
0&0\end{bmatrix}\begin{bmatrix}Z_{1}(t)&Z_{2}(t)\\\ Z_{3}(t)&Z_{4}(t)\\\
\end{bmatrix}\begin{bmatrix}A_{1}&0\\\ 0&A_{0}\end{bmatrix}$
$\displaystyle+\begin{bmatrix}A_{1}^{*}&0\\\
0&A^{*}_{0}\end{bmatrix}\begin{bmatrix}Z_{1}(t)&Z_{2}(t)\\\
Z_{3}(t)&Z_{4}(t)\\\ \end{bmatrix}\begin{bmatrix}E_{1}&0\\\ 0&0\end{bmatrix}$
$\displaystyle-\begin{bmatrix}E_{1}^{*}&0\\\
0&0\end{bmatrix}\begin{bmatrix}Z_{1}(t)&Z_{2}(t)\\\ Z_{3}(t)&Z_{4}(t)\\\
\end{bmatrix}\begin{bmatrix}B_{1}\\\
B_{0}\end{bmatrix}R^{-1}\begin{bmatrix}B_{1}^{*}&B_{0}^{*}\end{bmatrix}\begin{bmatrix}Z_{1}(t)&Z_{2}(t)\\\
Z_{3}(t)&Z_{4}(t)\\\ \end{bmatrix}\begin{bmatrix}E_{1}&0\\\ 0&0\end{bmatrix}$
$\displaystyle-\begin{bmatrix}E_{1}^{*}&0\\\
0&0\end{bmatrix}\begin{bmatrix}Z_{1}(t)&Z_{2}(t)\\\ Z_{3}(t)&Z_{4}(t)\\\
\end{bmatrix}\begin{bmatrix}B_{1}\\\
B_{0}\end{bmatrix}R^{-1}\begin{bmatrix}0&B_{0}^{*}\Pi_{0}P^{\mathcal{Z}_{0}}\end{bmatrix}+\begin{bmatrix}QP^{\mathcal{X}_{1}}&0\\\
0&0\end{bmatrix}=\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}.$

Similarly, use the assumption on $G$ in (18) to decompose the final condition
(79b) as

(85) $\displaystyle\begin{bmatrix}G_{1}&0\\\ 0&0\end{bmatrix}$
$\displaystyle=\begin{bmatrix}E_{1}^{*}&0\\\
0&0\end{bmatrix}\begin{bmatrix}Z_{1}(t_{f})&Z_{2}(t_{f})\\\
Z_{3}(t_{f})&Z_{4}(t_{f})\\\ \end{bmatrix}\begin{bmatrix}E_{1}&0\\\
0&0\end{bmatrix}$ (86)
$\displaystyle=\begin{bmatrix}E_{1}^{*}Z_{1}(t_{f})E_{1}&0\\\ 0&0\\\
\end{bmatrix}.$

The resulting 4 equations yield $Z_{3}(t)\equiv 0$, $Z_{4}(t)$ is arbitrary,
$Z_{2}(t)$ solves (80) and, recalling that $\Pi_{1}(t)$ is the unique solution
of (60),

(87) $\displaystyle Z_{1}(t)=E_{1}^{-*}\Pi_{1}(t)E_{1}^{-1}.$

2. This assertion can be shown by substituting for the general solution of (66) in equation (82). Using that $P^{\mathcal{Z}_{0}}E=0$, we arrive at the expression of the optimal control (64). Similarly, we can find the minimum cost from (83).

Equations (9) and the projections $P^{\mathcal{X}}$, $P^{\mathcal{Z}}$ were
used to derive (66). However, the restrictions of $Q$ on the subspaces
$\mathcal{X}_{1}$ and $\mathcal{X}_{0}$ are found, solving the system (66)
requires no knowledge of the projections.

When comparing system (66) with the optimality equations derived in finite-
dimensional space, it is apparent that system (66) differs from the ones
derived in [5, 3] since the knowledge of the restrictions such as $E_{1}$,
$A_{1}$, etc is not required here. Assuming no penalization on the
$L_{2}$-norm of the algebraic state, i.e. $Q_{0}P^{\mathcal{X}_{0}}=0$, system
(66) reduces to a single differential Riccati equation that mirrors the one
derived for finite-dimensional DAEs through the use of behaviors [35].

##  5 Numerical simulations

In order to illustrate the theoretical results, we study the following class
of coupled systems

(88a) $\displaystyle w_{t}(x,t)=$ $\displaystyle w_{xx}(x,t)-\rho
w(x,t)+\alpha v(x,t)+u(t),$ (88b) $\displaystyle 0=$ $\displaystyle
v_{xx}(x,t)-\gamma v(x,t)+\beta w(x,t)+u(t),$ with the boundary conditions
(88c) $\displaystyle w_{x}(0,t)=$ $\displaystyle 0,\quad w_{x}(1,t)=0,$ (88d)
$\displaystyle v_{x}(0,t)=$ $\displaystyle 0,\quad v_{x}(1,t)=0,$

where $x\in[0,1]$ and $t\geq 0$. Also, $w(x,0)=\sin(\pi x)$ and
$v(x,0)=\beta(\gamma-d_{xx})^{-1}\sin(\pi x).$ The system’s parameters are
$\rho=1$, $\gamma=\alpha=\beta=2$ . For all $n$, $\gamma\neq-(n\pi)^{2}$, it
then $\gamma$ is not an eigenvalue of the operator $\partial_{xx}$ [9, Example
8.1.8]. Consequently, defining $\mathrm{Z}=H_{2}(0,1)$ and $I$ to be the
identity operator on $\mathrm{Z}$, the operator $\gamma I-\partial_{xx}$ is
invertible. It will also prove useful to define

$A_{d}w=\frac{d^{2}}{dx^{2}}w,\;\;D(A_{d})=\\{w\in\mathrm{Z}:w_{x}(0)=w_{x}(1)=0\\},$

and

$\displaystyle x(t)=\begin{bmatrix}w(x,t)\\\
v(x,t)\end{bmatrix}\in\mathcal{X}=\mathrm{Z}\times\mathrm{Z}.$

System (88) can now be written in the form (1)

$\displaystyle\frac{d}{dt}\underbrace{\begin{bmatrix}I&0\\\
0&0\end{bmatrix}}_{E}x(t)$
$\displaystyle=\underbrace{\begin{bmatrix}A_{d}-\rho I&&\alpha I\\\ \beta
I&&A_{d}-\gamma I\end{bmatrix}}_{A}x(t)+\underbrace{\begin{bmatrix}I\\\
I\end{bmatrix}}_{B}u(t),$ $\displaystyle x(0)$
$\displaystyle=\begin{bmatrix}\sin(\pi x)&\beta(\gamma-d_{xx})^{-1}\sin(\pi
x)\end{bmatrix}^{*}$

System (88) was shown in [19, section 4 ] to be radial of degree 0. To
approximate this coupled system, use finite-element method with linear splines
to obtain a system of DAEs. The spatial interval $[0,1]$ is subdivided into 27
equal intervals. The dynamics of the parabolic and elliptic states without
control (i.e. $u(t)\equiv 0$) is given in Figure 2 a & b.

Define the cost functional

(89) $\displaystyle J(x_{i},u;6)=$ $\displaystyle\int_{0}^{6}\langle
x(s),\begin{bmatrix}I&0\\\ 0&0\end{bmatrix}x(s)\rangle_{\mathrm{Z}}+\langle
u(s),u(s)\rangle_{\mathcal{U}}ds.$

Comparing the previous cost with (14), it is clear that $t_{f}=6$,
$Q=\begin{bmatrix}I&0\\\ 0&0\end{bmatrix}$ , $G=\begin{bmatrix}0&0\\\
0&0\end{bmatrix}$. Note that projections $P^{\mathcal{X}}$ and
$P^{\mathcal{Z}}$, which were calculated in [19, section 4 ], can be used used
to obtain

(90) $\displaystyle QP^{\mathcal{X}_{1}}=\begin{bmatrix}I&0\\\
0&0\end{bmatrix},\quad QP^{\mathcal{X}_{0}}=\begin{bmatrix}0&0\\\
0&0\end{bmatrix}.$

To demonstrate the finding in the previous sections, we solve system (66) for
the optimal control (64). Since this system consists of operator equations on
an infinite-dimensional Hilbert space, it cannot be solved exactly. Therefore,
the control is calculated using an finite-dimensional approximation. The
convergence of the approximation method to the true optimal one and closed-
loop performance can be discussed by doing calculations of different
approximation order and following a similar approach as in [29, Chapter 4].
This task will be addressed in the future work. Without the need of
decomposing the state $x(t)$ or calculation operators $E_{1}$, $A_{1}$ etc, we
now solve the finite-dimensional approximation of system (66). This is done
via using “ode15s” which is based on a backward differential formula (BDF).
Consequently, we obtain an approximation of the optimal control (64). The
approximated control signal at $x=0$ (64) is given in Figure 1. Note that
control $u(t)$ ensures the consistency statement on the initial conditions,
which is

$\displaystyle u(x,0)$ $\displaystyle=\frac{(d_{xx}-\gamma
I)}{\beta}v(x,0)-\beta w(x,0)$ $\displaystyle=0.$

Figure (2) c & d illustrates the dynamics of the coupled system after applying
control (64).

Figure 1: LQ-optimal feedback control at $x=0$ . This control signal minimizes
the cost functional (89), and is derived by solving system (66) after
discretization.

(a) Uncontrolled $w(x,t)$

(b) Uncontrolled $v(x,t)$

(c) Controlled $w(x,t)$

(d) Controlled $v(x,t)$

Figure 2: A 3D landscape of the dynamics of a coupled parabolic-elliptic
system with initial condition $w_{0}=\sin(\pi x)$, $v_{0}=\beta(d_{xx}-\gamma
I)^{-1}\sin(\pi x)$, without and with control (64). The parameters of the
system are $\rho=1$, $\gamma=\alpha=\beta=2.$ The uncontrolled system is
unstable, but the use of LQ feedback control causes the states to decay
towards zero.

##  6 Conclusions

This paper extends the classical finite-time linear quadratic control problem
for finite-dimensional DAEs into the infinite-dimensional case. We showed the
existence of a continuous optimal control that ensures the consistency of the
initial conditions while minimizing the cost functional. Decomposing the PDAE
into a Weierstra$\beta$ canonical form was crucial in the proofs. However, the
optimal control can be calculated derived differential Riccati equation
without projecting of any operators other than the state weight. Future work
will consider the minimization problem on an infinite-time horizon.

The class of PDAEs considered here was restricted to those with radiality-
index zero. We aim to extend the results to higher-index PDAEs in future
research. This already been done [3] for finite-dimensional DAEs without the
use of behaviors. Another prospective research problem is to examine the
linear quadratic control problem for linear PDAEs in situations where the
operator associated with the penalization on the $L_{2}$-norm of the control
input is not necessarily invertible, since as illustrated in [35] a unique
solution can exist for the optimization problem in the finite-dimensional
situation. Finally, since the derived differential Riccati equation is an
operator equation on an infinite-dimensional Hilbert space, numerical
treatment for solving needs further attention.

## References

  * [1] J. Ahn and C. Yoon, Global well-posedness and stability of constant equilibria in parabolic–elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32 (2019), p. 1327.
  * [2] A. Alalabi and K. A. Morris, Boundary control and observer design via backstepping for a coupled parabolic-elliptic system, preprint, (2023).
  * [3] A. Alalabi and K. A. Morris, Linear quadratic control problem on a finite-horizon for a class of differential-algebraic equations, in the proceedings of the 2024 American Control Conference, (2024).
  * [4] A. Bartel and R. Pulch, A concept for classification of partial differential algebraic equations in nanoelectronics, in Progress in Industrial Mathematics at ECMI 2006, Springer, (2008), pp. 506–511.
  * [5] D. Bender and A. Laub, The linear-quadratic optimal regulator for descriptor systems, IEEE Transactions on Automatic Control, 32 (1987), pp. 672–688.
  * [6] D. Bender and A. Laub, The linear-quadratic optimal regulator for descriptor systems, IEEE Transactions on Automatic Control, 32 (1987), pp. 672–688.
  * [7] T. Breiten, K. Kunisch, and L. Pfeiffer, Feedback stabilization of the two-dimensional Navier–Stokes equations by value function approximation, Applied Mathematics & Optimization, 80 (2019), pp. 599–641.
  * [8] S. L. Campbell and W. Marszalek, The index of an infinite dimensional implicit system, Mathematical and Computer Modelling of Dynamical Systems, 5 (1999), pp. 18–42.
  * [9] R. Curtain and H. Zwart, Introduction to infinite-dimensional systems theory: a state-space approach, vol. 71, Springer Nature, (2020).
  * [10] G.-R. Duan, Analysis and design of descriptor linear systems, vol. 23, Springer Science & Business Media, (2010).
  * [11] M. Erbay, B. Jacob, K. Morris, T. Reis, and C. Tischendorf, Index concepts for linear differential-algebraic equations in finite and infinite dimensions, arXiv preprint arXiv:2401.01771, (2024).
  * [12] Z. Ge and D. Feng, Well-posed problem of nonlinear singular distributed parameter systems and nonlinear ge-semigroup, Science China Information Sciences, 56 (2013), pp. 1–14.
  * [13] Z. Ge, X. Ge, and J. Zhang, Approximate controllability and approximate observability of singular distributed parameter systems, IEEE Transactions on Automatic Control, 65 (2019), pp. 2294–2299.
  * [14] Z. Ge, G. Zhu, and D. Feng, Exact controllability for singular distributed parameter system in hilbert space, Science in China Series F: Information Sciences, 52 (2009), pp. 2045–2052.
  * [15] H. Gernandt and T. Reis, Linear-quadratic optimal control for abstract differential-algebraic equations, arXiv preprint arXiv:2402.08762, (2024).
  * [16] G. V. Grenkin, A. Y. Chebotarev, A. E. Kovtanyuk, N. D. Botkin, and K.-H. Hoffmann, Boundary optimal control problem of complex heat transfer model, Journal of Mathematical Analysis and Applications, 433 (2016), pp. 1243–1260.
  * [17] M. Günther, A joint dae/pde model for interconnected electrical networks, Mathematical and Computer Modelling of Dynamical Systems, 6 (2000), pp. 114–128.
  * [18] M. Heinkenschloss, D. C. Sorensen, and K. Sun, Balanced truncation model reduction for a class of descriptor systems with application to the oseen equations, SIAM Journal on Scientific Computing, 30 (2008), pp. 1038–1063.
  * [19] B. Jacob and K. Morris, On solvability of dissipative partial differential-algebraic equations, IEEE Control Systems Letters, 6 (2022), pp. 3188–3193.
  * [20] E. Kreyszig, Introductory functional analysis with applications, vol. 17, John Wiley & Sons, (1991).
  * [21] P. Kunkel and V. Mehrmann, Differential-algebraic equations: analysis and numerical solution, vol. 2, European Mathematical Society, (2006).
  * [22] R. Lamour, R. März, and C. Tischendorf, Differential-algebraic equations: a projector based analysis, Springer Science & Business Media, (2013).
  * [23] A. W. Leung, Systems of nonlinear partial differential equations: applications to biology and engineering, vol. 49, Springer Science & Business Media, (2013).
  * [24] P. Lin, A sequential regularization method for time-dependent incompressible navier–stokes equations, SIAM Journal on Numerical Analysis, 34 (1997), pp. 1051–1071.
  * [25] W. S. Martinson and P. I. Barton, A differentiation index for partial differential-algebraic equations, SIAM Journal on Scientific Computing, 21 (2000), pp. 2295–2315.
  * [26] V. Mehrmann and H. Zwart, Abstract dissipative hamiltonian differential-algebraic equations are everywhere, arXiv preprint arXiv:2311.03091, (2023).
  * [27] V. L. Mehrmann, The autonomous linear quadratic control problem: theory and numerical solution, Springer, (1991).
  * [28] K. A. Morris, Introduction to feedback control, Academic Press, Inc., (2000).
  * [29] K. A. Morris, Controller design for distributed parameter systems, Springer, (2020).
  * [30] H. Parada, E. Cerpa, and K. A. Morris, Feedback control of an unstable parabolic-elliptic system with input delay, preprint, (2020).
  * [31] M. Petreczky and S. Zhuk, Solutions of differential–algebraic equations as outputs of lti systems: Application to LQ control problems, Automatica, 84 (2017), pp. 166–173.
  * [32] A. A. Polezhaev, R. A. Pashkov, A. I. Lobanov, and I. B. Petrov, Spatial patterns formed by chemotactic bacteria escherichia coli, International Journal of Developmental Biology, 50 (2003), pp. 309–314.
  * [33] J. Rang and L. Angermann, Perturbation index of linear partial differential-algebraic equations, Applied numerical mathematics, 53 (2005), pp. 437–456.
  * [34] T. Reis, Controllability and observability of infinite-dimensional descriptor systems, IEEE transactions on automatic control, 53 (2008), pp. 929–940.
  * [35] T. Reis and M. Voigt, Linear-quadratic optimal control of differential-algebraic systems: the infinite time horizon problem with zero terminal state, SIAM Journal on Control and Optimization, 57 (2019), pp. 1567–1596.
  * [36] T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation, Linear Algebra and its Applications, 415 (2006), pp. 262–289.
  * [37] G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev type equations and degenerate semigroups of operators, de Gruyter, (2003).
  * [38] Y. Wagner, A further index concept for linear PDAEs of hyperbolic type, Mathematics and computers in simulation, 53 (2000), pp. 287–291.