Dissipative Hamiltonian representations of linear evolution equations

Introduction

Dissipative Hamiltonian representations of dynamical systems express the evolution of a system in terms of an energy operator and a dissipation operator. Besides making energy balances transparent, these representations provide a structural starting point for investigating well-posedness, stability, and the properties of related operator pencils. For finite-dimensional linear differential equations and differential-algebraic equations, the relationships between various Hamiltonian, dissipative, and relation-based descriptions have been studied extensively; see, for instance, [4, 10, 20, 22, 23]. Infinite-dimensional counterparts arise in the analysis of partial differential equations (PDEs) and abstract port-Hamiltonian systems. The formulation of PDEs as abstract dissipative Hamiltonian differential-algebraic equations on extended state spaces and the spectral theory of the associated operator pencils have been investigated in [21, 32]; see also [27, 28]. Explicit and implicit port-Hamiltonian formulations of particular distributed-parameter models have been compared in [6]. The existing literature approaches dissipation in infinite-dimensional systems from several complementary perspectives. Passive system theory emphasizes input-output behavior and storage functions [3, 29], while port-Hamiltonian formulations describe partial differential equations directly in terms of energy and interconnection structures [14, 27]. We take a complementary viewpoint: starting from a given semigroup generator, we ask whether its dynamics admit a dissipative Hamiltonian representation and which properties the energy operator must satisfy. Related work in the finite-dimensional setting includes [1, 5]; see [2] for preliminary results in infinite dimensions.

More precisely, we consider the linear evolution equation

where $A:\operatorname{dom}(A)\subset\mathcal{X}\to\mathcal{X}$ generates a strongly continuous semigroup $T=(T(t))_{t\geq0}$ on a Hilbert space $\mathcal{X}$. Throughout the paper, Hilbert spaces may be real or complex, and all spaces appearing in the same statement are understood to be over the same scalar field. We denote the norm and inner product on $\mathcal{X}$ by $\|\cdot\|_{\mathcal{X}}$ and $\langle\cdot,\cdot\rangle_{\mathcal{X}}$, respectively, and omit the subscripts whenever the underlying space is clear. We investigate when its dynamics can be represented in either of the two forms

with dissipative operators $D_{\mathrm{e}}$ and $D_{\mathrm{i}}$ and positive selfadjoint energy operators $H$ or $M$, respectively. The dissipative operators $D_{\mathrm{e}}$ and $D_{\mathrm{i}}$ need not coincide. Throughout the paper, we assume that $\ker H=\ker M=\{0\}$, but do not require either energy operator to be coercive.

The implicit formulation (3) is reminiscent of the mass-operator formulations arising in weak formulations of partial differential equations and their finite element discretizations. We therefore also refer to it as a mass representation.

The case of energy operators with nontrivial kernels is also of interest, but lies outside the scope of the present paper; see [32]. For sufficiently regular solutions, the quadratic energies of the explicit and implicit formulations satisfy, respectively,

In infinite dimensions, however, these formal energy balances do not automatically lead to well-defined operator representations. For the explicit equation (2), the product $D_{\mathrm{e}}H$ is defined only for states $x\in\operatorname{dom}(H)$ with $Hx\in\operatorname{dom}(D_{\mathrm{e}})$. Thus, recovering the original evolution requires not merely a formal factorization, but equality of the operators $A$ and $D_{\mathrm{e}}H$, including their domains.

The implicit formulation (3) involves a different difficulty. If $M$ is unbounded, then the term $M\dot{z}$ is meaningful only when $\dot{z}\in\operatorname{dom}(M)$, which need not hold for every classical solution of the original evolution equation. Even for bounded and injective $M$, the dissipative operator $D_{\mathrm{i}}=MA$ need not be closed. If $M$ is not surjective, then $\lambda M-D_{\mathrm{i}}$ is not surjective for any $\lambda$.

A further distinction arises when the energy is defined by a closed quadratic form. Its dissipation inequality then may yield contractive dynamics on a suitable energy space without providing an operator factorization on the original state space $\mathcal{X}$. We therefore distinguish exact operator representations, implicit equations with specified solution classes, and representations at the level of quadratic forms. In particular, we do not assume a decomposition of the dissipative operators into skew-adjoint and negative semidefinite selfadjoint parts. For unbounded operators, such a decomposition requires additional domain conditions.

A useful perspective comes from the renorming theory of semigroups. Every uniformly bounded semigroup on a Banach space is contractive with respect to the equivalent norm

More generally, every strongly continuous semigroup becomes quasi-contractive after an equivalent Banach-space renorming; see [8]. Even when the original space is a Hilbert space, such a norm need not be induced by an inner product. In fact, there exist exponentially stable Hilbert-space semigroups that are not similar, through a boundedly invertible operator, to a contraction semigroup; see [17, 24]. By contrast, a contractive equivalent Hilbertian norm is induced by a bounded, coercive positive operator. This observation motivates allowing positive energy operators that are non-coercive or unbounded, while retaining a Hilbert-space framework.

A natural way of constructing such energy operators is through intertwining operators. Suppose that a closed, densely defined, injective operator $E : \operatorname{dom}(E) \subset\mathcal{X} \to\mathcal{Z}$ intertwines the strongly continuous semigroup $T$ with a contraction semigroup $S$ on a Hilbert space $\mathcal{Z}$, in the sense that $\operatorname{dom}(E)$ is $T$-invariant and

The positive quadratic form $h[x] = \|Ex\|_{\mathcal{Z}}^2$ is then closed and nonincreasing along trajectories in its domain, and its associated positive selfadjoint operator is $H = E^*E$. Neither $E$ nor its inverse on $\operatorname{ran} E$ is required to be bounded. This form-theoretic construction is closely related to existing work on unbounded storage operators and pseudo-similarity, in particular [3]. It does not, without further domain assumptions, provide an operator identity of the form $A = DH$ or an equivalent implicit equation in the original state space. Understanding this additional step is central to the representation problem that we discuss in this paper.

A particularly useful realization of the intertwiner is furnished by output trajectories. For a suitable observation operator $C$ with values in a Hilbert space $\mathcal{Y}$, consider

where the domain consists of the initial states with square-integrable output trajectories. This mapping intertwines $T$ with the contractive left-shift semigroup on $L^2(0,\infty;\mathcal{Y})$. The shift approach has a substantial history [18, 31], as do admissibility and observability criteria [9, 11, 13, 26, 30] [12]. Closed, potentially unbounded output-trajectory operators are also treated in [3]. In particular, the existence of an infinite-time admissible and exactly observable output operator is characterized by strong stability together with similarity to a contraction semigroup [24 Theorem 8.11]. Here we use observation maps primarily as a construction tool: we allow one-sided estimates, or neither upper nor lower estimates, and seek the resulting dissipative Hamiltonian representations rather than new admissibility criteria.

Our central result is a fixed-energy representation criterion: under the domain condition $\operatorname{dom}(A) \subset\operatorname{dom}(H)$, the Lyapunov dissipation inequality is equivalent to an exact factorization $A = DH$ with a maximally dissipative operator $D$. In particular, maximal dissipativity of the factor can always be achieved by extension, without any additional range condition. For bounded $H$, the criterion is also equivalent to contractivity of the associated quadratic energy and yields an implicit representation. In particular, every exponentially stable semigroup has both representations with bounded energy. We then use pseudo-similarity and output-trajectory maps to construct energies that need not be bounded or coercive. For unbounded energy operators, a contractive quadratic form always determines a contraction semigroup on a canonical energy space, but a strong factorization on the original space demands additional domain compatibility. A dual construction and a variational interpretation complement the operator result. We stress that the aim of the paper is a unified representation framework, not a new general observability theorem.

The paper is organized as follows. § 2 introduces the representation concepts and pseudo-similarity. § 3 gives the fixed-energy criterion and bounded-energy consequences. § 4 treats closed intertwiners and the output-trajectory construction. § 5 discusses energy-space dynamics, duality, and variational realizations. § 6 provides examples and counterexamples, and § 7 a conclusion.

Preliminaries and representations

Throughout the paper, Hilbert-space inner products are linear in the first argument. All operators are linear, and equalities of operators are understood to include equality of their domains. For a Hilbert space $\mathcal{X}$, $\mathcal{L}(\mathcal{X})$ denotes the bounded operators on $\mathcal{X}$. If $A$ generates a strongly continuous semigroup $T$, then $A$ is densely defined and closed, and $T(t)\operatorname{dom}(A) \subset\operatorname{dom}(A)$ with $AT(t)x = T(t)Ax$ for $x \in\operatorname{dom}(A)$; see [8 Lemma II.1.3 and Theorem II.1.4]. A subspace $\mathcal{D} \subset\mathcal{X} \times\mathcal{X}$ is called dissipative if

An operator is dissipative if its graph is dissipative, and is maximally dissipative if its graph has no proper dissipative linear extension. In particular, if a dissipative operator $D$ satisfies $\operatorname{ran}(\lambda I-D)=\mathcal{X}$ for some $\lambda>0$, then it is maximally dissipative by the injectivity property for dissipative extensions [8 Proposition II.3.14(i)]. By the Lumer–Phillips theorem [8 Corollary II.3.20], it also generates a strongly continuous contraction semigroup; see also [19]. More general linear relations are useful for singular Hamiltonian descriptions [10].

An operator $H$ will be called positive definite if it is selfadjoint, $H \ge0$, and $\operatorname{ker} H = \{0\}$; note that this does not mean that $H \ge cI$ for some $c>0$. The latter property will be called coercivity. If $H$ is unbounded, its positive quadratic form has domain $\operatorname{dom}(H^{1/2})$.

Note that the specification of the solution class in (ii) is essential. In particular, writing $D=MA$ when $M$ is unbounded does not say that $MA$ is defined on $\operatorname{dom}(A)$. The same care is required for the product $DH$.

We may, and often will, replace $\mathcal{X}_2$ by $\overline{\operatorname{ran} E}$, which is invariant under $T_2$.

The word pseudo-similarity refers here to a directed intertwining relation; no boundedness of $E$ or $E^{-1}\colon\operatorname{ran} E \to\mathcal{X}_1$ is assumed. Various concepts of quasi- and pseudo-similarity are studied in operator theory; see, e.g., [3 Definition 1.8] and [24].

A fixed-energy representation criterion

We begin by studying the operator-theoretic core of the representation problem. Although the Hilbert-space Lyapunov inequality is classical, see e.g. [8, 9, 30], the purpose of the following construction is to identify the corresponding dissipative Hamiltonian factors, including their domains, within a single statement.

Proof. Assume (a). Injectivity of $H$ makes (3.2) well-defined, and

Thus, $D_{\mathrm{e}}$ is dissipative. Moreover, $D_{\mathrm{e}}H=A$, including equality of domains, by the injectivity of $H$.

We first show that $D_{\mathrm{e}}$ is densely defined. Choose $\mu>0$ such that $\mu I-A$ is bijective. Let $z\in(H\operatorname{dom}(A))^\perp$ and set $x=(\mu I-A)^{-1}z$. Then

Positive definiteness of $H$ implies that $x=0$, and hence $z=0$. Therefore $\overline{H}\,\operatorname{dom}(A)=\mathcal{X}$.

Since $D_e$ is densely defined and dissipative, its closure $D_0=\overline{D_e}$ is dissipative. The range $\mathcal{R}=\operatorname{ran}(I-D_0)$ is closed, since $D_0$ is closed and

Define an extension $D$ by

where $u\in\operatorname{dom}(D_0)$ and $v\in\mathcal{R}^{\perp}$. The sum is direct by the dissipativity of $D_0$. Since $(I-D_0)u\perp v$, we obtain

Furthermore,

Thus $D$ is maximally dissipative and generates a strongly continuous contraction semigroup. It remains to show that every dissipative extension $\widetilde{D}$ of $D_e$ satisfies $A=\widetilde{D}H$. Since $\widetilde{D}$ extends $D_e$, we have $A\subseteq\widetilde{D}H$. Let $x\in\operatorname{dom}(\widetilde{D}H)$. By the surjectivity of $\mu I-A$, there exists $y\in\operatorname{dom}(A)$ such that

With $w=x-y$ we obtain $w\in\operatorname{dom}(\widetilde{D}H)$ and $\widetilde{D}Hw=\mu w$. The dissipativity then yields that

and the positive definiteness of $H$ implies that $w=0$. Consequently, $x=y\in\operatorname{dom}(A)$, which proves that $A=\widetilde{D}H$.

We have shown that (a) implies (c). The implication (c) $\Rightarrow$ (b) is immediate, and (b) implies (a) by dissipativity.

Finally, suppose that $H$ is bounded. Differentiating $\langle HT(t)x,T(t)x\rangle$ for $x\in\operatorname{dom}(A)$, and using density, proves the equivalence of (a) and (3.3). The identity $D_i=HA$ is dissipative, and injectivity of $H$ gives the stated classical equivalence. On $\operatorname{ran}H^{1/2}$ define $S(t)H^{1/2}x=H^{1/2}T(t)x$. The inequality (3.3) then makes $S(t)$ contractive. It extends by density to a strongly continuous contraction semigroup; strong continuity follows from the boundedness of $H^{1/2}$ and the strong continuity of $T$.

Remark 3.2 (Maximality of the canonical factor). For every $\lambda>0$ one has

Therefore, the canonical factor $D_e$ is maximally dissipative if and only if $\operatorname{ran}(\lambda H-A)=\mathcal{X}$ for some $\lambda>0$.

If $A$ is bijective, then $HA^{-1}$ is closed and everywhere defined, hence bounded. Therefore

is bijective for all sufficiently small $\lambda>0$. In this case $D_e$ itself is maximally dissipative.

The range condition is not automatic, even when both $A$ and $H$ are bounded. The second construction in Subsection 6.2 shows that the canonical factor may have a proper dense, nonclosed domain, while its maximally dissipative extension still yields the same exact factorization.

Our next result discusses the implicit formulation.

Proof. The assumed inequality $\operatorname{Re}\langle MAx,x\rangle\le0$ for $x \in\operatorname{dom}(A)$ shows that $D_{\mathrm{i}}$ is dissipative. Fix $\lambda$ such that $\lambda I-A$ is bijective. Then $L(\lambda)x=M(\lambda I-A)x$ for every $x \in\operatorname{dom}(A)$. Since $M$ is injective and $\lambda I-A$ is bijective, this identity proves injectivity, the range formula, and (9). Classical equivalence follows from the injectivity of $M$.

Remark 3.4. The implicit factor $D_{\mathrm{i}}=HA$ need not be closed, let alone maximally dissipative, even though the two equations in (8) are classically equivalent. Theorem 3.3 shows why invertibility of the pencil $L(\lambda)$ on all of $\mathcal{X}$ is not the right criterion for the dynamics in this setting. For unbounded $H$, condition (5) is an actual restriction, not a technical convenience. The first construction in Subsection 6.2 exhibits a contractive closed energy form whose associated energy operator does not satisfy this domain condition. If the condition holds, form contractivity implies (a) in Theorem 3.1; see § 5.

As a corollary we obtain that every exponentially stable semigroup admits both dissipative Hamiltonian representations.

Proof. For the infinite-time Gramian, understood as a weak operator integral,

exponential stability implies boundedness, since $\lVert T(t)\rVert\le Ke^{-\omega t}$ gives $\lVert H\rVert\le K^{2}/(2\omega)$, and $\langle Hx,x\rangle=\int_{0}^{\infty}\lVert T(t)x\rVert^{2}\,\mathrm{d}t>0$ for $x\ne0$. Differentiating the energy for $x\in\operatorname{dom}(A)$ gives $2\operatorname{Re}\langle HAx,x\rangle=-\lVert x\rVert^{2}$. The integral representation in [8 Theorem II.1.10] yields $A^{-1}=-\int_{0}^{\infty}T(t)\,\mathrm{d}t$. Hence Theorem 3.1 and Theorem 3.2 apply.

The construction of the Gramian (3.6) and its connection to Lyapunov equations are described in [30 Theorem 5.1.1]. Conversely, if the identity observation has square-integrable trajectories for all $x \in\mathcal{X}$, the Datko–Pazy theorem yields exponential stability; see [7] and [8 Theorem V.1.8].

Pseudo-similarity and output trajectories

Closed energy forms from intertwiners

Proof. The form in (4.1) is closed because $E$ is closed; it is strictly positive because $E$ is injective. The form representation theorem for positive selfadjoint operators yields that $H = E^*E$ and the asserted square-root identity; see [15 Theorems VI.2.1 and VI.2.23]. By intertwining and the contractivity of $S$, we have

Finally the map $x \mapsto Ex$ identifies the completion in the energy norm with $\overline{\operatorname{ran} E}$, and the induced semigroup is precisely $S$ restricted to that closed invariant subspace.

The converse to the construction in Theorem 4.1 is also useful: a contractive closed form produces an intertwining operator without an output being specified beforehand.

Proof. The map $S(t)$ defined initially by $S(t)H^{1/2}x = H^{1/2}T(t)x$ is well-defined and contractive on the dense subspace $\operatorname{ran} H^{1/2}$; it extends uniquely to a contraction semigroup on $\mathcal{X}$. It remains to check strong continuity. Fix $x \in\operatorname{dom}(H^{1/2})$ and let $t_n \downarrow0$. The sequence $H^{1/2}T(t_n)x$ is bounded by the form inequality (4.2). Every weakly convergent subsequence has weak limit $H^{1/2}x$: indeed $T(t_n)x \to x$ in $\mathcal{X}$ and the graph of the closed operator $H^{1/2}$ is weakly closed. Moreover,

Weak lower semicontinuity forces convergence of these norms, and hence strong convergence to $H^{1/2}x$. Density and contractivity give strong continuity on $\mathcal{X}$.

The analysis in this subsection shows why closed energy forms provide a natural language for pseudo-similar contraction dynamics. However, they do not, by themselves, produce an everywhere valid operator product $DH$. We discuss this issue in § 5.

The output-trajectory transformation

An observation operator $C : \operatorname{dom}(A) \to\mathcal{Y}$ that maps to a Hilbert space $\mathcal{Y}$ is assumed to be bounded with respect to the graph norm $\lVert x\rVert_{\operatorname{dom}(A)}^2 = \lVert x\rVert_{\mathcal{X}}^2 + \lVert Ax\rVert_{\mathcal{X}}^2$. It is called finite-time admissible if there exist $\tau> 0$ and $\beta_\tau> 0$ such that

It is infinite-time admissible if the same estimate holds with $\tau= \infty$ and a finite constant $\beta_\tau$. An infinite-time admissible observation operator $C$ is called infinite-time exactly observable if there exists $\alpha> 0$ such that

See [26, 30] for these notions and related results.

We first consider $C \in\mathcal{L}(\mathcal{X},\mathcal{Y})$, for which finite-time admissibility is automatic. Define its possibly unbounded infinite-time output operator by

The left-shift semigroup on $L^2(0,\infty;\mathcal{Y})$ is

It is a strongly continuous contraction semigroup and it is strongly stable. The next proposition records elementary properties which also motivate more general output maps for graph-bounded $C$; see [18], [3 Proposition 3.1], and [30 Secs. 4.3 and 4.6].

Proof. Suppose that $x_n\in\operatorname{dom}(\mathcal{O}_C)$, $x_n\to x$ in $\mathcal{X}$, and $\mathcal{O}_Cx_n\to f$ in $L^2(0,\infty;\mathcal{Y})$. On every finite interval $[0,\tau]$, one has the convergence $CT(\cdot)x_n\to CT(\cdot)x$ uniformly. Hence, $f=CT(\cdot)x$ almost everywhere and $x\in\operatorname{dom}(\mathcal{O}_C)$, proving closedness. The intertwining identity follows from the semigroup property, and invariance follows since the tail of an $L^2$-function is in $L^2$. The last identity is obtained by subtracting the integrals over $[t,\infty)$ and $[0,\infty)$.

The upper admissibility estimate and the lower observability estimate have a particularly simple interpretation in terms of $\mathcal{O}_C$. Assume that $\mathcal{O}_C$ is densely defined and injective. Then

where the respective estimates hold for some $\alpha,\beta>0$ and all $x\in\operatorname{dom}(\mathcal{O}_C)$. The upper estimate is equivalent to infinite-time admissibility of $C$: by closedness and density it implies $\operatorname{dom}(\mathcal{O}_C)=\mathcal{X}$. Under this assumption, the lower estimate is precisely the infinite-time exact-observability inequality for $(A,C)$.

Without admissibility, the lower estimate may still hold on the possibly proper domain $\operatorname{dom}(\mathcal{O}_C)$. This is a lower observability estimate on the natural domain of the output transformation, but does not imply infinite-time admissibility. In this case $\operatorname{ran}\mathcal{O}_C$ is closed. Thus, bounded invertibility on the range of the output transformation does not require admissibility of the underlying observation operator.

When both estimates hold, the associated semigroup is similar to a contraction semigroup. A precise existence characterization, including the required strong stability, is given in [24 Theorem 8.11]; compare the earlier results in [3, 11]. We quote these results rather than reproving their observation-theoretic content. The connection can also be formulated intrinsically in terms of Lyapunov operators. A bounded injective output map induces a bounded positive definite $H = O_C^*O_C$ such that $T(t)^*HT(t) \le H$ and $\|H^{1/2}T(t)x\| \to0$ as $t \to\infty$. Conversely, given such an $H$, the nonnegative form

for all $x,y \in\operatorname{dom}(A)$, is bounded with respect to the graph norm on $\operatorname{dom}(A)$. Factoring this nonnegative form yields a graph-norm bounded observation operator $C : \operatorname{dom}(A) \to\mathcal{Y}$, for a suitable Hilbert space $\mathcal{Y}$, such that

for all $x \in\operatorname{dom}(A)$. In particular, $C$ is infinite-time admissible. If $H$ is additionally coercive, the pair $(A,C)$ is infinite-time exactly observable. These standard Lyapunov and admissibility constructions are discussed in [9, 24]; see also [30 Sec. 5.1].

Both one-sided output constructions for stable semigroups

There is a particularly simple way to construct the two one-sided cases without asserting any similarity to a contraction semigroup.

Consequently, every exponentially stable semigroup has both a bounded injective output transformation and a closed, bounded-below output transformation, even if it is not similar to a contraction semigroup.

Proof. Choose $K \ge1$ and $\omega> 0$ such that $\|T(t)\| \le K e^{-\omega t}$ for all $t \ge0$. Part (a) follows immediately from

For $x \in\operatorname{dom}(A)$, commutation with the semigroup and exponential stability show $O_1x \in L^2$ and

Moreover, differentiating the squared norm, integrating over $[0,\infty)$, and applying the Cauchy–Schwarz inequality yield

It follows that

To see closability, let $B$ be the generator of the left-shift semigroup, $Bf=f'$ on $H^1(0,\infty;\mathcal{X})$. For $x\in\operatorname{dom}(A)$ we have $\mathcal{O}_0x\in\operatorname{dom}(B)$ and $B\mathcal{O}_0x=\mathcal{O}_1x$. Since $\mathcal{O}_0$ is bounded and $B$ is closed, $B\mathcal{O}_0$ is closed. As $\mathcal{O}_1$ is its restriction to $\operatorname{dom}(A)$, it is closable. The estimate (13) extends to its closure, proving injectivity and boundedness of the inverse. The intertwining identity holds on $\operatorname{dom}(A)$ and passes to the closure using boundedness of $T(t)$ and $S(t)$.

Although $C_1=A:\operatorname{dom}(A)\to\mathcal{X}$ is graph-norm bounded, this does not ensure finite-time or infinite-time admissibility. For exponentially stable semigroups, these two notions of admissibility are equivalent; see [30 Sec. 4.6]. Its closed output transformation $\overline{\mathcal{O}}_1$ is bounded below. If it were also bounded, the associated semigroup would be similar to a contraction semigroup. Hence, for an exponentially stable semigroup not similar to a contraction, $\overline{\mathcal{O}}_1$ must be unbounded, while $\overline{\mathcal{O}}_0^{-1}$ is unbounded on its range.

The existence of a closed, bounded-below pseudo-similarity in Theorem 4.4 cannot be extended to arbitrary uniformly bounded strongly continuous semigroups. Indeed, there exist semigroups $T$ for which no bounded, injective, dense-range intertwiner $F$ from a strongly continuous contraction semigroup $S$ satisfies $T(t)F=FS(t)$ for all $t\ge0$. The norm bound of $T$ can even be chosen arbitrarily close to one [25 Theorem 1.4 and Remark 6.3]. In particular, such a semigroup cannot admit a closed, densely defined, bounded-below pseudo-similarity to a contraction semigroup: its inverse on the closed range would provide precisely such an intertwiner.

Unbounded energies and operator representations

Let $E:\operatorname{dom}(E)\subset\mathcal{X}\to\mathcal{Z}$ be a closed, densely defined, injective intertwiner from $T$ to a contraction semigroup $S$. Write

The transition to the energy space draws on closed-form theory [15 Chap. VI, Sec. 2] and the theory of pseudo-similarity and storage functions [29 Sec. 9.2]; see also [3 Secs. 1 and 7]. We record it to specify which operator is maximally dissipative.

Then $\mathcal{M}$ is bounded and injective, $\mathcal{D}$ is dissipative in the duality sense, and

is equivalent to $\dot{z}=A_{\mathcal{V}}z$ for $\mathcal{V}$-classical solutions.

Proof. Intertwining gives $\|T(t)x\|_{\mathcal{V}}^2\leq\|T(t)\|^2\|x\|_2^2+\|Ex\|_2^2$. Since $T(t)x\to x$ and $ET(t)x=S(t)Ex\to Ex$, the restricted semigroup is strongly continuous on $\mathcal{V}$. The generator formula follows from [30 Proposition 2.4.4]. Differentiating $ET_{\mathcal{V}}(t)x=S(t)Ex$ gives $BEx=EA_{\mathcal{V}}x$. For sufficiently large $\lambda$,

As $\operatorname{ran}E$ is dense in $\mathcal{K}$, the resolvent maps it onto a graph-norm dense subset of $\operatorname{dom}(B)$, giving the core property. Finally, $\mathcal{M}$ is bounded and injective since $E$ is, and

Injectivity of $\mathcal{M}$ yields the equivalence.

There is also a useful dual consequence of the lower observation estimate. It concerns the adjoint dynamics and must not be confused with an operator identity for $A$ on $\mathcal{X}$.

Proof. Closedness and the lower bound imply $\operatorname{ran}E=\mathcal{K}$. Thus $F=E^{-1}:\mathcal{K}\to\mathcal{X}$ is bounded, with dense range, and $T(t)F=FS(t)|_{\mathcal{K}}$. Moreover $P=FF^*$, by the polar decomposition of $E$. Therefore,

The last assertion follows by applying Theorem 3.1 to $A^*$ and $P$.

From energy forms to operators on $\mathcal{X}$. Contractivity of the energy form alone does not imply an operator factorization on the original state space. Indeed, if $H=E^*E$ and $x\in\operatorname{dom}(A)\cap\operatorname{dom}(H)$, then

Dividing by $t$ and taking the limit to zero gives $\operatorname{Re}\langle Ax,Hx\rangle\leq0$. If $\operatorname{dom}(A)\subset\operatorname{dom}(H)$, Theorem 3.1 therefore provides an exact representation $A=DH$ with maximally dissipative $D$. The domain inclusion is also necessary for an exact product representation with this particular $H$, as the theorem shows. The first construction in Subsection 6.2 shows that it may fail even for a closed, injective output-trajectory map.

Examples and limitations

The following examples illustrate the scope and limitations of the representation results.

All four boundedness combinations

Let $I\subset(0,\infty)$ be an interval, set $\mathcal{X}=L^2(I)$, and let

The semigroup $T(t)x(s)=e^{-st}x(s)$ is selfadjoint, contractive, and strongly stable. For the identity observation,

The corresponding energy operator is multiplication by $1/(2s)$. Consequently each of the following cases is realized:

$I$	$O_0$	$O_0^{-1}$ on $\operatorname{ran}O_0$
$(1,2)$	bounded	bounded
$(1,\infty)$	bounded	unbounded
$(0,1)$	unbounded	bounded
$(0,\infty)$	unbounded	unbounded

Thus, the first case satisfies both the upper and lower observation estimates, the second only the upper estimate, the third only the lower estimate on its natural domain, and the fourth neither. In particular, a lower observation estimate does not imply admissibility.

In every case, the output map is closed, densely defined, and injective. Notice that these four patterns can occur although the original semigroup is already contractive. The distinction concerns the particular energy representation, not just the contractivity of the semigroup itself.

Limitations of canonical factorizations

Let $\mathcal{X}=\ell^2(\mathbb{N})$ and $A=\operatorname{diag}(-1/n)$. Then $A$ is bounded and $T(t)=\operatorname{diag}(e^{-t/n})$ is contractive and strongly stable, but not exponentially stable. For $C=I$,

The output map is closed, densely defined, injective, and bounded below, but $H$ is unbounded. More precisely, $\operatorname{dom}(O_C)=\{x\in\ell^2:\sum_{n\geq1}n\lvert x_n\rvert^2<\infty\}$, whereas $\operatorname{dom}(H)=\{x\in\ell^2:\sum_{n\geq1}n^2\lvert x_n\rvert^2<\infty\}$. Since $\operatorname{dom}(A)=\mathcal{X}\not\subset\operatorname{dom}(H)$, it cannot yield any exact factorization $A=DH$ on $\mathcal{X}$, even though the energy is nonincreasing. The obstruction concerns this particular $H$; trivially, $A=AI$ is already a dissipative representation. The same semigroup also illustrates the distinction between the canonical dissipative factor and its maximally dissipative extensions. Indeed, $H_b = \operatorname{diag}(1/n)$ is bounded, positive definite, and a contractive energy, but $D_e = -I$ on $\operatorname{ran} H_b$ is not closed and hence not maximally dissipative. Nevertheless, $D = -I$ on $\mathcal{X}$ is a maximally dissipative extension of $D_e$, and $A = D H_b$ holds with equality of domains.

The Dirichlet heat equation

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain and let $A = \Delta_D$ be the Dirichlet Laplacian on $L^2(\Omega)$. Then $A$ is selfadjoint and there exists $\lambda_1 > 0$ such that

Consequently, the corresponding heat semigroup is exponentially stable. By the functional calculus for selfadjoint operators (see [15 Chap. VI, Sec. 5]), the identity observation yields

The bounded operator $H_0$ is positive definite but not coercive. The implicit factor is $D_i = -\frac{1}{2}I$ restricted to $\operatorname{dom}(A)$. With observation $C_1 = A$, we instead obtain

The closure of the output map initially defined by $O_1x = T(\cdot)Ax$ on $\operatorname{dom}(A)$ has domain $\operatorname{dom}((-A)^{1/2})$, whereas $\operatorname{dom}(H_1) = \operatorname{dom}(A)$. The energy operator $H_1$ is unbounded and coercive, but $H_1A = -\frac{1}{2}A^2$ is defined only on $\operatorname{dom}(A^2)$, not on all of $\operatorname{dom}(A)$. On the other hand, for $M = H_0$ and $D_i = H_0A = -\frac{1}{2}I|_{\operatorname{dom}(A)}$, the operator pencil $\lambda H_0 - D_i$ has range $\operatorname{ran} H_0 = \operatorname{dom}(A) \ne\mathcal{X}$ whenever $\lambda I - A$ is bijective, while (9) recovers the heat resolvent. Thus, both the explicit and implicit representations can be meaningful without a boundedly invertible energy operator.

Exponential stability without Hilbertian similarity

Exponentially stable analytic Hilbert-space semigroups need not be similar to contraction semigroups [16, 17, 24]. More strikingly, there exist strongly continuous semigroups that are nilpotent and immediately compact, as well as strongly continuous semigroups that are quasi-nilpotent, immediately compact, and bounded holomorphic of angle $\pi/2$. Neither class is similar even to a quasi-contraction semigroup. The construction uses a tensor-product splitting principle: similarity of a tensor product to a quasi-contraction semigroup forces the same property for its nonzero factors [25 Theorems 1.1 and 1.3].

Both classes of examples are exponentially stable. Theorem 3.5 therefore furnishes the bounded, positive definite Gramian $H_0$ from (10) and an exact factorization $A = D_eH_0$ with maximally dissipative $D_e$. For a nilpotent semigroup, the Gramian integral even terminates at a finite extinction time, and in both cases the strict identity $2\operatorname{Re}\{H_0Ax,x\} = -\lVert x\rVert^2$ holds for all $x \in\operatorname{dom}(A)$. Nevertheless, $H_0$ cannot be coercive: any bounded, coercive energy operator satisfying (7) would yield a boundedly invertible similarity to a contraction semigroup, contradicting the examples. Thus strict Lyapunov dissipation, even together with very strong stability and compactness, does not imply an equivalent contractive Hilbertian norm.

The one-sided output constructions in Theorem 4.4 reflect the same obstruction: the identity output is bounded with unbounded inverse on its range, whereas the closure of the generator output is unbounded but bounded below.

Concluding remarks

We have characterized exact dissipative Hamiltonian factorizations of semigroup generators in terms of a Lyapunov inequality and a domain inclusion. Whenever such a factorization exists, its dissipative factor can be chosen maximally dissipative without additional assumptions. The canonical factor need not itself be maximally dissipative, but every dissipative extension preserves the exact operator factorization. For bounded energy operators, the same Lyapunov inequality also yields an implicit representation.

Closed intertwiners and output-trajectory maps provide energy operators and contractive dynamics on associated energy spaces. Bounded-below intertwiners additionally yield a dual Lyapunov inequality and a dissipative Hamiltonian factorization of the adjoint generator. These constructions do not, in general, guarantee an exact factorization of the original generator with the constructed energy operator. Domain compatibility remains essential.

Acknowledgments. Both authors gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 531152215 – CRC 1701 “Port-Hamiltonian Systems”.