Hyperbolic partial differential equations on a one-dimensional spatial domain
are studied. This class of systems includes models of beams and waves as well
as the transport equation and networks of non-homogeneous transmission lines.
The main result of this paper is a simple test for $C_0$-semigroup generation
in terms of the boundary conditions. The result is illustrated with several
examples

Jacob, Birgit

Morris, Kirsten

Zwart, Hans

English

arXiv

Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of nonhomogeneous transmission lines. The main result of this paper is a simple test for $C_{0}$-semigroup generation in terms of the boundary conditions. The result is illustrated with several examples

Zwart, Heiko J.

NARCIS 

C0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

Birgit Jacob

Kirsten Morris

Hans Zwart

Crossref

Journal of Evolution Equations

C 0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

University of Twente Research Information

J. Evol. Equ. 15 (2015), 493–502© 2015 Springer Basel1424-3199/15/020493-10, published online January 10, 2015DOI 10.1007/s00028-014-0271-1Journal of EvolutionEquationsC0-semigroups for hyperbolic partial differential equationson a one-dimensional spatial domainBirgit Jacob, Kirsten Morris and Hans ZwartAbstract. Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. Thisclass of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. The main result of this paper is a simple test forC0-semigroup generationin terms of the boundary conditions. The result is illustrated with several examples.1. Introduction and main resultConsider the following class of partial differential equations∂x∂t(ζ, t) =(P1∂∂ζ+ P0)(H(ζ )x(ζ, t)), ζ ∈ [0, 1], t ≥ 0,x(ζ, 0) = x0(ζ ), (1)where P1 is an invertible n×nHermitianmatrix, P0 is a n×nmatrix andH(ζ ) is a pos-itive n×n Hermitian matrix for a.e. ζ ∈ (0, 1) satisfyingH,H−1 ∈ L∞(0, 1;Cn×n).This class of Cauchy problems covers in particular the wave equation, the trans-port equation and the Timoshenko beam equation, and also coupled beam and waveequations. These Cauchy problems are also known as Hamiltonian partial differentialequations or port-Hamiltonian systems, see [3,6] and in particular the Ph.D thesis [7].The boundary conditions are of the formW̃B[(Hx)(1,t)(Hx)(0,t)]= 0, (2)where W̃B is an n × 2n-matrix. DefineAx :=(P1ddζ+ P0)(x), x ∈ D(A), (3)on X p := L p(0, 1;Cn), 1 ≤ p < ∞, with the domainD(A) :={x ∈ W1,p(0, 1;Cn) | W̃B[x(1)x(0)]= 0}. (4)Keywords: C0-semigroups, Hyperbolic partial differential equations, Port-Hamiltonian differentialequations.494 B. Jacob et al J. Evol. Equ.Then, the partial differential equation (1) with the boundary conditions (2) can bewritten as the abstract differential equationẋ(t) = AHx(t), x(0) = x0.If we equip X2 with the energy norm 〈·,H·〉, then AH generates a contractionsemigroup (or an unitary C0-group) on (X2, 〈·,H·〉) if and only if A is dissipative on(X2, 〈·, ·〉)(or A and −A are dissipative on (X2, 〈·, ·〉), respectively) [1,3,4]. Matrixconditions to guarantee generation of a contraction semigroup or of a unitary grouphave been obtained [1,3,4]. The following theorem extends these results.THEOREM 1.1. Let WB := W̃B[P1 −P1I I]−1and  := [ 0 II 0 ].1. The following statements are equivalent:(a) AHwith domain D(AH) := {x ∈ X2 | Hx ∈ D(A)} = H−1D(A) generatesa contraction semigroup on (X2, 〈·,H·〉);(b) Re 〈Ax, x〉 ≤ 0 for every x ∈ D(A);(c) Re P0 ≤ 0 and u∗P1u − y∗P1y ≤ 0 for every[ uy] ∈ ker W̃B;(d) Re P0 ≤ 0, WBW ∗B ≥ 0 and rank W̃B = n.2. The following statements are equivalent:(a) AHwith domain D(AH) := {x ∈ X2 | Hx ∈ D(A)} = H−1D(A) generatesa unitary C0-group on (X2, 〈·,H·〉);(b) Re 〈Ax, x〉 = 0 for every x ∈ D(A);(c) Re P0 = 0 and u∗P1u − y∗P1y = 0 for every[ uy] ∈ ker W̃B;(d) Re P0 = 0, WBW ∗B = 0 and rank W̃B = n.Theorem 1.1 was proved in [3, Theorem 7.2.4] with the additional assumptions thatP∗0 = −P0 and rank W̃B = n. The extension to non-skew-adjoint matrices P0 is in[1]. However, the equivalence with (c) is not explicitly shown in the above references,and it is assumed that rank W̃B = n. A short proof of Theorem 1.1 is in the followingsection.By the assumptions on H, it is clear that the norm on (X2, 〈·,H·〉) is equivalent tothe standard norm on X2. Hence, if AH generates a contraction (or a unitary group)with respect to the energy norm for some H, then it will generate a C0-semigroup(C0-group) on X2 equipped with the standard norm as well.The following corollary follows immediately.COROLLARY 1.2. The following statements are equivalent:1. A generates a contraction semigroup on (X2, 〈·, ·〉),2. AH generates a contraction semigroup on (X2, 〈·,H·〉).Corollary 1.2 implies that whether or not AH generates a contraction semigroupon the energy space (X2, 〈·,H·〉) is independent of the Hamiltonian density H: A isthe generator of a contraction semigroup on (X2, 〈·, ·〉) if and only if AH generates acontraction semigroup on (X2, 〈·,H·〉). The condition of a contraction semigroup isessential here. For a counterexample, see Example 3.2 or [8, Section 6].Vol. 15 (2015) C0-semigroups for hyperbolic partial differential equations 495DEFINITION 1.3. An operatorA generates a quasi-contractive semigroup ifA−ωI generates a contraction semigroup for some ω ∈ R. COROLLARY1.4. If Re P0 ≤ 0, then AH generates aquasi-contractive semigroupon (X2, 〈·,H·〉) if and only if AH generates a contraction semigroup on (X2, 〈·,H·〉).The proof of Corollary 1.4 will be given in Sect. 2.Theorem 1.1 characterizes boundary conditions for which AH generates a contrac-tion semigroup or a unitary group. However, other boundary conditions may still leadto a C0-semigroup. To characterize those, we diagonalize P1H(ζ ). It is easy to seethat the eigenvalues of P1H(ζ ) are the same as the eigenvalues of H(ζ ) 12 P1H(ζ ) 12 .Hence, by Sylvester’s law of inertia, the number of positive and negative eigenval-ues of P1H(ζ ) equal those of P1. We denote by n1 the number of positive and byn2 = n − n1 the number of negative eigenvalues of P1. Hence, we can find matricessuch thatP1H(ζ ) = S−1(ζ )[(ζ) 00 (ζ)]S(ζ ), a.e. ζ ∈ (0, 1), (5)with (ζ) and (ζ) diagonal matrices of size n1 × n1 and n2 × n2, respectively.The main result of this paper is the following theorem that provides easily checkedconditions for when the operator AH generates a C0-semigroup on X p. These coverthe situation where AH may not generate a contraction semigroup.THEOREM 1.5. Assume that S,  and  in (5) are continuously differentiable on[0, 1] and that rank W̃B = n. Define Z+(ζ ) to be the span of eigenvectors of P1H(ζ )corresponding to its positive eigenvalues. Similarly, we define Z−(ζ ) to be the spanof eigenvectors of P1H(ζ ) corresponding to its negative eigenvalues. We write W̃B asW̃B =[W1 W0](6)with W1,W0 ∈ Cn×n. Then, the following statements are equivalent:1. The operator AH defined by (3)–(4) generates a C0-semigroup on X p.2. W1H(1)Z+(1) ⊕ W0H(0)Z−(0) = Cn.The proof of Theorem 1.5 will be given in the next section.REMARK 1.6. 1. In Kato [9, Chapter II], conditions on P1H are given guar-anteeing that S,  and  are continuously differentiable.2. In [2], a more restrictive version of Theorem 1.5 that applies when H = I andp = 2 was proven by a different approach. In [2] estimates for the growth boundare given.3. Theorem 1.5 implies that if AH generates a C0-semigroup on one X p, thenAH generates a C0-semigroup on every X p, 1 ≤ p < ∞. A similar statementdoes not hold for contraction semigroups. Example 3.3, given later in this paper,illustrates this point.496 B. Jacob et al J. Evol. Equ.2. Proof of Theorems 1.1 and 1.5 and Corollary 1.4Proof of Theorem 1.1: Since the proof of Part 2 is similar to that of Part 1 we onlypresent the details for Part 1.The implication (a) ⇒ (b) follows directly from the Lumer–Phillips theorem andLemma 7.2.3 in [3]. Next, we show the implication (b) ⇒ (c). It is easy to see thatRe〈Ax, x〉 = x(1)∗P1x(1) − x(0)∗P1x(0) + Re∫ 10x(ζ )∗P0x(ζ )dζ (7)holds for every x ∈ D(A). Choosing x ∈ W 1,2(0, 1;Cn) with x(0) = x(1) = 0, weobtain Re P0 ≤ 0. For every u, y ∈ Cn and every ε > 0, there exists a function inx ∈ W 1,2(0, 1;Cn) such that x(0) = u, x(1) = y and the L2-norm of x is less than ε.Choosing this function in Eq. (7) and letting ε go to zero implies the second assertionin (c), see also Lemma 2.4 of [1]. The implication (d) ⇒ (a) follows from Theorem2.3 of [1], see also [4]. Hence, it remains to show (c) ⇒ (d).We introduce the notation f1 = x(1) and f0 = x(0). Then, the condition in (c) canbe written as[f ∗1 f ∗0] [ P1 00 −P1] [f1f0]≤ 0, for[f1f0]∈ ker W̃B . (8)Since W̃B is an n × 2n matrix, its kernel has dimension 2n minus its rank. Hence, thisdimension will be larger or equal to n. Since P1 is an invertible Hermitian n×nmatrix,the matrix[P1 00 −P1]will have n positive and n negative eigenvalues. This implies thatif v∗[P1 00 −P1]v ≤ 0 for all v in a linear subspace, then that subspace has at mostdimension n. Combining these two facts, the dimension of the kernel of W̃B equals n,and so W̃B is a matrix of rank n.Defining[ y1y0] = [ P1 −P1I I ][f1f0], and using (8), an easy calculation showsy∗1 y0 + y∗0 y1 ≤ 0, for[y1y0]∈ kerWB . (9)We writeWB asWB = [W1 W2]. Now, it is easy to see thatW1 +W2 is invertible (werefer to page 87 in [3] for the details). Defining V := (W1 + W2)−1(W1 − W2), weobtainWB = 12(W1 + W2) [I + V, I − V ] .Let[fe] ∈ kerWB be arbitrary. By [3, Lemma 7.3.2], there exists a vector 	 such that[fe] = [ I−V−I−V ] 	. This implies0 ≥ f ∗e + e∗ f = 	∗(−2I + 2V ∗V )	, (10)Vol. 15 (2015) C0-semigroups for hyperbolic partial differential equations 497This inequality holds for any[fe] ∈ kerWB . Since the n × 2n matrix WB has rank n,its kernel has dimension n, and so the set of vectors 	 satisfying[fe] = [ I−V−I−V ] 	 forsome[fe] ∈ kerWB equals the whole space Cn . Thus, (10) implies that V ∗V ≤ I ,and by [3, Lemma 7.3.1] we obtain WBW ∗B ≥ 0. Proof of Corollary 1.4: As AH − ωI generates a contraction semigroup, Theorem1.1 implies WBW ∗B ≤ 0 and rank W̃B = n. Thanks to Re P0 ≤ 0 and Theorem 1.1,finally AH generates a contraction semigroup. The following proposition is needed for the proof of Theorem 1.5.PROPOSITION 2.1. ([8, Theorem 3.3] [3, Theorem 13.3.1] for p = 2 and [8,Theorem 3.3 and Section 7] for 1 ≤ p < ∞) Suppose K , Q ∈ Cn×n,  ∈ C1([0, 1];Cn1×n1) is a diagonal real matrix-valued function with (strictly) positive functions onthe diagonal and  ∈ C1([0, 1];Cn2×n2), n1 + n2 = n, is a diagonal real matrix-valued function with (strictly) negative functions on the diagonal. We split a functiong ∈ L p(0, 1;Cn) asg(ζ ) =[g+(ζ )g−(ζ )], (11)where g+(ζ ) ∈ Cn1 and g−(ζ ) ∈ Cn2 .Then, the operator Ã : D( Ã) ⊂ X p → X p defined byÃ[g+g−]= ddζ([ 00 ] [g+g−])(12)D( Ã) ={[g+g−]∈W 1,p(0, 1,Cn) | K[(1)g+(1)(0)g−(0)]+Q[(0)g+(0)(1)g−(1)]= 0}(13)generates a C0-semigroup on X p if and only if K is invertible.Proof of Theorem 1.5: We define the new state variable g := Sx . Since S definesa boundedly invertible operator on L p(0, 1;Cn), the operator AH generates a C0-semigroup if and only if SAHS−1 generates a C0-semigroup. We define :=[ 00 ].Then, we obtain(SAHS−1g)(ζ ) = ddζ((ζ )g(ζ )) + S(ζ )dS−1dζ(ζ )(ζ )g(ζ )+S(ζ )P0H(ζ )S−1(ζ )g(ζ )D(SAHS−1) ={g ∈ W 1,p(0, 1;Cn) | W̃B[(HS−1g)(1)(HS−1g)(0)]= 0}. (14)498 B. Jacob et al J. Evol. Equ.Since the last two operators in (14) are bounded, SAHS−1 generates a C0-semigroupif and only if the operatorASg = ddζ(g) (15)D(AS) ={g ∈ W 1,p(0, 1;Cn×n) | W̃B[(HS−1g)(1)(HS−1g)(0)]= 0}(16)generates a C0-semigroup on X p. We split the matrices W1(HS−1)(1) andW0(HS−1)(0) asW1(HS−1)(1) =[V1 V2]W0(HS−1)(0) =[U1 U2],where U1, V1 ∈ Cn×n1 and U2, V2 ∈ Cn×n2 , and as in (11) writeg(ζ ) =[g+(ζ )g−(ζ )], (17)where g+(ζ ) ∈ Cn1 and g−(ζ ) ∈ Cn2 . Then,0 = W̃B[(HS−1g)(1)(HS−1g)(0)]= [V1 V2][g+(1)g−(1)]+ [U1 U2][g+(0)g−(0)]= [V1 U2][g+(1)g−(0)]+ [U1 V2][g+(0)g−(1)]= [V1 U2][(1)−1 00 (0)−1] [(1)g+(1)(0)g−(0)]+ [U1 V2][(0)−1 00 (1)−1] [(0)g+(0)(1)g−(1)].Thus, by Proposition 2.1, the operator AS as defined in (15) and (16) generates aC0-semigroup if and only if the matrixK = [V1 U2][(1)−1 00 (0)−1]is invertible. Since the matrix[(1)−1 00 (0)−1]is invertible, AS generates aC0-semigroup if and only if[V1 U2]is invertible. Now,[V1 U2]is invertible if andonly if for every f ∈ Cn there exists x ∈ Cn1 and y ∈ Cn2 such thatf = [V1 U2][xy]= [V1 U2][xy]+ [U1 V2][00]= [V1 V2][x0]+ [U1 U2][0y]= W1(HS−1)(1)[x0]+ W0(HS−1)(0)[0y]. (18)Vol. 15 (2015) C0-semigroups for hyperbolic partial differential equations 499Referring, to Eq. (5) the columns of S−1(ζ ) are the eigenvectors of P1H(ζ ). Theeigenvectors corresponding to the positive eigenvalues forms the first n1 columns.Thus, S−1(1)[x0]is in Z+(1). Similarly, S−1(0)[0y]is in Z−(0). Thus,[V1 U2]isinvertible if and only ifW1H(1)Z+(1) ⊕ W0H(0)Z−(0) = Cn,which concludes the proof. 3. ExamplesThe following three examples are provided as illustration of Theorem 1.5.EXAMPLE 3.1. Consider the one-dimensional transport equation on the interval(0, 1):∂x∂t(ζ, t) = ∂Hx∂ζ(ζ, t), x(ζ, 0) = x0(ζ ),[w1 w0] [(Hx)(1, t)(Hx)(0, t)]= 0,where H ∈ C1[0, 1] withH(ζ ) > 0 for every ζ ∈ [0, 1].An easy calculation shows P1H = H and thus Z+(1) = C and Z−(0) = {0}. Thus,by Theorem 1.5 the corresponding operatorAHx = ∂∂ζ(Hx),D(AH) ={x ∈ W 1,p(0, 1) | [w1 w0][(Hx)(1)(Hx)(0)]= 0},generates a C0-semigroup on L p(0, 1) if and only if w1 = 0. Further, by Theorem1.1, AH generates a contraction semigroup (unitary C0-group) on L2(0, 1) equippedwith the scalar product 〈·,H·〉 if and only if w21 ≥ w20 (w21 = w20). EXAMPLE 3.2. An (undamped) vibrating string can be modeled by∂2w∂t2(ζ, t) = 1ρ(ζ )∂∂ζ(T (ζ )∂w∂ζ(ζ, t)), t ≥ 0, ζ ∈ (0, 1), (19)where ζ ∈ [0, 1] is the spatial variable, w(ζ, t) is the vertical position of the string atplace ζ and time t, T (ζ ) > 0 is the Young’s modulus of the string and ρ(ζ ) > 0 is themass density, which may vary along the string. We assume that T and ρ are positiveand continuously differentiable functions on [0, 1]. By choosing the state variables500 B. Jacob et al J. Evol. Equ.x1 = ρ ∂w∂t (momentum) and x2 = ∂w∂ζ (strain), the partial differential equation (19)can equivalently be written as∂∂t[x1(ζ, t)x2(ζ, t)]=[0 11 0]∂∂ζ([1ρ(ζ )00 T (ζ )] [x1(ζ, t)x2(ζ, t)])= P1 ∂∂ζ(H(ζ )[x1(ζ, t)x2(ζ, t)]), (20)where P1 =[0 11 0]and H(ζ ) =[ 1ρ(ζ )00 T (ζ )].The boundary conditions for (20) are[W1 W0] [(Hx)(1, t)(Hx)(0, t)]= 0,where[W1 W0]is a 2× 4-matrix with rank 2, or equivalently, the partial differentialequation (19) is equipped with the boundary conditions[W1 W0]⎡⎢⎢⎢⎢⎣ρ ∂w∂t (1, t)∂w∂ζ(1, t)ρ ∂w∂t (0, t)∂w∂ζ(0, t)⎤⎥⎥⎥⎥⎦ = 0.Defining γ = √T (ζ )/ρ(ζ ), the matrix function P1H can be factorized asP1H =[γ −γρ−1 ρ−1] [γ 00 −γ] [(2γ )−1 ρ/2−(2γ )−1 ρ/2].This implies Z+(1) = span[T (1)γ (1)]and Z−(0) = span[ −T (0)γ (0)]. Thus, by Theorem1.5 the corresponding operator(AHx)(ζ ) =[0 11 0]∂∂ζ([1ρ(ζ )00 T (ζ )]x(ζ ));D(AH) ={x ∈ W 1,p(0, 1;C2) | [W1 W0][(Hx)(1)(Hx)(0)]= 0},generates a C0-semigroup on L p(0, 1;C2) if and only ifW1[γ (1)T (1)]⊕ W0[−γ (0)T (0)]= C2,or equivalently if the vectors W1[γ (1)T (1)]and W0[ −γ (0)T (0)]are linearly independent.If W1 := I and W0 :=[ −1 00 1], then AH generates a C0-semigroup if and only if thevectors[γ (1)T (1)]and[γ (0)T (0)]are linearly independent. Thus, not only the nature of theboundary conditions but also Young’s modulus and the mass density on the interval[0, 1] affect whether or not AH generates a C0-semigroup. Vol. 15 (2015) C0-semigroups for hyperbolic partial differential equations 501EXAMPLE 3.3. Consider the following network of three transport equations onthe interval (0, 1):∂x j∂t(ζ, t) = ∂x j∂ζ(ζ, t), t ≥ 0, ζ ∈ (0, 1), j = 1, 2, 3,x j (ζ, 0) = x j,0(ζ ), ζ ∈ (0, 1), j = 1, 2, 3⎡⎣1 0 0 0 0 00 1 0 −1 0 −10 0 1 0 −1 0⎤⎦⎡⎢⎢⎢⎢⎢⎢⎢⎣x1(1, t)x2(1, t)x3(1, t)x1(0, t)x2(0, t)x3(0, t)⎤⎥⎥⎥⎥⎥⎥⎥⎦= 0, t ≥ 0.Writing x =[ x1x2x3], the corresponding operator A : D(A) ⊂ L p(0, 1;C3) →L p(0, 1;C3) is(Ax)(ζ ) = ∂x∂ζ(ζ ),D(A) =⎧⎨⎩x ∈ W 1,p(0, 1;C3) |⎡⎣1 0 0 0 0 00 1 0 −1 0 −10 0 1 0 −1 0⎤⎦[x(1)x(0)]= 0⎫⎬⎭ .In this example,H = P1 = I and P0 = 0 and therefore the assumptions on S,  and are satisfied. An easy calculation yieldsx∗(1)x(1) − x∗(0)x(0) = 2x1(0)x3(0)for every x ∈ D(A). Theorem 1.1 implies that A does not generate a contractionsemigroup on L2(0, 1;C3).However, by Theorem 1.5 A generates a C0-semigroup on L p(0, 1;C3) for 1 ≤p < ∞: In this example, Z+(ζ ) = C3, Z−(ζ ) = {0}, W1 = I and W0 =[ 0 0 0−1 0 −10 −1 0].Thus,W1Z+(1) ⊕ W0Z−(0) = C3.Finally, [5, Corollary2.1.6] implies that A generates a contraction semigroup onL1(0, 1;C3).Summarizing, A generates a C0-semigroup on L p(0, 1;C3) for 1 ≤ p < ∞ and infact a contraction semigroup on L1(0, 1;C3) but it does not generate a contractionsemigroup on L2(0, 1;C3). AcknowledgementsThe authors gratefully acknowledge support from the DFG (Grant JA 735/9-1), theNWO (Grant DN 63-261) and the RiP program in Oberwolfach. Further the secondauthor gratefully acknowledges support by a NSERC Discovery grant.502 B. Jacob et al J. Evol. Equ.REFERENCES[1] B. Augner and B. Jacob, Stability and stabilization of infinite-dimensional linear port-Hamiltoniansystems, Evolution Equations and Control Theory, 3(2) (2014), 207–229.[2] K.-J. Engel, Generator property and stability for generalized difference operators, Journal of Evolu-tion Equations, 13(2) (2013), 311–334.[3] B. Jacob and H.J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, OperatorTheory: Advances and Applications, 223 (2012), Birkhäuser, Basel.[4] Y. Le Gorrec, H. Zwart and B. Maschke, Dirac structures and boundary control systems associatedwith skew-symmetric differential operators, SIAM J. Control Optim., 44 (2005), 1864–1892.[5] E. Sikolya, Semigroups for flows in networks, Ph.D thesis, University of Tübingen, 2004.[6] A.J. van der Schaft and B.M. Maschke, Hamiltonian formulation of distributed parameter systemswith boundary energy flow, J. Geom. Phys., 42 (2002), 166–174.[7] J.A. Villegas, A port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D thesis, Univer-siteit Twente in Enschede, 2007. Available from: http://doc.utwente.nl/57842/1/thesis_Villegas.[8] H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas, Well-posedness and regularity of hyperbolicboundary control systems on a one-dimensional spatial domain, ESAIM Control Optim. Calc. Var.,16(4) (2010), 1077–1093.[9] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.B. JacobFachbereich C - Mathematik undNaturwissenschaften, ArbeitsgruppeFunktionalanalysis University ofWuppertal, Gaußstraße 20,42119 Wuppertal,GermanyE-mail: bjacob@uni-wuppertal.deK. MorrisDepartment of Applied Mathematics,University of Waterloo,Waterloo, ON N2L 3G1, CanadaE-mail: kmorris@uwaterloo.caH. ZwartDepartment of Applied Mathematics,Faculty of Electrical Engineering,Mathematics and Computer Science,University of Twente,P.O. Box 217, 7500 AE Enschede,The NetherlandsE-mail: h.j.zwart@utwente.nl

C<sub>0</sub>-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

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$C_0$-semigroups for hyperbolic partial differential equations on a
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$C_0$ -semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

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$C_0$ -semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain