Uniqueness for Riccati equations with application to the optimal boundary control of composite systems of evolutionary partial differential equations

Abstract

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

Appendix A. Instrumental results

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

• the input-to-state map L,

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

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

All of them stem from the Assumptions 2.3 or 2.8 on the (dynamics and control) operators A and B. The role played by the assertions of the novel Lemma A.2 and Lemma A.5 in the proofs of our uniqueness results is absolutely critical.

Initially, it is useful to recall from [2, 4] the basic regularity properties of the input-to-state map L. The first result pertains to the finite time horizon problem. The reader is referred to [2, Appendix B] for the details of the computations leading to the various statements in the following Proposition.

Proposition A.1

([2], Proposition B.3) Let $L_s$ be the operator defined by

$$\begin{aligned} L_s:u(\cdot ) \longrightarrow (L_su)(t): = \int _s^t e^{A(t-r}Bu(r)\,\textrm{d}r\,, \qquad 0\le s\le t\le T\,. \end{aligned}$$

(A.1)

Under the Assumptions 2.3, the following regularity results hold true.

• (1) If $p=1$, then $L_s \in {{\mathcal {L}}}(L^1(s,T;U),L^{1/\gamma }(s,T;[{{\mathcal {D}}}({A^*}^\epsilon )]')$;

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

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

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

Moreover, in all cases the norm of $L_s$ does not depend on s.

The space regularity in the last assertion can be actually enhanced. To be more precise, $L_s$ maps control functions $u(\cdot )$ which belong to $L^{q'}(s,T;U)$ into functions which take values in ${{\mathcal {D}}}(A^\epsilon )$ ($q'$ being the conjugate exponent of q in the Assumptions 2.3). We highlight this property— appparently left out of the work [2]—as a separate result, since it will be used throughout in the paper. The proof is omitted, as it is akin to (and somewhat simpler than) the one carried out to establish assertion (v) of the subsequent Proposition A.3.

Lemma A.2

Let $\epsilon$ and q be as in (iii) of the Assumptions 2.3. Then, for the operator $L_s$ defined in (A.1) we have

$$\begin{aligned} L_s \in {{\mathcal {L}}}(L^{q'}(s,T;U),C([s,T];{{\mathcal {D}}}(A^\epsilon )))\,. \end{aligned}$$

A counterpart of Proposition A.1 specific for the infinite time horizon problem was proved in [4, Proposition 3.6]. The collection of findings on the regularity of the input-to-state map L is recorded here for the reader’s convenience.

Proposition A.3

([4], Proposition 3.6) Let L be the operator defined by

$$\begin{aligned} L:u(\cdot ) \longrightarrow (L u)(t): = \int _0^t e^{A(t-r}Bu(r)\,\textrm{d}r\,, \qquad t\ge 0\,. \end{aligned}$$

Under the Assumptions 2.8, the following regularity results hold true.

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

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

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

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

• (v) $L \in {{\mathcal {L}}}(L^r(0,\infty ;U),C_b([0,\infty );{{\mathcal {D}}}(A^\epsilon ))$, for any $r\in [q',\infty ]$.

Because they occur in the present work, besides being central to the analysis of [4], we need to recall the $L^p$-spaces with weights. Set

$$\begin{aligned} L^p_g (0,\infty ;X) := \big \{f:(0,\infty ) \longrightarrow X\,, \; g(\cdot ) f(\cdot ) \in L^p(0,\infty ;X)\big \}\,, \end{aligned}$$

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

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

Remark A.4

As pointed out in [4, Remark 3.8], all the regularity results provided by the statements contained in the Propositions A.1 and A.3 extend readily to natural analogues involving $L^p_\delta$ spaces (rather than $L^p$ ones), maintaining the respective summability exponents p.

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

Lemma A.5

Let $\epsilon$ be as in (iii) of the Assumptions 2.3. If $Q(\cdot )\in {{\mathcal {Q}}}_T$ and $f\in C([0,T];{{\mathcal {D}}}(A^\epsilon ))$, then

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

Proof

We proceed along the lines of the proof of [2, Lemma A.3]. Let $Q(\cdot )\in {{\mathcal {Q}}}_T$ and let $t_0\in [0,T]$. By the definition of ${{\mathcal {Q}}}_T$, there exists $c_1>0$ such that

$$\begin{aligned} \Vert B^*Q(t)z\Vert _U\le c_1 \Vert z\Vert _{{{\mathcal {D}}}(A^\epsilon )}\qquad \forall t\in [0,T]\,, \; \forall z\in {{\mathcal {D}}}(A^\epsilon )\,. \end{aligned}$$

(A.2)

Since $f(t_0)\in {{\mathcal {D}}}(A^\epsilon )$, then $B^*Q(\cdot )f(t_0)\in C([0,T];U)$. Then

$$\begin{aligned} \begin{aligned}&\Vert B^*Q(t)f(t)- B^*Q(t_0)f(t_0)\Vert _U \\&\qquad \le \Vert B^*Q(t)[f(t)-f(t_0)]\Vert _U + \Vert B^*Q(t)f(t_0)- B^*Q(t_0)f(t_0)\Vert _U \\&\qquad \le c_1 \Vert f(t)-f(t_0)\Vert _{{{\mathcal {D}}}(A^\epsilon )}+ \Vert [B^*Q(t)- B^*Q(t_0)]f(t_0)\Vert _U =o(1)\,, \quad t\longrightarrow t_0\,. \end{aligned} \end{aligned}$$

$\square$

Essential as well in this work, and more specifically in the proof of Theorem 2.10, is a stronger property of the operator $B^* e^{A^*\cdot }{A^*}^{\epsilon }$, namely, (A.3) below, which holds true for appropriate $\delta$, under the Assumptions 2.8. Originally devised in [4], this result reveals that once the validity of iiic) of Assumptions 2.3 is ascertained on some bounded interval [0, T], then the very same regularity estimate extends to the half line, along with an enhanced summability of the function $B^* e^{A^*\cdot }{A^*}^{\epsilon }x$, $x\in Y$. The key to this is the exponential stability of the semigroup, i.e. (2.12); see [4, Proposition 3.2].

Proposition A.6

([4], Proposition 3.2) Let $\omega$, $\eta$ and $\epsilon$ like in the Assumptions 2.8. For each $\delta \in (0,\omega \wedge \eta )$ the map

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

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

$$\begin{aligned} B^* e^{A^*\cdot }{A^*}^{\epsilon }\in {{\mathcal {L}}}(Y,L^q_\delta (0,\infty ;U))\,. \end{aligned}$$

(A.3)

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

Lemma A.7

Under the Assumptions 2.8, the following regularity results are valid, for any $\delta \in (0,\omega \wedge \eta )$:

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

(A.4)

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

Proof

The regularity results in (A.4) are, in essence, dual properties of the regularity result in Proposition A.6 and of assertion A6. in Theorem 2.9, respectively. To infer (a), we introduce the notation S for the mapping from Y into $L^q(0,\infty ;U)$ defined by

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

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

$$\begin{aligned} \begin{aligned} \langle S^*h,z\rangle _Y&=\langle h,Sz\rangle _{L^{q'}(0,\infty ;U),L^q(0,\infty ;U)} = \int _0^\infty \big \langle h(t),e^{\delta t}B^* e^{A^*t}{A^*}^\epsilon z\big \rangle _U\,\hbox {d}t \\&= \Big \langle \int _0^\infty e^{\delta t}A^\epsilon e^{At} Bh(t)\,\hbox {d}t,z \Big \rangle _Y\,. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} S^*h = \int _0^\infty e^{\delta t}A^\epsilon e^{At} Bh(t)\,\hbox {d}t\,, \qquad h\in L^{q'}(0,\infty ;Y)\,. \end{aligned}$$

(A.5)

To achieve (b) of (A.4), we recall instead the assertion A6. in Theorem 2.9, which further tells us that

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

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

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

by construction, $T^*\in {{\mathcal {L}}}(L^{p'}(0,\infty ;Y),U)$ for all $p'>1/(1-\gamma )$. More precisely, for any $w\in U$ and any $g\in L^{p'}(0,\infty ;Y)$ we have

$$\begin{aligned} \begin{aligned} \langle T^*g,w\rangle _U&= \langle g,Tw\rangle _{L^{p'}(0,\infty ;Y),L^p(0,\infty ;Y)} =\int _0^\infty \big \langle g(t),e^{\delta t} A^{-\epsilon } e^{At} Bw\big \rangle _Y\,\hbox {d}t \\&= \Big \langle \int _0^\infty e^{\delta t}B^* e^{A^*t}{A^*}^{-\epsilon } g(t)\,\hbox {d}t, w\Big \rangle _U\,, \end{aligned} \end{aligned}$$

which establishes

$$\begin{aligned} T^*g=\int _0^\infty e^{\delta t}B^* e^{A^*t}{A^*}^{-\epsilon } g(t)\,\hbox {d}t\,. \qquad \forall g\in L^{p'}(0,\infty ;Y)\,. \end{aligned}$$

(A.6)

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