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Abstract and Applied Analysis An abstract setting for differential Riccati equations in optimal control problems for...
An abstract setting for differential Riccati equations in optimal control problems for hyperbolic/Petrowskitype P.D.E.'s with boundary control and slightly smoothing observation
Triggiani, R.এই বইটি আপনার কতটা পছন্দ?
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1996
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Abstract and Applied Analysis
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10.1155/s1085337596000243
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AN ABSTRACT SETTING FOR DIFFERENTIAL RICCATI EQUATIONS IN OPTIMAL CONTROL PROBLEMS FOR HYPERBOLIC/PETROWSKITYPE P.D.E.’S WITH BOUNDARY CONTROL AND SLIGHTLY SMOOTHING OBSERVATION R. TRIGGIANI Abstract. 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 Petrowskitype 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 [DLT.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 Petrowskitype 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 [DLT.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. Key words and phrases. Diﬀerential Riccati equations, optimal control, hyperbolic, Petrowskitype partial diﬀerential equations. Paper presented at the Minisymposium on “Monotonicity Methods in Nonlinear Analysis”, Second World Congress of Nonlinear Analysts, Athens, Greece, July 1017, 1996. Research partially supported by the National Science Foundation under Grant DMS9504822, and by the Army Research Oﬃce under Grant DAAH049610059. Received: December 6, 1996. c 1996 Mancorp Publishing, Inc. 435 436 R. TRIGGIANI 0. 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 [LT.2] (see also [LT.3], [LT.6], [FLT.1]) and has since been shown to be typical of mixed problems for hyperbolic and Petrowskitype partial diﬀerential equations (P.D.E.’s), see [LT.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 nonhomogeneous 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 nonnegative, selfadjoint 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 Petrowskitype 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 EulerBernoulli equations, to Schrödinger equations) are provided in a companion paper [T.1] and in a forthcoming book [LT.8, Chapter 10]. More on this will be said below. Diﬀerential Riccati Equations: Direct method [DLT.1] with R smoothing. At the abstract level, the present paper is a conceptual successor of [DLT.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 L2 boundary data, via the equivalent version (H.1∗ ) = (1.7). Following the socalled ‘direct method’—from the D.R.E. to the optimal control problem, via dynamic programming—[DLT.1] established wellposedness (existence and uniqueness) of the corresponding D.R.E. by local contraction plus global apriori bounds. Wellposedness of the D.R.E. is, in the present context, nontrivial, 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 [DLT.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 wellposedness 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 [DLT.1] requires, however, the hypothesis that the observation operator R in (1.3) be smoothing, in the sense that R∗ ReAt B : continuous U → L1 (0, T ; Y ). (0.1) DIFFERENTIAL RICCATI EQUATIONS 437 For instance, if A is (−∆) with homogeneous Dirichlet B.C., then (0.1) 1 amounts to the smoothing requirement that R∗ R is comparable to A−( 4 +) × 1 A−( 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 ∼ 1 1 A−( 4 +) × A−( 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 nonsmoothing. To achieve this quantum improvement over [DLT.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 Petrowskitype P.D.E.’s which we seek to cover. In the present paper, our approach is variational—from the optimal control problem to the wellposedness of the D.R.E., thus reversing the ‘direct method’ of [DLT.1]. In return for weakening the assumption on the observation operator R, we obtain a less regular theory, not unexpectedly. In contrast with [DLT.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 (Yδ− 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, selfadjoint solution of the D.R.E. (2.6). Nothing is said about uniqueness now (within a class of nonnegative, selfadjoint solutions satisfying the regularity property (2.4)): this is not surprising, and is akin to the situation in the abstract parabolic case [LT.6], [LT.7], [LT.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 Rn , with boundary control. Speciﬁcally, these include, but are not limited to, the following cases: (i) secondorder hyperbolic equations with Dirichlet 438 R. TRIGGIANI boundary control; (ii) nonsymmetric, nondissipative, ﬁrstorder hyperbolic systems with boundary control; (iii) (hyperbolic) Kirchoﬀ equations with ﬁnite speed of propagation, as well as (iv) (nonhyperbolic) EulerBernoulli 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 [LT.3] and [CL.1] respectively. Indeed, it was their successful treatment that stimulated the need of producing an allencompassing 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 nonhyperbolic, Petrowskitype [(iv), (v)] mixed problems (with various boundary conditions for (iii) and (iv)). We shall report them in bookform in [LT.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 L2 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, EulerBernoulli and Schrödinger equations, such interior and boundary higherorder 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 EulerBernoulli and Schrödinger equations, we refer to a companion paper [T.1] and [LT.8, Chapter 10]. In all these latter three classes, the derivation of higherlevel trace regularity results (in space) presents (unexpected) additional diﬃculties (see Remark 5.13.1) over the known cases of secondorder hyperbolic equations with Dirichlet control [LLT.1], [LT.3, Section 3]. In the case of ﬁrstorder, hyperbolic systems, these higher level results were given in [Rau.1] (see also [CL.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. DIFFERENTIAL RICCATI EQUATIONS 439 1. Mathematical Setting and problem statement Dynamical model. We consider the abstract diﬀerential equation ẏ = Ay + Bu on, say, [D(A∗ )] ; y(s) = y0 ∈ Y, (1.1) or its mild version y(t, s; y0 ) = eA(t−s) y0 + (Ls u)(t), (Ls u)(t) = t s (1.2a) eA(t−τ ) Bu(τ )dτ, (1.2b) where 0 ≤ s ≤ T < ∞, subject to the abstract hypotheses listed below. Optimal control problem on the interval [s, T ]. We introduce the cost functional J(u, y) = T s Ry(t) 2 Z + u(t) 2 U dt, (1.3) and the corresponding optimal control problem O.C.P. is then: Minimize J(u, y) over all u ∈ L2 (s, T ; U ), where y(t) = y(t, s; y0 ) is the solution of Eqn. (1.1) with initial condition y(s) = y0 . (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 eAt on Y, t ≥ 0; B is a (linear) continuous operator U → [D(A∗ )] , equivalently A−1 B ∈ L(U ; Y ). (1.5a) [without loss of generality, we take A−1 ∈ L(Y ). For otherwise we replace A−1 with the resolvent operator R(λ0 , A), λ0 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 y D(A∗ ) = A∗ y Y; y [D(A∗ )] = A−1 y Y. (1.5b) Via (1.5a), we let (B ∗ x, u)U = (x, Bu)Y for u ∈ U, x ∈ D(A∗ ), and then B ∗ ∈ L(D(A∗ ); U ). ∗ (H.1): (abstract trace regularity) the (closable) operator B ∗ eA t can be extended as a map ∗ B ∗ eA t : continuous Y → L2 (0, T ; U ); (1.6a) T ∗ A∗ t 2 B e x dt ≤ cT x 2Y , (1.6b) 0 U x ∈ Y. Consequently, as seen in [LT.3, Thm. 1.1], [FLT.1, Appendix A], it follows equivalently that the operator Ls in (1.2b) satisﬁes 440 R. TRIGGIANI (H.1∗ ) Ls : continuous L2 (s, T ; U ) → C([s, T ]; Y ) with a norm which may be made independent of s, i.e., Ls u Then, the operator C([s,T ];Y ) ≤ cT u L2 (s,T ;U ) uniformly L∗s , adjoint of Ls in the sense that in s. (1.7a) (1.7b) (Ls u, f )L2 (s,T ;Y ) = (u, L∗s f )L2 (s,T ;U ) , and thus given by (L∗s f )(t) = T t ∗ (τ −t) B ∗ eA f (τ )dτ (1.8) satisﬁes L∗s : continuous L1 (s, T ; Y ) → L2 (s, T ; U ) with a norm which may be made independent of s, i.e., L∗s f L2 (s,T ;U ) ≤ cT f L1 (s,T ;Y ) uniformly in s. (1.9a) (1.9b) 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 [DLT.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 [DLT.1], the assumption on the degree of smoothing of 1 1 the observation operator, from R∗ R ∼ A−( 4 +) × A−( 4 +) in [DLT.1] to R∗ R ∼ A− × A− in assumption (H.8) = (1.23) below, where, say, A is the Laplacian with Dirichlet B.C. Distinctive new hypotheses. Distinctive new hypotheses over [DLT.1] are as follows: There exist families of Hilbert spaces (which in applications to P.D.E.’s are Sobolev spaces) Uθ ; Yθ , 0 ≤ θ ≤ 12 + δ, for some 12 > δ > 0; θ = 12 for Yθ ; U0 = U ; Y0 = Y, δ henceforth kept ﬁxed, with the property that: injection Uθ2 → Uθ1 and Yθ2 → Yθ1 is compact, 0 ≤ θ1 < θ2 ≤ 12 + δ, and the interpolating property [Y 1 −δ , [Y 1 +δ ] ]θ= 1 −δ = Y, δ > 0, 2 2 (1.10) 2 ] duality of [Y 1 +δ with respect to Y , such that, setting 2 U θ [s, T ] ≡ L2 (s, T ; Uθ ) ∩ H θ (s, T ; U ) θ θ Y [s, T ] ≡ L2 (s, T ; Yθ ) ∩ H (s, T ; Y ), (1.11) (1.12) DIFFERENTIAL RICCATI EQUATIONS 441 where u ∈ H θ (s, T ; U ) means that the fractional time derivative Dtθ u ∈ L2 (s, T ; U ), as usual [LM.1], with norm u 2 U θ [s,T ] ≡ u 2 L2 (s,T ;Uθ ) + u 2 H θ (s,T ;U ) , (1.13) and similarly for Y θ [s, T ], then: (H.2) continuous U θ [s, T ] → Y θ [s, T ] ∩ C([s, T ]; Yθ ), 0 ≤ θ < 12 , with a norm which may be made independent of s, i.e., Ls : (1.14a) 1 uniformly in s; 0 ≤ θ < . (1.14b) 2 [For θ = 0, (H.2) = (1.14) specializes to (H.1∗ ) = (1.7), via (1.10).] Ls u ≤ cT,θ u Y θ [s,T ]∩C([s,T ];Yθ ) U θ [s,T ] Remark 1.1. In applications to mixed problems for P.D.E.’s [see Section 5 below, as well as [LT.3], [CL.1], [T.1]], one ﬁrst establishes (H.1) = (1.6), hence the regularity (H.1∗ ) = (1.7) for Ls [case θ = 0]; next, one establishes a regularity result for Ls for θ = 1 involving the spaces U1 and Y1 , which, however, requires a compatibility condition. In interpolating the two above cases θ = 0 and θ = 1 for θ < 12 , the compatibility condition is irrelevant, and one thus obtains (H.2) = (1.14). (H.3) L∗s : 1 + δ, θ = ; 2 with a norm which may be made independent of s, i.e., (1.15a) continuous L2 (s, T ; Yθ ) → U θ [s, T ], 0 ≤ θ ≤ L∗s f U θ [s,T ] ≤ CT,θ f L2 (s,T ;Yθ ) , 1 2 uniformly in s (1.15b) [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 Yδ− ≡ Yθ= 1 −δ ⊃ Yδ+ ≡ Yθ= 1 +δ (1.16) Uδ− ≡ Uθ= 1 −δ ⊃ Uδ+ ≡ Uθ= 1 +δ . (1.17) 2 2 2 2 [The values θ = 12 ± δ and θ = 12 − 2δ 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 Yδ− = Y 1 −δ to Yδ+ = Y 1 +δ may represent a jump across compatibility 2 2 conditions. The meaning of the assumption on R∗ R, made in (H.8) = (1.23) below, may be precisely this: to perform a passage to bypass compatibility conditions. (H.4) (complementing (1.6)) ∗ B ∗ eA t : continuous Yδ+ → C([0, T ]; U ); (H.5) (H.6) eAt (1.18) is also a s.c. semigroup on Yδ− and D(A) is dense in Yδ− in the Yδ− topology; (1.19) A : continuous Yδ− → [Yδ+ ] , (1.20) 442 R. TRIGGIANI where the duality [Yδ+ ] of Yδ+ is with respect to the space Y . (H.7) (complementing (1.5a)) B : continuous Uδ− → [Yδ+ ] (1.21) [which is automatically implied by A−1 B : continuous Uδ− → Yδ− (1.22) via assumption (H.6) = (1.20)]; (H.8) (assumption on smoothing observation) R ∈ L(Y ; Z) and R∗ R : continuous Yδ− → Yδ+ , (1.23) which then, by duality, implies R∗ R : continuous [Yδ+ ] → [Yδ− ] . (1.24) Remark 1.3. As already noted, in the applications [in the subsequent Section 5 as well as in [LT.3], [CL.1], and [T.1]] to P.D.E.’s with boundary control, the spaces Yδ− and Yδ+ are Sobolev spaces (which may coincide with domains of appropriate fractional powers of the basic diﬀerential operator) ∗ invariant under the action of the semigroups eAt and eA t . More insight on the impact of the smoothing assumption R∗ R ∈ L(Yδ− ; Yδ+ ) in (1.23) is provided in the orientation below. Needless to say, for the class of boundary control problems for P.D.E.’s for which this setting is intended [hyperbolic dynamics, platelike equations, see Section 5 below, as well as [LT.3], [CL.1], and [T.1]], all basic assumptions (H.1) through (H.7) on A and B are nothing but intrinsic dynamical properties. Preliminary, direct consequences of the assumptions. Some preliminary, direct consequences of the abstract assumptions, to be invoked in the sequel, are listed next. (C.1) Putting together (H.2) = (1.14) and (H.3) = (1.15) for θ = 12 − 2δ , we obtain via (1.11), (1.12), (1.16), 1 δ Ls L∗s : continuous L2 (s, T ; Y δ− ) → C([s, T ]; Y δ− ) ∩ H 2 − 2 (s, T ; Y ) 1 2 δ 2 ⊂ Y 2 − 2 [s, T ] with a norm which may be made independent of s, i.e., Ls L∗s f C([s,T ];Y δ− ) 2 + Ls L∗s f 1 δ Y 2 − 2 [s,T ] ≤ CT,δ f L2 (s,T ;Y δ− ) 2 (1.25a) , uniformly in s. (1.25b) (C.2) By (strongly) diﬀerentiating (1.2b) in t, we obtain (at least in [D(A∗ )] ) t dLs u (t) = A eA(t−τ ) Bu(τ )dτ + A−1 Bu(t) dt s = A(Ls u)(t) + Bu(t). (1.26a) (1.26b) DIFFERENTIAL RICCATI EQUATIONS 443 Of the possible regularity results which may be given on (1.26), we point out the following one, to be invoked below. By recalling (H.2) = (1.14) on Ls for θ = 12 − δ, and (H.6) = (1.20), (H.7) = (1.21) on A and B, respectively, we obtain via (1.26b) and (1.11). 1 dLs = ALs + B : continuous U 2 −δ [s, T ] → L2 (s, T ; [Yδ+ ] ) dt with a norm which may be made independent of s, i.e., (1.27a) dLs dt u L2 (s,T ;[Yδ+ ] ) ≤ CT,δ u 1 U 2 −δ [s,T ] , uniformly in s. (1.27b) (C.3) The assumption that D(A) is dense in Yδ− , made in (H.5) = (1.19), implies that: Given x ∈ Yδ− , there exists xn ∈ D(A) such that xn −x Y − → δ 0 and At e Axn C([0,T ];[Yδ+ ] ) = AeAt xn C([0,T ];[Yδ+ ] ) ≤ CT xn Yδ− , (1.28) by recalling (H.6) = (1.20) for A, and that eAt is a s.c. semigroup on Yδ− by (H.5) = (1.19). Then, by continuous extension, (1.28) yields eAt A : continuous Yδ− → C([0, T ]; [Yδ+ ] ) (1.29) under assumption (H.5) and (H.6). (C.4) Assumption (H.4) = (1.18), by duality, is equivalent to eAt B : continuous U → C([0, T ]; [Yδ+ ] ). (1.30) Indeed, for u ∈ U and y ∈ [Yδ+ ] we compute ∗ (eAt Bu, y)Y = (u, B ∗ eA t y)U ≤ CT u U y Yδ+ , (1.31) and (1.30) follows then from (1.31). We then see that assumption (1.30), equivalently (H.4) = (1.18), for eAt B = eAt AA−1 B is implied by the property A−1 B : continuous U → Yδ− , (1.32) along with (H.5) and (H.6), since these in turn imply (1.29). 2. Statement of the main results Our starting point is [DLT.1], [LT.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. Theorem 2.1. (Regularity of the optimal pair). Assume hypotheses (i), (H.1) = (1.6); (H.2) = (1.14); (H.3) = (1.15); (H.5) = (1.19); (H.6) = (1.20); (H.7) = (1.21); and (H.8) = (1.23) [actually the weaker requirement R∗ R : Y 1 −δ → Y 1 will suﬃce]. Then, the unique optimal pair 2 2 444 R. TRIGGIANI {u0 ( · , s; y0 ), y 0 ( · , s; y0 )} of the O.C.P. (1.4) for (1.1) [guaranteed by [DLT.1]], satisﬁes the following regularity properties: With y0 ∈ Yδ− deﬁned in (1.16), we have (i) y 0 ( · , s; y0 ) = [Is + Ls L∗s R∗ R]−1 eA( · −s) y 0 (2.1a) 1 ∈ C([s, T ]; Yδ− = Y 1 −δ ) ∩ H 2 −δ (s, T ; Y ) 2 (2.1b) 1 ⊂ Y 2 −δ [s, T ]; (ii) u0 ( · , s; y0 ) = −L∗s R∗ Ry 0 ( · , s; y0 ) ∈ L2 (s, T ; Uδ+ = U 1 +δ ) ∩ H 2 (2.2a) 1 +δ 2 (s, T ; U ) 1 (2.2b) ≡ U 2 +δ [s, T ], a fortiori, u0 ( · , s; y0 ) ∈ C([s, T ]; U ) (2.2c) (see Theorem 3.1.2 below). s. All the above results are with norms which may be made independent of Theorem 2.2. (Regularity of the gain operator B ∗ P (t)). Assume hypotheses (i), (H.1) = (1.6) through (H.8) = (1.23). Then, the operator P (t) deﬁned by P (t)x = T t ∗ (τ −t) eA R∗ Ry 0 (τ, t; x)dτ : continuous Y → C([0, T ]; Y ) (2.3a) (2.3b) satisﬁes the following regularity property B ∗ P (t) : continuous Yδ− → C([0, T ]; U ) (2.4) (see Theorem 3.2.1 below). Remark 2.1. The importance of (2.3) and (2.4) is, of course, that [LT.3] J 0 (y0 ) ≡ J(u0 ( · , s; y0 ), y 0 ( · , s; y0 ) = (P (s)y0 , y0 )Y , and u0 (t, s; y0 ) = −B ∗ P (t)y 0 (t, s; y0 ) ∈ L2 (s, T ; U ), y0 ∈ Y. Theorem 2.3. (D.R.E.). Assume (i), (H.1) = (1.6) through (H.8) = (1.23). Then, the operator P (t) deﬁned by Eqn. (2.3a) satisﬁes (P (t)x, Ay)Y , (P (t)Ax, y)Y ∈ C[0, T ], ∀ x, y ∈ Yδ− , (2.5) in the sense that the above quantities, originally deﬁned on D(A), can be extended on Yδ− ; and, moreover, the following Diﬀerential Riccati Equation DIFFERENTIAL RICCATI EQUATIONS 445 for all 0 ≤ t < T : d (P (t)x, y)Y = −(Rx, Ry)Z − (P (t)x, Ay)Y − (P (t)Ax, y)Y dt +(B ∗ P (t)x, B ∗ P (t)y)U , P (T ) = 0; ∀ x, y ∈ Yδ− ; (2.6) as well as the corresponding Integral Riccati Equation for all 0 ≤ t ≤ T : (P (t)x, y)Y = − T t T t ReA(τ −t) x, ReA(τ −t) y) Z dτ B ∗ P (τ )eA(τ −t) x, B ∗ P (τ )eA(τ −t) y U dτ, x, y ∈ Yδ− (2.7) (see Lemma 4.2.1 and Theorem 4.2.2 below). Orientation. Existence of a unique optimal pair {u0 ( · , s; y0 ), y 0 ( · , s; y0 } and formulas (2.1a), (2.2a) apply to the present situation [LT.3, pp. 890–891], and we seek to go beyond these preliminary results. Now, Eqns. (1.14) for Ls and (1.15) for L∗s show, by (1.11), (1.12), that—in the present setting, —the operators Ls and L∗s do not provide any smoothing in the (Sobolev spaces) Yθ and Uθ , 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; y0 ) with a “regular” initial datum y0 ∈ Yδ− , one needs to perform a critical bounded inversion of the operator [Is + Ls L∗s R∗ R], which describes y 0 ( · , s; y0 ) (see Eqn. (2.1a)) on the smoother space L2 (s, T ; Yδ− ), in fact on its subspace 1 1 Y 2 −δ [s, T ] ≡ L2 (s, T ; Yδ− ) ∩ H 2 −δ (s, T ; Y ), see (1.12). This bounded inversion would, however, be a serious problem, unless Ls L∗s R∗ R could be 1 asserted to be compact on Y 2 −δ [s, T ]. It is to this end that a “minimal” smoothing assumption on R∗ R, such as, e.g., (H.8) = (1.23), is then invoked [but even the weaker requirement R∗ R : continuous Yθ= 1 −δ → Yθ= 1 2 2 would do it, of course]. Once the bounded inversion of [Is + Ls L∗s R∗ R] on 1 Y 2 −δ [s, T ] is performed, then one obtains, along with assumptions (H.5) = (1.19), regularity properties of y 0 ( · , s; y0 ); and hence, via (H.3) = (1.15) applied to the optimality condition, see Eqn. (2.2a) regularity properties of u0 ( · , s; y0 ), for s ﬁxed. Next, however, in order to obtain the regularity property of the gain operator B ∗ P (t) : continuous Yδ− → C([0, T ]; U ) (via [Eqn. (2.3) and Remark 2.1]), we see that we need to reﬁne the preceding result by asserting that, in fact, [Is + Ls L∗s R∗ R] is boundedly invertible on 1 Y 2 −δ [s, T ], uniformly in s. The aforementioned regularity of B ∗ P (t) then justiﬁes the wellposedness 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. 446 R. TRIGGIANI (2) Second, in seeking to derive the Diﬀerential Riccati Equation (2.6) on Yδ− , one encounters the obstacle of performing the bounded inversion of the operator [Is + Ls L∗s R∗ R], this time, however, on the weaker space L2 (s, T ; [Yδ+ ] ); equivalently, by duality, the bounded inversion of [Is + R∗ RLs L∗s ] on the space L2 (s, T ; Yδ+ ). This task would, however, be again a serious problem, unless R∗ RLs L∗s could be asserted to be compact on L2 (s, T ; Yδ+ ). It is at this level that the smoothing assumption R∗ R: continuous Yδ− → Yδ+ in (H.8) = (1.23) is used in full force, as the operator Ls , by assumption (H.2) = (1.14), has a known regularity property only on Yθ , θ < 12 . Accordingly, the bounded inversion of [Is + Ls L∗s R∗ R] on L2 (s, T ; [Yδ+ ] ) is then performed for each s, a result suﬃcient in the derivation of the D.R.E. (2.6) in Section 4. 3. Proofs of theorems 2.1 and 2.2. 3.1 Bounded inversion of [Is + Ls L∗s R∗ R] 1 on the space Y 2 −δ [s, T ], uniformly in s. Proof of Theorem 2.1 The key preliminary result is the following. Theorem 3.1.1. Assume (H.1); (H.2), and (H.3) only for θ < 12 ; (H.5); (H.6); (H.7); (H.8) [though the weaker requirement R∗ R : Y 1 −δ → Y 1 will 2 2 suﬃce]. Then: (i) With reference to the spaces in (1.12) for θ = have the following estimate Ls L∗s R∗ Rf 1 δ Y 2 − 2 [s,T ] ≤ CT,δ f 1 Y 2 −δ [s,T ] , 1 2 − δ and θ = uniformly in s. 1 2 − 2δ , we (3.1.1) 1 (ii) For ﬁxed s, the operator Ls L∗s R∗ R : Y 2 −δ [s, T ] → itself, is compact, and, in fact, {Ls L∗s R∗ R} is a family (in s) of collectively compact op1 erators on Y 2 −δ [0, T ], once extended by zero on [0, s) (in the sense of [An.1, p. 3]). 1 (iii) For f ∈ Y 2 −δ [0, T ], indeed, even f ∈ L2 (0, T ; Yδ− ), the map s → 1 Ls L∗s R∗ Rf is continuous in Y 2 −δ [0, T ]. 1 (iv) The operator [Is + Ls L∗s R∗ R] is boundedly invertible on Y 2 −δ [s, T ], indeed uniformly with respect to s: [Is + Ls L∗s R∗ R]−1 1 L(Y 2 −δ [s,T ]) ≤ CT,δ , uniformly in s. (3.1.2) 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: DIFFERENTIAL RICCATI EQUATIONS Y 1 −δ 2 R∗ R [s, T ] continuous by (1.23) ✲ L2 (s, T ; Yδ+ ) continuous ✲ 447 L2 (s, T ; Y δ− ) 2 injection Ls L∗s Y 1 −δ 2 [s, T ] ✛ ❄ compact injection by (3.1.3) continuous by (1.25) Y 1 − 2δ 2 [s, T ]. In the ﬁrst step, we use (H.8) = (1.23) for R∗ R [but R∗ R: continuous Y 1 −δ → Y 1 would suﬃce]; followed by [the combination of (H.2) = (1.14) 2 2 and (H.3) = (1.15) for θ = 12 − 2δ culminating in] the regularity (C.1) = (1.25); followed in the last step by the 1 δ 1 compact injection Y 2 − 2 [s, T ] → Y 2 −δ [s, T ], (3.1.3) a consequence, via (1.12), of the compact injection Y 1 − δ → Y 1 −δ in (1.10) 2 2 2 and of T < ∞. (ii) A fortiori from the diagram, Ls L∗s R∗ R, extended by zero on [0, s) is a 1 compact operator on Y 2 −δ [0, T ], and the family {Ls L∗s R∗ R} is collectively 1 compact (in s [A.1, p. 4]) on Y 2 −δ [0, T ], by estimate (3.1.1). This means that 1 the union, over 0 ≤ s ≤ s0 , of the image [Ls L∗s R∗ R] (unit ball in Y 2 −δ [0, T ]) 1 is a relatively compact set in Y 2 −δ [0, T ]. 1 1 (iii) Step 1. Let g ∈ U 2 −δ [0, T ] ≡ L2 (0, T ; Uδ− )∩H 2 −δ (0, T ; U ). We shall ﬁrst show that, when Ls g is extended by zero on [0, s), then the map s → Ls g is continuous from [0, T ] to L2 (0, T ; Yδ− ). (3.1.4) In fact, with, say, t > s1 > s, recalling (1.2b), (Ls g)(t) − (Ls1 g)(t) Yδ− s 1 A(t−τ ) = e Bg(τ )dτ Yδ− s A(t−s ) s1 A(s −τ ) 1 1 = e e Bg(τ )dτ Yδ− s (by (1.19)) ≤ CT (Ls g)(s1 ) Yδ− 448 R. TRIGGIANI ≤ CT,δ g 1 U 2 −δ [s,s1 ] → 0 as [s − s1 ] → 0, (3.1.5) where in the last steps we have recalled (1.2b) as well as assumption (H.5) = (1.19) on eAt , and (H.2) = (1.14) with θ = 12 − δ. Thus, by (3.1.5), Ls g − Ls1 g C([s1 ,T ];Yδ− ) → 0 as [s1 − s] → 0, (3.1.6a) as well as Ls g − Ls1 g C([s,s1 ];Yδ− ) = Ls g C([s,s1 ];Yδ− ) → 0 as [s − s1 ] → 0. (3.1.6b) Then, (3.1.6a) and (3.1.6b) a fortiori imply (3.1.4). Step 2. Next, let f ∈ L2 (0, T ; Yδ− ). We then show that 1 the map s → L∗ f is continuous from [0, T ] to U 2 −δ [0, T ]. (3.1.7) In fact, the deﬁnition (1.8) implies, still with s1 > s, ∗ (L f − L∗ f 1 s s1 U 2 −δ [0,T ] (by (1.15b)) = L∗s f 1 U 2 −δ [s,s1 ] ≤ CT,δ f L2 (s,s1 ;Yδ− ) → 0 as [s − s1 ] → 0, (3.1.8) after using (H.3) = (1.15b), and (3.1.7) is proved. 1 Step 3. Next, with g = L∗s R∗ Rf ∈ U 2 −δ [0, T ] (conservatively) with f ∈ L2 (0, T ; Yδ− ), via (H.8) = (1.23) and (H.3) = (1.15), we recall (1.26b) and write d(Ls L∗s R∗ Rf ) d(Ls g) = (3.1.9) = ALs L∗s R∗ Rf + BL∗s R∗ Rf dt dt = ALs L∗ R∗ Rf + BL∗s R∗ Rf, (3.1.10) since, by the deﬁnitions (1.2b) and (1.8), we have readily Ls L∗s = Ls L∗ . With reference to (3.1.10), and with f ∈ L2 (0, T ; Yδ− ), we then have that the map s → ALs L∗ R∗ Rf continuous in L2 (0, T ; [Yδ+ ] ), (3.1.11) L∗ R∗ Rf by combining (3.1.4) with g = and (H.6) = (1.20) on A. Also, again − ∗ with f ∈ L2 (0, T ; Yδ ), hence R Rf ∈ L2 (0, T ; Yδ− ) a fortiori the map s → BL∗s R∗ Rf continuous in L2 (0, T ; [Yδ+ ] ), (3.1.12) by combining (3.1.7) and (H.7) = (1.21) on B. Using (3.1.11) and (3.1.12) in (3.1.10) we conclude that: If f ∈ L2 (0, T ; Yδ− ), then: d the map s → (Ls L∗s R∗ Rf ) continuous in L2 (0, T ; [Yδ+ ] ), dt as well as the map s → Ls L∗s R∗ Rf continuous inL2 (0, T ; Yδ− ), by (3.1.4). (3.1.13) (3.1.14) DIFFERENTIAL RICCATI EQUATIONS 449 Step 4. Hence, by interpolation between (3.1.13) and (3.1.14), we obtain, recalling the interpolation property in (1.10), θ= 12 −δ the map s → Dt Ls L∗s R∗ Rf is continuous in L2 0, T ; [Yδ− , [Yδ+ ] ]θ= 1 −δ = L2 (0, T ; Y ), 2 (3.1.15) via [LM, pp. 15, 23]. Then (3.1.14) and (3.1.5) together mean: If f ∈ L2 (0, T ; Yδ− ), then: 1 the map s → Ls L∗s R∗ Rf continuous inY 2 −δ [0, T ], (3.1.16) which proves the desired part (iii). (iv) We ﬁrst show that [Is + Ls L∗s R∗ R] is boundedly invertible on the set 1 Y 2 −δ [s, T ] for each s ﬁxed 1 [Is + Ls L∗s R∗ R]−1 ∈ L Y 2 −δ [s, T ] . (3.1.17) 1 Indeed, since Ls L∗s R∗ R is a compact operator on Y 2 −δ [s, T ] by part (ii), then a (necessary and) suﬃcient condition for (3.1.17) to hold true is that λ = 1 1 be not an eigenvalue of Ls L∗s R∗ R on Y 2 −δ [s, T ], which is certainly the case, for otherwise λ = 1 would also be an eigenvalue of Ls L∗s R∗ R on L2 (s, T ; Y ), thus contradicting [LT.3, p. 891], which asserts that [Is + Ls L∗s R∗ R]−1 ∈ L(L2 (s, T ; Y )). Thus, (3.1.7) is proved. Finally, to assert the uniform estimate (3.1.2), we simply invoke [LT.3, 1 δ Lemma 3.12] with Z1 ≡ Y 2 − 2 [0, T ] with compact injection into Z0 ≡ 1 Y 2 −δ [0, T ], see (3.1.3): 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 R∗ R: continuous Y 1 −δ → Y 1 would suﬃce. 2 2 Remark 3.1.2. With reference to (2.1a), setting Γs = [Is + Ls L∗s R∗ R], we obtain −1 −1 −1 Γ−1 s − Γs1 = Γs [Γs1 − Γs ]Γs1 (3.1.18) by the second resolvent equation. Hence, estimate (3.1.2) of Theorem 3.1.1 1 applied to (3.1.18) readily implies that, for each f ∈ Y 2 −δ [s, T ] ﬁxed, 1 −δ 2 [0, T ], the map s → Γ−1 s f is continuous in Y (3.1.19) a result which can be applied to y 0 ( · , s; y0 ) via (2.1a). See also Remark 3.1.3 below. As a corollary of Theorem 3.1.1, we shall prove Theorem 2.1 on the regularity of the optimal pair. 450 R. TRIGGIANI Theorem 3.1.2. Assume the hypotheses of Theorem 3.1.1: (H.1) through (H.3); (H.5) through (H.8). Then, the optimal pair {u0 ( · , s; y0 ), y 0 ( · , s; y0 )} guaranteed by [LT.3], satisﬁes the following regularity properties for y0 ∈ Yδ− : (i)y 0 ( · , s; y ) ≡ Φ( · , s)y ∈ C([s, T ]; Y − ) ∩ H 12 −δ (s, T ; Y ) 0 0 ⊂ Y 1 −δ 2 δ [s, T ], (3.1.20a) with norms which may be made independent of s: Φ( · , s) L(C([s,T ];Yδ− );Yδ− ) + Φ( · , s) 1 L(Y 2 −δ [s,T ];Yδ− ) ≤ CT,δ , uniformly in s; (3.1.20b) (ii) still for y0 ∈ Yδ− , 1 u0 ( · , s; y0 ) ∈ U 2 +δ [s, T ], (3.1.21a) with a norm which may be made independent of s, u0 ( · , s; y0 ) 1 L(U 2 +δ [s,T ];Yδ− ) ≤ CT,δ , uniformly in s. (3.1.21b) Proof. Step 1. We recall Eqn. (2.1a) and (3.1.20a), y 0 ( · , s; y0 ) ≡ Φ( · , s)y0 = [Is + Ls L∗s R∗ R]−1 [eA( · −s) y0 ]. (3.1.22) With y0 ∈ Yδ− , we apply (H.5) = (1.19), which gives that eA( · −s) is a s.c. semigroup on Yδ− , and ﬁnally invoke Theorem 3.1.1(iv), Eqn. (3.1.2), to 1 obtain (3.1.20b) for L(Y 2 −δ [s, T ]; Yδ− ). Step 2. We now recall the optimality condition u0 ( · , s; y0 ) = −L∗s R∗ Ry 0 ( · , s; y0 ), (3.1.23) from Eqn. (2.2a), to which we apply the diagram Y 1 −δ 2 [s, T ] R∗ R −→ by (1.23) L2 (s, T ; Yδ+ ) L∗s −→ by (1.15b) 1 U 2 +δ [s, T ], (3.1.24) 1 with y 0 ( · , s; y0 ) ∈ Y 2 −δ [s, T ] uniformly in s by (3.1.20b) just proved in Step 1. All the maps in the diagram are uniform with respect to s, the last one, L∗s , by (H.3) = (1.15b) with θ = 12 + δ. Then the above diagram and (3.1.23) prove part (ii), i.e., (3.1.21). Step 3. It remains to complete the proof of part (i), by showing the statement for C([s, T ]; Yδ− ). To this end, we use the optimal dynamics y 0 ( · , s; y0 ) = eA( · −s) y0 + Ls u0 ( · , s; y0 ), (3.1.25) with y0 ∈ Yδ− , hence eA( · −s) y0 ∈ C([s, T ]; Yδ− ) by (H.5) = (1.19), and ﬁnally Ls u0 ( · , s; y0 ) ∈ C([s, T ]; Yδ− ) by (H.2) = (1.14a) with θ = 12 − δ, 1 since a fortiori from part (ii), u0 ( · , s; y0 ) ∈ U 2 −δ [s, T ]. Then, y 0 ( · , s; y0 ) ∈ DIFFERENTIAL RICCATI EQUATIONS 451 C([s, T ]; Yδ− ) by (3.1.25). Moreover, all results are uniform in s. Theorem 3.1.2 is fully proved. Remark 3.1.3. As we have seen, e.g., in the proof of [LT.3, Lemma 2.1], continuity of Φ(t, s)x in the ﬁrst variable, as established by (3.1.20a) t → Φ(t, s)x continuous in Yδ− , for x ∈ Yδ− , T ≥ t ≥ s, (3.1.26) for s ﬁxed, combined with the uniform bound obtained in (3.2.20b) Φ(t, s) L(Yδ− ) ≤ CT uniformly in s ≤ t ≤ T (3.1.27) implies continuity of Φ(t, s)x in the second variable s → Φ(t, s)x continuous in Yδ− , for x ∈ Yδ− , s ≤ t. (3.1.28) 3.2 Proof of theorem 2.2 We restate Theorem 2.2 as Theorem 3.2.1. Assume hypotheses (H.1) = (1.6) through (H.8) = (1.23). Then, the operator P (t) deﬁned by Eqn. (2.3) satisﬁes B ∗ P (t) : continuous Yδ− → C([0, T ]; U ); max B ∗ P (t)x 0≤t≤T U ≤ CT x (3.2.1) Yδ− . Remark 3.2.1. The weaker statement B ∗ P (t) : continuous Yδ− → L∞ (0, T ; U ) (3.2.2) can be immediately proved, by applying (H.4) = (1.18), (H.8) = (1.23) and (3.1.20) of Theorem 3.1.2 (or (3.1.27)), to B ∗ P (t)x = T t We obtain with x ∈ Yδ− : B ∗ P (t)x U ≤ CT ∗ (τ −t) B ∗ eA T t Φ(τ, t)x R∗ RΦ(τ, t)x dτ. Yδ− dτ ≤ CT,δ x (3.2.3) Yδ− , (3.2.4) and (3.2.2) is proved. Proof of Theorem 3.2.1. Let t1 ∈ [0, T ) and let t > t1 . From (3.2.3), we compute after a change of variable, with x ∈ Yδ− : ∗ ∗ B P (t)x − B P (t1 )x = B ∗ T −t 0 −B ∗ T −t1 0 ∗ eA σ R∗ RΦ(t + σ, t)x dσ ∗ eA σ R∗ RΦ(t1 + σ, t1 )x dσ = I1 (t)x − I2 (t)x, (3.2.5) 452 R. TRIGGIANI where, after adding and subtracting, T −t I1 (t)x = 0 ∗ B ∗ eA σ R∗ R[Φ(t + σ, t)x − Φ(t1 + σ, t1 )x]dσ; T −t1 I2 (t)x = T −t (3.2.6) ∗ B ∗ eA σ R∗ RΦ(t1 + σ, t1 )x dσ. (3.2.7) As to I2 (t)x, we apply (H.4) = (1.18), (H.8) = (1.23) and (3.1.21b) of Theorem 3.1.2, or (3.1.27) to obtain I2 (t)x ≤ CT U T −t1 T −t Φ(t1 + σ, t1 )x ≤ CT,δ (t − t1 ) x Yδ− Yδ− dσ → 0 as t ↓ t1 . (3.2.8) As to I1 (t)x, we again apply (H.4) = (1.18) and (H.8) = (1.23) to obtain after adding and subtracting, Φ(t+σ, t1 )x = Φ(t+σ, t)Φ(t, t1 )x [recall [LT.3, Lemma 2.1]: I1 (t)x U ≤ CT ≤ CT + T −t 0 Φ(t + σ, t)x − Φ(t1 + σ, t1 )x T −t 0 dσ Φ(t + σ, t)x − Φ(t + σ, t)Φ(t, t1 )x 0 T −t Yδ− Yδ− dσ (3.2.9) Φ(t + σ, t1 )x − Φ(t1 + σ, t1 )x Yδ− dσ . As to the ﬁrst term on the righthand side of (3.2.9), we compute T −t Φ(t + σ, t)[x − Φ(t, t1 )x] 0 ≤ T −t 0 Φ(t + σ, t) Yδ− L(Yδ− ) ≤ CT,δ T x − Φ(t, t1 )x (by (3.1.27)) dσ Yδ− x − Φ(t, t1 )x → 0 as t ↓ t1 , Yδ− dσ x ∈ Yδ− , (3.2.10) after recalling the uniform bound (3.1.27), i.e., (3.1.21b), where convergence to zero attains because of the continuity property in (3.1.17), or (3.1.26). As to the second term in (3.2.9), the integrand, with [t + σ] − [t1 + σ] = t − t1 , is uniformly continuous and hence arbitrarily small as t − t1 is suﬃciently small. Thus lim t↓t1 T −t 0 Φ(t + σ, t1 )x − Φ(t1 + σ, t1 )x Yδ− dσ = 0. (3.2.11) Using (3.2.10) and (3.2.11) on the righthand side of (3.2.9), then yields lim I1 (t)x = 0, t↓t1 x ∈ Yδ− , (3.2.12) DIFFERENTIAL RICCATI EQUATIONS 453 as desired. Then, (3.2.8) for I2 (t)x and (3.2.12) for I1 (t)x, used in (3.2.5), complete the proof that lim B ∗ P (t)x − B ∗ P (t1 )x t↓t1 U x ∈ Yδ− . = 0, (3.2.13) A similar argument applies if t < t1 , and t ↑ t1 . We then obtain that B ∗ P (t)x ∈ C([0, T ]; Yδ− ), x ∈ Yδ− . (3.2.14) This, along with (3.2.4), shows (3.2.1), as desired. Remark 3.2.1. Recalling the pointwise relationship u0 (t, 0; y0 ) = −B ∗ P (t)y 0 (t, 0; y0 ), y0 ∈ Yδ− , (3.2.15) from Remark 2.1, and applying to it the continuity y 0 (t, 0; y0 ) ∈ C([0, T ]; Yδ− ) via (3.1.20) of Theorem 3.1.2, as well as (3.2.1) of Theorem 3.2.1, we reobtain that u0 (t, 0; y0 ) ∈ C([0, T ]; U ), a result afortiori contained in (2.2b), or (3.1.21); see (2.2c). 4. Proof of theorem 2.3 4.1. Bounded inversion of [Is + Ls L∗s R∗ R] on the space L2 (s, T ; [Yδ+ ] ). Consequences on Φ(t, s) We begin with the result which will serve our purposes in the sequel. Theorem 4.1.1. Assume (i), (H.1) = (1.6), (H.2) = (1.14), (H.3) = (1.15), and (H.8) = (1.23). Then, for s ﬁxed: (i) the operator R∗ RLs L∗s is compact on L2 (s, T ; Yδ+ ). (ii) The operator [Is + R∗ RLs L∗s ] is boundedly invertible on L2 (s, T ; Yδ+ ): [Is + R∗ RLs L∗s ]−1 ∈ L(L2 (s, T ; Yδ+ )). (4.1.1) (iii) The operator [Is +Ls L∗s R∗ R] is boundedly invertible on L2 (s, T ; [Yδ+ ] ): [Is + Ls L∗s R∗ R]−1 ∈ L(L2 (s, T ; [Yδ+ ] ). (4.1.2) Proof. (i) The proof of part (i) is a consequence of the following diagram 454 R. TRIGGIANI L2 (s, T ; Yδ+ ) L∗s injection ✲ U 12 +δ [s, T ] ✲ U 12 −δ [s, T ] continuous by (1.15) compact by (4.1.3) continuous by (1.14) Ls ✛ L2 (s, T ; Yδ+ ) Y continuous by (1.23) The above diagram uses (H.3) = (1.15) for θ = the 1 ❄ R∗ R 1 2 1 −δ 2 [s, T ]. + δ on L∗s ; followed by 1 injection U 2 +δ [s, T ] → U 2 −δ [s, T ] compact, (4.1.3) as a consequence of the compactness property U 1 +δ → U 1 −δ of the injection 2 2 contained in (1.10) and of T < ∞; followed by (H.2) = (1.14) for θ = 12 − δ on Ls ; followed by (H.8) = (1.23) on R∗ R. Thus, as a result, R∗ RLs L∗s is a compact operator on L2 (s, T ; Yδ+ ), as desired. (ii) Since R∗ RLs L∗s is compact on L2 (s, T ; Yδ+ ) 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∗ RLs L∗s on L2 (s, T ; Yδ+ ), which is certainly the case, for otherwise λ = 1 would also be an eigenvalue of R∗ RLs L∗s on L2 (s, T ; Y ), thus contradicting [LT.3, p. 891], which asserts that [Is + R∗ RLs L∗s ]−1 ∈ L(L2 (s, T ; Y )). Thus, (4.1.1) is proved. (iii) Part (iii), Eqn. (4.1.2), follows from part (ii), Eqn. (4.1.1) by duality. 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 Corollary 4.1.2. Assume preliminarily (i), (H.1) = (1.6) through (H.3) = (1.15) and (H.8) = (1.23). (a) Assume (H.5) = (1.19) on eAt , and (H.6) = (1.20). Then Φ( · , s)A : continuous Yδ− → L2 (s, T ; [Yδ+ ] ), (4.1.4) (ii) Assume (H.4) = (1.18). Then Φ(t, s)B : continuous U → L2 (s, T ; [Yδ+ ] ). (4.1.5) (iii) Assume (H.4) = (H.7). Then Φ(t, s)[A − BB ∗ P (s)] : continuous Yδ− → L2 (s, T ; [Yδ+ ] ). (4.1.6) DIFFERENTIAL RICCATI EQUATIONS 455 Proof. (i) Recalling (3.1.22), we have Φ( · , s)Ax = [Is + Ls L∗s R∗ R]−1 [eA( · −s) Ax], (4.1.7) where, for x ∈ Yδ− we have eA( · −s) Ax ∈ C([s, T ]; [Yδ+ ] ) by (C.3) = (1.29), i.e., by (H.5) and (H.6) continuously in x ∈ Yδ− . Finally, we invoke (4.1.2) of Theorem 4.1.1 and obtain Φ( · , s)Ax ∈ L2 (s, T ; [Yδ+ ] ), continuously in x ∈ Yδ− , from (4.1.7), as desired. (ii) Similarly, we have for u ∈ U , via (3.1.22), Φ( · , s)Bu = [Is + Ls L∗s R∗ R]−1 [eA( · −s) Bu], (4.1.8) where now eA( · −s) Bu ∈ C([s, T ]; [Yδ+ ] ) by consequence (C.4) = (1.30), i.e., duality on (H.4) = (1.18). Again, (4.1.2) then yields Φ( · , s)Bu ∈ L2 (s, T ; [Yδ+ ] ), continuously in u ∈ U , from (4.1.8), as desired. (iii) Regularity (4.1.6) is an immediate consequence of (4.1.4) and (4.1.5), via (3.2.1) of Theorem 3.2.1. Corollary 4.1.3. Assume (H.1) through (H.8). Then, (i) for x ∈ Yδ− , s ≤ t ≤ T, dΦ(t, s)x = [A − BB ∗ P (t)]Φ(t, s)x ∈ C([s, T ]; [Yδ+ ] ), dt (ii) for x ∈ Yδ− , x ∈ Yδ− ; (4.1.9) s ≤ t ≤ T, dΦ(t, s)x = −Φ(t, s)[A − BB ∗ P (t)]x ∈ L2 (s, T ; [Yδ+ ] ), ds x ∈ Yδ− . (4.1.10) Proof. (i) Eqn. (4.1.9) is simply the optimal dynamics in diﬀerential form via (3.2.15), and may be obtained by diﬀerentiation on its integral version (3.1.25), i.e., Φ(t, s)x = eA(t−s) x − t s eA(t−τ ) BB ∗ P (τ )Φ(τ, s)x dτ, (4.1.11) where the regularity in C([s, T ]; [Yδ+ ] ) in (4.1.9) is obtained by use of assumptions (H.6) = (1.20) and (H.7) = (1.21) on A and B, respectively, combined with the regularity properties of Φ(t, s)x ∈ C([s, T ]; Yδ− ) in (3.1.20) and (3.2.1) on B ∗ P (t). (ii) One way to derive (4.1.10) [in line with [LT.3] is to start from (3.1.22) rewritten as Φ(t, s)x + {Ls L∗s R∗ RΦ( · , s)x}(t) = eA(t−s) x, x ∈ Yδ− , (4.1.12a) or, explicitly via (1.2b) and (1.8) as Φ(t, s)x + t s A(t−τ ) e = eA(t−s) x, BB ∗ T τ ∗ (σ−τ ) eA R∗ RΦ(σ, s)xdσdτ (4.1.12b) 456 R. TRIGGIANI take the distributional derivative in s, to obtain [Is + Ls L∗s R∗ R] dΦ( · , s)x = −eA( ds · −s) [A − BB ∗ P (s)]x ∈ C([s, T ]; [Yδ+ ] ) ⊂ L2 (s, T ; [Yδ+ ] ), x ∈ Yδ− , (4.1.13) after invoking the deﬁnition of P (s) from Eqn. (2.3a). The regularity displayed at the righthand side of (4.1.13) is a consequence of (3.2.1) for B ∗ P (t); (C.3) = (1.28) for eAt A; (C.4) = (1.30) for eAt B. Then, applying (4.1.2) of Theorem 4.1.1 on (4.1.13) yields dΦ( , s)x ds = −[Is + Ls L∗s R∗ R]−1 eA( · −s) [A − BB ∗ P (s)]x (by (3.1.22)) = −Φ( · , s)[A − BB ∗ P (s)]x ∈ L2 (s, T ; [Yδ+ ] ), x ∈ Yδ− , (4.1.14) and (4.1.10) is proved. Alternatively, writing Φ(t, τ )x = Φ(t, s)Φ(s, τ )x, τ ≤ s ≤ t, x ∈ Yδ− , (4.1.15) by the evolution property of [LT.3, Lemma 2.1] or [DLT.1], we diﬀerentiate both sides of (4.1.15) in s, e.g., as a distributional derivative, obtaining 0= dΦ(t, τ )x dΦ(t, s) dΦ(s, τ )x = Φ(s, τ )x + Φ(t, s) , ds ds ds (4.1.16) or using (4.1.6) and (3.1.26) or (3.1.20a) dΦ(t, s) Φ(s, τ )x ds = −Φ(t, s)[A − BB ∗ P (s)]Φ(s, τ )x ∈ L2 (s, T ; [Yδ+ ] ), in t x ∈ Yδ− , (4.1.17) recalling the regularity of (4.1.6) combined with that of Φ(t, s) on Yδ− given by (3.1.20a). Since (4.1.17) is valid for all τ ≤ s, setting τ = s yields (4.1.10), as desired. 4.2 Derivation of the differential and integral riccati equations Lemma 4.2.1. Assume (i), (H.1) = (1.6) through (H.3) = (1.15), (H.5) = (1.19) through (H.8) = (1.23). Then, with reference to the nonnegative, selfadjoint operator P (t) ∈ L(Y ) deﬁned by Eqn. (2.3a), we have (i) A∗ P (t) : continuous Yδ− → C([0, T ]; [Yδ− ] ), (4.2.1) (ii) so that, for x, y ∈ Yδ− , we have the duality pairings (P (t)x, Ay)Y , (P (t)Ax, y)Y ∈ C[0, T ]; (4.2.2) DIFFERENTIAL RICCATI EQUATIONS 457 Proof. (i) We examine A∗ P (t)x = T t ∗ (τ −t) A∗ eA R∗ RΦ(τ, t)x dτ (4.2.3) for x ∈ Yδ− . Then, by (3.1.20), or (3.1.26), and by (H.8) = (1.23), we have R∗ RΦ(τ, t)x ∈ C([t, T ]; Yδ+ ), x ∈ Yδ− , (4.2.4) and by duality on (C.3) = (1.29), we have ∗ A∗ eA t : continuous Yδ+ → C([0, T ]; [Yδ− ] ). (4.2.5) Using (4.2.4) and (4.2.5) in (4.2.3) yields A∗ P (t)x ∈ [Yδ− ] . Actually, since by (3.1.28) and (4.2.5), respectively, t → Φ(τ, t)x continuous in Yδ− , for x ∈ Yδ− ; ∗ (τ −t) t → A∗ eA (4.2.6) y continuous in [Yδ− ] for y ∈ Yδ+ , (4.2.7) then, in fact, A∗ P (t)x ∈ C[0, T ]; [Yδ− ] ), x ∈ Yδ− , 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 selfadjoint on Y . Remark 4.2.1. Notice that we would have: P (t) : continuous Yδ− → C([0, T ]; Yδ+ ), (4.2.8) if and only if A is an isomorphism Yδ− onto [Yδ+ ], (4.2.9) a property for A which is generally false; see illustrations below. 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, selfadjoint operator P (t) deﬁned by (2.3a) satisﬁes the following Diﬀerential Riccati Equation for all 0 ≤ t < T : d (P (t)x, y)Y = dt −(Rx, Ry)Z − (P (t)x, Ay)Y − (P (t)Ax, y)Y +(B ∗ P (t)x, B ∗ P (t)y)U , ∀ x, y ∈ Yδ− (4.2.10) P (T ) = 0 Proof. Let x, y ∈ Yδ− . We diﬀerentiate in t the expression (P (t)x, y)Y = T t (R∗ RΦ(τ, t)x, eA(τ −t) y)Y dτ (4.2.11) 458 R. TRIGGIANI obtained from Eqn. (2.3a), thus obtaining d (P (t)x, y)Y dt = −(R∗ Rx, y)Y − − (by (4.1.10)) T t T t R∗ R ∂Φ(τ, t)x A(τ −t) y ,e ∂t R∗ RΦ(τ, t)x, eA(τ −t) Ay Y Y dτ dτ (4.2.12) = −(R∗ Rx, y)Y − T t R∗ RΦ(τ, t)[A − BB ∗ P (t)]x, eA(τ −t) y −(P (t)x, Ay)Y , x, y ∈ Yδ− , Y dτ (4.2.13) after substituting (4.1.10) in the second term on the righthand side of (4.2.12), as well as substituting (4.2.11) [with y replaced by Ay] in the third term on the righthand side of (4.2.12). We notice explicitly that each term in (4.2.12), or (4.2.13), is welldeﬁned at each t: the last term by (4.2.2), and the critical second term on the righthand side of (4.2.12), or (4.2.13), by the regularity in (4.1.10) for dφ(τ,t)x , combined with R∗ R: continuous dt − + [Yδ ] → [Yδ ] by (1.24), as well as with eA(τ −t) y ∈ C([t, T ]; Yδ− ) for y ∈ Yδ− , by (H.5) = (1.19). Thus, invoking again (4.2.11) on the second term on the righthand side of (4.2.13), we obtain d (P (t)x, y)Y dt = −(R∗ Rx, y)Y −(P (t)[A − BB ∗ P (t)]x, y)Y − (P (t)x, Ay)Y = −(R∗ Rx, y)Y − (P (t)Ax, y)Y − (P (t)x, Ay)Y +(B ∗ P (t)x, B ∗ P (t)y)U , x, y ∈ Yδ− , (4.2.14) each term being well deﬁned, by virtue of (3.2.1) and (4.2.2). Then, (4.2.14) proves (4.2.10), as desired. As a consequence of Theorem 4.2.2, we obtain that the operator P (t) satisﬁes the Integral Riccati Equation as well. Theorem 4.2.3. Assume (i), (H.1) = (1.6) through (H.8) = (1.23). Then, the nonnegative, selfadjoint operator P (t) of Theorem 4.2.2 satisﬁes DIFFERENTIAL RICCATI EQUATIONS 459 the following Integral Riccati Equation for all 0 ≤ t < T . T = (P (t)x, y)Y t T − t ReA(τ −t) x, ReA(τ −t) y Z dτ B ∗ P (τ )eA(τ −t) x, B ∗ P (τ )eA(τ −t) y U dτ, x, y ∈ Yδ− , (4.2.15) where all terms are well deﬁned by (1.19), (3.2.1), (1.23). Proof. For x, y ∈ Yδ− , we compute d A∗ (τ −t) e P (τ )eA(τ −t) x,y dτ Y d P (τ )eA(τ −t) x, eA(τ −t) y Y dτ ∂ = P (r)eA(τ −t) x, eA(τ −t) y Y ∂r = + P (τ )eA(τ −t) Ax, eA(τ −t) y A(τ −t) + P (τ )e A(τ −t) x, e Ay Y Y (4.2.16) r=τ , where by using the D.R.E. (4.2.10), we have ∂ P (r)eA(τ −t) x, eA(τ −t) y r=τ ∂r ∗ A(τ −t) A(τ −t) = − R Re x, e y − P (τ )eA(τ −t) x, AeA(τ −t) y Y A(τ −t) − P (τ )Ae ∗ x, e A(τ −t) + B P (τ )e A(τ −t) ∗ y Y A(τ −t) x, B P (τ )e y U Y (4.2.17) , x, y ∈ Yδ− . We note explicitly that each term of (4.2.16) and (4.2.17) is well deﬁned, indeed, we have eA(τ −t) x, eA(τ −t) y in C([t, T ]; Yδ− ), for x, y ∈ Yδ− , and hence: P (τ )eA(τ −t) x ∈ C([t, T ]; Yδ+ ), by (4.2.3), (4.2.18) AeA(τ −t) x ∈ C([0, T ]; [Yδ+ ] ), by (1.19), (1.20), (4.2.19) B ∗ P (τ )eA(τ −t) x ∈ C([t, T ]; U ), by (3.2.1), (1.19), (4.2.20) and similarly for y0 . Thus, (4.2.18)(4.2.20) and (1.23) show that each term in (4.2.16) and (4.2.17) is well deﬁned. Inserting (4.2.17) into (4.2.16) results in a cancellation of the last two terms of (4.2.16), hence d A∗ (τ −t) e P (τ )eA(τ −t) x, y = − ReA(τ −t) x, ReA(τ −t) y Y Z dτ + B ∗ P (τ )eA(τ −t) x, B ∗ P (τ )eA(τ −t) y U , x, y ∈ Yδ− . (4.2.21) 460 R. TRIGGIANI Integrating (4.2.21) in τ over [t, T ] and using P (T ) = 0 from the D.R.E. (4.2.10) results in (4.2.15), as desired. 5. Application: kirchoff equation with one boundary control. Regularity theory 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) Secondorder hyperbolic equations with Dirichlet boundary control, deﬁned on a smooth, bounded domain Ω ⊂ Rn , see [LT.3]; here one may take 1 1 Uθ ≡ H θ (Γ); Yθ ≡ H0θ (Ω) × H θ−1 (Ω), 0 ≤ θ ≤ + δ, θ = ; 2 2 U0 = U = L2 (Γ); Y0 = Y = L2 (Ω) × H −1 (Ω). 1 Uδ− = U 1 −δ = H 2 −δ (Γ); 2 1 1 Uδ+ = U 1 +δ = H 2 +δ (Γ); 2 1 Yδ− = Y 1 −δ = H 2 −δ (Ω) × H − 2 −δ (Ω); 2 1 Yδ+ = Y 1 +δ = H02 2 +δ 1 (Ω) × H − 2 +δ (Ω), etc. (ii) nonsymmetric, nondissipative, ﬁrstorder hyperbolic systems with boundary control, see [CL.1]; here one make take similarly deﬁned explicit Sobolev spaces for Uθ , Yθ , 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 secondorder hyperbolic equations of [LT.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 EulerBernoulli 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). DIFFERENTIAL RICCATI EQUATIONS 461 5.1. Problem formulation The dynamics. Let Ω be an open bounded domain in Rn with suﬃciently smooth boundary Γ, say, of class C 2 . The Kirchoﬀ equation is given by wtt − ρ∆wtt + ∆2 w = 0 w(0, · ) = w0 , wt (0, · ) = w1 wΣ = 0 ∆w = u Σ in (0, T ] × Ω ≡ Q; (5.1.1a) in Ω; (5.1.1b) in (0, T ] × Γ ≡ Σ, (5.1.1c) in Σ, (5.1.1d) where ρ > 0 is a constant (proportional to the square of the thickness in the 2dimensional plate model), and where u ∈ L2 (0, T ; L2 (Γ)) ≡ L2 (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 5.3.2 below, the cost functional which we seek to minimize over all u ∈ L2 (s, T ; L2 (Γ)) = L2 (ΣsT ) is taken to be T R J(u, w) = 0 2 w(t) + u(t) wt (t) H 2 (Ω)×H 1 (Ω) 2 L2 (Γ) dt, (5.1.2) with initial data {w0 , w1 } ∈ [H 2 (Ω)∩H01 (Ω)]×H01 (Ω), where the observation operator R ∈ L([H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω)) will be further speciﬁed below in (5.2.3). 5.2. 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. Theorem 5.2.1. (a) With the observation operator R in (5.1.2) only assumed to satisfy R ∈ L([H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω)). (5.2.1) the preliminary theory of [LT.3] applies to the optimal control problem (5.1.1), (5.1.2), with y(t) = w(t) wt (t) ; U = L2 (Γ); Y ≡ [H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω), (5.2.2) and yields a unique optimal pair {u0 ( · , s; y0 ), y 0 ( · , s; y0 )}, y0 = [w0 , w1 ] ∈ Y , satisfying, in particular, the pointwise feedback synthesis, and the optimal cost relation of Remark 2.1, as well as the other properties listed in (2.1a), (2.2a). (b) Assume, in addition to (5.2.1), that R satisﬁes the smoothing assumption R∗ R : continuous Yδ− → Yδ+ , (5.2.3) 462 R. TRIGGIANI R∗ adjoint of R in Y , where 5 3 Yδ− ≡ h ∈ H 2 −δ (Ω) : hΓ = 0 × h ∈ H 2 −δ (Ω) : hΓ = 0 5 (5.2.4a) 3 = H 2 −δ (Ω) ∩ H01 (Ω) × H 2 −δ (Ω) ∩ H01 (Ω) 5 3 Yδ+ ≡ h ∈ H 2 +δ (Ω) : hΓ = ∆hΓ = 0 × H 2 +δ (Ω) ∩ H01 (Ω) (5.2.4b) (5.2.5) (see Section 5.6), and where δ > 0 is an arbitrarily small constant, which is kept ﬁxed throughout. [An additional characterization of Yδ− and Yδ+ will be given below in (5.6.4), (5.6.5)]. Thus, the above choice centered at θ = 52 yields that assumption (5.2.3) jumps from Yδ− to Yδ+ across the new B.C. ∆hΓ = 0. (Refer to the paragraph before Section 5.1.) Then Theorems 2.1, 2.2, and 2.3 of Section 2 hold true, with Yδ− and Yδ+ given by (5.2.4) and (5.2.5), and with 1 Uδ− = H 2 −δ (Γ) 1 Uδ+ = H 2 +δ (Γ) (5.2.6) (see Section 5.6). In particular, explicitly: (b1 ) (regularity of the optimal pair) the optimal pair satisﬁes, for 5 3 y0 = [w0 , w1 ] ∈ H 2 −δ (Ω) ∩ H01 (Ω) × H 2 −δ (Ω) ∩ H01 (Ω) (5.2.7) the following regularity properties: (i) w0 ( · , s; y0 ) wt0 ( · , s; y0 ) = y 0 ( · , s; y0 ) 5 3 ∈ C [s, T ]; H 2 −δ (Ω) ∩ H01 (Ω) × H 2 −δ (Ω) ∩ H01 (Ω) ∩ 1 H 2 −δ (s, T ; [H 2 (Ω) ∩ H01 (Ω) × H01 (Ω)); (5.2.8) (ii) 1 1 u0 ( · , s; y0 ) ∈ L2 s, T ; H 2 +δ (Γ) ∩ H 2 +δ (s, T ; L2 (Γ)) 1 1 ≡ H 2 +δ, 2 +δ (ΣsT ), (5.2.9) ΣsT = [s, T ] × Γ, a fortiori, u0 ( · , s; y0 ) ∈ C([s, T ]; H δ (Γ)). (5.2.10) All the above results hold true uniformly in s, i.e., with norms which may be made independent of s. DIFFERENTIAL RICCATI EQUATIONS 463 (b2 ) (gain operator) The gain operator B ∗ P (t), P (t) deﬁned by (2.3a), satisﬁes the following regularity property (see (2.4)), P11 (t) P12 (t) P21 (t) P22 (t) B ∗ P (t)x = B ∗ = − : x1 x2 ∂ [P21 (t)x1 + P22 (t)x2 ] ∂ν 5 continuous H 2 −δ (Ω) ∩ H01 (Ω) (5.2.11a) (5.2.11b) 3 × H 2 −δ (Ω) ∩ H01 (Ω) → C([0, T ]; L2 (Γ)), (5.2.11c) where the adjoint B ∗ is computed with respect to the space Y , topologized, 1 1 however, as Y ≡ D A 2 × D Aρ4 , see below in (5.4.22). (b3 ) (Diﬀerential Riccati Equation) With A= 0 I A 0 , 1 A = I + ρA 2 −1 A as in (5.4.3) below, we have that P (t) satisﬁes the following D.R.E. d (P (t)x, y)Y = −(Rx, Ry)Y − (P (t)x, Ay)Y − (P (t)Ax, y)Y dt ∂ ∂ [P21 (t)x1 + P22 (t)x2 ], [P21 (t)y1 + P22 (t)y2 ] + ∂ν ∂ν L (Γ) 2 5 3 P (T ) = 0, ∀ x, y ∈ H 2 −δ (Ω) ∩ H01 (Ω) × H 2 −δ (Ω) ∩ H01 (Ω) . (5.2.12) Furthermore, P (t) satisﬁes the corresponding Integral Riccati Equation, as in (2.7), for all such x, y. 5.3. Regularity theory for problem (5.1.1) with u ∈ L2 (Σ) The following wellposedness Theorem 5.3.2 provides the critical regularity result, which justiﬁes as natural the selection of the cost functional J in (5.1.2), and which will permit us to verify assumption (H.1) = (1.6) below. To this end, as we shall see, it is expedient to associate with problem (5.1.1) the following boundaryhomogeneous version (which would be the corresponding adjoint problem (5.5.7) below, if the initial conditions were given at t = T , an inessential modiﬁcation since the problem is timereversible), φtt − ρ∆φtt + ∆2 φ = f in Q; (5.3.1a) φ(0, · ) = φ0 ; φt (0, · ) = φ1 in Ω; (5.3.1b) φΣ ≡ 0; ∆φΣ ≡ 0 (5.3.1c) in Σ. The following two key results are taken from [LT.5]. 464 R. TRIGGIANI Theorem 5.3.1. [LT.5] With reference to (5.3.1), assume {φ0 , φ1 } ∈ V × [H 2 (Ω) ∩ H01 (Ω)]; f ∈ L1 (0, T ; L2 (Ω)); (5.3.2) V = h ∈ H 3 (Ω) : hΓ = ∆hΓ = 0 . (5.3.3) Then, the unique solution of problem (5.3.1) satisﬁes, continuously, {φ, φt } ∈ C([0, T ]; V × [H 2 (Ω) × H01 (Ω)]); (5.3.4a) φtt ∈ L1 (0, T ; H01 (Ω)); or φtt ∈ C([0, T ]; H01 (Ω)) if f ≡ 0, (5.3.4b) and ∂(∆φ) ∈ L2 (0, T ; L2 (Γ)) ≡ L2 (Σ). ∂ν (5.3.5) Proof. The proof of ((5.3.5) of) Theorem 5.3.1 is by P.D.E.’s energy methods and will be indicated in Section 5.10 below. Remark 5.3.1. The trace regularity (5.3.5) is, of course, the key result of Theorem 5.3.1. It does not follow from the optimal interior regularity (5.3.4): indeed, (5.3.5) is “ 12 higher” in Sobolev space regularity on Ω over a formal application of trace theory to φ in (5.3.4). Theorem 5.3.2. [LT.5] With reference to the nonhomogeneous problem (5.1.1), assume {w0 , w1 } ∈ [H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω); u ∈ L2 (Σ). (5.3.6) Then, the unique solution of (5.1.1) satisﬁes {w, wt } ∈ C([0, T ]; [H 2 (Ω) ∩ H01 (Ω)]) × H01 (Ω)), wtt ∈ C([0, T ]; L2 (Ω)) (5.3.7) continuously; i.e., more precisely, {w, wt , wtt } 2 C([0,T ];[H 2 (Ω)∩H01 (Ω)])×H01 (Ω)×L2 (Ω)) ≤ CT u 2 L2 (Σ) + {w0 , w1 } 2 [H 2 (Ω)∩H01 (Ω)]×H01 (Ω) (5.3.8) Proof. The proof of Theorem 5.3.2 is a consequence by transposition of Theorem 5.3.1 and will be given in Section 5.11 below. 5.4. Abstract setting for problem (5.1.1) We follow [LT.5], [LT.6]. To put problems (5.1.1), (5.1.2) into the abstract model (1.1), (1.3), we introduce the following operators and spaces: (i) Ah = ∆2 h; A : D(A) → L2 (Ω); D(A) = {h ∈ H 4 (Ω) : hΓ = ∆hΓ = 0}; (5.4.1) DIFFERENTIAL RICCATI EQUATIONS 1 465 1 A 2 h = −∆h; D(A 2 ) = H 2 (Ω) ∩ H01 (Ω); 1 A = I + ρA 2 −1 1 (5.4.2) A; D(A) = D A 2 . (5.4.3) The operator A in (5.4.1) is positive selfadjoint on L2 (Ω). Furthermore, as usual, we shall freely extend A originally deﬁned in (5.4.1) [while maintaining the same symbol, with no fear of confusion] as A : L2 (Ω) → [D(A∗ )] = [D(A)] , duality with respect to the pivot space L2 (Ω). The following space identiﬁcations are known (with equivalent norms) [Gr.1], [BDDM.1] D(Aθ ) = h ∈ H 4θ (Ω) : hΓ = 0 , 5 1 <θ< ; 8 8 (5.4.4) 5 < θ ≤ 1. (5.4.5) 8 The following specializations thereof will be needed below: 1 1 θ = : D A 4 = H01 (Ω), and for g ∈ H01 (Ω); (5.4.6a) 4 D(Aθ ) = g h ∈ H 4θ (Ω) : hΓ = ∆hΓ = 0 , D A = A 14 g 1 L2 (Ω) = 4 Ω equivalent to the g in turn equivalent to Ω 2 2 1 [g + ρ∇g dΩ 2 = g ≡ g D 2 L2 (Ω) 1 Aρ4 1 2 ∇g dΩ 2 , H01 (Ω) norm, (5.4.6b) 1 1 2 + ρ A 4 g 2 L2 (Ω) = g 1 H0,ρ (Ω) , (5.4.6c) 1 1 (Ω)norm; the latter norm being denoted by D Aρ4 norm or H0,ρ θ= 1 1 : D A 2 = h ∈ H 2 (Ω) : hΓ = 0 = H 2 (Ω) ∩ H01 (Ω), 2 1 (5.4.7a) and for g ∈ D A 2 : g 1 = A 2 g 1 D A2 L2 (Ω) 14 2 A g 1 2 + ρ A 2 g = Ω 1 2 (∆g) dΩ 2 , (5.4.7b) , (5.4.7c) equivalent to L2 (Ω) 1 2 L2 (Ω) = g 1 D Aρ2 1 the latter norm being denoted as D Aρ2 norm; 3 3 θ = , D A 4 = h ∈ H 3 (Ω) : hΓ = ∆hΓ = 0 , 4 (5.4.8a) 466 R. TRIGGIANI 3 and for h ∈ D A 4 : g 3 = A 4 g 3 L2 (Ω) D A4 1 = A 4 ∆g L2 (Ω) = Ω 1 2 ∇(∆g) dΩ by (5.4.2) and (5.4.6b). Problem (5.3.1) can then be rewritten abstractly as 1 1 I + ρA 2 φtt + Aφ = f, or φtt = −Aφ + I + ρA 2 2 −1 , (5.4.8b) f, φ(0) = φ0 , φt (0) = φ1 , (5.4.9) recalling The operator in (5.4.3), rewritten by using twice (5.4.1)–(5.4.3). 1 1 1 R λ, A 2 A 2 = −I + λR λ, A 2 as 1 −A = − I + ρA 2 −1 1 1 A2 A2 = − 1 −1 1 1 1 I− I + ρA 2 A2 ρ ρ 1 1 −1 A2 1 1 =− , + 2 I − 2 I + ρA 2 ρ ρ ρ 1 A : D(A) = D A 2 → L2 (Ω), (5.4.10a) (5.4.10b) 1 being a bounded perturbation of the negative selfadjoint operator A 2 , is the generator of a s.c. cosine operator C(t), with corresponding “sine” operator S(t) = 0t C(τ )dτ , where the maps 1 t → A 4 S(t), C(t), are strongly continuous on L2 (Ω). (5.4.11) Accordingly, the unique solution of problem (5.4.9), or (5.3.1), is given explicitly by: φ(t) = C(t)φ0 + S(t)φ1 + t 0 φt (t) = −AS(t)φ0 + C(t)φ1 + 1 S(t − τ ) I + ρA 2 t 0 −1 1 C(t − τ ) I + ρA 2 f (τ )dτ, −1 (5.4.12) f (τ )dτ, (5.4.13) in appropriate function spaces, depending on {φ0 , φ1 , f }. Moreover, returning to (5.4.3), 1 A is a positive selfadjoint operator on the space D Aρ4 deﬁned by (5.4.6c), with respect to the corresponding inner product. (x, y) 1 D Aρ4 = I + ρA 1 2 x, y L2 (Ω) , x, y ∈ H01 (Ω). (ii) We introduce the Green map G2 by y = G2 v ⇐⇒ ∆2 y = 0 in Ω; yΓ = 0; ∆yΓ = v , (5.4.14a) (5.4.14b) (5.4.15) and by elliptic theory [LM.1, vol. I, p. 188] 5 G2 : H s (Γ) → H s+ 2 (Ω), s ∈ R. (5.4.16) DIFFERENTIAL RICCATI EQUATIONS 467 We have already shown in [LT.4][LT.6], that 1 G2 = −A− 2 D where y = Dv ⇐⇒ {∆y = 0 in Ω; yΓ = v}, (5.4.17) D being the Dirichlet map satisfying the regularity, 1 D : continuous H s (Γ) → H s+ 2 (Ω); in particular, 1 2 D : continuous L2 (Γ) → H (Ω) ⊂ H =D A 1 1 −2 2 1 − 2 8 (5.4.18a) (Ω) (5.4.18b) , > 0; 1 A 8 − 2 D ∈ L(L2 (Γ); L2 (Ω)); D∗ A 8 − 2 ∈ L(L2 (Ω); L2 (Γ)), (5.4.18c) with the property, 1 ∂ (5.4.19) G∗2 Ah = −D∗ A 2 h = − h, h ∈ D(A). ∂ν (iii) By (5.4.15), (5.4.1), problem (5.1.1) can be written abstractly, ﬁrst as 1 I + ρA 2 wtt + A(w − G2 u) = 0 on L2 (Ω), or wtt = −Aw + AG2 u on [D(A)] , (5.4.20) recalling (5.4.3), next as in (1.1); i.e., as yt = Ay + Bu on [D(A∗ )] , 1 1 Y ≡ D A 2 × D Aρ4 y(0) = y0 ∈ Y ; (5.4.21) = [H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω); U = L2 (Γ) (equivalent norms); y= w wt ; A= 0 AG2 u equivalently, Bu = 0 I −A 0 3 (5.4.22) 1 , D(A) = D A 4 × D A 2 → Y ; (5.4.23) : continuous U → [D(A∗ )] (duality w.r.t. Y ), (5.4.24) A−1 B : continuous U → Y, (5.4.25) since, in fact, for u ∈ L2 (Γ), recalling (5.4.23) and (5.4.17), A −1 Bu = 0 −A−1 I 0 0 AG2 u 1 −G2 u 0 = 5 = 1 A− 2 Du 0 A− 2 Du ∈ D A 8 − 2 . ∈Y (5.4.26) 1 It is property (5.4.14) for A that makes the choice of D Aρ4 as the second component space of Y particularly convenient. In fact, with such a choice, 1 1 we have that A in (5.4.23) is skewadjoint on Y = D A 2 × D Aρ4 , i.e., A∗ = −A, and so it generates a s.c. unitary group eAt on Y . 468 R. TRIGGIANI Starting from (5.4.24) we compute B ∗ with respect to Y topologized as 1 1 D A 2 × D Aρ4 , to obtain for v = [v1 , v2 ], recalling (5.4.14): = (AG2 u, v2 ) (Bu, v)Y 1 D Aρ4 = (AG2 u, v2 )L (Ω) 2 = (u, G∗2 Av2 )L2 (Γ) ; (5.4.27) i.e., by virtue of (5.4.19): B∗ v1 v2 ∂v2 : D(A∗ ) = D(A) → U. ∂ν = G∗2 Av2 = − (5.4.28) Eqn. (5.4.22), the solution of problem (5.1.1), or (5.4.21), with initial condition y(s) = y0 = [w0 , w1 ] is written abstractly as w(t, s; y0 ) wt (t, s; y0 (Ls u)(t) = t s A eA(t−τ ) Bu(τ )dτ = 1 2 − I + ρA = w0 w1 = eA(t−s) −1 1 −1 − I + ρA 2 1 A2 1 A t A2 A s S(t − τ )G2 u(τ )dτ C(t − τ )G2 u(τ )dτ (5.4.30) S(t − τ )Du(τ )dτ t s s (5.4.29) t st + (Ls u)(t); C(t − τ )Du(τ )dτ , after recalling, in the last step, (5.4.3) and (5.4.17), where A generates a s.c. group eAt on Y , t ∈ R, which is given by C(t) S(t) −AS(t) C(t) eAt = . (5.4.31) In (5.4.31), C(t) is even and S(t) is odd. By (5.4.28) and (5.4.31), since ∗ A is skewadjoint on Y, A∗ = −A, and so eA t = e−At , we compute with 1 1 x = [x1 , x2 ] ∈ Y = D A 2 × D Aρ4 : ∗ B ∗ eA t x = G∗2 A[AS(t)x1 + C(t)x2 ] 1 1 1 −1 = G∗2 AA 2 C(t)A− 2 x2 + S(t) I + ρA 2 (5.4.32) 1 A 2 x1 (5.4.33) ∂ ∆φ(t; φ0 , φ1 ), (5.4.34) ∂ν recalling (5.4.28), (5.4.2), and (5.4.12), where φ(t; φ0 , φ1 ) solves problem (5.3.1) with f ≡ 0 and 1 (by (5.4.28)) = G∗2 AA 2 [C(t)φ0 + S(t)φ1 ] = − 1 3 1 1 φ0 = A− 2 x2 ∈ D A 4 , for x2 ∈ D A 4 = D Aρ4 ; 1 φ1 = I + ρA 2 −1 1 1 A 2 x1 ∈ D A 2 1 for x1 ∈ D A 2 . (5.4.35) (5.4.36) DIFFERENTIAL RICCATI EQUATIONS 469 5.5. Verification of assumption (H.1) = (1.6) From (5.4.34)–(5.4.36), we see that (H.1) = (1.6) holds true T ∗ A∗ t 2 B e x dt ≤ CT x 2Y , U x ∈ Y, 0 (5.5.1) if and only if problem (5.3.1) with f ≡ 0 satisﬁes T ∂∆φ 2 dΣ ≤ CT {φ0 , φ1 } 2 3 (5.5.2) 1 , D(A 4 )×D(A 2 ) ∂ν Γ 0 which is precisely the trace regularity result, guaranteed by Theorem 5.3.1, Eqn. (5.3.5). Then, according to duality [LT.2][LT.6], estimate (5.5.1), i.e., (5.5.2), is, in turn, equivalent to the following property that t w(t; 0, 0) w t (t; 0, 0) s : continuous L2 (s, T ; L2 (Γ)) (Ls u)(t) = 1 eA(t−τ ) Bu(τ )dτ = 1 (5.5.3a) → C [s, T ]; D A 2 × D A 4 ≡ [H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω) (5.5.3b) uniformly in s, where w0 = w1 = 0 in problem (5.1.1). This is precisely conclusion (5.3.7) of Theorem 5.3.2, the additional statement of uniformity in s being an immediate consequence of formula (5.5.3a) for Ls (via a change of variable). Moreover, recalling (5.4.33), we have, by duality on (5.5.3) with v = [v1 , v2 ]: (L∗s v)(t) = B ∗ T t ∗ (τ −t) eA 1 (by (5.4.33)) = G∗2 AA 2 + T t T t v(τ )dτ, s ≤ τ ≤ T; 1 S(τ − t) I + ρA 2 1 C(τ − t)A− 2 v2 (τ )dτ −1 (5.5.4) 1 A 2 v1 (τ )dτ (5.5.5) 1 1 : continuous L1 s, T ; D A 2 × D A 4 → L2 (s, T ; L2 (Γ)) (5.5.6) uniformly in s. Now, let ψ(t; h) be the solution of the (adjoint) problem ψtt − ρ∆ψtt + ∆2 ψ = f in Q; (5.5.7a) ψ(T, · ) = 0; ψt (T ; · ) = 0 in Ω; (5.5.7b) ψΣ ≡ ∆ψΣ ≡ 0 (5.5.7c) rewritten abstractly via (5.4.3) as ψtt = −Aψ + h; 1 h = I + ρA 2 −1 in Σ, f; ψ(T ) = ψt (T ) = 0, (5.5.8) 470 R. TRIGGIANI hence given explicitly by t ψ(t; h) = T S(t − τ )h(τ )dτ ; ψt (t; h) = t T C(t − τ )h(τ )dτ. (5.5.9) Then, recalling (5.4.19) and (5.4.2), we see via (5.5.9) that the two terms in (5.5.5) can be rewritten, with 1 1 v = [v1 , v2 ] ∈ L1 0, T ; D A 2 × D A 4 as (L∗s v)(t) t = ∂∆ ∂ν = ∂∆ψ(t; h1 ) ∂∆ψt (t; h2 ) − ∂ν ∂ν T S(t − τ )h1 (τ )dτ − t T , C(t − τ )h2 (τ )dτ (5.5.10) 1 1 : continuous v = [v1 , v2 ] ∈ L1 0, T ; D A 2 × D A 4 → L2 (0, T ; L2 (Γ)). h1 = 1 I + ρA 2 −1 1 1 (5.5.11) 1 A 2 v1 ∈ L1 0, T ; D A 2 3 h2 = A− 2 v2 ∈ L1 0, T ; D A 4 ; . (5.5.12) (5.5.13) Notice that regularity (5.5.11) of the normal trace ∂∆ ∂ν ψ(t; h1 ), for ψ(t; h1 ) solution of (5.5.8) due to h = h1 given by (5.5.12), is precisely conclusion (5.3.5) of Theorem 5.3.1 for the timereversed 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 [LT.5] is by energy (P.D.E.’s)methods. Instead, regularity (5.5.9) for the normal trace ∂∆ ∂ν ψt (t; h2 ) for the time derivative ψt of the solution ψ(t; h2 ) of problem (5.5.8) due to h = h2 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 righthand side, nonhomogeneous term]. Thus, assumption (H.1) = (1.6) is veriﬁed. 5.6. Selection of spaces Uθ and Yθ IN (1.10) We select the spaces in (1.10) to be the following Sobolev spaces Uθ = H θ (Γ); U0 = U = L2 (Γ), 1 θ 0≤θ≤ 1 θ 1 1 + δ, θ = ; 2 2 Yθ = D A 2 + 4 × D A 4 + 4 ≡ D(Aθ ), 1 1 (5.6.1) Y0 = Y = D A 2 × D A 4 , (5.6.2) DIFFERENTIAL RICCATI EQUATIONS 471 where A is as in (5.4.23), in particular the critical spaces for θ = Uδ− = U 1 −δ = H 2 1 −δ 2 Uδ+ = U 1 +δ = H (Γ); 1 Yδ− = Y 1 −δ = D A 2 −δ = D A 2 5 1 +δ 2 2 5 δ − 8 4 (Γ) × D A8−4 3 = H 2 −δ (Ω) ∩ H01 (Ω) × H 2 −δ (Ω) ∩ H01 (Ω) Yδ+ = Y 1 +δ = D A 1 +δ 2 2 =D A 5 + 4δ 8 5 ± δ: (5.6.3) δ 3 1 2 ×D A (5.6.4) 3 + 4δ 8 3 = h ∈ H 2 +δ (Ω) : hΓ = ∆hΓ = 0 × H 2 +δ (Ω) ∩ H01 (Ω) , (5.6.5) recalling (5.4.4) and, respectively, (5.4.5), with equivalent 1 norms. 1 The spaces − + 2 [Yδ ] and [Yδ ] , duality with respect to Y = D A × D A 4 , are given by 3 δ 1 δ [Yδ− ] = D A 8 + 4 × D A 8 + 4 = 3 1 H 2 +δ (Ω) ∩ H01 (Ω) × H02 (5.6.6) H 2 −δ (Ω) ∩ H01 (Ω) × H 2 −δ (Ω). (5.6.7) +δ (Ω) 3 δ 1 δ [Yδ+ ] = D A 8 − 4 × D A 8 − 4 = 3 1 Thus, by (5.6.4) and (5.6.7) we verify the interpolation property 1 1 [Yδ− , [Yδ+ ] ]θ= 1 −δ = Y = D A 2 × D A 4 , 2 as required in (1.10), since 5 8 − δ 4 (1 − θ) + 3 8 δ 4 − (5.6.8) 1 2 1 2 θ = for the ﬁrst component; and 38 − 4δ (1 − θ) + 18 − 4δ θ = 14 for θ = − δ, for the second component space. Moreover, the injections Uθ1 /→ Uθ2 , Yθ1 /→ Yθ2 are compact, 0 ≤ θ2 < θ1 ≤ 12 + δ, as required in (1.10), since Ω is a bounded domain. Thus, the spaces in (1.11), (1.12) are in the present case as follows for 0 ≤ θ ≤ 12 + δ, θ = 12 : U θ [s, T ] = H θ,θ (ΣsT ) = L2 s, T ; H θ (Γ)) ∩ H θ (s, T ; L2 (Γ) 1 θ 1 θ 1 1 Y θ [s, T ] = L2 s, T ; D A 2 + 4 × D A 4 + 4 ∩ H θ s, T ; D A 2 × D A 4 = L2 s, T ; D Aθ 1 θ ∩ H θ (s, T ; Y ) 1 θ D(Aθ ) = D A 2 + 4 × D A 4 + 4 . (5.6.9) (5.6.10) (5.6.11) (5.6.12) 472 R. TRIGGIANI 5.7. Verification of assumption (H.2) = (1.14) The following regularity result is critical in verifying assumption (H.2) = (1.14). It is the main new P.D.E. result of this paper. Theorem 5.7.1. With reference to the nonhomogeneous problem (5.1.1), assume w0 ∈ H 3 (Ω) ∩ H01 (Ω), w1 ∈ H 2 (Ω) ∩ H01 (Ω); (5.7.2) with the compatibility relations w0 Γ = 0 and ∆w0 Γ = u(0); 1 u ∈ C [0, T ]; H 2 (Γ) ∩ H 1 (0, T ; L2 (Γ)) (5.7.3) [(5.7.2) is a fortiori guaranteed, if u ∈ H 1,1 (Σ), by [LM.1, I, Thm. 3.1, p. 19]. Then, the unique solution to problem (5.1.1) satisﬁes {w,wt , wtt } ∈ C [0, T ]; [H 3 (Ω) ∩ H01 (Ω)] × [H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω) , (5.7.4) continuously. Proof. The proof of Theorem 5.7.1 will be given in Section 5.12 below. Corollary 5.7.2. With reference to the nonhomogeneous problem (5.1.1), assume w0 = w1 = 0, and for 0 ≤ θ < 12 : u ∈ H θ,θ (Σ) = U θ [0, T ] = L2 0, T ; H θ (Γ) ∩ H θ (0, T ; L2 (Γ)) . (5.7.5) Then, the unique solution to problem (5.1.1) satisﬁes 1 θ +4 2+θ (Ω) ∩ H 1 (Ω) ; 2 w( · ; 0, 0) ∈ C [0, T ]; D A = H 0 1 θ w ( · ; 0, 0) ∈ C [0, T ]; D A 4 + 4 = H 1+θ (Ω) ∩ H01 (Ω) ; t wtt ( · ; 0, 0) ∈ C [0, T ]; D A θ4 = H θ (Ω) . 1 θ r , 0 ≤ r ≤ 1; Dtr w( · ; 0, 0) ∈ L2 0, T ; D A 2 + 4 − 4 1 D θ w( · ; 0, 0) ∈ L2 0, T ; D A 2 ; t 1 θ r , 0 ≤ r ≤ 1; Dtr wt ( · ; 0, 0) ∈ L2 0, T ; D A 4 + 4 − 4 1 D θ wt ( · ; 0, 0) ∈ L2 0, T ; D A 4 . t (5.7.6) (5.7.7) (5.7.8) (5.7.9) (5.7.10) (5.7.11) (5.7.12) DIFFERENTIAL RICCATI EQUATIONS 473 A fortiori, 1 θ 1 1 1 θ Lu ∈ Y θ [0, T ] = L2 0, T ; D A 2 + 4 × D A 4 + 2 = D(Aθ ) ∩ H θ 0, T ; D A 2 × D A 4 = Y . (5.7.13) Proof of Corollary 5.7.2. For θ < 12 , the compatibility relations in (5.7.1), which now read u(0) = ∆w0 Γ = 0; w0 Γ = 0, do not interfere, and we then interpolate between (5.3.6), (5.3.7), or (5.5.3b), for θ = 0 and (5.7.2), (5.7.3) for θ = 1, thereby obtaining (5.7.5)–(5.7.7), as desired. Next, application of the intermediate derivative theorem [LM.1, p. 15] to (5.7.5) and (5.7.6), as well as to (5.7.6) and (5.7.7), yields, respectively, (5.7.8) and (5.7.10), which then specialize to (5.7.9), and respectively, (5.7.11) for r = θ. Thus, (5.7.12) is a consequence of (5.7.5), (5.7.6), and (5.7.9), (5.7.11). Corollary 5.7.2 plainly veriﬁes assumption (H.2) = (1.1.4). 5.8. verification of assumption (H.3) = (1.15) Veriﬁcation of assumption (H.3) = (1.15) is based upon the following regularity result. Theorem 5.8.1. (i) With reference to the operator L∗ deﬁned by (5.5.4), we have for 0 ≤ r ≤ 1, (L∗s v)(t) = B ∗ T t ∗ (τ −t) eA v(τ )dτ, s≤t≤T 1 r 1 r : continuous L2 s, T ; D A 2 + 4 × D A 4 + 4 ≡ D(Ar ) → H r,r (ΣsT ) = U r [s, T ], (5.8.1) uniformly in s. Proof. The proof of Theorem 5.8.1 will be given in Section 5.13 below. Restricting (5.8.1) to 0 ≤ θ = r ≤ assumption (H.3) = (1.15). 1 2 + δ, θ = 12 , we obtain veriﬁcation of 5.9. Verification of assumptions (H.4) = (1.18) through (H.7) = (1.21) + Veriﬁcation of 5assumption 3 (H.4) = (1.18). For x = [x1 , x2 ] ∈ Yδ = 1 δ δ D A 2 +δ = D A 8 + 4 × D A 8 + 4 by (5.6.5), we compute starting from 474 R. TRIGGIANI 1 (5.4.33), and recalling G∗2 A = −D∗ A− 2 from (5.4.17): ∗ 1 B ∗ eA t x = G∗2 A C(t)x2 + S(t) I + ρA 2 1 −1 1 (by (5.4.17)) = −D∗ A 2 C(t)x2 + S(t) I + ρA 2 1 δ 3 δ 1 1 = −D∗ A 8 − 4 C(t)A 8 + 4 x2 + A 4 S(t) I + ρA 2 −1 Ax1 −1 Ax1 δ 9 A 8 + 4 x1 ∈ C([0, T ]; L2 (Γ)), (5.9.1) (5.9.2) (5.9.3) (5.9.4) where the desired regularity in (5.9.4) follows since, recalling (5.4.18c) and (5.4.11), 1 δ 1 D∗ A 8 − 4 ∈ L(L2 (Ω); L2 (Γ)); t → A 4 S(t), C(t) strongly continuous on L2 (Ω), (5.9.5) as well as, via the assumptions on [x1 , x2 ]. 3 δ A 8 + 4 x2 ∈ L2 (Ω); 1 I + ρA 2 −1 9 δ A 8 + 4 x1 ∈ L2 (Ω). (5.9.6) Thus, (5.9.4) veriﬁes assumption (H.4) = (1.18). Veriﬁcation of assumption (H.5) = (1.19). With 5 δ 3 δ Yδ− = D A 8 − 4 × D A 8 − 4 by (5.6.4), C(t) and S(t) are likewise s.c. cosine/sine operators on any space D(Aθ ), hence eAt in (5.4.31) is a s.c. group on Yδ− as well. The space D(A) in (5.4.23) is clearly dense in Yδ− . = (1.20). For x = [x1 , x2 ] ∈ Yδ− = Veriﬁcation of 3assumption 1 (H.6) 5 δ δ D A 8 − 4 × D A 8 − 4 = D A 2 −δ , we obtain via (5.4.23), (5.4.3), Ax = = 0 1 −1 − I + ρA 2 I A 0 x2 1 −1 − I + ρA 2 Ax1 x1 x2 ∈D A 3 − 4δ 8 ×D A 1 − 4δ 8 (5.9.7) ≡ [Yδ+ ] , recalling in the last step (5.6.7) [and A is, in fact, an isomorphism Yδ− onto [Yδ+ ] ]. Eqn. (5.9.7) veriﬁes assumption (H.6) = (1.20). Remark 5.9.1. Returning to (5.4.26), we see via (5.6.4) that, in the present case, A−1 B : continuous U → Yδ− , (5.9.8) which is property (1.32). Thus, as remarked below (1.32), property (5.9.8), along with (H.5) = (1.19) and (H.6) = (1.20) already veriﬁed, reprove (H.4) = (1.18). DIFFERENTIAL RICCATI EQUATIONS 475 Veriﬁcation of assumption (H.7) = (1.21). Let 1 u ∈ Uδ− = H 2 −δ (Γ), so that Du ∈ H 1−δ (Ω), (5.9.9) by (5.4.18a). Thus, recalling (5.4.3), (5.4.24), and (5.4.17), Bu = = 0 AG2 u = 0 − I + ρA 3 1 2 −1 ⊂ [Y + ] δ 1 2 A Du 1 H 2 −δ (Ω) ∩ H01 (Ω) × H 2 −δ (Ω), (5.9.10) using, in the last step, (5.6.7) for [Yδ+ ] , and (5.9.9). Thus, (5.9.10) shows B : continuous Uδ− → [Yδ+ ] , (5.9.11) as desired, and assumption (H.7) = (1.21) is veriﬁed. 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. Lemma 5.10.1. [LT.3] Let φ be a solution of Eqn. (5.3.1a) (with no boundary conditions imposed) for smooth data, say 3 1 {φ0 , φ1 , f } ∈ D(A) × D A 4 × L1 0, T ; D A 4 . (5.10.1) Then, the following identity holds true: Σ ∂(∆φ) h · ∇(∆φ)dΣ + ∂ν + + = ρ 2 Σ Q (∆φt )2 h · νdΣ − 1 2 Σ Q Q ∇(∆φ)2 h · νdΣ − Σ 1 2 Σ ∆φt 2 h · νdΣ φt ∆φt h · νdΣ H∇(∆φ) · ∇(∆φ)dQ + 1 + 2 + Σ ∂φt h · ∇φt dΣ ∂ν ∂φt φt div hdΣ − ∂ν Σ Q H∇φt · ∇φt dQ {∇φt 2 + ρ(∆φt )2 − ∇(∆φ)2 } div h dQ φt ∇(div h) · ∇φt dQ + Q f h · ∇(∆φ)dQ − [(φt , h · ∇(∆φ))Ω + ρ(∆φt , h · ∇(∆φ))Ω ]T0 , (5.10.2) 476 R. TRIGGIANI where ∂h1 ∂h1 ∂x1 , · · · ∂xn ··· H(x) = ∂hn ∂hn ∂x1 , · · · ∂xn ν(x) = , (5.10.3) outward unit normal vector at x ∈ Γ, (5.10.4) and h(x) = [h1 (x), h2 (x), . . . , hn (x)] ∈ C 2 (Ω̄) is a given vector ﬁeld. Proof of Lemma 5.10.1. The key is to multiply Eqn. (5.3.1a) by h · ∇(∆φ) and integrate by parts over Q, see [LT.5]. 5.11. Proof of theorem 5.3.2 We now provide the details, already contained in the preceding development, that the trace regularity for the homogeneous φproblem (5.3.1), 3 1 {φ0 , φ1 } ∈ D A 4 × D A 2 f ≡0 ⇒ ∂(∆φ) ∈ L2 (0, T ; L2 (Γ)) ∂ν (5.11.1) ≡ L2 (Σ), established in Theorem 5.3.1, implies by transposition the interior regularity u ∈ L2 (0, T ; L2 (Γ)) ≡ L2 (Σ) w0 = w1 = 0 ⇒ t A 0 S(t − τ )G2 u(τ )dτ ∈ C([0, T ]; Y ); t w(t) wt (t) = Y =D A 1 2 A 0 (5.11.2) C(t − τ )G2 u(τ )dτ 1 × D Aρ4 = [H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω) (5.11.3) (see (5.4.30) and (5.4.22)) for the nonhomogeneous wproblem (5.1.1). Operatortheoretic proof of Theorem 5.3.2. Step 1. We have already seen in Section 5.4, Eqns. (5.4.32)–(5.4.36), that 1 ∂∆φ − (5.11.4) (t; φ0 , φ1 ) = G∗2 A S(t)Ax1 + C(t) I + ρA 2 x2 ∂ν ∗ = B ∗ eA t x; 1 φ0 = A− 2 x2 ; Thus, if we take 1 (5.11.5) 1 φ1 = I + ρA 2 1 −1 1 A 2 x1 . (5.11.6) 1 x1 ∈ D A 2 , x2 ∈ D A 4 = D Aρ4 , hence 3 1 {φ0 , φ1 } ∈ D A 4 × D A 2 , (5.11.7) DIFFERENTIAL RICCATI EQUATIONS 477 we then see, via (5.11.4), (5.11.5), that implication (5.11.1) of Theorem 5.3.1 applies and yields the following operatortheoretic restatement, already noted in (5.5.1). Theorem 5.11.1. With reference to (5.11.4)–(5.11.6), (5.11.1), we have with Y as in (5.11.3) and A as in (5.4.3): ∗ B ∗ eA t : continuous Y → L2 (0, T ; L2 (Γ)) ≡ L2 (Σ); 1 G∗2 AA 2 S(t), G∗2 AS(t) : continuous L2 (Ω) → 3 1 G∗2 AA 4 C(t); G∗2 A 4 C(t) : continuous L2 (Ω) L2 (Σ); → L2 (Σ). (5.11.8) (5.11.9) (5.11.10) Step 2. The following result then stems from Theorem 5.11.1, by an application of [LT.3] or [FLT.1, Appendix A]. With reference to the (operator) explicit formulas (5.4.29), (5.4.30) for the solution {w(t), wt (t)} of the wproblem (5.1.1) with initial conditions w0 = w1 = 0 at t = s = 0, we have t w(t) eA(t−τ ) Bu(τ )dτ = (Lu)(t) = wt (t) 0 t = t A A 0 0 S(t − τ )G2 u(τ )dτ C(t − τ )G2 u(τ )dτ (5.11.11) : continuous L2 (Σ) → C([0, T ]; Y ), (5.11.12) Y as in (5.11.3). This establishes (5.11.2) and, in turn, yields the key part of Theorem 5.3.2 due to u. Then, recalling (5.4.20), (5.4.3), (5.4.17), we obtain wtt = −Aw + AG2 u = 1 I + ρA 2 −1 1 Aw − I + ρA 2 ∈ L2 (0, T ; L2 (Ω)), −1 1 A 2 Du (5.11.13) where the indicated 1 regularity stems from the established regularity w ∈ C [0, T ]; D A 2 of w in (5.11.12), and by the regularity of D in (5.4.18b). To get wtt ∈ L2 (0, T ; L2 (Ω)), we could also diﬀerentiate wt in (5.11.1). Finally, we omit the details for the regularity due to the initial conditions {w0 , w1 }, using (5.4.11). Theorem 5.3.2 is proved. A P.D.E. version of the duality or transposition argument may be given. 478 R. TRIGGIANI 5.12. Proof of theorem 5.7.1 With reference to the nonhomogeneous wproblem (5.1.1), we assume {w0 , w1 } ∈ [H 3 (Ω) ∩ H01 (Ω)] × [H 2 (Ω) ∩ H01 (Ω)]; 1 w0 Γ = 0; ∆w0 Γ = u(0) ∈ H 2 (Γ); (5.12.1) 1 u ∈ C [0, T ]; H 2 (Γ) ∩ H 1 (0, T ; L2 (Γ)), (5.12.2) and we must show that {w,wt , wtt } ∈ C [0, T ]; [H 3 (Ω) ∩ H01 (Ω)] × [H 2 (Ω) ∩ H01 (Ω)] × H01 (Ω) . (5.12.3) Operatortheoretic proof. We return to the explicit solution formula (5.4.30), i.e., w(t) = C(t)w0 + S(t)w1 + A t 0 S(t − τ )G2 u(τ )dτ, (5.12.4) and integrate by parts the integral term with u ∈ H 1 (0, T ; L2 (Γ)), thus obtaining ✒ w(t) = C(t)[w0 − G2 u(0)] + S(t)w1 + G2 u(t) − t 0 C(t − τ )G2 u̇(τ )dτ. (5.12.5) Here, by the ﬁrst Compatibility Condition in (5.12.1), and, respectively, u̇ ∈ L2 (Σ), we have w0 = G2 u(0); t 0 3 C(t − τ )G2 u̇(τ )dτ ∈ C [0, T ]; D A 4 , (5.12.6) respectively, recalling the deﬁnition of G2 in (5.4.15) 1 and, 3 the regularity 4 (5.11.2) (second component) Az ∈ D A ⇐ ⇒ z ∈ D A 4 . Likewise, by (5.4.11), 3 S(t)w1 ∈ C [0, T ]; D A 4 1 1 , with w1 ∈ D A 2 . (5.12.7) We now use that u ∈ C [0, T ]; H 2 (Γ) as well, from (5.12.2), so that by elliptic theory [(5.4.15), (5.4.16) with s = 12 ], G2 u(t) ∈ C [0, T ]; H 3 (Ω) ∩ H01 (Ω) , (5.12.8) since G2 u(t)Γ = 0 by deﬁnition (5.4.15). Thus, (5.12.6), (5.12.7), and (5.12.8) used in (5.12.5), show (5.12.3) for w, via (5.4.8a). As to wt , we diﬀerentiate (5.12.5), thus obtaining wt (t) = C(t)w1 + G2 u̇(t) − G2 u̇(t) + A 1 ∈ C [0, T ]; D A 2 . t 0 S(t − τ )G2 u̇(τ )dτ (5.12.9) DIFFERENTIAL RICCATI EQUATIONS 479 Conclusion (5.12.3) for wt now follows from (5.12.9) [where a cancellation of G2 u̇(t) occurs] via (5.11.2) [ﬁrst component] and (5.11.3). Similarly, one diﬀerentiates (5.12.9) in t and obtains (5.12.3) for wtt via (5.11.2) [second component] and (5.11.3). A P.D.E. proof may also be given, of course. 5.13. Proof of theorem 5.8.1 5.13.1. A preliminary trace result Theorem 5.13.1. (i) With reference to the operators in (5.11.8), (5.11.9), we have the following regularity properties for 0 ≤ r ≤ 1 which generalize the case r = 0 of Theorem 5.11.1: 1 r G∗2 AA 2 S(t), G∗2 AS(t) : continuous D A 4 → H r,r (Σ); 1 G∗2 A 4 C(t) : continuous D A 4 → H r,r (Σ); 3 G∗2 AA 4 C(t), ∗ r 1 r 1 r (5.13.1) (5.13.2) B ∗ eA t : continuous D(Ar ) ≡ D A 2 + 4 ×D A 4 + 4 → H r,r (Σ); (5.13.3) H r,r (Σ) ≡ L2 (0, T ; H r (Γ)) ∩ H r (0, T ; L2 (Γ)). (5.13.4) (ii) Equivalently, in P.D.E.’s terms (see (5.3.5) for r = 0) {φ0 , φ1 } → 3 ∂∆φ(t; φ0 , φ1 ) : continuous ∂ν r 1 r (5.13.5) D A1+r = D A 4 + 4 × D A 2 + 4 → H r,r (Σ), (5.13.6) where φ(t; φ0 , φ1 ) is the solution of problem (5.3.1) with f ≡ 0, and where we further recall (5.11.4)–(5.11.6) to justify the stated equivalence between parts (i) and (ii). Proof. Case r = 0. The case r = 0 is contained in Theorem 5.11.1 for (i) and in Theorem 5.3.1, Eqn. (5.3.5), for (ii). The stated equivalence uses (5.11.4)–(5.11.6). Thus, it is suﬃcient to prove the case r = 1, and interpolate to establish Theorem 5.13.1. Case r = 1. We ﬁrst show the time regularity 1 1 1 G∗2 AA 2 S(t)x, G∗2 AA 4 C(t)x ∈ H 1 (0, T ; L2 (Γ)), x ∈ D A 4 . 1 (5.13.7) Indeed, with x ∈ D A 4 , we compute, recalling from (5.4.9) and f.f. that −A is the inﬁnitesimal generator of C(t) 1 1 1 d ∗ G2 AA 2 S(t)x = G∗2 AA 4 C(t)A 4 x ∈ L2 (Σ); dt (5.13.8) 1 1 d ∗ (5.13.9) G AA 4 C(t)x = −G∗2 A2 S(t)A 4 x ∈ L2 (Σ), dt 2 where the regularity in (5.13.8) is a direct application of (5.11.10) (case r = 0), while the regularity of (5.13.9) is equivalent to (5.11.9) (case r = 0) 480 R. TRIGGIANI by the deﬁnition of A in (5.4.3). Thus, (5.13.7) is proved. To show the space regularity 1 1 1 G∗2 AA 2 S(t)x, G∗2 AA 4 C(t)x ∈ L2 (0, T ; H 1 (Γ)), x ∈ D A 4 , (5.13.10) we shall equivalently show, by (5.11.4)–(5.11.6) that {φ0 , φ1 } → ∂∆φ(t; φ0 , φ1 ) : continuous D(A2 ) ∂ν 3 = D(A) × D A 4 → L2 (0, T ; H 1 (Γ)) (5.13.11) with φ(t; φ0 , φ1 ) solutions of problem (5.3.1) with f ≡ 0. To this end, we introduce ∂ B = bi (x) = ﬁrstorder operator with ∂x i i (time independent) coeﬃcients bi smooth in Ω̄ and such that bi νi = 0 on Γ. B is tangent to Γ, i.e., (5.13.12) i Accordingly, we consider the problem φtt − ρ∆φtt + ∆2 φ ≡ 0 (5.13.13a) in Q φ(0, · ) = φ0 ; φt (0, · ) = φ1 in Ω φΣ ≡ ∆φΣ ≡ 0 in Σ 3 (5.13.13b) or φtt = −Aφ, (5.13.13c) {φ0 , φ1 } ∈ D(A) × D A 4 ⊂ H 4 (Ω) × H 3 (Ω), (5.13.14) whose solution is φ(t) = C(t)φ0 + S(t)φ1 ∈ C([0, T ]; D(A)); 3 φt (t) = −AS(t)φ0 + C(t)φ1 ∈ C [0, T ]; D A 4 1 φtt (t) = −AC(t)φ0 − AS(t)φ1 ∈ C [0, T ]; D A 2 (5.13.15) ; (5.13.16) . (5.13.17) We then introduce a new variable z = Bφ, (5.13.18) which, therefore, has apriori regularity from (5.13.12) and (5.13.15)–(5.13.17), given by ! z ∈ C [0, T ]; H 3 (Ω) ∩ H01 (Ω) ; ! zt ∈ C [0, T ]; H 2 (Ω) ∩ H01 (Ω) ; ! ztt ∈ C [0, T ]; H01 (Ω) . (5.13.19) (5.13.20) (5.13.21) Then proving (5.13.11) is equivalent to showing that ∂(∆Bφ) ∂ν Γ = ∂(∆z) ∂ν Γ ∈ L2 (Σ). (5.13.22) DIFFERENTIAL RICCATI EQUATIONS 481 The variable z satisﬁes the problem ztt − ρ∆ztt + ∆2 z = −[B, ∆2 ]φ + ρ[B, ∆]φtt in Q, (5.13.23a) zΣ ≡ 0 in Σ, (5.13.23b) ∆zΣ = −[B, ∆]φΓ in Σ, (5.13.23c) as one readily sees by (5.3.18), (5.3.13). Since the commutators [B, ∆2 ] = operator of order 1 + 4 − 1 = 4; (5.13.24a) [B, ∆] = operator of order 1 + 2 − 1 = 2, (5.13.24b) we see via the regularity (5.13.15) for φ and (5.13.17) for φtt that the righthand side term k in (5.13.23a) satisﬁes k ≡ [B, ∆2 ]φ + ρ[B, ∆]φtt ∈ C([0, T ]; L2 (Ω)), 1 (5.13.25) recalling D(A) ⊂ H 4 (Ω) and D A 2 ⊂ H 2 (Ω). Similarly, via (5.13.15), (5.13.16), (5.13.24), as well as by using trace theory, we see that the boundary term g in (5.13.23c) satisﬁes 3 g = −[B, ∆]φΓ ∈ C [0, T ]; H 2 (Γ) ; 1 (5.13.26) gt = −[B, ∆]φt Γ ∈ C [0, T ]; H 2 (Γ) . (5.13.27) Thus, by (5.13.23), (5.13.25), (5.13.26), (5.13.27), we see that the zproblem becomes: ztt − ρ∆ztt + ∆2 z = k ∈ C([0, T ]; L2 (Ω)), (5.13.28a) zΣ ≡ 0, ∆zΣ = g ∈ C [0, T ]; H 32 (Γ) ∩ C 1 [0, T ]; H 12 (Γ) (5.13.28b) (5.13.28c) with apriori interior regularity given by (5.13.19)–(5.13.21). We now return to the basic identity (5.10.2). Because of the apriori interior regularity (5.13.19), (5.13.20), for {z, zt } and that of k in (5.13.28a), the righthand side (R.H.S.) of identity (5.10.2) (with {φ, f } there replaced by {z, k} now) is well deﬁned. Thus, the lefthand side of identity (5.10.2) is well deﬁned. Taking the vector ﬁeld h such that hΓ = ν = outward unit normal vector on Γ, we have that on Γ : h · ∇(∆z) = ∇(∆z) · ν = (∆zt )2 h · ν = gt2 ; ∂∆z ; ∂ν h · ∇zt = ∂zt = ∇zt 2 by (5.13.28b); ∂ν ∂φt ; ∂ν (5.13.29) (5.13.30) 482 R. TRIGGIANI ∇(∆z) = = ∂(∆z) ∂(∆z) ν+ τ, ∂ν ∂ν τ = tangential unit vector on Γ ∂(∆z) ∂g ν+ τ ∂ν ∂τ (5.13.31) ∂(∆z) 2 ∂g 2 . (5.13.32) + ∂τ ∂ν Thus, the lefthand side (L.H.S.) of identity (5.10.2) can be rewritten, in the new variable z as 1 ∂(∆z) 2 ∂zt 2 1 L.H.S. of (5.10.2) = dΣ dΣ + 2 Σ ∂ν 2 Σ ∂ν ∇(∆z)2 = + ρ 2 Σ gt2 dΣ − 1 2 Σ ∂g ∂τ 2 dΣ (5.13.33) = well deﬁned by R.H.S. of (5.10.2), since the last two integral terms on the L.H.S. of (5.10.2) vanish due to the B.C. (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 t containing ∂z ∂ν is well deﬁned by (5.13.20) and trace theory. We conclude that the remaining boundary term in (5.13.33) containing ∂(∆z) is well deﬁned, ∂ν ∂(∆z) i.e., ∂ν ∈ L2 (Σ), and thus (5.13.22) is established, as desired. The proof of Theorem 5.13.1 is complete. Remark 5.13.1. In Theorem 5.3.1, the required interior regularity {φ, φt } ∈ C([0, T ]; H 3 (Ω) × H 2 (Ω))needed to guarantee that the righthand side of identity (5.10.2) is well deﬁned—is ensured by the assumed regularity 1 of the data {φ0 , φ1 , f } as in (5.10.1), in particular f ∈ L1 0, T ; D A 2 , whereby then the (positive) lefthand side of identity (5.10.2) establishes that ∂(∆φ) ∈ L2 (Σ). By contrast, in the zproblem (5.13.28), the right∂ν hand side k is only in C([0, T ]; L2 (Ω)). However, the required regularity {z, zt } ∈ C([0, T ]; H 3 (Ω)∩H 2 (Ω)) for the righthand side of identity (5.10.2) is guaranteed by the apriori regularity (5.13.19), (5.13.20), which is a consequence of the regularity (5.13.15), (5.13.16) of {φ, φt } via the change of variable z = Bφ in (5.13.18). Thus, for the zproblem (5.13.28), k is only required to have the regularity that makes the term Q kh · ∇(∆z)dQ on the righthand side of (5.10.2) well deﬁned, i.e., say k ∈ L1 (0, T ; L2 (Ω)), and we still obtain ∂(∆z) ∈ L2 (Σ). The above contrast between the φproblem in ∂ν Theorem 5.3.1 and the zproblem in (5.13.28) did not occur in the case of the wave equation of [LT.3, Section 3], while instead is typical for the other illustrating examples: EulerBernoulli equation, Schrödinger equations, etc., see [T.1], [LT.8, Chapter 10]. DIFFERENTIAL RICCATI EQUATIONS 483 5.13.2. Completion of the proof of theorem 5.8.1 Space regularity. To show space regularity (L∗ v)(t) = T t ∗ (τ −t) B ∗ eA v(τ )dτ (5.13.34) 1 r 1 r : continuous L2 0, T ; D(Ar ) ≡ D A 2 + 4 × D A 2 + 4 → L2 (0, T ; H r (Γ)), 0 ≤ r ≤ 1, (5.13.35) we simply invoke [LT.3], [FLT.1, Appendix A], which is permissible by virtue of the regularity (5.13.3) of Theorem 5.13.1. Time regularity. It suﬃces to show the case r = 1, since the case r = 0 is contained in (5.13.35), or in (5.5.6), and then interpolate. Thus, diﬀerentiating (5.13.34) in t for v ∈ L2 (0, T ; D(A)) or A∗ v ∈ L2 (0, T ; Y ), (5.13.36) since A is skewadjoint on Y (see below (5.4.26)) yields d(L∗ v) (t) = − B ∗ v(t) dt − T t ∗ (τ −t) B ∗ eA A∗ v(τ )dτ ∈ L2 (0, T ; L2 (Γ)), (5.13.37) as desired, by (5.13.36) and B ∗ A∗−1 ∈ L(Y ; U ) in (1.5a), Y and U as in (5.4.22). Then (5.13.37) shows L∗ : continuous L2 (0, T ; D(A)) → H 1 (0, T ; L2 (Γ)), (5.13.38) as required. 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