Research article

Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation

  • Received: 14 November 2023 Revised: 17 December 2023 Accepted: 23 January 2024 Published: 05 February 2024
  • 35B35, 93D05, 93C73, 93C10

  • We study the well-posedness and stability for a nonlinear Euler-Bernoulli beam equation modeling railway track deflections in the framework of input-to-state stability (ISS) theory. More specifically, in the presence of both distributed in-domain and boundary disturbances, we prove first the existence and uniqueness of a classical solution by using the technique of lifting and the semigroup method, and then establish the Lr-integral input-to-state stability estimate for the solution whenever r[2,+] by constructing a suitable Lyapunov functional with the aid of Sobolev-like inequalities, which are used to deal with the boundary terms. We provide an extensive extension of relevant work presented in the existing literature.

    Citation: Panyu Deng, Jun Zheng, Guchuan Zhu. Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation[J]. Communications in Analysis and Mechanics, 2024, 16(1): 193-216. doi: 10.3934/cam.2024009

    Related Papers:

    [1] Yang Liu, Xiao Long, Li Zhang . Long-time dynamics for a coupled system modeling the oscillations of suspension bridges. Communications in Analysis and Mechanics, 2025, 17(1): 15-40. doi: 10.3934/cam.2025002
    [2] Shuyue Ma, Jiawei Sun, Huimin Yu . Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term. Communications in Analysis and Mechanics, 2023, 15(2): 245-266. doi: 10.3934/cam.2023013
    [3] Pierluigi Colli, Jürgen Sprekels . On the optimal control of viscous Cahn–Hilliard systems with hyperbolic relaxation of the chemical potential. Communications in Analysis and Mechanics, 2025, 17(3): 683-706. doi: 10.3934/cam.2025027
    [4] Anthony Bloch, Marta Farré Puiggalí, David Martín de Diego . Metriplectic Euler-Poincaré equations: smooth and discrete dynamics. Communications in Analysis and Mechanics, 2024, 16(4): 910-927. doi: 10.3934/cam.2024040
    [5] Yang Liu . Global attractors for a nonlinear plate equation modeling the oscillations of suspension bridges. Communications in Analysis and Mechanics, 2023, 15(3): 436-456. doi: 10.3934/cam.2023021
    [6] Wenqian Lv . Ground states of a Kirchhoff equation with the potential on the lattice graphs. Communications in Analysis and Mechanics, 2023, 15(4): 792-810. doi: 10.3934/cam.2023038
    [7] Shu Wang . Global well-posedness and viscosity vanishing limit of a new initial-boundary value problem on two/three-dimensional incompressible Navier-Stokes equations and/or Boussinesq equations. Communications in Analysis and Mechanics, 2025, 17(2): 582-605. doi: 10.3934/cam.2025023
    [8] Caojie Li, Haixiang Zhang, Xuehua Yang . A new $ \alpha $-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation. Communications in Analysis and Mechanics, 2024, 16(1): 147-168. doi: 10.3934/cam.2024007
    [9] Isaac Neal, Steve Shkoller, Vlad Vicol . A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy. Communications in Analysis and Mechanics, 2025, 17(1): 188-236. doi: 10.3934/cam.2025009
    [10] Yao Sun, Pan Wang, Xinru Lu, Bo Chen . A boundary integral equation method for the fluid-solid interaction problem. Communications in Analysis and Mechanics, 2023, 15(4): 716-742. doi: 10.3934/cam.2023035
  • We study the well-posedness and stability for a nonlinear Euler-Bernoulli beam equation modeling railway track deflections in the framework of input-to-state stability (ISS) theory. More specifically, in the presence of both distributed in-domain and boundary disturbances, we prove first the existence and uniqueness of a classical solution by using the technique of lifting and the semigroup method, and then establish the Lr-integral input-to-state stability estimate for the solution whenever r[2,+] by constructing a suitable Lyapunov functional with the aid of Sobolev-like inequalities, which are used to deal with the boundary terms. We provide an extensive extension of relevant work presented in the existing literature.



    The notion of input-to-state stability (ISS) was originally introduced by Sontag in 1989 during the study of nonlinear systems governed by ordinary differential equations (ODEs) [1]. It was mainly used to quantify the influence of external inputs or disturbances on the stability of control systems, and has been proved to be a powerful tool for describing the robustness of nonlinear systems in control theory and applications. In numerous physical and engineering scenarios, external inputs or disturbances often exhibit unbounded characteristics. To provide a more comprehensive understanding of the stability of nonlinear systems subjected to such external influences, Sontag introduced a variation of ISS in 1998, known as the integral input-to-state stability (iISS); see [2]. The iISS offers a description of stability in a sense weaker than the ISS, specifically permitting unbounded inputs with finite energy. In recent years, the ISS, iISS, and their variations have been developed as the ISS theory and have found extensive applications in various fields; see, e.g., [3] for comprehensive surveys.

    In the past decade, ISS theory for ODE systems has been widely extended and applied to systems described by partial differential equations (PDEs). For instance, ISS-Lyapunov characterizations were provided for abstract infinite-dimensional systems including PDEs [4,5,6,7,8,9,10]; different ISS estimates were established for PDE systems with various types of disturbances [9,11,12,13,14,15,16,17,18,19,20,21,22]; and the ISS was applied to PDEs arising in engineering, such as multi-agent control [23], the railway track model [24], and bridge vibrations [25], etc. We refer [9,14,22] for summaries on this topic.

    For PDE systems, external disturbances typically manifest within the interior of the domain, on the boundary of the domain, or simultaneously within the interior and on the boundary of the domain. Regarding PDEs with in-domain disturbances, the Lyapunov method is the most common approach for establishing the ISS in various norms; see [6,17,20,26,27,28]. However, studying PDE systems with boundary disturbances is much more challenging. This is because when a PDE system has boundary disturbances, it is not easy to handle the boundary terms without involving time derivatives of the disturbances. To overcome this obstacle, different solutions have been proposed for different PDE systems. For instance:

    (i) In the context of linear PDEs with boundary disturbances, the approach of spectral decomposition and finite difference schemes can be effectively employed for the ISS analysis of systems governed by Sturm-Liouville operators [12,13], while the Riesz-spectral approach is suitable for establishing the ISS of Riesz-spectral systems [15,16];

    (ii) Regarding nonlinear PDEs with boundary disturbances, various approaches have been proposed for assessing the ISS of PDE systems with Dirichlet type boundary disturbances, such as the monotonicity method [29], the technique of De Giorgi iteration [30], and the maximum principle-based approach [21,31], etc., while the Lyapunov method remains the primary one for the ISS analysis in the systems with only Robin or Neumann type boundary disturbances [9,18,20,28].

    In this paper, we intend to investigate the well-posedness and stability in the framework of ISS theory for a nonlinear Euler-Bernoulli beam equation with both in-domain and boundary disturbances. It is worth mentioning that, as one of the representative PDEs, the Euler-Bernoulli beam equation and its variations have attracted a lot of attention in the past few decades; see, e.g., [32,33,34,35]. In particular, a railway track model governed by a class of nonlinear Euler-Bernoulli beam equations was studied in [24,36,37], and the ISS for the system was established when only in-domain disturbances appeared (see [24]), whereas the effect of boundary disturbances on the stability of the system has not been considered. In addition, the iISS corresponding to the integrals of in-domain or boundary disturbances has not been studied for such a system, while it is worthy of probing. Motivated by these facts and based on the model considered in [24], we focus on the situation where the nonlinear Euler-Bernoulli beam equation involves both in-domain and boundary disturbances. Within the framework of ISS theory, we will prove first the well-posedness for the system and then establish both the ISS and the iISS estimates for the solution to characterize the influence of the Lr-integral (w.r.t. t) of disturbances on the stability of the system whenever r[2,+].

    It is worth noting that, as previously mentioned, the presence of boundary disturbances in PDE systems leads to significant complexity in the well-posedness and stability analysis. Consequently, the problem addressed in this paper represents more of a challenge compared to the one tackled in [24]:

    (i) Regarding the well-posedness analysis, due to the fact that the nonlinear Euler-Bernoulli beam equation considered in [24] is subject to homogeneous boundary conditions and the nonlinear term solely depends on the state, classical and mild solutions can be directly obtained by using the semigroup method after transforming the specific system into an abstract system. However, in the presence of boundary disturbances, the abstract system involves an unbounded operator, making it non-trivial to prove the well-posedness. To overcome this difficulty, in this paper, we will employ the technique of lifting to transform the original system with non-homogeneous boundary conditions into an equivalent system with homogeneous boundary conditions. Nevertheless, after applying lifting, more nonlinear terms will arise in the equivalent system and depend simultaneously on both the time variable and the state, introducing complexity when verifying properties of these nonlinear terms.

    (ii) For the stability analysis, it is worth noting that analyzing the stability of nonlinear systems is inherently more challenging compared to linear ones. Nevertheless, in [24], owing to the homogeneous boundary conditions of the considered nonlinear system, the authors were able to construct a suitable ISS-Lyapunov functional and employ various technical lemmas, such as Poincaré's inequality, Young's inequality, and Gronwall's inequality, to establish the ISS for the system with only in-domain disturbances. For the nonlinear equation under consideration in this paper, the presence of boundary disturbances leads to a challenge. Indeed, after applying the technique of lifting, if an equivalent system with homogeneous boundary conditions is considered and stability analysis is performed by using the Lyapunov method as in [24], the resulting stability estimates must contain the time derivatives of the boundary disturbances, which do not strictly adhere to the real ISS or iISS property as pointed out in [12,13,20]. To address this issue, we will directly deal with the original system with non-homogeneous boundary conditions for the stability analysis, but handling the boundary terms is a non-trivial task. Indeed, this requires more techniques than those presented in [24], thereby amplifying the complexity of the problem under consideration.

    The outline of this paper is as follows: Section 2 introduces notations and auxiliary results used in the paper. Section 3 presents the problem formulation and the main result, which is divided into two propositions stated in Section 4 and Section 5, respectively. More specifically, the first proposition is concerned with the well-posedness of the considered system and is presented in Section 4, while the second one states the result on the ISS and iISS for the considered system and is presented in Section 5.

    In this paper, let R:=(,+), R0:=[0,+), and R>0:=(0,+).

    The following sets of comparison functions are defined in the standard way; see, e.g., [3,A.1]:

    K:={γ:R0R0|γ is continuous, strictly increasing, and γ(0)=0},K:={γK|γ is unbounded},L:={γ:R0R0|γ is continuous, strictly decreasing, and limtγ(t)=0},KL:={β:R0×R0R0|β is continuous, β(,t)K,β(r,)L,tR0,rR>0}.

    For a given operator A, its range and resolvent set are denoted by R(A) and ρ(A), respectively. The kernel of A is denoted by ker(A). For given subsets C and D in normed linear spaces, the set of all bounded linear operators from C to D is defined in the standard way as in, e.g., [38,Definition A.3.1], and is denoted by L(C;D). In particular, let L(C;C):=L(C).

    For a given function u:[0,1]×R0R, we use the notation u[t] to denote the profile at certain tR0, i.e., u[t](x)=u(x,t) for all x[0,1].

    Let AC([0,1]) denote the set of all absolutely continuous functions defined on [0,1]. The following Sobolev-like inequalities, which can be proved as in [20], play an important role in dealing with boundary terms when establishing the ISS and iISS for PDE systems with boundary disturbances.

    Lemma 2.1. Suppose that υAC([0,1]). Then, the following inequalities hold true:

    (i) υ2(c)2υ2L2(0,1)+υx2L2(0,1), c[0,1];

    (ii) υ2L2(0,1)υ2(c)+12υx2L2(0,1) for c=0 or c=1.

    We provide the concept of the Fréchet derivative, which can be found in, e.g., [38,Definition A.5.25,p. 629].

    Definition 2.2. Consider the mapping F from the Banach space Y to the Banach space Z. Given y0Y, if a linear bounded operator 𝕕F(y0) exists such that

    limhY0=F(y0+h)F(y0)𝕕F(y0)hZhY=0,

    then F is said to be Fréchet differentiable at y0, and 𝕕F(y0) is said to be the Fréchet derivative of F at y0.

    In this paper, we study the well-posedness and stability of the following nonlinear Euler-Bernoulli beam equation in the framework of ISS theory:

    wtt+(awxx+bwtxx)xx+cwt+kw+lw3=f(x,t),(x,t)[0,1]×R0, (3.1a)
    w(0,t)=0,tR0, (3.1b)
    wx(1,t)=0,tR0, (3.1c)
    (awxx+bwtxx)(0,t)=d1(t),tR0, (3.1d)
    (awxx+bwtxx)x(1,t)=d2(t),tR0, (3.1e)
    w(x,0)=φ1(x),x[0,1], (3.1f)
    wt(x,0)=φ2(x),x[0,1], (3.1g)

    where

    a,b,c,kR>0andk,lR0,

    the function f represents the distributed in-domain disturbance, the functions d1,d2 represent the boundary disturbances, and φ1,φ2 are given initial data. It is worth mentioning that equation (3.1a) with c=k=l=0 is well-known as the one-dimensional Euler-Bernoulli beam equation [39,40]; equation (3.1a) with k=l=0 is a model of flexible aircraft wing with Kelvin-Voigt damping [19,41]; while equation (3.1a) with l=0 or the general case of equation (3.1a) can be used to model railway track deflections [24,36,42,43].

    Before defining a solution, and the ISS and iISS for the system (3.1), we would like to reformulate (3.1) in an abstract form. More specifically, we introduce first the Hilbert space

    H2[0](0,1):={wH2(0,1)|w(0)=wx(1)=0},

    which is endowed with the inner product

    w1,w2H2[0](0,1):=10aw1xxw2xxdx,w1,w2H2[0](0,1),

    and the norm

    wH2[0](0,1):=awxxL2(0,1),wH2[0](0,1),

    respectively. Define H:=H2[0](0,1)×L2(0,1), which is also a Hilbert space endowed with the inner product

    (w1,v1),(w2,v2)H:= w1,w2H2[0](0,1)+v1,v2L2(0,1),(w1,v1),(w2,v2)H,

    and the norm

    (w,v)H:=(awxx2L2(0,1)+v2L2(0,1))12,(w,v)H,

    respectively.

    Let the linear operator A:D(A)HH be defined by

    A(w,v):=(v,(awxx+bvxx)xx)

    with the domain

    D(A):={(w,v)H|vH2[0](0,1),awxx+bvxxH2(0,1),awxx+bvxxAC([0,1]),(awxx+bvxx)xAC([0,1])},

    which is dense in H. Let the nonlinear operator A1 be defined by

    A1(w,v):=(0,cvkwlw3),(w,v)D(A1):=H.

    Let the boundary operator B be defined by

    B(w,v):=((awxx+bvxx)(0),(awxx+bvxx)x(1)),(w,v)D(B):=D(A).

    Throughout this paper, for the in-domain disturbance f, the boundary disturbances d1,d2, and the initial data φ1,φ2, we always assume that

    (H1) fC1([0,1]×R0) and d1,d2C2(R0);

    (H2) (φ1,φ2)D(A) and satisfies the compatibility condition B(φ1,φ2)=(d1(0),d2(0)).

    Now, let X(t):=(w[t],wt[t]) be the state of system (3.1), and X0:=(φ1,φ2) be the corresponding initial datum. Let F(t):=(0,f[t]). Then, system (3.1) can be written in the following abstract form:

    ˙X(t)=(A+A1)X(t)+F(t), (3.2a)
    BX(t)=(d1(t),d2(t)), (3.2b)
    X(0)=X0D(A)withBX0=(d1(0),d2(0)). (3.2c)

    Definition 3.1. For any TR>0, if XC([0,T];D(A))C1((0,T);H) satisfies equation (3.2a) for all t(0,T), boundary condition (3.2b) for all t[0,T], and intitial-value condition (3.2c), then X is said to be a classical solution to system (3.2).

    Definition 3.2. For certain r[1,+], system (3.2) is said to be Lr-integral input-to-state stable (Lr-iISS) in the norm of H w.r.t. the in-domain disturbance f and the boundary disturbances d1,d2 if there exist functions μKL and γ1,γ2,γ3K such that the solution to system (3.2) satisfies

    X(t)Hμ(X0H,t)+γ1(d1Lr(0,t))+γ2(d2Lr(0,t))+γ3(fLr((0,t);L2(0,1))),tR0. (3.3)

    In particular, system (3.2) is said to be input-to-state stable (ISS) in the norm of H w.r.t. the in-domain disturbance f and the boundary disturbances d1,d2 if inequality (3.3) is fulfilled with r=+.

    Remark 3.3. The notions of ISS and iISS provide a powerful tool of characterizing the robustness of nonlinear systems in presence of disturbances. For instance, inequality (3.3) implies that the disturbance-free system (3.2) is asymptotically stable, while the state remains bounded when bounded external disturbances are involved. In particular, the state becomes smaller when external disturbances become smaller in a certain sense.

    The main result obtained in this paper is stated as follows:

    Theorem 3.4. System (3.2) admits a unique classical solution. Moreover, for any r[2,+], system (3.2) is Lr-iISS in the norm of H w.r.t. the in-domain disturbance f and the boundary disturbances d1,d2.

    In the following, we will divide Theorem 3.4 into two propositions and provide their proofs in Section 4 and Section 5, respectively.

    In order to prove the well-posedness of system (3.2), we employ the technique of lifting to transform the original system into an equivalent one, which has homogeneous boundary conditions. Indeed, letting

    g1(x,t):=(x22x)(1bt0d1(s)eab(ts)ds),g2(x,t):=(16x312x)(1bt0d2(s)eab(ts)ds),

    and

    ˜w:=w+g1+g2,

    we transform system (3.1) into

    ˜wtt+(a˜wxx+b˜wtxx)xx+c˜wt+k˜w=f+g1tt+g2tt+(a(g1xx+g2xx)+b(g1txx+g2txx))xx+c(g1t+g2t)+k(g1+g2)l(˜wg1g2)3, (4.1a)
    ˜w(0,t)=0, (4.1b)
    ˜wx(1,t)=0, (4.1c)
    (a˜wxx+b˜wtxx)(0,t)=0, (4.1d)
    (a˜wxx+b˜wtxx)x(1,t)=0, (4.1e)
    ˜w(x,0)=˜φ1(x), (4.1f)
    ˜wt(x,0)=˜φ2(x). (4.1g)

    Define A2L(H) by

    A2(˜w,˜v):=(0,c˜vk˜w),(˜w,˜v)H,

    and ˜A:D(˜A)HH by

    ˜A:=A|D(˜A)

    with the domain D(˜A):=D(A)ker(B), respectively.

    For the given functions f, g1, and g2, we define the nonlinear functional ˜F:R0×HH by

    ˜F(t,Z)(x):=(0,f+g1tt+g2tt+(a(g1xx+g2xx)+b(g1txx+g2txx))xx+c(g1t+g2t)+k(g1+g2)l(˜wg1g2)3),Z:=(˜w,˜v)H. (4.2)

    Let ˜X(t):=(˜w[t],˜wt[t]) and ˜X0:=(˜φ1,˜φ2) be the state and the corresponding initial datum of system (4.1), respectively. Then, system (4.1) can be written in the following abstract form:

    ˙˜X(t)=(˜A+A2)˜X(t)+˜F(t,˜X(t)), (4.3a)
    B˜X(t)=(0,0), (4.3b)
    ˜X(0)=˜X0D(˜A). (4.3c)

    Definition 4.1. For any TR>0, if a function ˜XC(R0;D(˜A))C1(R0;H) satisfies equation (4.3a) for all t(0,T), boundary condition (4.3b) for all t[0,T], and initial-value condition (4.3c), then ˜X is said to be a classical solution to system (4.3).

    In order to prove the well-posedness of system (4.1), it suffices to prove the well-posedness of system (4.3). Indeed, we have the following result:

    Proposition 4.2. System (4.3) admits a unique classical solution, and hence system (3.2) admits a unique classical solution.

    Before proving Proposition 4.2, we present first some auxiliary results.

    Lemma 4.3. The inverse of the linear operator ˜A, denoted by ˜A1, exists and is bounded, namely, ˜A1L(H;D(˜A)). Thus, 0ρ(˜A) and ˜A is a closed operator.

    Proof. Let us show first that ˜A is surjective, namely, for any (ˆw,ˆv)H, we need to find (˜w,˜v)D(˜A) such that ˜A(˜w,˜v)=(ˆw,ˆv). Indeed, for any (ˆw,ˆv)H, we consider the solution to the following equations

    ˜v=ˆw, (4.4a)
    (a˜wxx+b˜vxx)xx=ˆv, (4.4b)
    (a˜wxx+b˜vxx)(0)=0, (4.4c)
    (a˜wxx+b˜vxx)x(1)=0. (4.4d)

    For any y[0,1], integrating both sides of equation (4.4b) over the interval [y,1], we obtain

    1y(a˜wzz(z)+b˜vzz(z))zzdz=1yˆv(z)dz. (4.5)

    By integrating by parts and using boundary condition (4.4c), we obtain

    1y(a˜wzz(z)+b˜vzz(z))zzdz=(a˜wzz(z)+b˜vzz(z))z|z=1z=y=(a˜wyy(y)+b˜vyy(y))y,

    which, along with equality (4.5), yields

    (a˜wyy(y)+b˜vyy(y))y=1yˆv(z)dz,y[0,1]. (4.6)

    Analogously, integrating both sides of equation (4.6) and using boundary condition (4.4d), we obtain

    a˜wxx(x)+b˜vxx(x)=x01yˆv(z)dzdy,x[0,1]. (4.7)

    In view of equation (4.4b) and ˆwH2[0](0,1), equation (4.7) is equivalent to

    a˜wxx(x)=x01yˆv(z)dzdybˆwxx(x):=M(x),x[0,1],

    which implies that

    ˜w(x)=1a0x1qM(p)dpdq,x[0,1]. (4.8)

    It is clear that (˜w,˜v)D(˜A) and ˜A(˜w,˜v)=(ˆw,ˆv). This proves that ˜A is surjective.

    Next, we show that ˜A is injective. Noting that ˜A is linear, it suffices to prove the implication

    ˜A(˜w,˜v)=0(˜w,˜v)=(0,0).

    Indeed, setting (ˆw,ˆv)=(0,0) in (4.4), and in view of equalities (4.4a) and (4.8), we get (˜w,˜v)=(0,0) immediately.

    It has been shown that ˜A is bijective. Thus, the inverse of ˜A, i.e., ˜A1:HD(˜A)H, exists.

    Now, we prove that ˜A1L(H;D(˜A)). For any (ˆw,ˆv)H, due to the fact that there is (˜w,˜v)D(˜A)H satisfying ˜A1(ˆw,ˆv)=(˜w,˜v), it follows that (ˆw,ˆv)=˜A(˜w,˜v), or, equivalently,

    ˜v=ˆw, (4.9a)
    (a˜wxx+b˜vxx)xx=ˆv. (4.9b)

    Applying Lemma 2.1(ii) to ˆw and using equality (4.9a), we have

    ˜v2L2(0,1)=ˆw2L2(0,1)14ˆwxx2L2(0,1)=14˜vxx2L2(0,1).

    Since (˜w,˜v)D(˜A), we also have

    a˜wxx+b˜vxx2L2(0,1)14(a˜wxx+b˜vxx)xx2L2(0,1).

    Then, we deduce by equation (4.9b) that

    ˜A1(ˆw,ˆv)H=(˜w,˜v)H=(10(a˜w2xx+˜v2)dx)12(10(a˜w2xx+14˜v2xx)dx)12C1(10((a˜wxx+b˜vxx)2+a˜v2xx)dx)12C2(10(a˜v2xx+((a˜wxx+b˜vxx)xx)2)dx)12=C2(10(aˆw2xx+ˆv2xx)dx)12=C2(ˆw,ˆv)H,

    where C1 and C2 are positive constants depending only on a and b. Therefore, ˜A1L(H;D(˜A)), and hence 0ρ(˜A) and ˜A is a closed operator (see [38,Theorem A.3.46,p. 596]).

    Lemma 4.4. The operator ˜A:D(˜A)H is dissipative w.r.t. ,H.

    Proof. Indeed, in view of the definitions of ˜A and ,H, and by integrating by parts, we have

    ˜A(˜w,˜v),(˜w,˜v)H=10(a˜wxx˜vxx˜v(a˜wxx+b˜vxx)xx)dx=10a˜wxx˜vxxdx10˜vxx(a˜wxx+b˜vxx)dx=b10˜v2xxdx0,(˜w,˜v)D(˜A),

    which indicates the dissipativity of ˜A.

    Lemma 4.5. The operator ˜A+A2 generates a C0-semigroup of contractions on ,H.

    Proof. We prove first that the linear operator ˜A generates a C0-semigroup of contractions on ,H. Indeed, we see from Lemma 4.4 that ˜A is dissipative. In view of Lemma 4.3, ˜A is closed, and thus the resolvent set ρ(˜A) is open (see [38,Lemma A.4.8,p. 612]). Since 0ρ(˜A), there must be a positive number λ0 such that R(λ0I˜A)=H, where I denotes the identity operator defined on D(˜A). According to the Lumer-Philips Theorem (see [44,Theorem 4.3,p. 14]), the linear operator ˜A generates a C0-semigroup of contractions on ,H.

    Next, we prove that A2 is bounded. Indeed, for any (˜w,˜v)H, due to the fact that ˜w2L2(0,1)14˜wxx2L2(0,1), we have

    A2(˜w,˜v)H=(10(c˜v+k˜w)2dx))12(102(c2˜v2+k2˜w2)dx)12(102(c2˜v2+14k2˜w2xx)dx)12C3(10(a˜w2xx+˜v2)dx)12=C3(˜w,˜v)H,

    where C3 is a positive constant depending only on a,c,k when k>0, and only on c when k=0, respectively. Therefore, the linear operator ˜A2 is bounded.

    Finally, according to [38,Theorem 3.2.1,p. 110], we conclude that ˜A+A2 generates a C0-semigroup of contractions on ,H.

    Lemma 4.6. For any TR>0, the nonlinear functional ˜F:[0,T]×HH is Fréchet differentiable.

    Proof. Let g(x,t):=g1(x,t)+g2(x,t). Recalling the definition of ˜F (see (4.2)), for any t[0,T] and Z:=(˜w,˜v)H, we have

    ˜F(t,Z)(x)=(0,f+gtt+(agxx+bgtxx)xx+cgt+kgl(˜wg)3).

    By virtue of the regularity of f and g, it suffices to show that (0,(˜wg)3) is Fréchet differentiable on Y:=[0,T]×HH. Furthermore, due to the fact that

    (0,(˜wg)3)=(0,˜w3)(0,3˜w2g)+(0,3˜wg2)(0,g3),

    it suffices to show that (0,˜w3), (0,˜w2g), and (0,˜wg2) are Fréchet differentiable on Y. Since the proof can proceed in a standard way (see, e.g., [24]), we only show that ˜F1(t,Z)(x):=(0,˜w2g) is Fréchet differentiable on Y in the following. More specifically, for any y0:=(t0,Z0)Y with Z0:=(˜w0,˜v0), we would like to prove that the Fréchet derivative of ˜F1 at y0 is given by

    𝕕˜F1(y0)h:=(0,2˜w0g[t0]˜w+˜w20gt[t0]t),

    where h:=(t,Z) with t[0,T] and Z:=(˜w,˜v)H, namely, we shall prove that

    limhY0˜F1(y0+h)˜F1(y0)𝕕˜F1(y0)hHhY=0,

    or, equivalently,

    lim|t|+˜wH2[0](0,1)+˜vL2(0,1)0(˜w+˜w0)2g[t+t0]˜w20g[t0]˜w20gt[t0]t2˜w0g[t0]˜wL2(0,1)|t|+˜wH2[0](0,1)+˜vL2(0,1)=0. (4.10)

    Indeed, we deduce that

    lim|t|+˜wH2[0](0,1)+˜vL2(0,1)0(˜w+˜w0)2g[t+t0]˜w20g[t0]˜w20gt[t0]t2˜w0g[t0]˜wL2(0,1)|t|+˜wH2[0](0,1)+˜vL2(0,1)limt0,˜wH2[0](0,1)0(˜w+˜w0)2g[t+t0]˜w20g[t0]˜w20gt[t0]t2˜w0g[t0]˜wL2(0,1)|t|+˜wH2[0](0,1)limt0,˜wH2[0](0,1)0˜w20(g[t+t0]g[t0])˜w20gt[t0]tL2(0,1)|t|+˜wH2[0](0,1)+limt0,˜wH2[0](0,1)0˜w2g[t+t0]L2(0,1)|t|+˜wH2[0](0,1)+limt0,˜wH2[0](0,1)02˜w˜w0g[t+t0]2˜w0g[t0]˜wL2(0,1)|t|+˜wH2[0](0,1)limt0˜w20(g[t+t0]g[t0])˜w20gt[t0]tL2(0,1)|t|+limt0,˜wH2[0](0,1)0˜w2g[t+t0]L2(0,1)˜wH2[0](0,1)+limt0,˜wH2[0](0,1)02˜w˜w0g[t+t0]2˜w0g[t0]˜wL2(0,1)|t|. (4.11)

    Now we assess each term on the right-hand side of inequality (4.11). First, we have

    limt0˜w20(g[t+t0]g[t0])˜w20gt[t0]tL2(0,1)|t|=limt0˜w20g[t+t0]g[t0]tt˜w20gt[t0]tL2(0,1)|t|=limt0˜w20g[t+t0]g[t0]t˜w20gt[t0]L2(0,1)=limt0˜w20gt[t0]˜w20gt[t0]L2(0,1)=0. (4.12)

    Second, noting that ˜w2L2(0,1)=˜w2L4(0,1) and applying the Sobolev embedding result H2[0](0,1)L4(0,1) with ˜wL4(0,1)C4˜wH2[0](0,1) and some positive constant C4, we infer that

    limt0,˜wH2[0](0,1)0˜w2g[t+t0]L2(0,1)˜wH2[0](0,1)gL((0,1)×(0,T))lim˜wH2[0](0,1)0˜w2L2(0,1)˜wH2[0](0,1)=gL((0,1)×(0,T))lim˜wH2[0](0,1)0˜w2L4(0,1)˜wH2[0](0,1)C4gL((0,1)×(0,T))lim˜wH2[0](0,1)0˜w2H2[0](0,1)˜wH2[0](0,1)=0. (4.13)

    Third, we have

    limt0,˜wH2[0](0,1)02˜w˜w0g[t+t0]2˜w0g[t0]˜wL2(0,1)|t|=limt0,˜wH2[0](0,1)02˜w˜w0g[t+t0]g[t0]tL2(0,1)=lim˜wH2[0](0,1)02˜w˜w0gt[t0]L2(0,1)2gtL((0,1)×(0,T))˜w0L2(0,1)lim˜wH2[0](0,1)0˜wL2(0,1)2gtL((0,1)×(0,T))˜w0L2(0,1)lim˜wH2[0](0,1)0˜wH2[0](0,1)=0. (4.14)

    Finally, combining inequality (4.11), equality (4.12), inequality (4.13), and inequality (4.14), we obtain equality (4.10).

    Proof of Proposition 4.2. For any TR>0, in view of Lemma 4.5 and Lemma 4.6, it is guaranteed by [45,Theorem 6.1.4 & 6.1.5,pp. 185-187] that system (4.3) admits a unique local classical solution ˜X on an interval [0,Tmax] with some Tmax(0,T). Moreover, it is guaranteed by [45,Theorem 6.1.4,pp. 185-186] that the local classical solution can be extended to the whole interval [0,T] if the solution satisfies

    limtTmax˜X[t]H<+,

    which is included in Section 5.

    The ISS and iISS results stated in Theorem 3.4 are re-formulated as in the following proposition:

    Proposition 5.1. For any r[2,+], system (3.2) is Lr-iISS w.r.t the in-domain disturbance f and the boundary disturbances d1,d2, having the following estimate for all tR0:

    X(t)HCeΛt(X0H+X02H)+C(fLr((0,t);L2(0,1))+d1Lr(0,t)+d2Lr(0,t)), (5.1)

    where C and Λ are positive constants depending only on a,b,c,k,l, and r when r[2,+), and depending only on a,b,c,k, and l when r=+, respectively.

    Lemma 5.2. For any positive constant m satisfying m<min{4a,1}, there are positive constants cl,cu, and ch depending only on a,k,l, and m such that

    cl(wxx2L2(0,1)+v2L2(0,1))1210(aw2xx+kw2+l2w4+v2+2mwv)dx (5.2)
    cu(wxx2L2(0,1)+v2L2(0,1))+ch(wxx2L2(0,1)+v2L2(0,1))2,(w,v)H. (5.3)

    Proof. For any positive constant m satisfying m<min{4a,1}, and (w,v)H, let

    E:=1210(aw2xx+kw2+l2w4+v2+2mwv)dx.

    It is clear that

    |102mwvdx|m10(w2+v2)dx. (5.4)

    Since wH2[0](0,1), by virtue of Lemma 2.1(ii), we have

    w2L2(0,1)14wxx2L2(0,1). (5.5)

    Therefore, it holds that

    E1210(aw2xx+(km)w2+(1m)v2)dx1210(aw2xxmw2+(1m)v2)dx1210((am4)w2xx+(1m)v2)dx,

    which implies that inequality (5.2) holds true with cl:=12min{am4,1m}.

    We deduce by inequalities (5.4) and (5.5) that

    E1210(aw2xx+(k+m)w2+l2w4+(1+m)v2)dx1210(aw2xx+14(k+m)w2xx+l2w4+(1+m)v2)dx (5.6)
    cu(wxx2L2(0,1)+v2L2(0,1))+l410w4dx (5.7)

    with cu:=12max{a+14(k+m),1+m}.

    In view of H2(0,1)L4(0,1), there is a positive constant Ce such that wL4(0,1)CewH2(0,1). It follows that

    l410w4dxl4C4e(10(w2+w2x+w2xx)dx)2l4C4e(7410w2xxdx)249l64C4e(wxx2L2(0,1)+v2L2(0,1))2:=ch(wxx2L2(0,1)+v2L2(0,1))2. (5.8)

    We deduce by inequalities (5.7) and (5.8) that inequality (5.3) holds true.

    Proof of Proposition 5.1. Let X(t):=(w[t],v[t]) be the state of system (3.2). We proceed with the proof in three steps.

    Step 1: We prove that there are positive constants m,λ, and C5 depending only on a,b,c,k, and l, such that

    E(t)E(0)eλt+C5t0(f[s]2L2(0,1)+d21(s)+d22(s))eλ(ts)ds,t[0,Tmax), (5.9)

    where

    E(t):=1210(aw2xx(x,t)+kw2(x,t)+l2w4(x,t)+v2(x,t)+2mw(x,t)v(x,t))dx,

    and TmaxR>0 is the maximal time for the existence of a solution.

    Indeed, in view of inequality (5.6), we deduce first that

    E(t)1210(aw2xx+14(k+m)w2xx+l2w4+(1+m)v2)dx12max{1+14a(k+m),1+m}10(aw2xx+l2w4+v2)dxλ010(aw2xx+kw2+l2w4+v2+bv2xx)dx, (5.10)

    where λ0:=12max{1+14a(k+m),1+m}. By equation (3.1a) and direct computations, we have

    ddtE(t)=10(awxxvxx+kwv+lw3v+vvt+mvv+mwvt)dx=10(awxxvxx+kwv+lw3v+mv2)dx10v((awxx+bvxx)xx+cv+kw+lw3f)dx10mw((awxx+bvxx)xx+cv+kw+lw3f)dx=10(awxxvxx+mv2cv2cmwvkmw2lmw4)dx10(awxx+bvxx)xx(v+mw)dx+10f(v+mw)dx. (5.11)

    Note that

    10(awxx+bvxx)xx(v+mw)dx=(awxx+bvxx)x(v+mw)|x=1x=0+(awxx+bvxx)(vx+mwx)|x=1x=010(awxx+bvxx)(vxx+mwxx)dx=(awxx+bvxx)x(1,t)(v+mw)(1,t)+(awxx+bvxx)x(0,t)(v+mw)(0,t)+(awxx+bvxx)(1,t)(vx+mwx)(1,t)(awxx+bvxx)(0,t)(vx+mwx)(0,t)10(awxx+bvxx)(vxx+mwxx)dx=d1(t)(vx(0,t)+mwx(0,t))d2(t)(v(1,t)+mw(1,t))10(awxx+bvxx)(vxx+mwxx)dx.

    Thus, equation (5.11) becomes

    ddtE(t)=d1(t)(vx(0,t)+mwx(0,t))d2(t)(v(1,t)+mw(1,t))10(amw2xx+kmw2+lmw4+(cm)v2+bv2xx)dx10(cmwv+bmwxxvxx)dx+10f(v+mw)dx. (5.12)

    Note that Young's inequality yields

    |10cmwvdx|cm10(ε02w2+12ε0v2)dx, (5.13a)
    |10bmwxxvxxdx|bm10(ε12w2xx+12ε1v2xx)dx, (5.13b)
    |10f(v+mw)dx|10(12ε2f2+ε2v2+ε2m2w2)dx, (5.13c)

    where ε0,ε1,ε2R>0 will be determined later.

    In addition, by using Young's inequality and Lemma 2.1(i) and (ii), we have

    |d1(t)(vx(0,t)+mwx(0,t))+d2(t)(v(1,t)+mw(1,t))|12ε3d21(t)+ε3v2x(0,t)+m2ε3w2x(0,t)+12ε4d22(t)+ε4v2(1,t)+m2ε3w2(1,t)12ε3d21(t)+ε3(2vx2L2(0,1)+vxx2L2(0,1))+m2ε3(2wx2L2(0,1)+wxx2L2(0,1))+12ε4d22(t)+ε4(2v2L2(0,1)+vx2L2(0,1))+m2ε4(2w2L2(0,1)+wx2L2(0,1))12ε3d21(t)+2ε3vxx2L2(0,1)+2m2ε3wxx2L2(0,1)+12ε4d22(t)+ε4vxx2L2(0,1)+m2ε4wxx2L2(0,1)=12ε3d21(t)+12ε4d22(t)+m2(2ε3+ε4)wxx2L2(0,1)+(2ε3+ε4)vxx2L2(0,1), (5.14)

    where ε3,ε4R>0 will be determined later.

    Now we discuss the two cases.

    Case 1: k>0. Combining equation (5.12), inequality (5.13), and inequality (5.14), we deduce that

    ddtE(t)12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t)10(abε122mε3mε4)mw2xxdx10(kcε02mε2)mw2dx10lmw4dx10(cmcm2ε0ε2)v2dx10(1m2ε12ε3bε4b)bv2xxdx=12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t)m(1bε12a2mε3amε4a)10aw2xxdxm(1cε02kmε2k)10kw2dxm10lw4dxm(cm1c2ε0ε2m)10v2dxm(1m12ε12ε3mbε4mb)10bv2xxdx. (5.15)

    Define the constant

    λ1:=mmin{1bε12a2mε3amε4a,1cε02kmε2k,cm1c2ε0ε2m,1m12ε12ε3mbε4mb}.

    To ensure that λ1 is positive, we first let ε0,ε1R>0 satisfy

    1bε12a>0and1cε02k>0.

    Then, we let mR>0 satisfy m<min{4a,1} and

    1m12ε1>0andcm1c2ε0>0.

    In addition, we choose sufficiently small ε2R>0 such that

    1cε02kmε22k>0andcm1c2ε0ε2m>0.

    Furthermore, we choose sufficiently small ε3,ε4R>0 such that

    1bε12amε32amε4a>0and1m12ε12ε3mbε4mb>0.

    It is clear that λ1>0 for such choices of ε0,ε1,ε2,ε3,ε4, and m. Therefore, inequality (5.15) becomes

    ddtE(t)12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t)λ110(aw2xx+kw2+l2w4+v2+bv2xx)dx. (5.16)

    It follows from inequalities (5.16) and (5.10) that

    ddtE(t)λ1λ0E(t)+12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t). (5.17)

    Applying Gronwall's inequality to equation (5.17), we deduce that inequality (5.9) holds true with λ:=λ1λ0 and C5:=max{12ε2,12ε3,12ε4}.

    Case 2: k=0. It suffices to note that inequality (5.15) becomes

    ddtE(t)12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t)10(abε122mε3mε4)mw2xxdx+10(cε02+mε2)mw2dx10lmw4dx10(cmcm2ε0ε2)v2dx10(1m2ε12ε3bε4b)bv2xxdx12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t)10(abε122mε3mε4)mw2xxdx+1014(cε02+mε2)mw2xxdx10lmw4dx10(cmcm2ε0ε2)v2dx10(1m2ε12ε3bε4b)bv2xxdx=12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t)m(1cε08abε12amε24a2mε3amε4a)10aw2xxdx10lmw4dxm(cm1c2ε0ε2m)10v2dxm(1m12ε12ε3mbε4mb)10bv2xxdx%.12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t)λ110(aw2xx+l2w4+v2+bv2xx)dx (5.18)

    with a positive constant

    λ1:=mmin{1cε08abε12amε24a2mε3amε4a,cm1c2ε0ε2m,1m12ε12ε3mbε4mb},

    which is ensured by selecting sufficiently small ε0,ε1,m,ε2,ε3,ε4R>0 in order.

    Indeed, we first let ε0,ε1R>0 satisfy

    1cε08abε12a>0,

    and mR>0 satisfy m<min{4a,1} and

    1m12ε1>0andcm1c2ε0>0,

    respectively. Then, we choose sufficiently small ε2R>0 such that

    1cε08abε12amε24a>0andcm1c2ε0ε2m>0.

    Finally, we choose sufficiently small ε3,ε4R>0 such that

    1cε08abε12amε24a2mε3amε4a>0and1m12ε12ε3mbε4mb>0.

    It follows from inequalities (5.16) and (5.10) with k=0 that

    ddtE(t)λ1λ0E(t)+12ε2f[t]2L2(0,1)+12ε3d21(t)+12ε4d22(t),

    which implies that inequality (5.9) holds true with λ:=λ1λ0 and C5:=max{12ε2,12ε3,12ε4}.

    Step 2: We show that estimate (5.1) holds true for all t[0,Tmax).

    Indeed, in view of the choice of m, we infer from Lemma 5.2 and inequality (5.9) that

    cl(wxx[t]2L2(0,1)+v[t]2L2(0,1))E(t)E(0)eλt+C5t0(f[s]2L2(0,1)+d21(s)+d22(s))eλ(ts)dscueλt(φ1xx2L2(0,1)+φ22L2(0,1))+cheλt(φ1xx2L2(0,1)+φ22L2(0,1))2+C5t0(f[s]2L2(0,1)+d21(s)+d22(s))eλ(ts)ds,

    where cl,cu,ch are positive constants depending only on a,k,l, and m.

    Therefore, it holds that

    wxx[t]2L2(0,1)+v[t]2L2(0,1)cucleλt(φ1xx2L2(0,1)+φ22L2(0,1))+chcleλt(φ1xx2L2(0,1)+φ22L2(0,1))2+C5clt0(f[s]2L2(0,1)+d21(s)+d22(s))eλ(ts)ds. (5.19)

    In the following, we shall estimate

    I(t):=t0(f[s]2L2(0,1)+d21(s)+d22(s))eλ(ts)ds.

    For simplicity, let F(s):=f[s]2L2(0,1). For any pi,qi[1,+] satisfying 1pi+1qi=1, i=1,2,3, we deduce by Hölder's inequality that

    I1(t):=t0F(s)eλ(ts)dseλ(t)Lp1(0,t)FLq1(0,t),I2(t):=t0eλ(ts)d21(s)dseλ(t)Lp2(0,t)d21Lq2(0,t),I3(t):=t0eλ(ts)d22(s)dseλ(t)Lp3(0,t)d22Lq3(0,t).

    We discuss the three cases.

    Case 1: pi,qi(1,+), i=1,2,3. By direct computations, we have

    I1(t)eλ(t)Lp1(0,t)FLq1(0,t)=(1λp1(1eλtp1))1p1(t0f[s]2q1L2(0,1)ds)1q1(1λp1)1p1(t0f[s]2q1L2(0,1)ds)1q1=(1λp1)1p1f2Lr1((0,t);L2(0,1)), (5.20)

    where r1:=2q1(2,+).

    Analogously, it holds that

    I2(t)+I3(t)(1λp2)1p2d12Lr2(0,t)+(1λp3)1p3d22Lr3(0,t), (5.21)

    where ri:=2qi(2,+), i=2,3.

    Letting CM:=max{(1λp1)1p1,(1λp2)1p2,(1λp3)1p3}, and combining inequalities (5.20) and (5.21), we have

    I(t)=I1(t)+I2(t)+I3(t)CM(f2Lr1((0,t);L2(0,1))+d12Lr2(0,t)+d22Lr3(0,t)). (5.22)

    Setting q1=q2=q3, or, equivalently, r=r1=r2=r3, we infer from inequalities (5.19) and (5.22) that estimate (5.1) holds true for t[0,Tmax) with r=r1(1,+), Λ=λ2, and some positive constant C depending only on cl,cu,ch,C5, and CM.

    Case 2: pi=1,qi=+, i=1,2,3. Similarly, we have

    I1(t)eλ(t)L1(0,t)FL(0,t)=f2L((0,t);L2(0,1))t0eλ(ts)ds1λf2L((0,t);L2(0,1)), (5.23)

    and

    I2(t)+I3(t)1λ(d12L(0,t)+d22L(0,t)). (5.24)

    We infer from inequalities (5.23), (5.24), and (5.19) that estimate (5.1) holds true for t[0,Tmax) with r=+, Λ=λ2, and some positive constant C depending only on cl,cu,ch,C5, and 1λ.

    Case 3: pi=+,qi=1, i=1,2,3. By direct computations, it holds that

    I1(t)eλ(t)L(0,t)FL1(0,t)=t0f[s]2L2(0,1)ds=f2L2((0,t);L2(0,1)), (5.25)

    and

    I2(t)+I3(t)d12L2(0,t)+d22L2(0,t). (5.26)

    We infer from inequalities (5.25), (5.26), and (5.19) that estimate (5.1) holds true for t[0,Tmax) with r=2, Λ=λ2, and some positive constant C depending only on cl,cu,ch, and C5.

    Step 3: Conclusion. In view of the regularity f,d1,d2,X0, and X, if Tmax<+, then estimate (5.1) ensures that

    limtTmaxX[t]H<+.

    We conclude that there must be Tmax=+. Therefore, estimate (5.1) holds true for all tR0.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank the editors and anonymous reviewers for their valuable suggestions and comments that contribute to improving the quality of the paper. This work is supported in part by the NSFC under grant 11901482 and in part by the NSERC under grant RGPIN-2018-04571.

    The authors declare there is no conflict of interest.



    [1] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435–443. https://doi.org/10.1109/9.28018 doi: 10.1109/9.28018
    [2] E. D. Sontag, Comments on integral variants of ISS, Systems Control Lett., 34 (1998), 93–100. https://doi.org/10.1016/S0167-6911(98)00003-6 doi: 10.1016/S0167-6911(98)00003-6
    [3] A. Mironchenko, Input-to-State Stability: Theory and Applications, Springer, 2023. https://doi.org/10.1007/978-3-031-14674-9
    [4] H. Damak, Input-to-state stability and integral input-to-state stability of non-autonomous infinite-dimensional systems, Internat. J. Systems Sci., 52 (2021), 2100–2113. https://doi.org/10.1080/00207721.2021.1879306 doi: 10.1080/00207721.2021.1879306
    [5] H. Damak, Input-to-state stability of non-autonomous infinite-dimensional control systems, Math. Control Relat. Fields, 13 (2023), 1212–1225. https://doi.org/10.3934/mcrf.2022035 doi: 10.3934/mcrf.2022035
    [6] S. Dashkovskiy, A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Math. Control Signals Systems, 25 (2013), 1–35. https://doi.org/10.1007/s00498-012-0090-2 doi: 10.1007/s00498-012-0090-2
    [7] B. Jacob, A. Mironchenko, J. R. Partington, F. Wirth, Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems, SIAM J. Control Optim., 58 (2020), 2952–2978. https://doi.org/10.1137/19M1297506 doi: 10.1137/19M1297506
    [8] B. Jacob, R. Nabiullin, J. R. Partington, F. L. Schwenninger, Infinite-dimensional input-to-state stability and Orlicz spaces, SIAM J. Control Optim., 56 (2018), 868–889. https://doi.org/10.1137/16M1099467 doi: 10.1137/16M1099467
    [9] A. Mironchenko, C. Prieur, Input-to-state stability of infinite-dimensional systems: recent results and open questions, SIAM Rev., 62 (2020), 529–614. https://doi.org/10.1137/19M1291248 doi: 10.1137/19M1291248
    [10] A. Mironchenko, F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Automat. Control, 63 (2018), 1692–1707. https://doi.org/10.1109/tac.2017.2756341 doi: 10.1109/tac.2017.2756341
    [11] B. Jayawardhana, H. Logemann, E. P. Ryan, Infinite-dimensional feedback systems: the circle criterion and input-to-state stability, Commun. Inf. Syst., 8 (2008), 413–444. https://doi.org/10.4310/CIS.2008.v8.n4.a4 doi: 10.4310/CIS.2008.v8.n4.a4
    [12] I. Karafyllis, M. Krstic, ISS with respect to boundary disturbances for 1-D parabolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 3712–3724. https://doi.org/10.1109/TAC.2016.2519762 doi: 10.1109/TAC.2016.2519762
    [13] I. Karafyllis, M. Krstic, ISS in different norms for 1-D parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 55 (2017), 1716–1751. https://doi.org/10.1137/16M1073753 doi: 10.1137/16M1073753
    [14] I. Karafyllis, M. Krstic, Input-to-state stability for PDEs, Springer-Verlag, Cham, 2019. https://doi.org/10.1007/978-3-319-91011-6
    [15] H. Lhachemi, D. Saussié, G. C. Zhu, R. Shorten, Input-to-state stability of a clamped-free damped string in the presence of distributed and boundary disturbances, IEEE Trans. Automat. Control, 65 (2020), 1248–1255. https://doi.org/10.1109/tac.2019.2925497 doi: 10.1109/tac.2019.2925497
    [16] H. Lhachemi, R. Shorten, ISS property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems, Automatica, 109 (2019), 108504. https://doi.org/10.1016/j.automatica.2019.108504 doi: 10.1016/j.automatica.2019.108504
    [17] A. Mironchenko, H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: an iISS approach, SIAM J. Control Optim., 53 (2015), 3364–3382. https://doi.org/10.1137/14097269X doi: 10.1137/14097269X
    [18] F. L. Schwenninger, Input-to-state stability for parabolic boundary control: linear and semilinear systems, in Kerner J., Laasri H., and Mugnolo D. (eds) Control Theory of Infinite-Dimensional Systems. Operator Theory: Advances and Applications, Birkhäuser, Cham, 2020, 83–116. https://doi.org/10.1007/978-3-030-35898-3_4
    [19] J. Zheng, H. Lhachemi, G. C. Zhu, D. Saussié, ISS with respect to boundary and in-domain disturbances for a coupled beam-string system, Math. Control Signals Systems, 30 (2018), 21. https://doi.org/10.1007/s00498-018-0228-y doi: 10.1007/s00498-018-0228-y
    [20] J. Zheng, G. C. Zhu, Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations, Automatica, 97 (2018), 271–277. https://doi.org/10.1016/j.automatica.2018.08.007 doi: 10.1016/j.automatica.2018.08.007
    [21] J. Zheng, G. C. Zhu, Input-to-state stability for a class of one-dimensional nonlinear parabolic PDEs with nonlinear boundary conditions, SIAM J. Control Optim., 58 (2020), 2567–2587. https://doi.org/10.1137/19M1283720 doi: 10.1137/19M1283720
    [22] J. Zheng, G. C. Zhu, ISS-like estimates for nonlinear parabolic PDEs with variable coefficients on higher dimensional domains, Systems Control Lett., 146 (2020), 104808. https://doi.org/10.1016/j.sysconle.2020.104808 doi: 10.1016/j.sysconle.2020.104808
    [23] L. Aguilar, Y. Orlov, A. Pisano, Leader-follower synchronization and ISS analysis for a network of boundary-controlled wave PDEs, IEEE Control Syst. Lett., 5 (2021), 683–688. https://doi.org/10.1109/LCSYS.2020.3004505 doi: 10.1109/LCSYS.2020.3004505
    [24] M. S. Edalatzadeh, K. A. Morris, Stability and well-posedness of a nonlinear railway track model, IEEE Control Syst. Lett., 3 (2019), 162–167. https://doi.org/10.1109/LCSYS.2018.2849831 doi: 10.1109/LCSYS.2018.2849831
    [25] Y. Y. Jiang, J. Li, Y. Liu, J. Zheng, Input-to-state stability of a variable cross-section beam bridge under moving loads (in Chinese), Math. Pract. Theory, 51 (2021), 177–185.
    [26] F. B. Argomedo, C. Prieur, E. Witrant, S. Bremond, A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients, IEEE Trans. Automat. Control, 58 (2013), 290–303. https://doi.org/10.1109/TAC.2012.2209260 doi: 10.1109/TAC.2012.2209260
    [27] F. Mazenc, C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Math. Control Relat. Fields, 1 (2011), 231–250. https://doi.org/10.3934/mcrf.2011.1.231 doi: 10.3934/mcrf.2011.1.231
    [28] J. Zheng, G. C. Zhu, Approximations of Lyapunov functionals for ISS analysis of a class of higher dimensional nonlinear parabolic PDEs, Automatica, 125 (2021), 109414. https://doi.org/10.1016/j.automatica.2020.109414 doi: 10.1016/j.automatica.2020.109414
    [29] A. Mironchenko, I. Karafyllis, M. Krstic, Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 57 (2019), 510–532. https://doi.org/10.1137/17M1161877 doi: 10.1137/17M1161877
    [30] J. Zheng, G. C. Zhu, A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers' equation with boundary and in-domain disturbances, IEEE Trans. Automat. Control, 64 (2019), 3476–3483. https://doi.org/10.1109/TAC.2018.2880160 doi: 10.1109/TAC.2018.2880160
    [31] I. Karafyllis, M. Krstic, ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs, ESAIM Control Optim. Calc. Var., 27 (2021), 57. https://doi.org/10.1051/cocv/2021053 doi: 10.1051/cocv/2021053
    [32] R. Díaz, O. Vera, Asymptotic behaviour for a thermoelastic problem of a microbeam with thermoelasticity of type III, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1–13. https://doi.org/10.14232/ejqtde.2017.1.74 doi: 10.14232/ejqtde.2017.1.74
    [33] B. W. Feng, B. Chentouf, Exponential stabilization of a microbeam system with a boundary or distributed time delay, Math. Methods Appl. Sci., 44 (2021), 11613–11630. https://doi.org/10.1002/mma.7518 doi: 10.1002/mma.7518
    [34] F. F. Jin, B. Z. Guo, Boundary output tracking for an Euler-Bernoulli beam equation with unmatched perturbations from a known exosystem, Automatica, 109 (2019), 108507. https://doi.org/10.1016/j.automatica.2019.108507 doi: 10.1016/j.automatica.2019.108507
    [35] Z. H. Luo, B. Z. Guo, Stability and stabilization of infinite dimensional systems with applications, Springer-Verlag, London, 1999. https://doi.org/10.1007/978-1-4471-0419-3
    [36] M. Ansari, E. Esmailzadeh, D. Younesian, Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads, J. Sound Vib., 330 (2011), 1455–1471. https://doi.org/10.1016/j.jsv.2010.10.005 doi: 10.1016/j.jsv.2010.10.005
    [37] H. Ding, L. Q. Chen, S. P. Yang, Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J. Sound Vib., 331 (2012), 2426–2442. https://doi.org/10.1016/j.jsv.2011.12.036 doi: 10.1016/j.jsv.2011.12.036
    [38] R. F. Curtain, H. Zwart, An introduction to infinite-dimensional linear systems theory, Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-4224-6
    [39] F. F. Jin, B. Z. Guo, Lyapunov approach to output feedback stabilization for the Euler-Bernoulli beam equation with boundary input disturbance, Automatica, 52 (2015), 95–102. https://doi.org/10.1016/j.automatica.2014.10.123 doi: 10.1016/j.automatica.2014.10.123
    [40] Ö. Civalek, Ç. Demir, Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory, Appl. Math. Model., 35 (2011), 2053–2067. https://doi.org/10.1016/j.apm.2010.11.004 doi: 10.1016/j.apm.2010.11.004
    [41] H. Lhachemi, D. Saussié, G. C. Zhu, Boundary feedback stabilization of a flexible wing model under unsteady aerodynamic loads, Automatica, 97 (2018), 73–81. https://doi.org/10.1016/j.automatica.2018.07.029 doi: 10.1016/j.automatica.2018.07.029
    [42] A. Barari, H. D. Kaliji, M. Ghadimi, G. Domairry, Non-linear vibration of Euler-Bernoulli beams, Latin Amer. J. Solids Struct., 8 (2011), 139–148. https://doi.org/10.1590/S1679-78252011000200002 doi: 10.1590/S1679-78252011000200002
    [43] A. D. Senalp, A. Arikoglu, I. Ozkol, V. Z. Dogan, Dynamic response of a finite length Euler-Bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force, J. Mech. Sci. Technol., 24 (2010), 1957–1961. https://doi.org/10.1007/s12206-010-0704-x doi: 10.1007/s12206-010-0704-x
    [44] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1
    [45] M. S. Edalatzadeh, K. A. Morris, Optimal actuator design for semilinear systems, SIAM J. Control Optim., 57 (2019), 2992–3020. https://doi.org/10.1137/18M1171229 doi: 10.1137/18M1171229
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1524) PDF downloads(152) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog