AN ABSTRACT SETTING FOR DIFFERENTIAL RICCATI EQUATIONS IN OPTIMAL CONTROL PROBLEMS FOR HYPERBOLIC/PETROWSKI-TYPE P.D.E.’S WITH BOUNDARY CONTROL AND SLIGHTLY SMOOTHING OBSERVATION

We study, by the variational method, the Differential Riccati Equation which arises in the theory of quadratic optimal control problems for ‘abstract hyperbolic’ equations (which encompass hyperbolic and Petrowski-type partial differential equations (P.D.E.) with boundary control). We markedly relax, at the abstract level, the original assumption of smoothing required of the observation operator by the direct method of [D-L-T.1]. This is achieved, by imposing additional higher level regularity requirements on the dynamics, which, however, are always satisfied by the class of hyperbolic and Petrowski-type mixed P.D.E. problems which we seek to cover. To appreciate the additional level of generality, and related technical difficulties associate with it, it suffices to point out that in the present treatment—unlike in [D-L-T.1]—the gain operator B∗P (t) is no longer bounded between the state space Y and the control space U . The abstract theory is illustrated by its application to a Kirchoff equation with one boundary control. This requires establishing new higher level interior and boundary regularity results. 1991 Mathematics Subject Classification. Primary 93C20; Secondary 35L, 49JK.


Introduction. Literature
This paper presents, in its first part (Sections 1 through 4), a general and unifying abstract treatment of the optimal control problem with quadratic cost functional, over a finite (time) horizon, for the abstract differential equation (1.1) below.Here, A is the generator of a s.c.semigroup and B is a (highly) unbounded control operator, satisfying the 'trace regularity' condition (H.1) = (1.6)below.This condition was introduced in [L-T.2](see also [L-T.3],[L-T.6],[F-L-T.1])and has since been shown to be typical of mixed problems for hyperbolic and Petrowski-type partial differential equations (P.D.E.'s), see [L-T.6].By duality, the (abstract) 'trace' regularity (H.1) = (1.6)below for the homogeneous problem (1.1) with u = 0 is converted into an 'interior' regularity result of the non-homogeneous problem (1.1), see (H.1 * ) = (1.7).In this paper, focus and emphasis are placed on the Differential Riccati Equation (D.R.E.) associated with the optimal control problem (1.1)-(1.4),whose non-negative, self-adjoint solution P (t) provides the value of the optimal control problem, as well as the pointwise (a.e. in time) synthesis of the optimal pair, as pointed out by Remark 2.1 below.Applications to mixed problems for hyperbolic and Petrowski-type P.D.E.'s are an integral and essential part of the present study.In Section 5, we illustrate the theory established in Sections 1-4, as it applies to Kirchoff equations.Because of space restrictions, additional P.D.E.'s applications (to Euler-Bernoulli equations, to Schrödinger equations) are provided in a companion paper [T.1] and in a forthcoming book [L-T.8,Chapter 10].More on this will be said below.
Differential Riccati Equations: Direct method [D-L-T.1]with R smoothing.At the abstract level, the present paper is a conceptual successor of [D-L-T.1]:this work assumed only hypothesis (H.1) = (1.6) on the dynamics (1.1), which in P.D.E.'s applications amounts to a basic level regularity result with L 2 -boundary data, via the equivalent version (H.1 * ) = (1.7).Following the so-called 'direct method'-from the D.R.E. to the optimal control problem, via dynamic programming-[D-L-T.1]established well-posedness (existence and uniqueness) of the corresponding D.R.E. by local contraction plus global a-priori bounds.Well-posedness of the D.R.E.is, in the present context, non-trivial, due to the high degree of unboundedness of the control operator B, expressed by (1.5a) below, which in P.D.E.'s applications models the action from the boundary to the interior.In [D-L-T.1]success of the direct strategy was based, among other technical issues, on a trick of performing a suitable change of (operator) variable, which made application of the contraction argument more amenable.Once well-posedness of the D.R.E. is established, one then recovers the optimal control problem by dynamic programming.A key point is that the theory of [D-L-T.1]requires, however, the hypothesis that the observation operator R in (1.3) be smoothing, in the sense that R * Re At B : continuous U → L 1 (0, T ; Y ).
(0.1) For instance, if A is (−∆) with homogeneous Dirichlet B.C., then (0.1) amounts to the smoothing requirement that R * R is comparable to A −( 1 4 + ) × A −( 1 4 + ) , > 0 arbitrary.In return, the critical gain operator B * P (t), which occurs in the quadratic term of the D.R.E.(2.6) (as well as in the optimal synthesis of Remark 2.1) is bounded, at each t, from the state space Y to the control space U .Differential Riccati Equation: Variational approach with R almost the identity.One goal of the present paper is to markedly weaken the smoothing requirement (0.1) assumed on R * R, in fact, from R * R ∼ A −( 1 4 + ) × A −( 1 4 + ) to R * R ∼ A − × A − , A defined above, see hypothesis (H.8) = (1.23)below, > 0 arbitrary; i.e., -away from the ideal situation with R = Identity where the observation operator is then non-smoothing.To achieve this quantum improvement over [D-L-T.1],we require in this paper additional abstract regularity assumptions on the dynamics (1.1) [in addition to the basic level (H.1) = (1.6), or (H.1 * ) = (1.7)],which amount to a higher level regularity: smoother data imply smoother solutions, in specifically required spaces.In this respect, we hasten to add that: all assumed hypotheses on (1.1) are nothing but actual regularity properties displayed by the 'concrete' classes of hyperbolic and Petrowski-type P.D.E.'s which we seek to cover.In the present paper, our approach is variational-from the optimal control problem to the well-posedness of the D.R.E., thus reversing the 'direct method' of [D-L-T.1].In return for weakening the assumption on the observation operator R, we obtain a less regular theory, not unexpectedly.In contrast with [D-L-T.1],under the present weakened smoothing assumption on R such as (H.8) = (1.23), it turns out that: (1) The gain operator B * P (t) is not bounded any longer from the state space Y to the control space U , but only densely defined on Y ; indeed, its domain is constant in t and coincides with an explicitly identified subspace (Y − δ below) of the state space Y ; see (2.4) for the technical statement.
(2) The present variational approach provides (constructively) existence of the D.R.E., indeed with the operator P (t) defined by (2.3a) [hence expressible directly in terms of the problem data via (2.1a)] being a nonnegative, self-adjoint solution of the D.R.E.(2.6).Nothing is said about uniqueness now (within a class of non-negative, self-adjoint solutions satisfying the regularity property (2.4)): this is not surprising, and is akin to the situation in the abstract parabolic case [L-T.6],[L-T.7], [L-T.8].
Applications to P.D.E.mixed problems.The setting of the present paper-although abstract-is in reality motivated by, and ultimately directed to, numerous classes of mixed problems for hyperbolic and Petrowskitype partial differential equations defined on a bounded domain Ω of R n , with boundary control.Specifically, these include, but are not limited to, the following cases: (i) second-order hyperbolic equations with Dirichlet boundary control; (ii) non-symmetric, non-dissipative, first-order hyperbolic systems with boundary control; (iii) (hyperbolic) Kirchoff equations with finite speed of propagation, as well as (iv) (non-hyperbolic) Euler-Bernoulli equations with infinite speed of propagation with one boundary control (such as they arise in linear elasticity in the special cases where dim Ω = 1, 2); (v) Schrödinger equations with Dirichlet boundary control.In cases (iii) and (iv), various choices among the two associated boundary conditions are possible, each leading to a different function space setting.So far, the abstract setting for the optimal control problem of the present paper has been successfully applied to all classes (i) through (v).The 'concrete' cases (i) and (ii) were studied in isolation in [L-T.3] and [C-L.1]respectively.Indeed, it was their successful treatment that stimulated the need of producing an all-encompassing abstract framework, by lifting and extracting the essential features common to all these (and other) dynamical P.D.E.'s classes, (i) through (v).These have resulted in seven dynamical assumptions, (H.1) through (H.7) below.We emphasize once more: needless to say, all these assumptions have been verified to hold true for the above classes of hyperbolic [(i)-(iii)] and non-hyperbolic, Petrowski-type [(iv), (v)] mixed problems (with various boundary conditions for (iii) and (iv)).We shall report them in book-form in [L-T.8,Chapter 10].Indeed, in all these cases these abstract assumptions are, in fact, nothing but distinctive interior and boundary (traces) regularity properties.To be sure, their verification is not a trivial or classical matter, and requires P.D.E.energy methods (not functional analysis techniques), which have been brought to bear only very recently on these mixed P.D.E.'s problems, with emphasis on the basic level with L 2 -boundary data.However, as already pointed out above, our present abstract setting requires also higher level regularity results, both interior and boundary, see e.g., assumptions (H.2) and (H.3) below.In the case of Kirchoff, Euler-Bernoulli and Schrödinger equations, such interior and boundary higher-order regularity results were not available in the literature, and it was our task to provide them.In the case of Kirchoff equations, they are given in Section 5 below, particularly the proof of Theorem 5.8.1 in Section 5.13.In the case of the Euler-Bernoulli and Schrödinger equations, we refer to a companion paper [T.1] and [L-T.8,Chapter 10].In all these latter three classes, the derivation of higher-level trace regularity results (in space) presents (unexpected) additional difficulties (see Remark 5.13.1) over the known cases of second-order hyperbolic equations with Dirichlet control [L-L-T.1],[L-T.3,Section 3].In the case of first-order, hyperbolic systems, these higher level results were given in [Rau.1](see also [C-L.1]), after the basic level regularity result in the fundamental paper [K.1].We expect that the present setting for the optimal control problem will also apply to additional P.D.E.'s mixed problems, such as the system of elasticity, and the Maxwell equation.or its mild version where 0 ≤ s ≤ T < ∞, subject to the abstract hypotheses listed below.
Optimal control problem on the interval [s, T ].We introduce the cost functional and the corresponding optimal control problem O.C.P. is then: Minimize J(u, y) over all u ∈ L 2 (s, T ; U ), where y(t) = y(t, s; y 0 ) is the solution of Eqn.(1.1) with initial condition y(s) = y 0 . (1.4) We now list the abstract assumptions of the present paper.
Abstract assumptions.We first group together in (i) below some standing preliminary basic assumptions: (i) U, Y , and Z are Hilbert spaces; A is the generator of an s.c.semigroup e At on Y, t ≥ 0; B is a (linear) continuous operator U → [D(A * )] , equivalently [without loss of generality, we take A −1 ∈ L(Y ).For otherwise we replace A −1 with the resolvent operator R(λ 0 , A), λ 0 a point of the resolvent set of A. However, A −1 will streamline the notation throughout, e.g., in (1.5b) below, where one would otherwise take the graph norm on D(A * ).]In (1.1),A * is the Y -adjoint of A, and [D(A * )] is the Hilbert space dual to the space D(A * ) ⊂ Y with respect to the Y -topology, with norms Consequently, as seen in [L-T.
Then, the operator L * s , adjoint of L s in the sense that (L s u, f ) L 2 (s,T ;Y ) = (u, L * s f ) L 2 (s,T ;U ) , and thus given by a norm which may be made independent of s, i.e., (1.9a) As stated in the introduction, the above assumption (H.1) was the only hypothesis (in addition to (i)) on the dynamics (1.1), or (1.2), required by the treatment of [D-L-T.1].The following additional hypotheses (H.2) through (H.7) on the dynamics (1.1) [all verified to be true for the hyperbolic/Petrowski class of P.D.E.'s we intend to cover] will allow us to drastically reduce over [D-L-T.1], the assumption on the degree of smoothing of the observation operator, from and the interpolating property where u ∈ H θ (s, T ; U ) means that the fractional time derivative and similarly for Y θ [s, T ], then: (H.2) , with a norm which may be made independent of s, i.e., (1.14a) [For θ = 0, (H.2) = (1.14)specializes to (H.1 * ) = (1.7),via (1.10).] Remark 1.1.In applications to mixed problems for P.D.E.'s [see Section 5 below, as well as [L-T.3],[C-L.1],[T.1]], one first establishes (H.1) = (1.6),hence the regularity (H.1 * ) = (1.7)for L s [case θ = 0]; next, one establishes a regularity result for L s for θ = 1 involving the spaces U 1 and Y 1 , which, however, requires a compatibility condition.In interpolating the two above cases θ = 0 and θ = 1 for θ < 1 2 , the compatibility condition is irrelevant, and one thus obtains (H.2) = (1.14). (H.3) with a norm which may be made independent of s, i.e., (1.15a) [for θ = 0, (H.3) = (1.15) is contained in (1.9)].Henceforth, we shall fix once and for all a number δ > 0 arbitrarily small and set for convenience (1.16) (1.17) [The values θ = 1 2 ± δ and θ = 1 2 − δ 2 will be the only values of θ where the assumptions (H.2) = (1.14) and (H.3) = (1.15)will be used.]Remark 1.2.In applications to mixed problems for P.D.E.'s, passage from [which is automatically implied by via assumption (H.6) = (1.20)];(H.8) (assumption on smoothing observation) which then, by duality, implies Preliminary, direct consequences of the assumptions.Some preliminary, direct consequences of the abstract assumptions, to be invoked in the sequel, are listed next.
) with a norm which may be made independent of s, i.e., (1.27a) , uniformly in s.

Statement of the main results
Our starting point is [D-L-T.1],[L-T.3], which applies by virtue of the assumptions (i), (H.1) = (1.6) and R ∈ L(Y ; Z).In the present setting, a far richer and complete theory becomes available. (2.2a) All the above results are with norms which may be made independent of s.

Orientation.
Existence of a unique optimal pair {u 0 ( • , s; y 0 ), y 0 ( • , s; y 0 } and formulas (2.1a), (2.2a) apply to the present situation [L-T.3,pp.890-891], and we seek to go beyond these preliminary results.Now, Eqns.(1.14) for L s and (1.15) for L * s show, by (1.11), (1.12), that-in the present setting, -the operators L s and L * s do not provide any smoothing in the (Sobolev spaces) Y θ and U θ , i.e., in what in P.D.E.'s applications will be "the space variable."Thus, in order to achieve a complete theory, which in particular includes the derivation of a Differential Riccati Equation, two main problems of similar nature arise: (1) First, in seeking regularity properties for the optimal trajectory y 0 ( • , s; y 0 ) with a "regular" initial datum y 0 ∈ Y − δ , one needs to perform a critical bounded inversion of the operator [ .12).This bounded inversion would, however, be a serious problem, unless is performed, then one obtains, along with assumptions (H.5) = (1.19),regularity properties of y 0 ( • , s; y 0 ); and hence, via (H.3) = (1.15)applied to the optimality condition, see Eqn. (2.2a) regularity properties of u 0 ( • , s; y 0 ), for s fixed.Next, however, in order to obtain the regularity property of the gain operator 3) and Remark 2.1]), we see that we need to refine the preceding result by asserting that, in fact, [ uniformly in s.The aforementioned regularity of B * P (t) then justifies the well-posedness of the critical quadratic term, which occurs in the Differential Riccati Equation (2.6).All this summarizes the content of Section 3, which provides the proof of the regularity Theorems 2.1 and 2.2.
(2) Second, in seeking to derive the Differential Riccati Equation (2.6) on Y − δ , one encounters the obstacle of performing the bounded inversion of the operator this time, however, on the weaker space L 2 (s, T ; [Y + δ ] ); equivalently, by duality, the bounded inversion of . This task would, however, be again a serious problem, unless R * RL s L * s could be asserted to be compact on L 2 (s, T ; Y + δ ).It is at this level that the smoothing assumption ) is used in full force, as the operator L s , by assumption (H.2) = (1.14),has a known regularity property only on Y θ , θ < 1 2 .Accordingly, the bounded inversion of [ ) is then performed for each s, a result sufficient in the derivation of the D.R.E.(2.6) in Section 4.

Bounded inversion of
The key preliminary result is the following.(i) With reference to the spaces in (1.12) for θ = 1 2 − δ and θ = 1 2 − δ 2 , we have the following estimate , uniformly in s.
(3.1.1)(ii) For fixed s, the operator , indeed uniformly with respect to s: Proof.(i) The proof of estimate (3.1.1)is a consequence of part of the following diagram, where all continuity maps are uniform in s: In the first step, we use ( where in the last steps we have recalled (1.2b) as well as assumption (H.5) = (1.19) on e At , and (H.2) = (1.14) with θ = 1 2 − δ.Thus, by (3.1.5), as well as In fact, the definition (1.8) implies, still with s 1 > s, after using (H.3) = (1.15b), and (3.1.7)is proved.
(iv) We first show that [ , then a (necessary and) sufficient condition for (3.1.17)to hold true is that λ = 1 be not an eigenvalue of which is certainly the case, for otherwise λ = 1 would also be an eigenvalue of Finally, to assert the uniform estimate (3.1.2),we simply invoke [L-T.3,Lemma 3.12] with Z 1 ≡ Y 3): this is legal by virtue also of (3.1.1) of part (i), (3.1.16) of part (iii), and (3.1.17) of part (iv).Theorem 3.1.1 is proved.
Remark 3.1.1.In the preceding diagram the weaker requirement by the second resolvent equation.Hence, estimate (3.1.2) of Theorem 3.1.1applied to (3.1.18)readily implies that, for each f ∈ Y a result which can be applied to y 0 ( • , s; y 0 ) via (2.1a).See also Remark 3.1.3below.
Remark 3.2.1.Recalling the pointwise relationship ). Consequences on Φ(t, s) We begin with the result which will serve our purposes in the sequel.
The above diagram uses (H.3) = (1.15)for θ = 1 2 + δ on L * s ; followed by the injection U as a consequence of the compactness property U 1 2 +δ → U 1 2 −δ of the injection contained in (1.10) and of T < ∞; followed by (H.2) = (1.14)for θ = 1 2 − δ on L s ; followed by (H.8) = (1.23) on R * R. Thus, as a result, R * RL s L * s is a compact operator on L 2 (s, T ; Y + δ ), as desired.(ii) Since R * RL s L * s is compact on L 2 (s, T ; Y + δ ) by part (i), then a (necessary and) sufficient condition for (4.1.1)to hold true is that λ = 1 be not an eigenvalue of R * RL s L * s on L 2 (s, T ; Y + δ ), which is certainly the case, for otherwise λ = 1 would also be an eigenvalue of R * RL s L * s on L 2 (s, T ; Y ), thus contradicting [L-T.3,p. 891], which asserts that [I s + R * RL s L * s ] −1 ∈ L(L 2 (s, T ; Y )).Thus, (4.1.1)is proved.
We can now draw some consequences of Theorem 4.1.1 on properties of the evolution operator Φ(t, s) in (3.1.20),to be invoked in the sequel where now e A(   take the distributional derivative in s, to obtain  Proof.(i) We examine Then,by (3.1.20),or (3.1.26),and by (H.8) = (1.23),we have (4.2.4) and by duality on (C.3) = (1.29),we have  We can finally establish that P (t) satisfies the D.R.E.
Theorem 4.2.2.Assume (i), (H.1) through (H.8).Then the nonnegative, self-adjoint operator P (t) defined by (2.3a) satisfies the following Differential Riccati Equation for all 0 ≤ t < T :  (4.2.13) after substituting (4.1.10)in the second term on the right-hand side of (4.2.12), as well as substituting (4.2.11) [with y replaced by Ay] in the third term on the right-hand side of (4.2.12).We notice explicitly that each term in (4.2.12), or (4.2.13), is well-defined at each t: the last term by (4.2.2), and the critical second term on the right-hand side of (4.2.12), or (4.2.13), by the regularity in (4.1.10)for dφ (τ,t)x dt , combined with R * R: continuous by (1.24), as well as with Thus, invoking again (4.2.11) on the second term on the right-hand side of (4.2.13), we obtain  (4.2.15)where all terms are well defined by (1.19), (3.2.1), (1.23).−t) x, e A(τ −t) y Y r=τ where by using the D.R.E.(4.2.10), we have −t) x, e A(τ −t) (4.2.17) We note explicitly that each term of (4.2.16) and (4.2.17) is well defined, indeed, we have e A(τ −t) x, e A(τ −t) y in C([t, T ]; Y − δ ), for x, y ∈ Y − δ , and hence:  (τ −t) x, y −t) x, B * P (τ )e All dynamical abstract hypotheses (H.1) = (1.6)through (H.7) = (1.21) of Section 1 have already been shown to hold true in the following two cases: (i) Second-order hyperbolic equations with Dirichlet boundary control, defined on a smooth, bounded domain Ω ⊂ R n , see [L-T.3]; here one may take (ii) non-symmetric, non-dissipative, first-order hyperbolic systems with boundary control, see [C-L.1]; here one make take similarly defined explicit Sobolev spaces for U θ , Y θ , etc.
In this section we consider an optimal quadratic cost problem over a finite horizon for a Kirchoff equation, subject only to one control acting in the "moment" boundary condition.[The physical bending moment in the 2dimensional Kirchoff plate model is actually a modification of the boundary condition (5.1.1d)below.]The Kirchoff equation is hyperbolic with finite speed of propagation, and displays a behavior similar to that of the wave equation.In the case of the Kirchoff mixed problem, we shall show likewise that all abstract system's assumptions (H.1) = (1.6)through (H.7) = (1.21) of Section 1 are automatically satisfied in a natural mathematical setting.Many such settings can be chosen, and we shall select a particular interesting one where, as in the case of second-order hyperbolic equations of [L-T.5], the observation R * R jumps across a boundary condition, see (5.2.3)-(5.2.5) below.Accordingly, Theorems 2.1, 2.2, and 2.3 of Section 2 are then applicable to the present class, for any observation operator R with "minimal" smoothing as in (H.8) = (1.23).In a companion paper [T.1], we show that the Euler-Bernoulli equation [Eqn. (5.1.1a)below with ρ = 0], which is not hyperbolic, also satisfies assumptions (H.1) through (H.7) in explicitly identified Sobolev spaces (different from the Kirchoff equation case).

Problem formulation
The dynamics.Let Ω be an open bounded domain in R n with sufficiently smooth boundary Γ, say, of class C 2 .The Kirchoff equation is given by where ρ > 0 is a constant (proportional to the square of the thickness in the 2-dimensional plate model), and where u ∈ L 2 (0, T ; L 2 (Γ)) ≡ L 2 (Σ) is the control function acting in the "moment" B.C. (5.1.1d).
The optimal control problem on [s, T ].Consistently with the (optimal) regularity theory for problem (5.1.1)presented in Theorem 5.3.2below, the cost functional which we seek to minimize over all u ∈ L 2 (s, T ; ) will be further specified below in (5.2.3).

Main results
As a specialization to problem (5.1.1),(5.1.2) of the abstract theory presented in Theorems 2.1, 2.2, and 2.3 of Section 2, in the present section we establish the following results.
(5.2.12)Furthermore, P (t) satisfies the corresponding Integral Riccati Equation, as in (2.7), for all such x, y.
(5.5.13)Notice that regularity (5.5.11) of the normal trace ∂∆ ∂ν ψ(t; h 1 ), for ψ(t; h 1 ) solution of (5.5.8) due to h = h 1 given by (5.5.12), is precisely conclusion (5.3.5) of Theorem 5.3.1 for the time-reversed problem ψ in (5.3.1) with initial data at t = 0, rather than t = T as for ψ, an inessential modification.The proof of Theorem 5.3.1 in [L-T.5] is by energy (P.D.E.'s)-methods.Instead, regularity (5.5.9) for the normal trace ∂∆ ∂ν ψ t (t; h 2 ) for the time derivative ψ t of the solution ψ(t; h 2 ) of problem (5.5.8) due to h = h 2 given by (5.5.13) is obtained by duality via operator methods as in (5.5.6)-(5.5.11) [while it appears that purely P.D.E.methods will require a time regularity assumption of the right-hand side, non-homogeneous term].Thus, assumption (H.1) = (1.6) is verified.We select the spaces in (1.10) to be the following Sobolev spaces (5.6.1) where A is as in (5.4.23), in particular the critical spaces for θ = 1 2 ± δ: (5.6.7)Thus, by (5.6.4) and (5.6.7)we verify the interpolation property since Ω is a bounded domain.Thus, the spaces in (1.11), (1.12) are in the present case as follows for 0 (5.6.9) (5.6.10) (5.6.12)
5.10.Proof of (5.3.5) of theorem 5.3.1 Key to this end is the following result.It is reported here because it will be critically invoked in Section 5.13.(5.10.1) Then, the following identity holds true:  We now provide the details, already contained in the preceding development, that the trace regularity for the homogeneous φ-problem (5.x 2 (5.11.4) = B * e A * t x; (5.11.5) (5.11.6)Thus, if we take of w in (5.11.12), and by the regularity of D in (5.4.18b).To get w tt ∈ L 2 (0, T ; L 2 (Ω)), we could also differentiate w t in (5.11.1).Finally, we omit the details for the regularity due to the initial conditions {w 0 , w 1 }, using (5.4.11).Theorem 5.3.2 is proved.
A P.D.E.version of the duality or transposition argument may be given.(5.13.28b).The two boundary terms containing g in (5.13.33) are well defined by the regularity of g in (5.13.28c), while the boundary term containing ∂zt ∂ν is well defined by (5.13.20) and trace theory.We conclude that the remaining boundary term in (5.13.33) containing ∂(∆z)   ∂ν is well defined, i.e., ∂(∆z)  ∂ν ∈ L 2 (Σ), and thus (5.13.22) is established, as desired.The proof of Theorem 5.13.1 is complete.

ρ
component space of Y particularly convenient.In fact, with such a choice, we have that A in (5.4.23) is skew-adjoint on Y = D A , i.e., A * = −A, and so it generates a s.c.unitary group e At on Y .
At and e A * t .More insight on the impact of the smoothing assumption R * R ∈ L(Y − At Bu, y) Y = (u, B * e A * t y) U ≤ C T u U y Y + as desired.The closed graph theorem then yields (4.2.1).
Part (ii), Eqn.(4.2.2), is an immediate consequence of part (i) and of P (t) being self-adjoint on Y .a property for A which is generally false; see illustrations below.