We study the exponential stability of Euler-Bernoulli beam with interior time delays and boundary damping. At first, we prove the well-posedness of the system by the C0 semigroup theory. Next we study the exponential stability of the system by constructing appropriate Lyapunov functionals. We transform the exponential stability issue into the solvability of inequality equations. By analyzing the relationship between delays parameters α and damping parameters β, we describe (β,α)-region for which the system is exponentially stable. Furthermore, we obtain an estimation of the decay rate λ⁎.
1. Introduction
It is well known that the time delay always exists in real system, which may be caused by acquisition of response and excitation data, online data processing, and computation of control forces. Since time delay may destroy stability [1, 2] even if it is very small, the stabilization problem of systems with time delays has been a hot topic in the mathematical control theory and engineering. In recent years, the systems described by PDEs with time delays have been an active area of research; see [3–7] and references therein. Generally speaking, there are mainly three kinds of time delay in the system, one is the interior time delay of the system (also called structural memory), one is the input delay (control delay), and the third is the output delay (measurement delay). Many scholars have made great efforts to minimize the negative effects of time delays although time delay cannot be eliminated due to its inherent nature, for example, [8–10] for boundary control with delays, [11, 12] for internal control delays, and [13] for output delays.
In past several years, the research on the Euler-Bernoulli beam with time delay has made great progress. For example, Park et al. [14] considered the stabilization problem of an Euler-Bernoulli beam with structural memory; Liang et al. [15] introduced the modified Smith predictor to Euler-Bernoulli beam with the boundary control and the delayed boundary measurement; Shang et al. [16–18] investigated the stabilization problem of the Euler-Bernoulli beam with boundary input delay; Yang et al. [19, 20] solved the stabilization problem of constant and variable coefficients Euler-Bernoulli beam with delayed observation and boundary control; at the same time, Jin and Guo [21] solved the output feedback stabilization of Euler-Bernoulli beam by Lyapunov approach. However, few people investigate the influence of an Euler-Bernoulli beam with interior delays and boundary damping on the system stability. In this paper we mainly study the exponential stability of a system described by the Euler-Bernoulli beam with interior delays and boundary damping. More precisely, we consider the following system, whose dynamic behavior is governed by the Euler-Bernoulli beam: (1)yttx,t+yxxxxx,t-2αytx,t-τ=0,x∈0,1,t>0,y0,t=yx0,t=yxx1,t=0,yxxx1,t=βyt1,t,yx,0=y0x,ytx,0=y1x,x∈0,1,ytx,s=h0x,s,x∈0,1,s∈-τ,0,with β,α,τ>0, where yt(x,t)=∂y/∂t, yx(x,t)=∂y/∂x, and τ is the delay time. We mainly investigate its exponential stability.
The rest is organized as follows: In Section 2, we at first formulate problem (1) into an appropriate Hilbert space H and then study the well-posedness of the system by the semigroup theory. In Section 3, we construct a Lyapunov functional for system (1) and prove the exponential stability under certain conditions. By optimization parameters we obtain a complicated relationship between the decay rate λ and delay time τ. Finally, in Section 4, we give a brief conclusion.
2. Well-Posedness of the System
In this section, we will discuss the well-posedness and some basic properties of system (1). For the purpose, firstly we formulate system (1) into an appropriate Hilbert space.
Set(2)zx,ρ,t=ytx,t-τρ,x∈0,1,ρ∈0,1,t>0.Clearly, z(x,ρ,t) satisfies (3)τztx,ρ,t+zρx,ρ,t=0,x∈0,1,ρ∈0,1,zx,0,t=ytx,t,zx,1,t=ytx,t-τ.Thus, system (1) is equivalent to the following:(4)yttx,t+yxxxxx,t-2αzx,1,t=0,x∈0,1,τztx,ρ,t+zρx,ρ,t=0,x∈0,1,ρ0,1,y0,t=yx0,t=yxx1,t=0,yxxx1,t=βyt1,t,zx,0,t=ytx,t,yx,0=y0x,ytx,t=y1x,zx,ρ,0=z0x,ρ=h0x,-τρ.
Set (5)HE20,1=y∈H20,1∣y0=y′0=0,where Hk(0,1) is the usual Sobolev space of order k. We take the state space as(6)H=HE20,1×L20,1×L20,1×0,1equipped with the following inner product, for any Yi=(fi,gi,hi)T∈H,i=1,2:(7)Y1,Y2H=∫01f1,xxxf2,xxx¯+g1xg2x¯dx+τ∬01h1x,ρh2x,ρ¯dρdx.Obviously (H,·H) is a Hilbert space.
We define an operator A in H by(8)Afgh=gx-fxxxxx+2αhx,1-1τhρx,ρwith domain (9)DA=f,g,hT∈Hf∈H40,1∩HE20,1,g∈HE20,1,h∈H10,1f′′1=0,f′′′1=βg1,hx,0=gx⬄⬄. With the assistance of operator A, we can rewrite (4) as an evolution equation in H: (10)dYtdt=AYt,t>0,Y0=Y0,where Y(t)=(y(x,t),yt(x,t),z(x,ρ,t))T and Y0=(y0(x),y1(x),z0(x,ρ))T.
For operator A, we have the following result.
Lemma 1.
Let A be defined as (8) and (9). Then A is a closed and densely defined linear operator in H. For any β>0 and α>0, 0∈ρ(A) and A-1 is compact on H. Hence σ(A) consists of all isolated eigenvalues of finite multiplicity.
Proof.
It is easy to check that A is a closed and densely defined linear operator in H; the detail of the verification is omitted.
Let α>0, β>0 and for any F≜(μ,ν,ω)T∈H we consider the equation AY=F where Y=(f,g,h)∈D(A); that is, (11)gx=μx,-fxxxxx+2αhx,1=νx,-1τhρx,ρ=ωx,ρ,with boundary condition (12)f0=f′0=f′′1=0,f′′′1=βg1,hx,0=gx.By a complex calculation, we get the solution (13)gx=μx,hx,ρ=μx-∫0ρτωx,sds,fx=-∫0x∫0r∫1q∫1p2αμs-2α∫01τωs,kdk-υsdsdpdqdr+16βμ1x3.Let f, g, h be given as (13). Then we have AY=F and Y=(f,g,h)∈D(A). The closed operator theorem asserts that 0∈ρ(A), and A-1:H→D(A) is a bounded linear operator. Since D(A)⊂HE4(0,1)×H2(0,1)×H1(0,1), the Sobolev Embedding Theorem asserts that A-1 is a compact operator on H. Hence, by the spectral theory of compact operator, σ(A) consists of all isolated eigenvalues of finite multiplicity.
Theorem 2.
Let A and H be defined as before. Then A generates a C0 semigroup on H. Hence, system (10) is well posed.
Proof.
For any real Y=(f,g,h)T∈D(A), we calculate(14)AY,YH=∫01g′′xf′′xdx+∫01gx-fxxxxx+2αhx,1dx-∬01hρx,ρhx,ρdxdρ=-g1f′′′1+2α∫01hx,1gxdx-12∫01h2x,1-h2x,0dx=-βg21+2α∫01hx,1gxdx-12∫01h2x,1dx+12∫01g2xdx.Since β>0, we have (15)AY,YH≤-12∫01hx,1-2αgx2dx+12∫014α2+1g2xdx≤MY,YH,where M=2α2+1/2, which shows that A-MI is a dissipative operator. This together with Lemma 1 shows that A-MI satisfies the conditions of Lumer-Phillips theorem [22]. So A generates a C0 semigroup on H.
3. Exponential Stability of the System
In this section, we consider the exponential stability issue of system (1) based on Lyapunov method.
The energy function of system (1) is defined as(16)Et=12∫01yxx2x,t+yt2x,tdx.
In what follows, we will give some lemmas that are the foundation of our method.
Lemma 3 (see [23]).
Let E(t) be a nonnegative function on R+. If there exists a function V(t) and some positive numbers c1 and λ such that the conditions(17)Vt>c1eλtEt,∀t≥0,(18)V˙t≤0∀t≥0hold, then E(t) decays exponentially at rate λ.
In order to construct a function V(t) satisfying the conditions in Lemma 3, we set (19)Gt=η∫01xyxx,tytx,tdx,where η is a constant and satisfies 0<η<2.
We can establish an equivalence relation between G(t) and E(t) via the following Lemma.
Lemma 4.
Let E(t) and G(t) be defined as before. Then there exist positive constants c2 and c3 such that(20)c2Et≤Gt+Et≤c3Et,∀t≥0,holds.
Proof.
Let y(x,t) be the solution of (1). Applying Young’s and Poincaré’s inequalities (21)∫01xyxx,tytx,tdx≤δ2∫01yt2x,tdx+12δ∫01x2yx2x,tdx≤δ2∫01xyt2x,tdx+12δ∫01x2dx∫0xyxx2s,tds<δ2∫01yt2x,tdx+18δ∫01yxx2x,tdx,∀δ>0,t≥0.Taking δ=1/2, we get(22)η∫01xyxx,tytx,tdx<η2Et,t≥0.
Since 0<η<2, we can set c1=1-η/2 and c2=1+η/2; then(23)c2Et≤Gt+Et≤c3Et.The desired inequality follows.
Let λ>0. We define a function V(t) by(24)Vt=V1t+V2t,where (25)V1t=e2λt12∫01yxx2x,t+yt2x,tdx+η∫01xyxx,tytx,tdx,V2t=2αe-λτ∫01∫t-τte2λs+τyt2x,sdsdx.Noting that α>0, according to Lemma 4 we can see that the following result is true.
Lemma 5.
Let V(t) defined as before. Then V(t) satisfies condition (17); that is,(26)Vt>c2e2λtEt,t≥0.
In what follows, we calculate V˙(t). For V1(t) we have the following result.
Lemma 6.
Let V1(t) be defined as before and let y(x,t) be the solution of (1). Then(27)V˙1t≤e2λtλ-3η2+λη2+βη2+αη24eλτ∫01yxx2dx+λ-η2+λη2+αeλτ∫01yt2x,tdx+e2λt-β+η2+βη2yt21,t+2αe-λτeλt∫01yt2x,t-τdx.
Proof.
By definition, we see that(28)V1t=e2λtEt+Gt,where E(t) and G(t) are defined as before.
So(29)V˙1t=e2λt2λEt+Gt+E˙t+G˙t.In what follows, we will calculate E˙(t) and G˙(t).
Using integration by parts and the boundary condition, we have(30)E˙t=∫01yxxx,tyxxtx,t+ytx,tyttx,tdx=yxxx,tytxx,t01-∫01yxxxx,tytxx,tdx-∫01ytx,tyxxxxx,tdx+∫012αytx,tytx,t-τdx=yxxx,tytxx,t01-yxxxx,tytx,t01+∫01yxxxxx,tytx,tdx-∫01ytx,tyxxxxx,tdx+2α∫01ytx,tytx,t-τdx=-βyt21,t+2α∫01ytx,tytx,t-τdx,G˙t=η∫01xytxx,tytx,tdx+∫01xyxx,tyttx,tdx=η∫01xytxx,tytx,tdx+2α∫01xyxx,tytx,t-τdx-∫01xyxx,tyxxxx,tdx=η12yt21,t-12∫01yt2x,tdx-32∫01yxx2x,tdx-βyx1,tyt1,t+2α∫01xyxx,tytx,t-τdx,where we have used equalities(31)∫01xytxx,tytx,tdx=12yt21,t-12∫01yt2x,tdx∫01xyxxxxx,tyxx,tdx=32∫01yxx2x,tdx+βyx1,tyt1,t.Summarizing the above all, we have(32)V˙1=e2λtλ∫01yxx2x,t+yt2x,tdx+2λη∫01xyxx,tytx,tdx+e2λt-βyt21,t+η2yt21,t-βηyx1,tyt1,t-η2∫01yt2x,tdx-3η2∫01yxx2x,tdx+2α∫01ytx,tytx,t-τdx+2αη∫01xyxx,tytx,t-τdx=e2λtλ-3η2∫01yxx2x,tdx+λ-η2∫01yt2x,tdx+e2λt2λη∫01xyxx,tytx,tdx+2α∫01ytx,tytx,t-τdx+2αη∫01xyxx,tytx,t-τdx+e2λtη-2β2yt21,t-βηyx1,tyt1,t.Since (33)∫01xyxx,tytx,tdx≤14∫01yxx2dx+14∫01yt2x,tdx,-βηyx1,tyt1,t≤βη2yx21,t+βη2yt21,t≤βη2∫01yxx2x,tdx+βη2yt21,t,we have(34)V˙1≤e2λtλ-3η2+λη2∫01yxx2x,tdx+λ-η2+λη2∫01yt2x,tdx+e2λt2α∫01ytx,tytx,t-τdx+2αη∫01xyxx,tytx,t-τdx+e2λtη-2β2yt21,t+e2λtβη2∫01yxx2x,tdx+βη2yt21,t=e2λtλ-3η2+λη2+βη2∫01yxx2x,tdx+λ-η2+λη2∫01yt2x,tdx+e2λt2α∫01ytx,tytx,t-τdx+2αη∫01xyxx,tytx,t-τdx+e2λtη-2β+βη2yt21,t.We now estimate the integral terms with time delay. Applying Young’s and Poincaré’s inequalities, we have(35)∫01ytx,tytx,t-τdx≤δ12∫01yt2x,t-τdx+12δ1∫01yt2x,tdx,∫01xyxx,tytx,t-τdx≤δ22∫01yt2x,t-τdx+18δ2∫01yxx2x,tdx.Thus,(36)V˙1t≤e2λtλ-3η2+λη2+βη2+αη4δ2∫01yxx2dx+λ-η2+λη2+αδ1∫01yt2x,tdx+e2λt-β+η2+βη2yt21,t+αδ1+αηδ2e2λt∫01yt2x,t-τdx.Taking δ1=e-λτ, δ2=e-λτ/η, we obtain (37)V˙1t≤e2λtλ-3η2+λη2+βη2+αη24eλτ∫01yxx2dx+λ-η2+λη2+αeλτ∫01yt2x,tdx+e2λt-β+η2+βη2yt21,t+2αe-λτe2λt∫01yt2x,t-τdx.The desired inequality follows.
Since(38)V2t=2αe-λτ∫01∫t-τte2λs+τyt2x,sdsdxwe have(39)V˙2t=2αe-λτ∫01e2λt+τyt2x,tdx-2αe-λτ∫01e2λtyt2x,t-τdx.Employing the estimate, we have(40)V˙t=V˙1t+V˙2t≤e2λtλ-3η2+λη2+βη2+αη24eλτ∫01yxx2dx+λ-η2+λη2+3αeλτ∫01yt2x,tdx+e2λt-β+η2+βη2yt21,t.Clearly, if the parameters η, β, α, λ, and τ are such that the inequalities(41)λ-3η2+λη2+βη2+αη2eλτ4≤0,λ-η2+λη2+3αeλτ≤0,-β+η2+βη2≤0hold, then we have V˙(t)≤0.
Summarizing discussion above, we have the following result.
Theorem 7.
Let y(x,t) be the solution of (1), and let 0<η<2 and λ>0. If inequalities (41) hold, then the energy function E(t) decays exponentially at rate 2λ.
We now are in a proposition to study the solvability of inequalities (41). Noting that η is not a system parameter, it is only a middle parameter which is introduced in the multiplier term. From the third inequality of (41) we see that η and β have a relationship: (42)0<η<2β1+β.Taking η=2β/(1+β), (41) is equivalent to (43)λ-3β1+β+λβ1+β+β21+β+αβ2eλτ1+β2≤0,λ-β1+β+λβ1+β+3αeλτ≤0.
Theorem 8.
Set η=2β/(1+β). If α and β satisfy the inequality(44)α<min3β+2-β,β31+β,then there exists λ∗>0 such that for λ∈(0,λ∗] inequality (41) holds true.
Proof.
If (44) holds, then (45)α<3β+2-β,α<β31+β,or equivalently (46)-3β1+β+β21+β+αβ21+β2<0,-β1+β+λβ1+β+3α<0.
Set (47)fx=x-3β1+β+xβ1+β+β21+β+αβ2exτ1+β2,x≥0,gy=y-β1+β+yβ1+β+3αeyτ,y≥0.Since(48)f0=-3β1+β+β21+β+αβ21+β2<0,g0=-β1+β+λβ1+β+3α<0,f3=3+β21+β+αβ2e3τ1+β2>0,g1=1+3αeτ>0,there exist x∗∈(0,3) and y∗∈(0,1) such that f(x∗)=0 and g(y∗)=0.
Set (49)x∗=minx∗∈0,3,fx∗=0,y∗=miny∗∈0,1,gy∗=0,and λ∗=min{x∗,y∗}. Clearly, when λ∈(0,λ∗], we have f(λ)≤0 and g(λ)≤0, so (43) holds. Hence (41) holds true.
In what follows, we discuss the property of the function(50)Gβ=min3β+2-β,β31+β,β≥0.
We consider equation(51)3β+2-β=β31+β,and it is equivalent to (52)9+15β+2β2-3β3=0.This equation has three real roots β1<β2<0<2<β3<3. So we have (53)Gβ=β31+β,β∈0,β33β+2-β,β∈β3,33β+2-β<0,β≥3,(54)maxβ>0Gβ=β331+β3.
3.1. (β,α)-Region of the Exponential Stability
According to (52) we determine β3≃2.818. And according to (53) we draw the (β,α)-region.
The picture of G(β) is given as Figure 1. (β,α)-region is given by(55)∑β,α=β,α∣α<Gβ,β∈0,3.
The graph of function G, which gives the relationship between α and β.
Figure 1 gives the graph of function G(β) that gives the relationship between α and β with which system (1) is exponentially stable. From this picture we see that if α is larger, we cannot stabilize it by the boundary damping. α has upper bound α∗=β3/3(1+β3)≃0.246.
3.2. The Best Decay Rate λ∗
Suppose that α<G(β) with β∈(0,3). According to (43) we determine the best decay rate λ∗. Note that λ∗=min{x∗,y∗}∈(0,1), where x∗ and y∗ solve the following equation, respectively: (56)x∗-3β1+β+x∗β1+β+β21+β+αβ2ex∗τ1+β2=0,y∗-β1+β+y∗β1+β+3αey∗τ=0.Firstly, let(57)px=x-3β1+β+xβ1+β+β21+β+αβ2exτ1+β2,qx=x-β1+β+xβ1+β+3αexτ.Obviously, p(x) and q(x) both are monotonic function. For comparing to the two equations of (56), we define the function f(x):(58)fx=px-qx=x-3β1+β+xβ1+β+β21+β+αβ2exτ1+β2-x-β1+β+xβ1+β+3αexτ=β2-2β1+β+αβ21+β2-3exτ.Since β,α,τ>0, we have α[β2/(1+β)2-3]exτ<0. When β∈(0,2], we get (β2-2β)/(1+β)<0; that is, f(x)<0. Therefore, when β∈(0,2], λ∗=y∗.
An example is given: Assume that β=1, α=0.1, τ=0.1 (i.e., they are constants). We obtain the approximative zero points of p(x) and q(x) from Figure 2. That is, x∗≃0.6489 and y∗≃0.1307. Therefore, we get the best decay rate λ∗=y∗≃0.1307.
β=1, α=0.1, and τ=0.1.
Next, we consider β∈(2,β3). By researching (58), we can obtain (β2-2β)/(1+β)>0 and [β2/(1+β)2-3]exτ≤β2/(1+β)2-3≤0. We have f(x)≤(β2-2β)/(1+β)+α[β2/(1+β)2-3].
Let (59)H=β2-2β1+β+αβ21+β2-3<0.We have α>(2β-β2)(1+β)/(β2-3(1+β)2)>0. According to (β,α)-region of the exponential stability, we obtain α<β/3(1+β). After comparison, (60)2β-β21+ββ2-31+β2<α<β31+β.
To summarize, β∈(2,β3], when (2β-β2)(1+β)/(β2-3(1+β)2)<α<β/3(1+β), we can get f(x)≤H<0. Therefore, the best decay λ∗=y∗.
Then, we give two examples.
Assume that β=2.1∈(2,β3], α=0.1∈(0.0267,0.2258), τ=0.1 (i.e., they are constants); we can obtain Figure 3. We get the best decay rate λ∗=y∗.
β=2.1, α=0.1, and τ=0.1.
Assume that β=2.1∈(2,β3], α=0.2∈(0.0267,0.2258), τ=0.1 (i.e., they are constants); we can obtain Figure 4. We get the best decay rate λ∗=y∗.
β=2.1, α=0.2, and τ=0.1.
Finally, we consider β∈(β3,3). We can also obtain α>(2β-β2)(1+β)/(β2-3(1+β)2)>0, when f(x)≤H<0. Here, α<3/β+2-β. After comparison, (61)2β-β21+ββ2-31+β2>3β+2-β.
Therefore, Not all f(x)<0 were always correct.
Summarizing the above all, the best decay λ∗ is easy to determine, when β∈(0,2). The best decay λ∗ is not easy to determine, when β∈(2,3). We have two conclusion:
β∈(0,2], the best decay λ∗=y∗;
β∈(2,β3], when (2β-β2)(1+β)/(β2-3(1+β)2)<α<β/3(1+β), the best decay λ∗=y∗.
4. Conclusions
In this paper, using the Lyapunov functional approach we discussed the exponential stabilization of an Euler-Bernoulli beam equation with interior delays and boundary damping. Different from the earlier papers, we added a multiplier term e2λt to the Lyapunov function so as to transform the exponential stability. By solving the inequality equations, we give the exponential stability region of the system.
We note that the method used in this paper also can apply to the investigation of the exponential stability of other model. In the future, we will study the boundary feedback control anti-interior time delay for other models.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This research is supported by the Natural Science Foundation of China (Grant nos. NSFC-61174080, 61573252, and 61503275).
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