STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS WITH VARIABLE COEFFICIENTS UNDER DELAYED BOUNDARY OUTPUT FEEDBACK

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1 Electronic Journal of Differential Equations, Vol. 25 (25), No. 75, pp. 4. ISSN: URL: or ftp ejde.math.txstate.edu STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS WITH VARIABLE COEFFICIENTS UNDER DELAYED BOUNDARY OUTPUT FEEDBACK KUN-YI YANG, JING-JING LI, JIE ZHANG Abstract. In this article, we study the stabilization of an Euler-Bernoulli beam equation with variable coefficients where boundary observation is subject to a time delay. To resolve the mathematical complexity of variable coefficients, we design an observer-predictor based on the well-posed open-loop system: the state of system is estimated with available observation and then predicted without observation. We show that the closed-loop system is stable exponentially under estimated state feedback by a numerical simulation illustrating our results.. Introduction The phenomenon of time delay is commonly observed in modern engineering and scientific research [3, 4, 5, 6, 7, 9, 2, 9. Much attention has been devoted to the stability of control systems with time delay. Nevertheless, even a small delay may break the system s stability [3, 4, 5, 6, 7,. It is indicated in [8 that for distributed parameter control systems, time delay in observation and control can cause complications. Stimulated by the work in [4, we solve the stabilization problem with delayed observation and boundary control, for the one-dimensional Euler-Bernoulli beam equation [6. In this article, we focus on the boundary stabilization of an Euler-Bernoulli beam equation with variable coefficients where boundary observation contains a fixed time delay. This is a generalization of the similar work such as [6 for the beam equation with constant coefficients. It is obvious that variable coefficients present more mathematical challenges, making the stabilization problems of the system much more complicated since it is difficult to construct the Lyapunov functions and estimate the eigenvalues and eigenfunctions by asymptotic analysis. Consider the following nonuniform Euler-Bernoulli beam equation with linear boundary feedback control: 2 Mathematics Subject Classification. 35J, 93C2, 93C25. Key words and phrases. Euler-Bernoulli beam equation; variable coefficients; time delay; observer; feedback control; exponential stability. c 25 Texas State University - San Marcos. Submitted December 3, 24. Published March 24, 25.

2 2 K.-Y. YANG, J.-J. LI, J. ZHANG EJDE-25/75 ρ(x)w tt (x, t) + (EI(x)w xx (x, t)) xx =, < x <, t >, w(, t) = w x (, t) = w xx (, t) =, t, (EI(x)w xx ) x (, t) = u(t), t, y(t) = w t (, t τ), t > τ, w(x, ) = w (x), w t (x, ) = w (x), x, (.) where x stands for the position and t the time, w is the state, u is the boundary controller input, (w, w ) T is the initial value, τ > is a known constant time delay, and y is the delayed observation(or output) which suffers from a given time delay τ. EI(x)(> ) C 2 [, is the flexural rigidity of the beam, and ρ(x)(> ) C[, is the mass density at x. The system above is considered in the energy state space H = H 2 E(, ) L 2 (, ), H 2 E(, ) = {f H 2 (, ) : f() = f () = }. The energy of the system is E (t) = 2 [EI(x)w 2 xx(x, t) + ρ(x)w 2 t (x, t)dx. As noted in [4 (where EI(x) = ρ(x) = ), even a small amount of time delay in the stabilizing boundary output feedback schemes destabilizes the system. Therefore, it is important to design stabilizing controllers that are robust to time delay for systems described in (.). The next section shows the well-posedness of the considered open-loop system. In section 3, we design the observer and predictor for the system. The asymptotic stability of the closed-loop system under the estimated state feedback control is then studied in section 4. Section 5 illustrates the simulation results and concludes the paper. 2. Well-posedness of the open-loop system We introduce a new variable z(x, t) = w t (, t xτ). Then the system (.) becomes ρ(x)w tt (x, t) + (EI(x)w xx (x, t)) xx =, < x <, t >, w(, t) = w x (, t) = w xx (, t) =, t, (EI(x)w xx ) x (, t) = u(t), t, τz t (x, t) + z x (x, t) =, < x <, t, z(, t) = w t (, t), t, w(x, ) = w (x), w t (x, ) = w (x), x, z(x, ) = z (x), x, y(t) = z(, t), t τ, (2.) where z is the initial value of the variable z. We consider the system (2.) in the energy state space H = H L 2 (, ), with the state variable (w(, t), w t (, t), z(, t)) T for which the inner product induced norm is defined as following: E (t) = 2 (w(, t), w t(, t), z(, t)) T 2 H

3 EJDE-25/75 STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS 3 = 2 [EI(x)w 2 xx(x, t) + ρ(x)w 2 t (x, t) + z 2 (x, t)dx. The input space and the output space are the same U = Y = C. Theorem 2.. System (2.) is well-posed: For any (w, w, z ) T H and u L 2 loc (, ), there exists a unique solution of (2.) such that (w(, t), w t(, t), z(, t)) T belongs to C(, ; H); and for any T >, there exist a constant C T > such that (w(, T ), w t (, t), z(, T )) T 2 H + T y(t) 2 dt T C T [ (w, w, z ) T 2 H + u(t) 2 dt. Proof. Firstly, we represent the system ρ(x)w tt (x, t) + (EI(x)w xx (x, t)) xx =, < x <, t, as a second-order system in H, w(, t) = w x (, t) = w xx (, t) =, t, (EI(x)w xx ) x (, t) = u(t), t, y w (t) = w t (, t), t, w tt (, t) + Aw(, t) + Bu(t) =, < x <, t, y ω (t) = B w t (, t), t, where A is a self-adjoint operator in H and B is the input operator: Af = ρ(x) (EI(x)f ), f D(A) = {f H 4 (, ) H 2 E(, ) : f () = (EIf ) () = }, B = δ(x ). (2.2) (2.3) (2.4) Here δ( ) denote the Dirac distribution. It was shown in [3 that system (2.3) and (2.4) is well-posed in the sense of Salamon [2: for any u L 2 loc (, ) and (w, w ) T H, there exists a unique solution (w(, t), w t (, t)) T C(, ; H) to (2.3) and for any T >, there exists a constant D T > such that (w(, T ), w t (, T )) T 2 H + D T [ (w, w ) T 2 H + T T y w (t) 2 dt u(t) 2 dt. Then the following inequality can be shown similarly as those in [7: T (w(, T ), w t (, T ), z(, T )) T 2 H + y(t) 2 dt T C T [ (w, w, z ) T 2 H + u(t) 2 dt, for a constant C T >. The details are omitted. (2.5)

4 4 K.-Y. YANG, J.-J. LI, J. ZHANG EJDE-25/75 Theorem 2. illustrates that, for any initial value in the state space, the output belongs to L 2 loc (τ, ) as long as the input u belongs to L2 loc (, ). This fact is particularly necessary to the solvability of observer shown in the next section ( [3, 4). 3. Observer and predictor design For any fixed time delay τ >, and when t > τ, we propose a two-step method to estimate the state of (.) by designing the observer and predictor systems. Step. From the known observation signal {y(s + τ) : s [, t τ, t > τ}, we construct an observer system to estimate the state {w(x, s) : s [, t τ, t > τ} which satisfies ρ(x)w ss (x, s) + (EI(x)w xx (x, s)) xx =, < x <, < s < t τ, t > τ, w(, s) = w x (, s) = w xx (, s) =, s t τ, t > τ, (EI(x)w xx ) x (, s) = u(s), s t τ, t > τ, y(s + τ) = w s (, s), s t τ, t > τ. Then a Luenberger observer naturally can be constructed for the system (3.), ρ(x)ŵ ss (x, s) + (EI(x)ŵ xx (x, s)) xx =, < x <, < s < t τ, t > τ, ŵ(, s) = ŵ x (, s) = ŵ xx (, s) =, s t τ, t > τ, (3.) (EI(x)ŵ xx ) x (, s) = u(s) + k [ŵ s (, s) y(s + τ), s t τ, t > τ, k >, ŵ(x, ) = ŵ (x), ŵ s (x, ) = ŵ (x), x, (3.2) where (ŵ, ŵ ) T is an arbitrary assigned initial state of the observer. For (3.2) to be an observer for (3.), we have to show its convergence. To do this, we set ε(x, s) = ŵ(x, s) w(x, s), s t τ, t > τ. (3.3) Then by (3.) and (3.2), ε satisfies ρ(x)ε ss (x, s) + (EI(x)ε xx (x, s)) xx =, < x <, < s < t τ, t > τ, ε(, s) = ε x (, s) = ε xx (, s) =, s t τ, t > τ, (EI(x)ε xx ) x (, s) = k ε s (, s), s t τ, t > τ, k >, The system above can be written as where ε(x, ) = ŵ (x) w (x), x, ε s (x, ) = ŵ (x) w (x), x. d ds ( ε(, s) ε s (, s) ) = B B(f, g) T = (g, ρ(x) (EI(x)f (x)) ) T, (3.4) ( ) ε(, s), (3.5) ε s (, s) D(B) = { (f, g) (H 4 (, ) H 2 E(, )) H 2 E(, ) : f () =, (EIf ) () = k g() }, and B generates an exponentially stable C -semigroup on H satisfying: (3.6) e Bs Me ωs, s, (3.7)

5 EJDE-25/75 STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS 5 for some positive constants M, ω. Hence, for any (w, w ) T H and (ŵ, ŵ ) T H, there exists a unique solution to (3.4) such that (ε(, s), ε s (, s)) T H Me ωs (ŵ w, ŵ w ) T H, (3.8) for all s [, t τ and all t > τ. Step 2. Predict {(w(x, s), w s (x, s)) T, s (t τ, t, t > τ} by {(ŵ(x, s), ŵ s (x, s)) T, s [, t τ, t > τ}. This is done by solving (.) with estimated initial value (ŵ(x, t τ), ŵ s (x, t τ)) T obtained from (3.2): ρ(x)ŵ ss (x, s, t) + (EI(x)ŵ xx (x, s, t)) xx =, < x <, t τ < s < t, t > τ, ŵ(, s, t) = ŵ x (, s, t) = ŵ xx (, s, t) =, t τ s t, t > τ, (EI(x)ŵ xx ) x (, s, t) = u(s), t τ s t, t > τ, ŵ(x, t τ, t) = ŵ(x, t τ), ŵ s (x, t τ, t) = ŵ s (x, t τ), x, t τ s t, t > τ. We finally get the estimated state variable by which is assured by Theorem 3. below. Theorem 3.. For all t > τ, we have (3.9) w(x, t) = ŵ(x, t, t), t > τ, (3.) (w(, t) w t (, t), w t (, t) w t (, t)) T H Me ω(t τ) (ŵ w, ŵ w ) T H, (3.) where (ŵ, ŵ ) T is the initial state of observer (3.2), (w, w ) T is the initial state of original system (.), M, ω are constants in (3.7). Proof. Let Then ε(x, s, t) satisfies ε(x, s, t) = ŵ(x, s, t) w(x, s), t τ s t, t > τ. (3.2) ρ(x)ε ss (x, s, t) + (EI(x)ε xx (x, s, t)) xx =, < x <, t τ < s < t, t > τ; ε(, s, t) = ε x (, s, t) = ε xx (, s, t) = (EI(x)ε xx ) x (, s, t) =, t τ s t, t > τ; ε(x, t τ, t) = ε(x, t τ), ε s (x, t τ, t) = ε s (x, t τ), x, t τ s t, t > τ; (3.3) which is a conservative system (ε(, s, t), ε s (, s, t)) T H = (ε(, t τ), ε s (, t τ)) T H. (3.4) Collecting (3.8), (3.) and (3.4) gives (3.).

6 6 K.-Y. YANG, J.-J. LI, J. ZHANG EJDE-25/75 4. Stabilization by the estimated state feedback Since the feedback u(t) = k 2 w t (, t) = k 2 ŵ s (, t, t) (k 2 > ) stabilizes exponentially the system (.), and we have the estimation w t (, t) of w t (, t), it is natural to design the estimated state feedback control law of the following: u (t) = { k 2 w t (, t) = k 2 ŵ s (, t, t), t > τ, k 2 >,, t [, τ. (4.) The closed-loop system becomes a system of partial differential equations (4.2)-(4.3) via applying the control law above: ρ(x)w tt (x, t) + (EI(x)w xx (x, t)) xx =, < x <, t >, w(, t) = w x (, t) = w xx (, t) =, t, (EI(x)w xx ) x (, t) = u (t), t, w(x, ) = w (x), w t (x, ) = w (x), x, (4.2) and ρ(x)ŵ ss (x, s) + (EI(x)ŵ xx (x, s)) xx =, < x <, < s < t τ, t > τ, ŵ(, s) = ŵ x (, s) = ŵ xx (, s) =, s t τ, t > τ, (EI(x)ŵ xx ) x (, s) = u (s) + k [ŵ s (, s) w s (, s), s t τ, t > τ, k >, and ŵ(x, ) = ŵ (x), ŵ s (x, ) = ŵ (x), x, ρ(x)ŵ ss (x, s, t) + (EI(x)ŵ xx (x, s, t)) xx =, < x <, t τ < s < t, t > τ, ŵ(, s, t) = ŵ x (, s, t) = ŵ xx (, s, t) =, t τ s t, t > τ, (EI(x)ŵ xx ) x (, s, t) = u (s), t τ s t, t > τ, ŵ(x, t τ, t) = ŵ(x, t τ), ŵ s (x, t τ, t) = ŵ s (x, t τ), x, t > τ. (4.3) We consider the closed-loop system (4.2)-(4.3) in the state space X = H 3. Obviously the system (4.2)-(4.3) is equivalent to the system (4.4)-(4.6) for t > τ: ρ(x)w tt (x, t) + (EI(x)w xx (x, t)) xx =, < x <, t > τ, w(, t) = w x (, t) = w xx (, t) =, t > τ, (EI(x)w xx ) x (, t) = k 2 [w t (, t) + ε s (, t, t), t > τ, k 2 >, w(x, ) = w (x), w t (x, ) = w (x), x, (4.4) and ρ(x)ε ss (x, s) + (EI(x)ε xx (x, s)) xx =, < x <, < s < t τ, t > τ, ε(, s) = ε x (, s) = ε xx (, s) =, s t τ, t > τ, (EI(x)ε xx ) x (, s) = k ε s (, s), s t τ, t > τ, k >, ε(x, ) = ŵ (x) w (x), ε s (x, ) = ŵ (x) w (x), x, (4.5)

7 EJDE-25/75 STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS 7 and ρ(x)ε ss (x, s, t) + (EI(x)ε xx (x, s, t)) xx =, < x <, t τ < s < t, t > τ; ε(, s, t) = ε x (, s, t) = ε xx (, s, t) = (EI(x)ε xx ) x (, s, t) =, ε(x, t τ, t) = ε(x, t τ, t), t τ s t, t > τ; x, t > τ; ε s (x, t τ, t) = ε s (x, t τ), where ε(x, s) and ε(x, s, t) are given by (3.3) and (3.2) respectively. (4.6) Theorem 4.. Let t > τ, for any (w, w ) T H, (ŵ, ŵ ) T H, there exists a unique solution of systems (4.4)-(4.6) such that (w(, t), w t (, t)) T C(τ, ; H), (ε(, s), ε s (, s)) T C(, t τ; H), (ε(, s, t), ε s (, s, t)) T C([t τ, t [τ, ); H) for any (ŵ w, ŵ w ) T D(B), where B is defined by (3.6), system (4.4) decays exponentially in the sense that (w(, t), w t (, t)) T H M e ω(t τ) (w, w ) T H + L CMM e ωτ ( ) e ω t 2 + e ωτ e ωt 2 B ( ε(, ), ε s (, ) ) T H. 2ω (4.7) Proof. For any (w, w ) T H, (ŵ, ŵ ) T H, since B defined by (3.6) generates an exponentially stable C semigroup on H, there is a unique solution (ε(, s), ε s (, s)) T C(, t τ; H) to (4.5) such that (3.8) holds. Now, for any given time t > τ, write (4.6) as ( ) ( ) d ε(, s, t) ε(, s, t) = A, (4.8) ds ε s (, s, t) ε s (, s, t) where A is defined by A(f, g) T = (g, ρ(x) (EI(x)f ) ) T, D(A) = {(f, g) T (H 4 (, ) H 2 E(, )) H 2 E(, ) : f () = (EIf ) () = }. (4.9) Then A is skew-adjoint in H and hence generates a conservative C -semigroup on H. For any (ε(, t τ), ε s (, t τ)) T H that is determined by (4.5), there exists a unique solution to (4.6) such that (ε(, s, t), ε s (, s, t)) T H = (ε(, t τ), ε s (, t τ)) T H, (4.) for all s [t τ, t. So, (ε(, s, t), ε s (, s, t)) C([t τ, t [τ, ); H). Moreover, since A is skew-adjoint with compact resolvent, the solution of (4.6) can be, in terms of s, represented as ( ) ε(x, s, t) ( ) = a ε s (x, s, t) n (t)e λns φ n (x) ( ) + b φ n (x) n (t)e λns φ n (x) (4.) φ n (x) n= where (± λφ(x), φ(x)) is a sequence of all ω-linearly independent approximated normalized orthogonal eigenfunctions of A corresponding to eigenvalues ±λ satisfies: n= φ (4) (x) + 2EI (x) EI(x) φ (x) + EI (x) EI(x) φ (x) + λ 2 ρ(x) φ(x) =, EI(x) φ() = φ () = φ () = φ () =. (4.2)

8 8 K.-Y. YANG, J.-J. LI, J. ZHANG EJDE-25/75 Set h = ( ρ(τ) EI(τ) )/4 dτ and = β 2 n/h 2, then from the reference [2, when n is large enough the solutions of the equations above can be represented as φ n (x) = e 4 β n = (n )π( + i) + O( ), n R z a(τ)dτ 2(i )[ sin ((n + ) 2 )πz cos ((n + ) 2 )πz + e (n+ 2 )πz + ( ) n e (n+ )π( z) 2 + O ( ), n βn 2 φ n(x) = ( ρ(x) ) /2e R z h 2 4 a(τ)dτ 2( + i)[ cos ((n + ) EI(x) 2 )πz sin ((n + ) 2 )πz + e (n+ 2 )πz + ( ) n e (n+ )π( z) 2 From (4.), For (4.6) we have + O ( ). n ε s (, t, t) = (4.3) [a n (t)e λnt + b n (t)e λnt φ n (). (4.4) n= l n a n (t)e λn(t τ) ( ) ( ε(, t τ) ) =, φ n ( ) ε s (, t τ) φ n ( ) = ( ε(, t τ) ε s (, t τ) = ( ε(, t τ) ε s (, t τ) ), A ) (, ( H φ n ( ) φ n ( ) ) H φ n ( ) ρ( ) (EI( )φ n( )) = [ EI(x)ε xx (x, t τ)φ λ n(x)dx ε s (x, t τ)(ei(x)φ n λ n(x)) dx n = [ (EI(x)ε xx (x, t τ)) x φ λ n(x)dx + ε sx (x, t τ)(ei(x)φ n λ n(x)) dx n = [ (EI(x)ε xx ) x (, t τ)φ n () + (EI(x)ε xx (x, t τ)) xx φ n (x)dx ε sxx (x, t τ)ei(x)φ n(x)dx = [ k ε s (, t τ)φ n () + ε sxx (x, t τ)ei(x)φ n(x)dx ) H (EI(x)ε xx (x, t τ)) xx φ n (x)dx = { k ε s (, t τ)φ n () + (EI(x)ε xx (x, t τ)) xx [e R z 4 a(τ)dτ 2(i ) ( sin ( (n + 2 )πz) cos ( (n + 2 )πz) + e (n+ 2 )πz + ( ) n e (n+ )π( z)) 2 dx

9 EJDE-25/75 STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS 9 ε sxx (x, t τ) EI(x) ρ(x)e 4 R z a(τ)dτ 2( + i)[ cos ( (n + 2 )πz) sin ( (n + 2 )πz) + e (n+ 2 )πz + ( ) n e (n+ )π( z) 2 dx + O ( ) }. n By the expression of φ n (x), there exists a constant c > such that Notice that ε s (, t τ) = ε sx (x, t τ)dx = φ n () c. (4.5) x ε sxx (y, t τ)dydx [ x /2dx [ /2 ε 2 sxx(y, t τ)dy ε 2 sxx(x, t τ)dx (4.6) [ /2 EI(x)ε 2 m sxx(x, t τ)dx where m = min ( x ) {EI(x)}. Then l n a n (t) { [ c k ε s (, t τ) + 8 ( ) /2 [ /2 ρ(x) dx + 8 EI(x)ε 2 sxx(x, t τ)dx [ c k ( m + 8 ) /2 ρ(x) dx + 8 ρ(x)dx Similarly, B(ε(, t τ), ε s (, t τ)) T H. l n b n (t)e λn(t τ) ( ) ( ) ε(, t τ) =, φ n ( ) ε s (, t τ) φ n ( ) = ( ε(, t τ) ε s (, t τ) = ( ε(, t τ) ε s (, t τ) ) (, A ) (, H φ n ( ) φ n ( ) ) H φ n ( ) ρ( ) (EI( )φ n( )) = [ EI(x)ε xx (x, t τ)φ λ n(x)dx + n = [ (EI(x)ε xx (x, t τ))dφ n (x) + = [ (EI(x)ε xx ) x (, t τ)φ n () + + ε sxx (x, t τ)ei(x)φ n(x)dx, = { k ε s (, t τ)φ n () + /2 ρ(x)(ei(x)ε xx (x, t τ)) 2 xxdx ) H } ρ(x)dx ε s (x, t τ)(ei(x)φ n(x)) dx (4.7) ε sxx (x, t τ)ei(x)φ n(x)dx (EI(x)ε xx (x, t τ)) xx φ n (x)dx (EI(x)ε xx (x, t τ)) xx [e R z 4 a(τ)dτ 2(i )

10 K.-Y. YANG, J.-J. LI, J. ZHANG EJDE-25/75 ( sin ( (n + 2 )πz) cos ( (n + 2 )πz) + e (n+ 2 )πz + ( ) n e (n+ )π( z)) 2 dx + ε sxx (x, t τ) EI(x) ρ(x)e 4 R z a(τ)dτ 2( + i)[ cos ( (n + 2 )πz) sin ( (n + 2 )πz) + e (n+ 2 )πz + ( ) n e (n+ )π( z) 2 dx + O ( ) }. n Then l n b n (t) { [ c k ε s (, t τ) + 8 ( ) /2 [ /2 ρ(x) dx + 8 EI(x)ε 2 sxx(x, t τ)dx [ c k ( m + 8 ) /2 ρ(x) dx + 8 ρ(x)dx B(ε(, t τ), ε s (, t τ)) T H. /2 ρ(x)(ei(x)ε xx (x, t τ)) 2 xxdx Collecting (4.4), (4.7), (4.8), and the expression of gives } ρ(x)dx (4.8) ε s (, t, t) C B(ε(, t τ), ε s (, t τ)) T H (4.9) for some constant C > independent of t. Now by (3.8) and C semigroup theory, we have B(ε(, t τ), ε s (, t τ)) H Me ω(t τ) B(ε(, ), ε s (, )) T H (4.2) for any t [τ, + ), where M, ω are given by (3.7). We finally get ε s (, t, t) CMe ω(t τ) B(ε(, ), ε s (, )) T H. (4.2) Furthermore, the equation (4.4) can be written as ( ) ( ) d w(, t) w(, t) = A dt w t (, t) + B w t (, t) ε s (, t, t) (4.22) where A (f, g) T = (g, ρ(x) (EI(x)f ) ) T, (f, g) T D(A ) = { (f, g) T (H 4 (, ) H 2 E(, )) H 2 E(, ) : A direct computation shows that f () =, (EIf ) () = k 2 g() }, ( ) B =. δ(x ) (4.23) B A (f, g)t = f(), (f, g) T H, (4.24) which means B A is bounded. For the energy E (t) of the system (4.4), simple computations tells us that which shows that Ė (t) = k 2 w 2 t (, t), (4.25) T k 2 w t (, t) 2 dt E (), (4.26)

11 EJDE-25/75 STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS for any T >. This inequality together with (4.24) illustrates that B is admissible for e At. Therefore, there exists a unique solution to (4.22) such that (w(, t), w t (, t)) T C(τ, ; H). Since A generates an exponentially stable C - semigroup, it follows from [24, Proposition 2.5 and (4.2) that and t t/2 t/2 τ e A(t/2 s) B ε s (, s, s)ds H L ε s (,, ) L2 (τ,t/2) e A(t s) B ε s (, s, s)ds H t L CM 2ω B(ε(, ), ε s (, )) T H, e A(t s) B ( ε s (, s, s))ds H t/2 L ε s (,, ) L 2 (t/2,t) L CMe ωτ e ωt 2 B(ε(, ), ε s (, )) T H, t, 2ω for some constant L > that is independent of ε s (, t, t), and (u v)(t) = τ { u(t), t τ, v(t), t > τ. On the other hand, the solutions of the systems (4.22) can be represented as (w(, t), w t (, t)) T = e A(t τ) (w(, τ), w t (, τ)) T + t τ e A(t s τ) B ε s (, s, s)ds t/2 = e A(t τ) (w(, τ), w t (, τ)) T + e A(t/2 τ) e A(t/2 s) B ε s (, s, s)ds t + e Aτ e A(t s) B ε s (, s, s)ds. t/2 τ (4.27) Since A generates an exponentially stable C -semigroup, there exists two positive constants M, ω such that e At M e ωt, which together with (4.27) and the conservative property of the system (4.2) for u (t) = lead to (w(, t), w t (, t)) T H M e ω(t τ) (w(, τ), w t (, τ)) T H + L CMM e ωτ ( e ω t 2 + e ωτ e ) ωt 2 B(ε(, ), εs (, )) T H 2ω = M e ω(t τ) (w(, ), w t (, )) T H + L CMM e ωτ ( e ω t 2 + e ωτ e ) ( ωt 2 B ε(, ), εs (, ) ) T H. 2ω

12 2 K.-Y. YANG, J.-J. LI, J. ZHANG EJDE-25/75 Figure. Displacement w(x, t) (top), and velocity w t (x, t) (bottom) of the solution 5. Simulation results In this section, using the finite difference method we present the numerical simulation for the closed-loop system (4.4)-(4.6). Here we choose the space grid size N = 3, time step dt =.3 and time span [, 4. Parameters and coefficients respectively are chosen to be τ = k = k 2 =, ρ(x) = +.2 sin(x), EI(x) = +.2 cos(x). For the initial values: w (x) = x 2, w (x) =,

13 EJDE-25/75 STABILIZATION OF EULER-BERNOULLI BEAM EQUATIONS 3 ε(x, ) = x 2, ε s (x, ) =, x [,, the displacement w(x, t) and velocity w t (x, t) are plotted in Figure. It shows clearly that the system is very stable with small displacement under time-variable coefficients. This simple simulation illustrates that the observer-predictor based scheme is useful to make the unstable system exponentially stable for the Euler- Bernoulli beam equation with variable coefficients. Acknowledgments. This work was supported by the National Natural Science Foundation of China under Grant 62358, and the training program for outstanding young teachers of North College University of Technology. References [ F. M. Callier, C. A. Desoer; Linear System Theory, Springer-Verlag, Berlin, (99). [2 R. F. Curtain; The Salamon-Weiss class of well-posed infite-dimensional linear systems: a survey, IMA Journal of Mathematical Control and Information, 4 (997), [3 R. Datko, J. Lagnese, M. P. Polis; An example on the effect of time delays in boundary feedback stabilization of wave equation, SIAM Journal on Control Optimization, 24 (986), [4 R. Datko; Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 26 (988), [5 R. Datko; Two questions concerning the boundary control of certain elastic systems, Journal of Differential Equations, 92 (99), [6 R. Datko; Two examples of ill-posedness with respect to small time delays in stabilized elastic systems, IEEE Transactions on Automatic Control, 38 (993), [7 Y. A. Fiagbedzi, A. E. Pearson; Feedback stabilization of linear autonomous time lag systems, IEEE Transactions on Automatic Control, 3 (986), [8 W. H. Fleming (Editor); Future directions in Control Theory, Philadelphia: SIAM, (988). [9 E. Fridman, Y.Orlov; Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45 (29), [ I. Gumowski, C. Mira, Optimization in control theory and practice, Cambridege University Press, Cambridge, (968). [ B. Z. Guo; Riesz basis approach to the stabilization of a flexibel beam with a tip mass, SIAM Journal on Control and Optimization, 39 (2), [2 B. Z. Guo; Biesz basis property and exponentail stsbility of controlled Euler-Bernoulli beam equations with variable coefficients, SIAM Journal on Control and Optimization, 4 (22), [3 B. Z. Guo, Y. H. Luo; Controllability and stability of a second order hyperbolic system with collocated sensor/actuator, Systems and Control Letters, 46(22), [4 B. Z. Guo, C. Z. Xu; The stabilization of a one-dimensional wave eqution by boundary feedback with noncollocated observation, IEEE Transactions on Automatic Control, 52 (27), [5 B. Z. Guo, C. Z. Xu, H. Hammouri; Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation, ESAIM: Control, Optimization and Calculus of Variations, 8 (22), [6 B. Z. Guo, K. Y. Yang; Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation; Automatica, 45 (29), [7 B. Z. Guo, K. Y. Yang; Output feedback stabilization of a one-dimensional Schrodinger equation by boundary observation with time delay, IEEE Transactions on Automatic Control, 55(2), [8 B. Z. Guo, H. C. Zhou, A. S. AL-Fhaid, A. M. Younas; Stabilization of Euler-Bernoulli beam equation with boundary moment control and disturbance by active disturbance rejection control and sliding model control approach, Journal of Dynamical and Control Systems, 2(24), [9 M. Krstic, A.Smyshlyaev; Backstepping boundary control for first-order hyperbolic PDEs and applications to systems with actuator and sensor delays, Systems and Control Letters, 57 (28),

14 4 K.-Y. YANG, J.-J. LI, J. ZHANG EJDE-25/75 [2 I. Lasiecka, R. Triggiani; Control theory for partial differential equations: continuous and approximation theories II abstract hyperbolic-link systems over a finite time horizon, Cambridge University Press, Cambridge (2). [2 H. Logemann, R. Rebarber, G.Weiss; Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop, SIAM Journal on Control and Optimization, 34(996), [22 M. A. Naimark; Linear Differential Operators, Vol.I, Ungar, New York, (967). [23 J. M. Wang, B. Z. Guo, M. Krstic; Wave equation stabilization by delays equal to even multiples of the wave propagation times, SIAM Journal on Control and Optimization, 49 (2), [24 G. Weiss; Admissibiligy of unbounded control operators, SIAM Journal on Control and Optimization, 27 (989), [25 K. Y. Yang, C. Z. Yao; Stabilization of one-dimensional Schrodinger equation with variable coefficient under delayed boundary output feedback, Asian Jounral of Control, 5 (23), Kun-Yi Yang (corresponding author) College of Science, North China University of Technology, Beijing 44, China address: kyy@amss.ac.cn Jing-Jing Li College of Science, North China University of Technology, Beijing 44, China address: @qq.com Jie Zhang College of Science, North China University of Technology, Beijing 44, China address: jzhang26@ncut.edu.cn