Well-Posedness of the Bochner Integral Form of Operator-Valued Riccati Equations

James Cheung Millennium Space Systems, A Boeing Company. 2265 E. El Segundo Blvd, El Segundo, CA, 90245.

Abstract.

In this short paper, we prove that the Bochner integral form of the operator-valued Riccati equation has a unique solution if and only if its mild form has a unique solution. This implies that the mild and Bochner integral forms of this equation are equivalent. The result is obtained through an operator representation argument.

1. Introduction

Let $H$ be a separable Hilbert space equipped with the inner product $(\cdot,\cdot)_{H}$ . We define $\mathcal{L}(H)$ to be the space of bounded linear operators defined on $H$ . We will then denote $A:\mathcal{D}(A)\rightarrow H$ as the generator of a $C_{0}$ -semigroup $S(t)\in\mathcal{L}(H)$ for all $t\in[0,\tau]$ , where $\tau>0$ and $\mathcal{D}(A)$ is the domain of $A$ defined densely in $H$ . The solution space of interest in this work is $\mathcal{C}([0,\tau],\mathcal{L}(H))$ , which defines the space of bounded operators that are norm-continuous with respect to $t\in[0,\tau]$ .

In the differential form, the operator-valued Riccati equation is given by

(1)

\left\{\begin{aligned} \frac{d}{dt}\Sigma(t)&=A\Sigma(t)+\Sigma(t)A^{*}+\Sigma(t)G\Sigma(t)-F\\ \Sigma(0)&=\Sigma_{0},\end{aligned}\right.

for all $t\in[0,\tau]$ , where $F,\Sigma_{0}\in\mathcal{L}(H)$ are self-adjoint operators, and $G$ is an unbounded self-adjoint operator whose domain is dense in $H$ . The mild form of this equation is then given by

(2)

\Sigma(t)\phi=S(t)\Sigma_{0}S^{*}(t)\phi+\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)S^{*}(t-s)\phi ds

for all $\phi\in H$ and $t\in[0,\tau]$ . From the results presented in [1], we know that it is generally known that (1) and (2) are equivalent, meaning that there exists a unique $\Sigma(\cdot)\in\mathcal{C}([0,\tau],\mathcal{L}(H))$ that satisfies both equations. In this paper, we demonstrate that $\Sigma(\cdot)\in\mathcal{C}([0,\tau],\mathcal{L}(H))$ satisfying (2) also satisfies

(3)

\Sigma(t)=S(t)\Sigma_{0}S^{*}(t)+\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)S^{*}(t-s)ds

for all $t\in[0,\tau]$ .

The well-posedness of the Bochner integral form of the operator-valued Riccati equation plays an important part in determining theoretical error bounds for approximations to this equation [2, 4]. The previously known result presented in [3] indicates that the Bochner integral form of the operator-valued Riccati equation is well-posed if the operators $F,G$ in (3) are compact. This result was derived through an approximation argument. This work extends well-posedness to cases where $G$ is not necessarily bounded. We proceed to prove this result in the following section.

2. Analysis

In the analysis, we will use an operator representation argument to demonstrate that the mild form and the Bochner integral form of the operator-valued Riccati equation are equivalent. To this end, we will utilize the following corollary to the Riesz Representation Theorem found in [5, Theorem A.63].

Lemma 1.

If $q(\cdot):H\rightarrow\mathbb{R}$ is a bounded quadratic form on $H$ , then there exists a unique self-adjoint operator $Q\in\mathcal{L}(H)$ such that

q(\phi)=\left({\phi,Q\phi}\right)_{H}

for all $\phi\in H$ .

We now move to prove the main result of this work given in the following.

Theorem 1.

Let $S(t)\in\mathcal{L}(H)$ be a $C_{0}$ -semigroup defined on $t\in[0,\tau]$ . Now, suppose that there exists an unique time-dependent self-adjoint operator $\Sigma(\cdot)\in\mathcal{C}([0,\tau],\mathcal{L}(H))$ that satisfies the following mild form of the operator-valued Riccati equation

(4)

\Sigma(t)\phi=S(t)\Sigma_{0}S^{*}(t)\phi+\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)S^{*}(t-s)\phi ds

for all $\phi\in H$ and $t\in[0,\tau]$ , where $F,\Sigma_{0}\in\mathcal{L}(H)$ are bounded self-adjoint operators and $G$ is a generally unbounded self-adjoint operator whose domains is dense in $H$ . Then $\Sigma(\cdot)\in\mathcal{C}([0,\tau],\mathcal{L}(H))$ satisfies (4) if and only if it satisfies also the following Bochner integral form of the operator-valued Riccati equation

(5)

\Sigma(t)=S(t)\Sigma_{0}S^{*}(t)+\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)S^{*}(t-s)ds

for all $t\in[0,\tau]$ .

Proof.

Since $\Sigma(\cdot)\in\mathcal{C}([0,\tau],\mathcal{L}(H))$ satisfies (4), then we must have that

(6)

\left({\phi,\Sigma(t)\phi}\right)_{H}=\left({\phi,S(t)\Sigma_{0}S^{*}(t)\phi}\right)_{H}+\left({\phi,\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)(s)S^{*}(t-s)\phi ds}\right)_{H}

for all $\phi\in H$ and $t\in[0,\tau]$ .

Defining

	$\displaystyle q^{1}_{t}(\phi)$	$\displaystyle:=\left({\phi,\Sigma(t)\phi}\right)_{H}$
	$\displaystyle q^{2}_{t}(\phi)$	$\displaystyle:=\left({\phi,S(t)\Sigma_{0}S^{*}(t)\phi}\right)_{H}$
	$\displaystyle q^{3}_{t}(\phi)$	$\displaystyle:=\left({\phi,\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)(s)S^{*}(t-s)\phi ds}\right)_{H}$

as quadratic forms defined for all $\phi\in H$ and $t\in[0,\tau]$ . The boundedness of $q^{1}_{t}(\cdot)$ follows from the observation that $\Sigma(t)\phi\in H$ for all $\phi\in H$ and $t\in[0,\tau]$ . Equation (4) then requires that $S(t)\Sigma_{0}S^{*}(t)\phi\in H$ and that $\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)(s)S^{*}(t-s)\phi ds\in H$ for all $\phi\in H$ and $t\in[0,\tau]$ , which implies the boundedness of $q^{2}_{t}(\cdot),q^{3}_{t}(\cdot)$ . Applying Lemma 1 then implies that there exists unique operators $Q^{1}_{t},Q^{2}_{t},Q^{3}_{t}\in\mathcal{L}(H)$ so that

	$\displaystyle q^{1}_{t}(\phi)=\left({\phi,Q^{1}_{t}\phi}\right)_{H}$
	$\displaystyle q^{2}_{t}(\phi)=\left({\phi,Q^{2}_{t}\phi}\right)_{H}$
	$\displaystyle q^{3}_{t}(\phi)=\left({\phi,Q^{3}_{t}\phi}\right)_{H}$

for all $\phi\in H$ and $t\in[0,\tau]$ .

It then follows from (6) that

q^{1}_{t}(\phi)=q^{2}_{t}(\phi)+q^{3}_{t}(\phi)

for all $\phi\in H$ and $t\in[0,\tau]$ . This can only be true if

Q^{1}_{t}=Q^{2}_{t}+Q^{3}_{t}

for all $t\in[0,\tau]$ . Then, by the definition of $q^{1}_{t},q^{2}_{t},q^{3}_{t}$ and the uniqueness of $Q^{1}_{t},Q^{2}_{t},Q^{3}_{t}$ (implied by Lemma 1) associated with their respective quadratic forms, we have necessarily that

	$\displaystyle Q^{1}_{t}$	$\displaystyle=\Sigma(t)$
	$\displaystyle Q^{2}_{t}$	$\displaystyle=S(t)\Sigma_{0}S^{*}(t)$
	$\displaystyle Q^{3}_{t}$	$\displaystyle=\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)(s)S^{*}(t-s)ds,$

for all $t\in[0,\tau]$ . Hence, $\Sigma(\cdot)\in\mathcal{C}\left({[0,\tau],\mathcal{L}(H)}\right)$ must also satisfy

\Sigma(t)=S(t)\Sigma_{0}S^{*}(t)+\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)(s)S^{*}(t-s)ds

for all $t\in[0,\tau]$ . Thus we have proven the “if” part of the theorem. The proof in the other direction follows by testing (5) with any $\phi\in H$ . ∎

Remark 1.

We would like to point out that the analysis presented in the proof above implies that the operator-valued integral

\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)(s)S^{*}(t-s)ds

is the unique representation of the bounded self-adjoint time-dependent linear operator $Q_{t}\in\mathcal{C}([0,\tau],\mathcal{L}(H))$ that satisfies

\left({\phi,Q_{t}\phi}\right)_{H}:=\left({\phi,\int_{0}^{t}S(t-s)\left({F-\Sigma G\Sigma}\right)(s)S^{*}(t-s)\phi ds}\right)_{H}

for all $\phi\in H$ and $t\in[0,\tau]$ . This indicates that the operator-valued integral used in the Bochner integral form of the operator-valued Riccati equation is well-defined.

3. Discussion

We have demonstrated above that the mild and Bochner integral forms of the operator-valued Riccati equation are equivalent. Instead of using an approximation argument, as done in [3], we have utilized an operator representation argument to achieve this result. This simpler proof then allows us to extend the known well-posedness results for the Bochner integral form to cases where the coefficient operator $G$ in the equation are unbounded.

In following works, the author will utilize the result presented in this paper to determine error bounds for approximation methods to operator-valued Riccati equations for cases where the coefficient operators $G$ in the equation are defined by boundary and point control/observation operators.

References

[1] Alain Bensoussan, Giuseppe Da Prato, Michel C Delfour, and Sanjoy K Mitter. Representation and control of infinite dimensional systems, volume 1. Springer, 2007.
[2] John A Burns and James Cheung. Optimal convergence rates for galerkin approximation of operator riccati equations. Numerical Methods for Partial Differential Equations, 2022.
[3] John A Burns and Carlos N Rautenberg. Solutions and approximations to the riccati integral equation with values in a space of compact operators. SIAM Journal on Control and Optimization, 53(5):2846–2877, 2015.
[4] James Cheung. On the approximation of operator-valued riccati equations in hilbert spaces. arXiv preprint arXiv:2308.10130, 2023.
[5] Brian C Hall. Quantum theory for mathematicians. Springer, 2013.