Boundary Stabilization and Observation of an Unstable Heat Equation in a General Multi-dimensional Domain
aa r X i v : . [ ee ss . S Y ] F e b Boundary Stabilization and Observation of anUnstable Heat Equation in a GeneralMulti-dimensional Domain ∗ Hongyinping Feng † , Pei-Hua Lang and Jiankang Liu School of Mathematical SciencesShanxi University, Taiyuan, Shanxi, 030006, P.R. China
Abstract
In this paper, we consider the exponential stabilization and observation of an un-stable heat equation in a general multi-dimensional domain by combining the finite-dimensional spectral truncation technique and the recently developed dynamics com-pensation approach. In contrast to the unstable one-dimensional partial differentialequation (PDE), such as the transport equation, wave equation and the heat equation,that can be treated by the well-known PDE backstepping method, stabilization of un-stable PDE in a general multi-dimensional domain is still a challenging problem. Wetreat the stabilization and observation problems separately. A dynamical state feed-back law is proposed firstly to stabilize the unstable heat equation exponentially andthen a state observer is designed via a boundary measurement. Both the stability ofthe closed-loop system and the well-posedness of the observer are proved. Some of thetheoretical results are validated by the numerical simulations.
Keywords:
Dynamics feedback, Multi-dimensional heat equation, Observer, Stabilization, Un-stable system.
Since the backstepping approach was first introduced into the systems described by the partialdifferential equations (PDEs) in [12], [17], and [18], the landscape of one-dimensional PDEs controlhas completely changed. This can be seen from its success in stabilizing the unstable [19, 20] or ∗ This work is supported by the National Natural Science Foundation of China (61873153,11901365). † Corresponding author. Email: [email protected]. ⊂ R n ( n ≥
2) is a bounded domain with C -boundary Γ, Γ is a non-emptyconnected open set in Γ, Γ = Γ \ Γ and Γ = ∅ . Let ν be the unit outward normal vector of Γ and let ∆ be the usual Laplacian which is defined by∆ f = n X i =1 ∂ f∂x i , ∀ f ∈ H (Ω) . (1.1)We consider the following heat equation: w t ( x, t ) = ∆ w ( x, t ) + µw ( x, t ) , x ∈ Ω , t > ,w ( x, t ) = 0 , x ∈ Γ , ∂w ( x, t ) ∂ν = u ( x, t ) , x ∈ Γ , t ≥ ,y ( x, t ) = w ( x, t ) , x ∈ Γ , t ≥ , (1.2)where w ( · , t ) is the state, µ > u is the control and y is the output. System (1.2) is a generalheat equation with interior convection. In physics and engineering contexts, it describes the flowof heat in a homogeneous and isotropic medium, with w ( x, t ) being the temperature at the point x and time t . The more detailed physical interpretation of the heat equation can be found in [6].By a simple computation, we can see that there are some eigenvalues of the open-loop system(1.2) (with u ( · , t ) ≡
0) in the right-half plane provided µ is sufficiently large. This shows that theopen-loop system (1.2) is unstable for large µ . The lower-order term µw ( · , t ) of (1.2) is usuallyreferred to as source term or unstable term. Heat equations with unstable term or source term havebeen extensively studied by the method of PDE backstepping. Examples can be found in [1], [13],[7], [8], and [19], to name just a few. The PDE backstepping method is powerful and is still valid toother one-dimensional distributed parameter systems such as the wave equation [10], Schr¨odingerequation [5], the first order hyperbolic equation [9] as well as some special Euler-Bernoulli beam[21]. However, the application of backstepping method seems to stop in front of unstable PDEsin the general multi-dimensional domain. There still exist formidable obstacles to applying thisapproach to general multi-dimensional PDEs. 2n this paper, we combine the newly developed dynamics compensation approach [3, 4] andthe finite-dimensional spectral truncation technique [2, 16] to cope with the unstable system (1.2).The control objective is to stabilize the system exponentially by virtue of the measurement output.Owing to the separation principle of the linear systems, the output feedback will be available oncewe address the following two problems: (i), stabilize system (1.2) by a full state feedback; (ii),design a state observer in terms of the measurement output. We will consider these two problemsseparately.We consider system (1.2) in the state space L (Ω). Let Af = ∆ f, ∀ f ∈ D ( A ) = (cid:26) f | f ∈ H (Ω) ∩ H (Ω) , ∂f∂ν (cid:12)(cid:12) Γ = 0 (cid:27) , (1.3)where H (Ω) = { f ∈ H (Ω) | f = 0 on Γ } . Then A generates an exponentially stable analyticsemigroup on L (Ω). It is well known (e.g. [11, p.668]) that D (( − A ) / ) = H (Ω) and ( − A ) / isa canonical isomorphism from H (Ω) onto L (Ω). Moreover, the following Gelfand triple compactinclusions are valid: H (Ω) = D (( − A ) / ) ֒ → L (Ω) = [ L (Ω)] ′ ֒ → [ D (( − A ) / )] ′ = H − (Ω) , (1.4)where H − (Ω) is the dual space of H (Ω) with the pivot space L (Ω). An extension ˜ A ∈L ( H (Ω) , H − (Ω)) of A is defined by h ˜ Ax, z i H − (Ω) ,H (Ω) = −h ( − A ) / x, ( − A ) / z i L (Ω) , ∀ x, z ∈ H (Ω) . (1.5)Since A is strictly negative, self-adjoint in L (Ω), and is the inverse of a compact operator, theoperator A has the infinite sequence of negative eigenvalues { λ j } ∞ j =1 and a corresponding sequenceof eigenfunctions { φ j ( · ) } ∞ j =1 that forms an orthonormal basis for L (Ω). Without loss of generality,we always assume that Assumption 1.1.
Let the operator A be given by (1.3) and µ > . Suppose that the eigenpairs { ( φ j ( · ) , λ j ) } ∞ j =1 of A satisfy > λ > λ > · · · > λ k > · · · → −∞ , (1.6) and ∆ φ k = λ k φ k , k φ k k L (Ω) = 1 ,φ k ( x ) = 0 , x ∈ Γ , ∂φ k ( x ) ∂ν = 0 , x ∈ Γ , k = 1 , , · · · . (1.7) Suppose that N is an integer that satisfies λ k + µ < , ∀ k > N. (1.8)Define the Neumann map Υ ∈ L ( L (Γ ) , H / (Ω)) ([11, p. 668]) by Υ u = ψ if and only if ∆ ψ = 0 in Ω ,ψ | Γ = 0 , ∂ψ∂ν (cid:12)(cid:12) Γ = u. (1.9)3sing the Neumann map, one can write (1.2) in H − (Ω) as˙ w ( · , t ) = ∆ w ( · , t ) − ∆ ψ + µw ( · , t ) = ∆( w ( · , t ) − ψ ) + µw ( · , t )= ˜ A ( w ( · , t ) − ψ ) + µw ( · , t ) = ( ˜ A + µ ) w ( · , t ) − ˜ A Υ u ( · , t ) . (1.10)That is ˙ w ( · , t ) = ( ˜ A + µ ) w ( · , t ) + Bu ( · , t ) in H − (Ω) , (1.11)where B ∈ L ( L (Γ ) , H − (Ω)) is given by Bu = − ˜ A Υ u, ∀ u ∈ L (Γ ) . (1.12)Define B ∗ ∈ L ( H (Ω) , L (Γ )) by h B ∗ f, u i L (Γ ) = h f, Bu i H (Ω) ,H − (Ω) , ∀ f ∈ H (Ω) , u ∈ L (Γ ) . (1.13)Then, for any f ∈ D ( A ) = D ( A ∗ ) and u ∈ L (Γ ), it follows from (1.5), (1.3), (1.13) and (1.9) that h B ∗ f, u i L (Γ ) = h f, − ˜ A Υ u i H (Ω) ,H − (Ω) = h A ∗ f, − Υ u i L (Ω) = h ∆ f, − ψ i L (Ω) = h∇ f, ∇ ψ i L (Ω) = Z Γ f ( x ) u ( x ) dx, (1.14)which, together with the denseness of D ( A ) in H (Ω), implies that B ∗ f = f | Γ , ∀ f ∈ H (Ω) . (1.15)Using the operators A , B and B ∗ , the control plant (1.2) can be written abstractly ˙ w ( · , t ) = ( ˜ A + µ ) w ( · , t ) + Bu ( · , t ) , t > ,y ( · , t ) = B ∗ w ( · , t ) , t ≥ . (1.16)The rest of the paper is organized as follows: In Section 2, we give a spectral truncationstabilizer that will be used in the full state feedback design in Section 3. The exponential stabilityof the closed-loop system is also proved in Section 3. Section 4 gives some preliminary resultsabout the observer design. Section 5 is devoted to the observer design and its well-posedness proof.Section 6 presents some numerical simulations, followed up conclusions in Section 7. For the sakeof readability, some results that are less relevant to the feedback or observer design are arrangedin the Appendix.Throughout the paper, the identity matrix on the space R n will be denoted by I n . The spaceof bounded linear operators from X to X is denoted by L ( X , X ). The space of bounded linearoperators from X to itself is denoted by L ( X ). The spectrum, resolvent set and the domain of theoperator A are denoted by σ ( A ), ρ ( A ), and D ( A ), respectively.4 A spectral truncation stabilizer
This section is devoted to the preliminaries on the state feedback design. Suppose that p ∈ L (Γ )such that Z Γ p ( x ) φ j ( x ) dx = 0 , j = 1 , , · · · , N, (2.1)where φ j is given by (1.7) and N is an integer that satisfies (1.8). The existence of such a function p is trivial and is given by Lemma 8.1 in Appendix. In terms of the function p , we can define theoperator P p : R → L (Ω) by P p θ = ζ p , ∀ θ ∈ R , (2.2)where ζ p is the solution of the following system: ∆ ζ p = θζ p in Ω ,ζ p ( x ) = 0 , x ∈ Γ , ∂ζ p ( x ) ∂ν = p ( x ) , x ∈ Γ . (2.3) Lemma 2.1.
In addition to Assumption 1.1, suppose that p ∈ L (Γ ) satisfies (2.1) and supposethat θ ∈ R satisfies θ = λ j , j = 1 , , · · · , N. (2.4) Then, the operator P p defined by (2.2) satisfies h P p θ, φ j i L (Ω) = 0 , j = 1 , , · · · , N. (2.5) Proof.
It follows from (1.7), (2.2) and (2.3) that θ Z Ω ζ p ( x ) φ j ( x ) dx = Z Ω ∆ ζ p ( x ) φ j ( x ) dx = Z Γ p ( x ) φ j ( x ) dx − Z Ω ∇ ζ p ( x ) ∇ φ j ( x ) dx = Z Γ p ( x ) φ j ( x ) dx + λ j Z Ω ζ p ( x ) φ j ( x ) dx, (2.6)which yields Z Ω ζ p ( x ) φ j ( x ) dx = 1 θ − λ j Z Γ p ( x ) φ j ( x ) dx = 0 , j = 1 , , · · · , N. (2.7)The proof is complete due to (2.1).For any θ ∈ R , we consider the stabilization of system ( A + µ, P p θ ) that is associated with thefollowing system: z t ( x, t ) = ∆ z ( x, t ) + µz ( x, t ) + ( P p θ )( x ) u ( t ) , x ∈ Ω , t > ,z ( x, t ) = 0 , x ∈ Γ , ∂z ( x, t ) ∂ν = 0 , x ∈ Γ , (2.8)5here p ∈ L (Γ ) satisfies (2.1) and u is a scalar control. Since { φ j ( · ) } ∞ j =1 defined by Assumption1.1 forms an orthonormal basis for L (Ω), the function P p θ and the solution z ( · , t ) of (2.8) can berepresented respectively as P p θ = ∞ X k =1 f k φ k , f k = Z Ω ( P p θ )( x ) φ k ( x ) dx, k = 1 , , · · · (2.9)and z ( · , t ) = ∞ X k =1 z k ( t ) φ k ( · ) , z k ( t ) = Z Ω z ( x, t ) φ k ( x ) dx, k = 1 , , · · · . (2.10)Inspired by [2, 16] and similarly to [3], system (2.8) can be stabilized by the finite-dimensionalspectral truncation technique. Actually, by a simple computation, it follows that˙ z k ( t ) = Z Ω z t ( x, t ) φ k ( x ) dx = Z Ω [∆ z ( x, t ) + µz ( x, t ) + ( P p θ )( x ) u ( t )] φ k ( x ) dx = ( λ k + µ ) z k ( t ) + f k u ( t ) . (2.11)Since z k ( t ) is stable for all k > N , where N is given by (1.8), it is therefore sufficient to consider z k ( t ) for k ≤ N , which satisfy the following finite-dimensional system:˙ Z N ( t ) = Λ N Z N ( t ) + F N u ( t ) , Z N ( t ) = ( z ( t ) , · · · , z N ( t )) ⊤ , (2.12)where Λ N and F N are defined by Λ N = diag( λ + µ, · · · , λ N + µ ) ,F N = ( f , f , · · · , f N ) ⊤ . (2.13)In this way, the stabilization of system (2.8) amounts to stabilizing the finite-dimensional system(2.12). Lemma 2.2.
In addition to Assumption 1.1, suppose that p ∈ L (Γ ) satisfies (2.1) and supposethat θ ∈ R satisfies (2.4). Then, there exists an L N = ( l , l , · · · , l N ) ∈ L ( R N , R ) such that Λ N + F N L N is Hurwitz, where Λ N and F N are defined by (2.13). Moreover, the operator A + µ +( P p θ ) K generates an exponentially stable C -semigroup on L (Ω) , where P p θ is given by (2.2) and K is given by K : g → Z Ω g ( x ) " N X k =1 l k φ k ( x ) dx, ∀ g ∈ L (Ω) . (2.14) Proof.
Owing to (2.1), it follows from Lemma 2.1 that (2.5) holds. By Lemma 8.2 in Appendix,the pair (Λ N , F N ) is controllable and hence, there exists a vector L N = ( l , l , · · · , l N ) such thatΛ N + F N L N is Hurwitz.Since A + µ generates an analytic semigroup e ( A + µ ) t on L (Ω) and ( P p θ ) K ∈ L ( L (Ω)), it followsfrom [15, Corollary 2.3, p.81] that A + µ + ( P p θ ) K also generates an analytic semigroup on L (Ω).As a result, the proof will be accomplished if we can show that σ ( A + µ +( P p θ ) K ) ⊂ { s | Re( s ) < } .6or any λ ∈ σ ( A + µ + ( P p θ ) K ), we consider the characteristic equation ( A + µ + ( P p θ ) K ) g = λg with g = 0.When g ∈ Span { φ , φ , · · · , φ N } , there exist g , g , · · · , g N ∈ R such that g = P Nj =1 g j φ j . Thecharacteristic equation becomes N X j =1 g j ( A + µ ) φ j + P p θ N X j =1 g j Kφ j = N X j =1 λg j φ j . (2.15)Since ( A + µ ) φ j = ( λ j + µ ) φ j and Kφ j = Z φ j ( x ) " N X k =1 l k φ k ( x ) dx = l j , j = 1 , , · · · , N, (2.16)the equation (2.15) takes the form N X j =1 g j ( λ j + µ ) φ j + P p θ N X j =1 g j l j = N X j =1 λg j φ j . (2.17)Take the inner product with φ k , k = 1 , , · · · , N on equation (2.17) to obtain g k ( λ k + µ ) + f k N X j =1 g j l j = λg k , k = 1 , , · · · , N, (2.18)which, together with (2.13), leads to( λ − Λ N − F N L N ) g g ... g N = 0 . (2.19)Since ( g , g , · · · , g N ) = 0, we have Det( λ − Λ N − F N L N ) = 0 . (2.20)Hence, λ ∈ σ (Λ N + F N L N ) ⊂ { s | Re( s ) < } , since Λ N + F N L N is Hurwitz.When g / ∈ Span { φ , φ , · · · , φ N } , there exists a j > N such that Z g ( x ) φ j ( x ) dx = 0. Takethe inner product with φ j on equation ( A + µ + P p θK ) g = λg to get( λ j + µ ) Z g ( x ) φ j ( x ) dx = λ Z g ( x ) φ j ( x ) dx, (2.21)which implies that λ = λ j + µ <
0. Therefore, λ ∈ σ ( A + µ + ( P p θ ) K ) ⊂ { s | Re( s ) < } . Theproof is complete. 7 State feedback
This section is devoted to the stabilization of system (1.2). Inspired by [3], we consider the followingdynamics feedback: u ( x, t ) = v ( x, t ) , x ∈ Γ ,v t ( · , t ) = − αv ( · , t ) + B v u v ( t ) in L (Γ ) , t ≥ , (3.1)where α > u v ( t ) ∈ R is a new scalar control to be designed and theoperator B v ∈ L ( R , L (Γ )) is given by B v c = cp ( · ) , ∀ c ∈ R , (3.2)with p ∈ L (Γ ) satisfying (2.1). Under the controller (3.1), the control plant (1.16), or equivalently(1.2), turns to be ˙ w ( · , t ) = ( ˜ A + µ ) w ( · , t ) + Bv ( · , t ) in H − (Ω) ,v t ( · , t ) = − αv ( · , t ) + B v u v ( t ) in L (Γ ) . (3.3)Since (3.3) is a cascade system, the “ v -part” can be regarded as the actuator dynamics of thecontrol plant w -system. As a result, we can stabilize system (3.3) by the newly developed actuatordynamics compensation approach in [3]. To demonstrate the key idea of controller design clearly,we first consider the following finite-dimensional example. Example 3.1.
Consider the following system in the state space R n × R : ˙ x ( t ) = Ax ( t ) + Bx ( t ) , ˙ x ( t ) = − αx ( t ) + B u ( t ) , α > , (3.4)where A ∈ R n × n , B ∈ R n , B ∈ R and u ( t ) is the control. By [3], if we choose S specially such that AS + αS = B, (3.5)then system (3.4) can be decoupled by the block-upper-triangular transformation: I n S ! A B − α ! I n S ! − = A − α ! . (3.6)Hence, the controllability of the following pairs is equivalent: A B − α ! , B !! and A − α ! , SB B !! . (3.7)Owing to the block-diagonal structure, the stabilization of the second system of (3.7) is much easierthan the first one. As a consequence of this fact, the controller u ( t ) in (3.4) can be designed bystabilizing system ( A, SB ): u ( t ) = ( K, I n S ! x ( t ) x ( t ) ! = KSx ( t ) + Kx ( t ) , (3.8)8here K ∈ R × n is chosen to make A + SB K Hurwitz. Under the feedback (3.8), we obtain theclosed-loop of system (3.4): ˙ x ( t ) = Ax ( t ) + Bx ( t ) , ˙ x ( t ) = ( B KS − α ) x ( t ) + B Kx ( t ) , (3.9)which is stable due to the Hurwitz matrix A + SB K and the similarity A BB K B KS − α ! ∼ A + SB K B K − α ! . (3.10)To sum up, the feedback of system (3.4) can be designed by the following scheme: (i), solvethe equation (3.5) to get S ; (ii), choose K such that A + SB K is Hurwitz; (iii), let u ( t ) = KSx ( t ) + Kx ( t ).Now, we return to the feedback design of system (3.3). Inspired by Example 3.1, the controllercan be designed as u v ( t ) = Kw ( · , t ) + KSv ( t ) , (3.11)where S ∈ L ( L (Γ ) , L (Ω)) solves the Sylvester equation( ˜ A + µ ) S + αS = B, (3.12)and K ∈ L ( L (Ω) , R ) stabilizes system ( A + µ, SB v ) exponentially in the sense of [22]. Lemma 3.1.
Let A and B be given by (1.3) and (1.12), respectively. Suppose that B v ∈ L ( R , L (Γ )) is given by (3.2) with p ∈ L (Γ ) satisfying (2.1) and suppose that α + µ ∈ ρ ( − A ) . (3.13) Then, the solution of Sylvester equation (3.12) satisfies Sg = − ϕ g ∈ L (Ω) , ∀ g ∈ L (Γ ) , (3.14) where ϕ g is given by ∆ ϕ g = ( − α − µ ) ϕ g in Ω ,ϕ g ( x ) = 0 , x ∈ Γ , ∂ϕ g ( x ) ∂ν = g ( x ) , x ∈ Γ . (3.15) Moreover, for any c ∈ R , we have SB v c = − cP p θ with θ = − α − µ, (3.16) where P p : R → L (Ω) is given by (2.2). roof. Owing to (3.13), we solve (3.12) to get S = ( α + µ + ˜ A ) − B. (3.17)By a straightforward computation, it follows that( α + µ + ˜ A ) ϕ g = ( α + µ + ˜ A ) ϕ g − ˜ A Υ g + ˜ A Υ g = ( α + µ ) ϕ g + ˜ A ( ϕ g − Υ g ) + ˜ A Υ g = ( α + µ ) ϕ g + ∆( ϕ g − Υ g ) + ˜ A Υ g = ˜ A Υ g = − Bg, (3.18)which, together with (3.17), leads to (3.14) easily.By (3.2) and (3.14), we have SB v c = − cϑ , where ∆ ϑ = ( − α − µ ) ϑ in Ω ,ϑ ( x ) = 0 , x ∈ Γ , ∂ϑ ( x ) ∂ν = p ( x ) , x ∈ Γ . (3.19)In view of (2.3) and letting θ = − α − µ , we can obtain (3.16) easily. The proof is complete.By Lemmas 2.2 and 3.1, the operator − K ∈ L ( L (Ω) , R ) defined by (2.14) stabilizes system( A + µ, SB v ) exponentially. As a result, the controller (3.11) turns to be u v ( t ) = − Z Ω [ w ( x, t ) − ϕ v ( x, t )] " N X k =1 l k φ k ( x ) dx, (3.20)where ∆ ϕ v ( · , t ) = ( − α − µ ) ϕ v ( · , t ) in Ω ,ϕ v ( x, t ) = 0 , x ∈ Γ , ∂ϕ v ( x, t ) ∂ν = v ( x, t ) , x ∈ Γ . (3.21)By (3.20) and (3.3), we obtain the closed-loop system ˙ w ( · , t ) = ( ˜ A + µ ) w ( · , t ) + Bv ( · , t ) in Ω ,v t ( · , t ) = − αv ( · , t ) − B v Z Ω [ w ( x, t ) − ϕ v ( x, t )] " N X k =1 l k φ k ( x ) dx in Γ , ∆ ϕ v ( · , t ) = ( − α − µ ) ϕ v ( · , t ) in Ω ,ϕ v ( x, t ) = 0 , x ∈ Γ , ∂ϕ v ( x, t ) ∂ν = v ( x, t ) , x ∈ Γ . (3.22)10ombining (3.2), (1.12) and (1.3), system (3.22) turns to be w t ( x, t ) = ∆ w ( x, t ) + µw ( x, t ) , x ∈ Ω ,w ( x, t ) = 0 , x ∈ Γ , ∂w ( x, t ) ∂ν = v ( x, t ) , x ∈ Γ ,v t ( · , t ) = − αv ( · , t ) − p ( · ) Z Ω [ w ( x, t ) − ϕ v ( x, t )] " N X k =1 l k φ k ( x ) dx in Γ , ∆ ϕ v ( · , t ) = ( − α − µ ) ϕ v ( · , t ) in Ω ,ϕ v ( x, t ) = 0 , x ∈ Γ , ∂ϕ v ( x, t ) ∂ν = v ( x, t ) , x ∈ Γ . (3.23) Theorem 3.1.
In addition to Assumption 1.1, suppose that p ∈ L (Γ ) satisfies (2.1) and α + µ + λ j = 0 , j = 1 , , · · · , N. (3.24) Then, there exists an L N = ( l , l , · · · , l N ) ∈ L ( R N , R ) such that Λ N + F N L N is Hurwitz, where Λ N and F N are defined by (2.13). Moreover, for any ( w ( · , , v ( · , ⊤ ∈ L (Ω) × L (Γ ) , system (3.23)admits a unique solution ( w, v ) ⊤ ∈ C ([0 , ∞ ); L (Ω) × L (Γ )) that decays to zero exponentially in L (Ω) × L (Γ ) as t → ∞ . Moreover, if the initial state ( w ( · , , v ( · , ⊤ ∈ D ( A ) × L (Γ ) , thesolution ( w, v ) ⊤ ∈ C ([0 , ∞ ); L (Ω) × L (Γ )) is classical.Proof. Notice that (2.14), the closed-loop system (3.22) can be written as the abstract form: ddt ( w ( · , t ) , v ( · , t )) ⊤ = A ( w ( · , t ) , v ( · , t )) ⊤ , (3.25)where the operator A : D ( A ) ⊂ L (Ω) × L (Γ ) → L (Ω) × L (Γ ) is defined by A = A + µ B − B v K − B v KS − α ! with D ( A ) = D ( A ) × L (Γ ) . (3.26)As proposed in [3] and similarly to (3.6), we introduce the following transformation: S ( f, g ) ⊤ = ( f + Sg, g ) ⊤ , ( f, g ) ⊤ ∈ L (Ω) × L (Γ ) , (3.27)where S ∈ L ( L (Γ ) , L (Ω)) solves the Sylvester equation (3.12). By a simple computation, S ∈L ( L (Ω) × L (Γ )) is invertible and its inverse is S − ( f, g ) ⊤ = ( f − Sg, g ) ⊤ , ( f, g ) ⊤ ∈ L (Ω) × L (Γ ) . (3.28)Moreover, (see, e.g., [3, Theorem 5.1]) S A S − = A S , D ( A S ) = S D ( A ) , (3.29)where A S = A + µ − SB v K B v K − α ! , (3.30)11 B v is given by (3.16) and K is given by (2.14). Since SB v = − P p θ with θ = − α − µ , it follows fromLemma 2.2 that the operator A + µ + ( P p θ ) K = A + µ − SB v K generates an exponentially stable C -semigroup on L (Ω). Owing to the block-triangle structure and [3, Lemma 3.2], the operator A S generates an exponentially stable C -semigroup e A S t on L (Ω) × L (Γ ). As a result, the operator A generates an exponentially stable C -semigroup on L (Ω) × L (Γ ) due to the similarity (3.29). This section is devoted to the preliminaries on the observer design. Let q ∈ L (Γ ) satisfy Z Γ q ( x ) φ j ( x ) = 0 , j = 1 , , · · · , N, (4.1)where φ j is given by (1.7) and N is an integer that satisfies (1.8). For any γ ∈ R , define theoperator J γq : L (Ω) → R by J γq ( g ) = − Z Γ q ( x ) ξ g ( x ) dx, ∀ g ∈ L (Ω) , (4.2)where ξ g is given by ∆ ξ g = γξ g + g in Ω ,ξ g ( x ) = 0 , x ∈ Γ , ∂ξ g ( x ) ∂ν = 0 , x ∈ Γ . (4.3) Lemma 4.1.
Let { ( φ j , λ j ) } ∞ j =1 be given by (1.7) and N be an integer that satisfies (1.8). Supposethat q ∈ L (Γ ) satisfies (4.1) and suppose that γ ∈ R satisfies γ = λ j , j = 1 , , · · · , N. (4.4) Then, the operator J γq defined by (4.2) satisfies J γq ( φ j ) = 0 , j = 1 , , · · · , N. (4.5) Proof.
Let η q be a solution of the following system ∆ η q = γη q in Ω ,η q ( x ) = 0 , x ∈ Γ , ∂η q ( x ) ∂ν = q ( x ) , x ∈ Γ . (4.6)Then, for any g ∈ L (Ω), it follows from (4.3) and (4.6) that γ h η q , ξ g i L (Ω) = h ∆ η q , ξ g i L (Ω) = Z Γ ∂η q ( x ) ∂ν ξ g ( x ) dx − h∇ η q , ∇ ξ g i L (Ω) = Z Γ q ( x ) ξ g ( x ) dx + γ h η q , ξ g i L (Ω) + h g, η q i L (Ω) , (4.7)which yields J γq ( φ j ) = h φ j , η q i L (Ω) , j = 1 , , · · · , N. (4.8)12n the other hand, λ j h φ j , η q i L (Ω) = h ∆ φ j , η q i L (Ω) = −h∇ φ j , ∇ η q i L (Ω) = − Z Γ ∂η q ( x ) ∂ν φ j ( x ) dx + h ∆ η q , φ j i L (Ω) = − Z Γ q ( x ) φ j ( x ) dx + γ h η q , φ j i L (Ω) , j = 1 , , · · · , N. (4.9)That is Z Γ q ( x ) φ j ( x ) dx = ( γ − λ j ) h η q , φ j i L (Ω) , j = 1 , , · · · , N. (4.10)Combining (4.8), (4.10) and (4.1), we obtain (4.5) easily.Next, we will find K to detect system ( A + µ, J γq ) exponentially in the sense of [22]. Define therow vector J N = ( J γq ( φ ) , J γq ( φ ) , · · · , J γq ( φ N )) , (4.11)where φ i is given by (1.7), i = 1 , , · · · , N and N is an integer that satisfies (1.8). By Lemma 4.1and Lemma 8.2 in Appendix, the finite-dimensional system (Λ N , J N ) is observable, where Λ N isgiven by (2.13). As a result, there exists a vector K N = ( k , k , · · · , k N ) ⊤ such that Λ N + K N J N is Hurwitz. Lemma 4.2.
Suppose that the operator A is given by (1.3), the eigenpairs { ( φ j ( · ) , λ j ) } ∞ j =1 satisfy(1.7), q ( · ) ∈ L (Γ ) satisfies (4.1), the integer N satisfies (1.8) and µ > . For any γ ∈ R satisfying(4.4), let J γq : L (Ω) → R be given by (4.2) and J N be given by (4.11). Then, there exists a vector K N = ( k , k , · · · , k N ) ⊤ such that Λ N + K N J N is Hurwitz, where Λ N is given by (2.13). Moreover,the operator A + µ + KJ γq generates an exponentially stable C -semigroup on L (Ω) , where theoperator K : R → L (Ω) is given by Kc = c N X j =1 k j φ j ( · ) , ∀ c ∈ R . (4.12) Proof.
Owing to (4.1), it follows from Lemma 4.1 that (4.5) holds. By Lemma 8.2 in Appendix, thepair (Λ N , J N ) is observable and there exists a vector K N = ( k , k , · · · , k N ) ⊤ such that Λ N + K N J N is Hurwitz.Since A + µ generates an analytic semigroup e ( A + µ ) t on L (Ω) and KJ γq is bounded, it followsfrom [15, Corollary 2.3, p.81] that A + µ + KJ γq also generates an analytic semigroup on L (Ω).The proof will be accomplished if we can show that σ ( A + µ + KJ γq ) ⊂ { s | Re( s ) < } . For any λ ∈ σ ( A + µ + KJ γq ), we consider the characteristic equation ( A + µ + KJ γq ) g = λg with g = 0.When g ∈ Span { φ , φ , · · · , φ N } , set g = P Nj =1 g j φ j . The characteristic equation becomes N X j =1 ( λ j + µ ) g j φ j + N X j =1 g j J γq ( φ j ) N X j =1 k j φ j = N X j =1 λg j φ j . (4.13)13ake the inner product with φ i , i = 1 , , · · · , N on equation (4.13) to obtain( λ i + µ ) g i + k i N X j =1 g j J γq ( φ j ) = λg i , i = 1 , , · · · , N, (4.14)which, together with (2.13) and (4.11), leads to( λ − Λ N − K N J N ) g g ... g N = 0 . (4.15)Since ( g , g , · · · , g N ) = 0, we have Det( λ − Λ N − K N J N ) = 0 . (4.16)Hence, λ ∈ σ (Λ N + K N J N ) ⊂ { s | Re( s ) < } , since Λ N + K N J N is Hurwitz.When g / ∈ Span { φ , φ , · · · , φ N } , there exists a j > N such that Z g ( x ) φ j ( x ) dx = 0. Takethe inner product with φ j on equation ( A + µ + KJ γq ) g = λg to get( λ j + µ ) Z g ( x ) φ j ( x ) dx = λ Z g ( x ) φ j ( x ) dx, (4.17)which, together with (1.8), implies that λ = λ j + µ <
0. Therefore, λ ∈ σ ( A + µ + KJ γq ) ⊂{ s | Re( s ) < } . The proof is complete. This section is devoted to the observer design by the newly proposed approach in [4]. Instead ofthe system (1.2), we design the observer for the following system: w t ( x, t ) = ∆ w ( x, t ) + µw ( x, t ) , x ∈ Ω ,w ( x, t ) = 0 , x ∈ Γ , ∂w ( x, t ) ∂ν = u ( x, t ) , x ∈ Γ ,v t ( x, t ) = − βv ( x, t ) + QB ∗ w ( x, t ) , x ∈ Γ ,y v ( t ) = Z Γ v ( x, t ) dx, (5.1)where β > v ( · , t ) is an extended state, y v is a new output, B ∗ is given by(1.15) and Q ∈ L ( L (Γ )) is given by( Qg )( x ) = q ( x ) g ( x ) , x ∈ Γ , ∀ g ∈ L (Γ ) (5.2)with q ∈ L (Γ ) satisfying (4.1). By (1.3) and (1.12), system (5.1) can be written as w t ( · , t ) = ( ˜ A + µ ) w ( · , t ) + Bu ( · , t ) ,v t ( · , t ) = − βv ( · , t ) + QB ∗ w ( · , t ) ,y v ( t ) = C v v ( · , t ) , (5.3)14here C v : L (Γ ) → R is defined by C v h = Z Γ h ( x ) dx, ∀ h ∈ L (Γ ) . (5.4)Now we demonstrate the key idea of the observer design via a simple finite-dimensional example. Example 5.1.
Consider the following system in the state space R n × R : ˙ x ( t ) = Ax ( t ) + Bu ( t ) , ˙ x ( t ) = − βx ( t ) + QB ∗ x ( t ) ,y ( t ) = x ( t ) , β > , (5.5)where A ∈ R n × n is the system matrix, B ∈ R n is the control matrix, Q ∈ R is a constant, u ( t ) isthe control and y ( t ) is the measurement. The Luenberger observer of system (5.5) is designed as ˙ˆ x ( t ) = A ˆ x ( t ) + F [ x ( t ) − ˆ x ( t )] + Bu ( t ) , ˙ˆ x ( t ) = − β ˆ x ( t ) + QB ∗ ˆ x ( t ) − F [ x ( t ) − ˆ x ( t )] , (5.6)where F ∈ R n and F ∈ R are the gain parameters to be determined. To demonstrate the key ideaof the observer design for the infinite-dimensional system (5.1), we will find a new way to choose F and F rather than the conventional pole placement theorem. Let˜ x j ( t ) = x j ( t ) − ˆ x j ( t ) , j = 1 , , (5.7)then the error is governed by ˙˜ x ( t ) = A ˜ x ( t ) − F ˜ x ( t ) , ˙˜ x ( t ) = − β ˜ x ( t ) + QB ∗ ˜ x ( t ) + F ˜ x ( t ) . (5.8)If we pick F and F properly such that system (5.8) is stable, then ( x , x ) can be estimated inthe sense that k (ˆ x ( t ) − x ( t ) , ˆ x ( t ) − x ( t )) k R n × R → t → ∞ . (5.9)Inspired by [4], the F and F can be chosen easily by decoupling the system (5.8) as a cascadesystem. Consider the following transformation I n P ! A − F QB ∗ F − β ! I n P ! − = A + F P − F P ( A + F P ) + QB ∗ − ( F − β ) P F − β − P F ! , (5.10)where P ∈ R × n to be determined. If we choose F = P F and P A + QB ∗ + βP = 0 , (5.11)15hen the matrix on the right side of (5.10) is reduced to A + F P − F − β ! , (5.12)which is obviously a Hurwitz matrix provided A + F P is Hurwitz. To sum up, the tuning parameters F and F can be chosen by the following scheme: (i), solve the equation P A + QB ∗ + βP = 0 toget P ; (ii), choose F such that A + F P is Hurwitz; (iii), let F = P F .Now, we return to design an observer for system (5.3). Inspired by Example 5.1, the observerof system (5.3) can be designed as ˆ w t ( x, t ) = ∆ ˆ w ( x, t ) + µ ˆ w ( x, t ) + K [ C v v ( · , t ) − C v ˆ v ( · , t )] , x ∈ Ω , ˆ w ( x, t ) = 0 , x ∈ Γ , ∂ ˆ w ( x, t ) ∂ν = u ( x, t ) , x ∈ Γ , ˆ v t ( · , t ) = − β ˆ v ( · , t ) + QB ∗ ˆ w ( · , t ) − L [ C v v ( · , t ) − C v ˆ v ( · , t )] in Γ , (5.13)where K and L are tuning parameters that can be chosen by the following scheme: • Solve the following equation βP + P ( A + µ ) + QB ∗ = 0 (5.14)to get P ∈ L ( L (Ω) , L (Γ )); • Find K to detect system ( A + µ, C v P ); • Let L = P K .By a straightforward computation, the solution of (5.14) is found to be P = − QB ∗ ( β + µ + A ) − ∈ L ( L (Ω) , L (Γ )) . (5.15)By (1.15), (5.4), (5.2) and (4.2), we have C v P = J γq ∈ L ( L (Ω) , R ) with γ = − β − µ. (5.16)By Lemma 4.2, (4.11) and (2.13), the operator K can be chosen by (4.12), where ( k , k , · · · , k N ) ⊤ is a vector such that Λ N + ( k , k , · · · , k N ) ⊤ J N is Hurwitz. As a result of (4.12), (5.2) and (5.15), L = P K = N X j =1 k j P φ j = − N X j =1 k j QB ∗ ( β + µ + A ) − φ j = − N X j =1 k j q ( x ) ξ j ( x ) , x ∈ Γ , (5.17)where ( β + µ + ∆) ξ j = φ j in Ω ,ξ j ( x ) = 0 , x ∈ Γ , ∂ξ j ( x ) ∂ν = 0 , x ∈ Γ . j = 1 , , · · · , N. (5.18)16ombining (4.12) and (5.17), the observer (5.13) turns to be ˆ w t ( x, t ) = ∆ ˆ w ( x, t ) + µ ˆ w ( x, t ) + [ C v v ( · , t ) − C v ˆ v ( · , t )] N X j =1 k j φ j ( x ) , x ∈ Ω , ˆ w ( x, t ) = 0 , x ∈ Γ , ∂ ˆ w ( x, t ) ∂ν = u ( x, t ) , x ∈ Γ , ˆ v t ( x, t ) = − β ˆ v ( x, t ) + QB ∗ ˆ w ( x, t ) + N X j =1 k j q ( x ) ξ j ( x )[ C v v ( · , t ) − C v ˆ v ( · , t )] , x ∈ Γ , (5.19)where ξ j is given by (5.18), j = 1 , , · · · , N . By (4.12) and (5.17), the observer can be written asthe abstract form: ddt ( ˆ w ( · , t ) , ˆ v ( · , t )) ⊤ = A ( ˆ w ( · , t ) , ˆ v ( · , t )) ⊤ + ( K, − L ) ⊤ C v v ( · , t ) , (5.20)where the operator A : D ( A ) ⊂ L (Ω) × L (Γ ) → L (Ω) × L (Γ ) is defined by A = A + µ − KC v QB ∗ LC v − β ! with D ( A ) = D ( A ) × L (Γ ) . (5.21) Theorem 5.1.
Suppose that the operator A is given by (1.3), B ∗ is given by (1.15), the eigenpairs { ( φ j ( · ) , λ j ) } ∞ j =1 are given by (1.7) and Q is given by (5.2) with q ( · ) ∈ L (Γ ) satisfying (4.1). Letthe integer N satisfy (1.8) and µ, β > satisfy − β − µ = λ j , j = 1 , , · · · , N. (5.22) Then, for any ( w ( · , , v ( · , , ˆ w ( · , , ˆ v ( · , ⊤ ∈ [ L (Ω) × L (Γ )] and u ∈ L ([0 , ∞ ); L (Γ )) , theobserver (5.19) of system (5.1) admits a unique solution ( ˆ w, ˆ v ) ⊤ ∈ C ([0 , ∞ ); L (Ω) × L (Γ )) suchthat e ωt k ( w ( · , t ) − ˆ w ( · , t ) , v ( · , t ) − ˆ v ( · , t )) k L (Ω) × L (Γ ) → as t → ∞ , (5.23) where ω is a positive constant that is independent of t .Proof. For any ( w ( · , , v ( · , ⊤ ∈ L (Ω) × L (Γ ) and u ∈ L ([0 , ∞ ); L (Γ )), it is well knownthat the control plant (5.1) admits a unique solution ( w, v ) ⊤ ∈ C ([0 , ∞ ); L (Ω) × L (Γ )) such that y v ∈ L [0 , ∞ ). Let ˜ w ( x, t ) = w ( x, t ) − ˆ w ( x, t ) , x ∈ Ω , ˜ v ( s, t ) = v ( s, t ) − ˆ v ( s, t ) , s ∈ Γ , t ≥ . (5.24)Then, the errors are governed by ˜ w t ( x, t ) = ∆ ˜ w ( x, t ) + µ ˜ w ( x, t ) − C v ˜ v ( · , t ) N X j =1 k j φ j ( x ) , x ∈ Ω , ˜ w ( x, t ) = 0 , x ∈ Γ , ∂ ˜ w ( x, t ) ∂ν = 0 , x ∈ Γ , ˜ v t ( x, t ) = − β ˜ v ( x, t ) + QB ∗ ˜ w ( x, t ) − C v ˜ v ( · , t ) N X j =1 k j q ( x ) ξ j ( x ) , x ∈ Γ . (5.25)17y (5.21), (5.17) and (4.12), system (5.25) can be written abstractly ddt ( ˜ w ( · , t ) , ˜ v ( · , t )) ⊤ = A ( ˜ w ( · , t ) , ˜ v ( · , t )) ⊤ . (5.26)Inspired by [4], we introduce the following transformation P ( f, g ) ⊤ = ( f, g + P f ) ⊤ , ( f, g ) ⊤ ∈ L (Ω) × L (Γ ) , (5.27)where P ∈ L ( L (Ω) , L (Γ )) is the solution of system (5.14). Then P is invertible and its inverse isgiven by P − ( f, g ) ⊤ = ( f, g − P f ) ⊤ , ( f, g ) ⊤ ∈ L (Ω) × L (Γ ) . (5.28)Moreover, a simple computation shows that (see, e.g., [4, Theorem 6.1]) P A P − = A P , D ( A P ) = P D ( A ) , (5.29)where A P = A + µ + KC v P − KC v − β ! with D ( A P ) = D ( A ) × L (Γ ) . (5.30)By Lemma 4.2 and (5.16), the operator A + µ + KC v P generates an exponentially stable C -semigroup on L (Ω). Thanks to the block-triangle structure and [3], the operator A P generatesan exponentially stable C -semigroup e A P t on L (Ω) × L (Γ ). By virtue of the similarity (5.29),the operator A also generates an exponentially stable C -semigroup e A t on L (Ω) × L (Γ ). As aresult, the error system with initial state ( ˜ w ( · , , ˜ v ( · , ⊤ = ( w ( · , − ˆ w ( · , , v ( · , − ˆ v ( · , ⊤ ∈ L (Ω) × L (Γ ) admits a unique solution ( ˜ w, ˜ v ) ⊤ ∈ C ([0 , ∞ ); L (Ω) × L (Γ )) such that e ωt k ( ˜ w ( · , t ) , ˜ v ( · , t )) k L (Ω) × L (Γ ) → t → ∞ , (5.31)where ω is a positive constant that is independent of t . Let( ˆ w ( · , t ) , ˆ v ( · , t )) = ( w ( · , t ) − ˜ w ( · , t ) , v ( · , t ) − ˜ v ( · , t )) . (5.32)Then, a straightforward computation shows that such a defined ( ˆ w, ˆ v ) ⊤ ∈ C ([0 , ∞ ); L (Ω) × L (Γ ))is a solution of system (5.20) or equivalently, system (5.19). Moreover, (5.23) holds due to (5.31)and (5.24). Owing to the linearity of system (5.19), the solution is unique. In this section, we present some numerical simulations for the closed-loop system (3.23) to demon-strate the theoretical results visually. In order to avoid the difficulty of numerical discretization, weconsider the unstable heat system in the rectangular domain Ω = (cid:8) ( x, y ) ∈ R | < x < , < y < (cid:9) .The actuator is installed on the boundaryΓ = (cid:8) ( x, y ) ∈ R | x = 1 , ≤ y ≤ (cid:9) ∪ (cid:8) ( x, y ) ∈ R | y = 1 , ≤ x ≤ (cid:9) . = (cid:8) ( x, y ) ∈ R | x = 0 , ≤ y < (cid:9) ∪ (cid:8) ( x, y ) ∈ R | y = 0 , ≤ x < (cid:9) . We adopt the finite difference scheme to discretize system (3.23) directly. The numerical resultsare programmed in Matlab. Inspired by [14] where the uniform exponential decay with respect tothe mesh size is obtained by the finite difference method, the space step h and time step τ aretaken as h = τ = 0 .
05. The initial state and tuning parameters are chosen as w ( x, y,
0) = x sin 2 πy, v ( x, y,
0) = 0 ,p ( x, y ) = sin x sin y, µ = 6 , α = 3 , N = 1 , l = 15 . (6.1)By a simple numerical computation, the largest eigenvalue of the operator (1.3) on Ω is λ ≈− . λ + µ > -10-0.500.51 y time =0 x (a) w ( x, y, w ( x, y, Figure 1: The initial state and the final state. (a) w ( x, . , t ) with control. (b) w ( x, . , t ) without control. Figure 2: State trace with control and state trace without control.order to demonstrate the dynamic evolution of the closed-loop system, the state trace w ( x, . , t ) isplotted in Figure 2(a). The same state trace without control is plotted in Figure 2(b) for comparison.19he distributed control traces v ( x, y, t ) are plotted in Figure 3(a) and 3(b). To demonstrate thedecay rate, the logarithmic state norm k w ( · , t ) k L (Ω) decay curve and the curve of the state normitself are plotted in Figure 4(a) and Figure 4(b), respectively. From these Figures 1-4 we observe (a) v ( x, , t ). (b) v (1 , y, t ). Figure 3: Controller traces. t -5 -4 -3 -2 -1 Loga r i t h m o f L no r m o f w ( x , y ,t ) Logarithm of L norm of w(x,y,t) (a) log( k w ( · , t ) k L (Ω) ). t L no r m o f w ( x , y ,t ) L norm of w(x,y,t) (b) k w ( · , t ) k L (Ω) . Figure 4: Decays of the state norm.that the state of the control plant are stabilized effectively despite the presence of the unstablesource term µw ( x, y, t ). Moreover, the dynamic evolution is smooth. Figure 4 implies that thestate norm decays to zero exponentially. So all the convergence in the closed-loop system is veryfast. In this paper, we consider the stabilization and observation for the unstable heat equation in ageneral multi-dimensional domain. The newly developed dynamics compensation approach andthe finite-dimensional spectral truncation technique are exploited to treat the difficulties caused byinstability. Both the full state feedback law and the state observer are designed. The closed-loopsystem and the observation error are convergent to zero exponentially as t → ∞ . The developed20ethod in this paper provides a new choice, in addition to the PDE backstepping method, fordealing with unstable PDEs, especially for multi-dimensional unstable PDEs. It is very interestingto extend this new method to other unstable or anti-stable PDEs such as the multi-dimensionalwave equation and Euler-Bernoulli beam equation, which are our future works. References [1] A. Baccoli, A. Pisano, and Y. Orlov, Boundary control of coupled reaction-diffusion processeswith constant parameters,
Automatica , 54(2015), 80-90.[2] J.M. Coron and E. Tr´elat, Global steady-state controllability of one-dimensional semilinear heatequations,
SIAM Journal on Control and Optimization , 43(2004), 549-569.[3] H. Feng, X.H. Wu and B.Z. Guo, Actuator dynamics compensation in stabilization of abstractlinear systems, arXiv:2008.11333 , https://arxiv.org/abs/2008.11333.[4] H. Feng, X.H. Wu and B.Z. Guo, Dynamics compensation in observation of abstract linearsystems, arXiv:2009.01643 , https://arxiv.org/abs/2009.01643.[5] B.Z. Guo and J.J. Liu, Sliding mode control and active disturbance rejection control to thestabilization of one-dimensional Schr¨odinger equation subject to boundary control matcheddisturbance,
International Journal of Robust and Nonlinear Control , 24(2014), 2194-2212.[6] D.W. Hahn and M.N. ¨Ozisik,
Heat Conduction , John Wiley & Sons, Inc., New Jersey, 2012.[7] M. Krstic, Systematization of approaches to adaptive boundary stabilization of PDEs,
Int. J.Robust Nonlinear Control , 16(2006), 801-818.[8] M. Krstic and A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs-Part I:Lyapunov Design,
IEEE Trans. Automat. Control , 53(2008), 1575-1591.[9] M. Krstic, A. Smyshlyaev, Backstepping boundary control for first order hyperbolic PDEs andapplication to systems with actuator and sensor delays,
Systems Control Lett. , 57(2008), 750-758.[10] M. Krstic, Adaptive control of an anti-stable wave PDE,
Dyn. Contin. Discrete Impuls. Syst.Ser. A Math. Anal. , 17(2010), 853-882.[11] I. Lasiecka and R. Triggiani,
Control Theory for Partial Differential Equations: Continuousand Approximation Theories , Vol. II, Cambridge University Press, Cambridge, 2000.[12] W. Liu, Boundary feedback stabilization of an untable heat equation,
SIAM J. Control Optim. ,42(2003), 1033-1043. 2113] T. Meurer,
Control of Higher Dimensional PDEs: Flatness and Backstepping Designs ,Springer, Berlin, 2012.[14] M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1-D wave equation,
Systems Control Lett. , 48(2003), 261-279.[15] A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations ,Springer-Verlag, New York, 1983.[16] C. Prieur and E. Tr´elat, Feedback stabilization of a 1-D linear reaction-diffusion equation withdelay boundary control,
IEEE Transactions on Automatic Control , 64(2018), 1415-1425.[17] A. Smyshlyaev and M. Krstic, Closed-form boundary state feedbacks for a class of 1-D partialintegro-differential equations,
IEEE Trans. Automat. Control , 49(2004), 2185-2202.[18] A. Smyshlyaev and M. Krstic, backstepping observers for a class of parabolic PDEs,
SystemsControl Lett. , 54(2005), 613-625.[19] A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. II.Estimation-based designs,
Automatica , 43(2007), 1543-1556.[20] A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs, III.Output feedback examples with swapping identifiers,
Automatica , 43(2007), 1557-1564.[21] A. Smyshlyaev, B.Z. Guo, and M. Krstic, Arbitrary decay rate for Euler-Bernoulli beam bybackstepping boundary feedback,
IEEE Trans. Automat. Control , 54(2009), 1134-1140.[22] G. Weiss and R. Curtain, Dynamic stabilization of regular linear systems,
IEEE Transactionson Automatic Control , 42(1997), 4-21.
Lemma 8.1.
For any positive integer N , there exists a function p ∈ L (Γ ) such that (2.1) holds.Proof. Choose p ( x ) = φ ( x ) for any x ∈ Γ . Then h p , φ i L (Γ ) = 0. If h φ , φ i L (Γ ) = 0, thenlet p = p . Otherwise, let p ( x ) = p ( x ) + φ ( x ) . (8.1)Then, h p , φ i L (Γ ) = 0 and h p , φ i L (Γ ) = 0. Suppose that we have obtain p N − such that h p N − , φ j i L (Γ ) = 0 , j = 1 , , , · · · , N − . (8.2)If h p N − , φ N i L (Γ ) = 0, we choose p N = p N − . Otherwise, p N ( x ) = p N − ( x ) + γφ N ( x ) , (8.3)22here γ small enough such that h p N − , φ j i L (Γ ) + γ h φ N , φ j i L (Γ ) = 0 , j = 1 , , · · · , N. (8.4)Therefore, the proof is complete due to the mathematical induction. Lemma 8.2.
For any positive integer N , define Λ N = diag( λ , λ , · · · , λ N ) (8.5) and B N = ( b , b , · · · , b N ) ⊤ , (8.6) where b k = 0 , k = 1 , , · · · , N and λ i = λ j , i = j, i, j = 1 , , · · · , N. (8.7) Then, system (Λ N , B N ) is controllable.Proof. By a simple computation, the controllability matrix of system (Λ N , B N ) is P c = b λ b · · · λ N − b b λ b · · · λ N − b ... ... · · · ... b N λ N b N · · · λ N − N b N . (8.8)Furthermore, | P c | = b b · · · b N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ · · · λ N − λ · · · λ N − ... ... · · · ...1 λ N · · · λ N − N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = b b · · · b N Y ≤ i