A time-delay equation: well-posedness to optimal control

Kenan Yildirim; Sertan Alkan

doi:10.1515/phys-2016-0026

Article Open Access

A time-delay equation: well-posedness to optimal control

Kenan Yildirim and Sertan Alkan

Published/Copyright: July 8, 2016

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From the journal Open Physics Volume 14 Issue 1

Abstract

In this paper, well-posedness, controllability and optimal control for a time-delay beam equation are studied. The equation of motion is modeled as a time-delayed distributed parameter system(DPS) and includes Heaviside functions and their spatial derivatives due to the finite size of piezoelectric patch actuators used to suppress the excessive vibrations based on displacement and moment conditions. The optimal control problem is defined with the performance index including a weighted quadratic functional of the displacement and velocity which is to be minimized at a given terminal time and a penalty term defined as the control voltage used in the control duration. Optimal control law is obtained by using Maximum principle and hence, the optimal control problem is transformed the into a boundary-, initial and terminal value problem.The explicit solution of the control problem is obtained by eigenfunction expansions of the state and adjoint variables. Numerical results are presented to show the effectiveness and applicability of the piezoelectric control.

Keywords: Wellposedness; Time-delay; Vibration; Control; Maximum Principle

PACS: 64.30.Ef; 62.20.dj; 62.20.fg; 62.30.+d

1 Introduction

Time-delay is a widespread phenomenon in applied sciences, e.g. space engineering, chemical kinetics, fluid dynamics, etc. Such delays occur as a result of the finite time-response of actuators used in the implementation of control law [11]. Since time delay leads to a decrement in the performance of the actuator, the state variable cannot reflect the changes in the system [18]. Active vibration control of mechanical systems, which are modeled as DPS without time delays, has been excessively studied in the literature by several authors, such as, but not limited to[3,7, 12–16]; however, the optimal control of DPS with time delay has not received considerable attention yet. Some studies of the control of DPS with time delay can be summarized by [4–6, 11, 18]. Although undamped free vibrations in mechanical systems are often studied from an optimal control point of view, controlling other forms of vibrations caused by the thermal effect or an external excitation has not received enough attention. Vibrations caused by the thermal effect and external excitation are also modeled as displacement or moment boundary conditions. The contribution of material damping to the piezoelectric control of a system without time delay in control function has been studied in the [18] and it was observed that neglecting natural damping leads to under-estimation of the structural behavior [12].

In the literature, according to author's best knowledge, there is no study dealing with the well-posedness, controllability and optimal control of a time-delay beam equation. Therefore, this paper presents an original contribution to the literature of the study of the well-posedness, controlability and optimal control of a time-delay beam equation is studied and results are simulated using MATLAB. The contribution of material damping to the system is also considered. Moreover, in order to achieve a better observation for the beam, three other conditions(time delay, internal damping and boundary effects) are taken into account.

This paper studies the dynamic response of a beam, modeled as an Euler-Bernoulli beam with Kelvin-Voigt damping and a time delay in the control function, subject to displacement and moment boundary conditions. The control is applied by means of piezoelectric actuators bonded onto the beam.

Due to the presence of the damping term, the governing equation does not admit a variational principle in the classical sense, making an approach based on variational calculus impractical. Also, the equation of motion includes Heaviside functions and their spatial derivatives due to the finite size of the piezoelectric patch actuators used. Because of this, it is not sensible to use the optimal linear quadratic regulator or linear quadratic Gaussian control algorithms for solving the system. Therefore, optimal control law is obtained by using the Maximum principle and hence, the optimal control problem is transformed into a boundary-, initial and terminal value problem. The performance index, to be minimized with minimum expenditure of voltage applied to the piezoelectric patch actuators, consists of a weighted quadratic function of displacement and velocity of the beam. It also includes a penalty function of control voltage. The solution of the control problem is sought by the eigenfunction expansion method. To illustrate the capability and efficiency of the control law, introduced, two numerical examples are presented.

2 Mathematical Formulation of the Control Problem

The time-delay optimal control problem of a smart beam is considered. The beam shown in Fig. 1, initially un-deformed, has internal damping modeled as Kelvin-Voigt damping and is subject to displacement and moment boundary conditions. The equation of motion of the beam is controlled by piezoelectric patch actuators, bonded on both sides of the beam, with a control voltage C(t - τ) is given by [1]

Figure 1

Beam diagram with piezoelectric pathces

ϑυt¯t¯+2ξ¯ρυt¯x¯x¯x¯x¯+ρυx¯x¯x¯x¯=C¯(t¯−τ)(H′′(x¯−x¯1)−H′′(x¯−x¯2))(1)

where υ is the transversal displacement, x ∈ (0, ℓ) is the space variable, ℓ is the length of the beam, t ∈(0, t_f is the time variable, t_f is the terminal time, τ ≥ 0 is the known constant delay in the control function, ϑ is the mass per unit area, ξ is the damping coefficient, H is the Heaviside function and (x₁, x₂) is the location of a piezoelectric actuator. The flexural stiffness of the beam, ρ is defined as follows:

ρ=Eh312(1−ϱ2)

in which E is Young's modulus, h is the elastic thickness of the plate and ϱ is Poisson's ratio. Eq. (1) is subject to the following boundary conditions:

υ(0, t¯)=υ(ℓ,t¯)=ζ¯(t¯), υx¯x¯(0,t¯)=υx¯x¯(ℓ,t¯)=η¯(t¯)(2)

where ζ(t) is the displacement boundary condition, η(t) is the moment boundary condition and the initial conditions are:

υ(x¯,0)=υ0(x¯), υt¯(x¯,0)=υ1(x¯).(3)

Theorem 1

(Second Order Linear Picard-Lindelof Existence-Uniqueness Theorem:) Let the coefficientsa(x), b(x), c(x), f(x)be continuous on an intervalJcontainingx = x0. Assumea(x) ≠ 0 onJ. Letg₁andg₂be real constants. The initial value problem

a(x)y′′+b(x)y′+c(x)y=f(x), y(x0)=g1, y′(x0)=g2(4)

has a solution and this solution is unique [20].

The system defined by Eqs. (1–3) can be reduced to an ordinary differential equation by using the Galerkin expansion method like in Eq. (42)section 3. Please note that all coefficients and functions are analytic functions in Eq. (42) near the 0. Then, Eq. (42) with initial conditions satisfies the Second Order Linear Picard-Lindelof Existence-Uniqueness theorem. Namely, the system has a unique solution.

The uniqueness of the optimal control function will be discussed in the next section of this paper. For convenience, let us introduce non-dimensional variables as

u=υℓ, x=x¯ℓ, t=1ℓ2ρϑt¯, ξ=ξ¯ℓ2ρϑ,C(t−τ)=ℓρC(t¯−τ), (x,t)∈𝓢=(0,1)×(0,tf)

where t_f is the terminal time. Substituting new parameters into Eq. (1), one obtains the non-dimensional equation of the motion subject to the following boundary and initial conditions, respectively.

utt+2ξutxxxx+uxxxx=C(t−τ)(H′′(x−x1)−H′′(x−x2))(5)

u(0,t)=u(1,t)=ζ(t), uxx(0,t)=uxx(1,t)=η(t),(6)

u(x,0)=u0(x), ut(x,0)=u1(x).(7)

The aim of the optimal control problem is to determine an optimum voltage function C(t - τ) to minimize the performance index of the beam at t_f with the minimum expenditure of control voltage. Therefore, the performance index is defined by the weighted dynamic response of the beam and the expenditure of control over (0, t_f). The set of admissible control functions is given by

Cad={C(t)|C(t)∈L2(0,tf), |C(t)| ≤c0<∞}(8)

and the performance index of the controlled system is defined as follows;

J(C(t))=∫01[μ1u2(x,tf)+μ2ut2(x,tf)]dx+∫0tfμ3C2(t)dt(9)

where μ₁, μ₂ ≥ 0, μ₁ + μ₂ ≠ 0 and μ₃ > 0 are weighting constants. The first integral in Eq. (9) is the modified dynamic response of the beam and last integral represents the measure of the total voltage energy that accumulates over (0, t_f). The optimal control of a beam with a time delay in the control function is expressed as

J(C°(t))=minC(t)∈CadJ(C(t))(10)

subject to the Eqs. 5–7. In order to achieve the maximum principle, let us introduce an adjoint variable ν(x, t) satisfying the following equation

νtt−2ξνtxxxx+νxxxx=0(11)

and subject to the following homogeneous boundary conditions

ν(x,t)=νxx(x,t)=0 at x=0,1(12)

and terminal conditions

νt(x,t)−2ξνxxxx(x,t)=2μ1u(x,t), ν(x,t)=−2μ2ut(x,t) at t=tf.(13)

A maximum principle, in terms of the Hamiltonian function is derived as a necessary condition for the optimal control function. It is proved in [3] that under some convexity assumption, which is satisfied by Eq. (9), on the performance index function, the maximum principle is also a sufficient condition for the optimal control function. Note that υ is the unique solution to the system defined by Eqs. 1–3. By considering uniqueness of the solution, it can be concluded that when υ° is the unique solution to the system, the corresponding control function C° must be unique. In this situation, system has a unique control function and solution and the system is called observable. The Hilbert uniqueness method proved that observable implies the system under consideration is controllable [21, 22]. The maximum principle gives an explicit expression for the optimal control function and implicitly relates optimal control to the state variable. Then, the maximum principle can be given as follows:

Theorem 2

(Maximum principle) The maximization problem states that if

H[t;ν°,C°(t)]=maxC(t)∈CadH[t;ν,C(t)](14)

in which ν = ν(x, t) satisfies the adjoint system given by Eqs. 11–13 and the Hamiltonian function is defined by

H[t;ν,C(t)]=C(t)[νx(x2,t)−νx(x1,t)]−μ3C2(t),(15)

then

J[C°(t)]≤J[C(t)], ∀C(t) ∈Cad(16)

where C°(t) is the optimal control function.

proof. Before starting the proof, let us introduce an operator and its adjoint operator as follows:

Y(u)=utt+uxxxx+2ξutxxxx,Y∗(ν)=νtt+νxxxx−2ξνtxxxx.(17)

The deviations are given by ∆u = u - u°, ∆u_t = u_t - u°_t in which u° is the optimal displacement. The operator Y(∆u) = ∆C(t)(H″(x - x₁) - H″(x - x₂)) is subject to the following boundary conditions

Δu(x,t)=Δuxx(x,t)=0 at x=0,1(18)

and initial conditions

Δu(x,t)=Δut(x,t)=0 at t=0.(19)

Consider the following functional

∫01∫0tf{νY(Δu)−ΔuY∗(ν)}dt dx=∫01∫0tf{νΔC(t)(H′′(x−x1)−H′′(x−x2))}dt dx.(20)

Integrating the left side of Eq. (20) twice integration by parts with respect to t and four times integration by parts with respect to x, using Eqs. (18–19), one observes the following relation:

∫01∫0tf{νY(Δu)−ΔuY∗(ν)}dt dx=∫01(ν(x,tf)Δut(x,tf)−Δu(x,tf)[νt(x,tf)−2ξνxxxx(x,tf)])dx.(21)

In view of Eq. (13), Eq. (21) becomes

∫01∫0tf{νY(Δu)−ΔuY∗(ν)}dt dx=−2∫01(μ1u(x,tf)Δu(x,tf)+μ2ut(x,tf)Δut(x,tf))dx.(22)

For the right side of Eq. (20), recall the properties of dirac-delta function

H′′(x−θ)=δ′(x−θ),∫01δ′(x−θ)ς(x)dx=−ς′(θ), θ∈(0,1).(23)

In the light of Eq. (23), the right side of Eq. (20) is obtained as follows:

∫0tf∫01ν(x,t)ΔC(t)[δ′(x−x1)−δ′(x−x2)]dx dt=∫0tfΔC(t)(νx(x2,t)−νx(x1,t))dt.(24)

Consider the difference of the performance index

ΔJ[C(t)]=J[C(t)]−J[C°(t)] =∫01{μ1[u2(x,tf)−u°2(x,tf)] + μ2[ut2(x,tf)−ut°2(x,tf)]}dx + ∫0tfμ3[C2(t)−C°2(t)]dt(25)

Let us expand u²(x, t_f) and ut2(x,tf) to Taylor series around u°2(x,tf) and ut°2(x,tf), respectively. Then, one observes the following

u2(x,tf)−u°2(x,tf)=2u°(x,tf)Δu(x,tf)+r,(26a)

ut2(x,tf)−ut°2(x,tf)=2ut°(x,tf)Δut(x,tf)+rt(26b)

where r = 2(∆u)² + higher order terms > 0 and r_t = 2(∆u_t)² + higher order terms > 0. Substituting Eq. (26) into Eq. (25) gives

ΔJ[C(t)]=∫01{μ1[2u°(x,tf)Δu(x,tf)+r]+ μ2[2ut°(x,tf)Δut(x,tf)+rt]}dx+∫0tfμ3[C(t)2−C(t)°2]dt.(27)

From Eq. (22) and because of μ₁r + μ₂r_t > 0, one obtains

ΔJ[C(t)]≥−∫0tf{(C(t)[νx(x2,t)−νx(x1,t)]+μ3C2(t))−(C°(t)[νx°(x2,t)−νx°(x1,t)]+μ3C°2(t))}dt≥0(28)

which leads to

C(t)[νx(x1,t)−νx(x2,t)]+μ3C(t)2≥C°(t)[νx°(x1,t)−νx°(x2,t)]+μ3C°2(t),(29)

that is,

H[t;ν°,C]≥H[t;ν,C].

Hence, we obtain

J[C]≥J[C°], ∀C∈Cad

Therefore, the optimal control function is given by

C(t)=νx°(x2,t)−νx°(x1,t)2μ3.(30)

□

The existence and uniqueness of the solution to the adjoint system, defined by Eqs. (11–13), can be obtained in a similar way to Eqs. (1–3). Then, the state system given by Eqs. (1–3) is controllable.

3 Solution Method

The solution of the optimal control problem is sought as follows: Let the adjoint variable ν(x, t) satisfying Eqs. (11–13) be expanded in Fourier sine series as

ν(x,t)=∑n=1∞Zn(t)φn(x)(31)

where the orthonormal eigenfunctions

φn(x)=2sin(λnx), λn=nπ(32)

satisfy boundary conditions given by Eq. (12). Substituting Eq. (31) into Eq. (11), multiplying both sides with φ_n(x) and integrating both sides over (0,1) leads to the following lumped parameter system(LPS) in time

Z¨n−2ξλn4Z˙n+λn4Zn=0, for n=1,2,….(33)

The general solution of the LPS given by Eq. (33) is given by

Zn(t)=anκn(t)+bnιn(−t),(34)

where

κn(t)=exp((ϖn+ξλn4)t), ιn(t)=exp((ϖn−ξλn4)t),ϖn=λn2ξ2λn4−1(35)

and a_n and b_n are constants to be determined. Next, we solve the equation of the optimal motion. In order to convert the non-homogeneous boundary conditions to homogeneous ones, let us define the following relation

w=u−ζ(t)−β(x)η(t), β(x)=x2−x2.(36)

Then, the system given by Eqs. (5–7) becomes

wtt+2ξwtxxxx+wxxxx=C(t−τ)(H′′(x−x1) −H′′(x−x2))−ζ′′(t)−β(x)η′′(t)(37)

subject to the new homogeneous boundary conditions

w(x,t)=wxx(x,t)=0 at x=0,1(38)

and initial conditions

w(x,0)=u0(x)−ζ(0)−β(x)η(0),wt(x,0)=u1(x)−ζ′(0)−β(x)η′(0).(39)

Due to Eq. (36), one observes the terminal conditions of the adjoint equation Eq. (13) as follows:

νt(x,tf)−2ξνxxxx(x,tf) =2μ1[w(x,tf)+ζ(tf)+β(x)η(tf)],(40a)

ν(x,tf)=−2μ2[wt(x,tf)+ζt(tf)+β(x)ηt(tf)].(40b)

Now, let us obtain the solution of the motion equation by using Fourier sine series

w(x,t)=∑n=1∞Ωn(t)φn(x)(41)

in which Ω_n(t) satisfies the following LPS

Ω¨n+2ξλn4Ω˙n+λn4Ωn=γ1ζ′′(t)+γ2η′′(t)+γ3C(t−τ),(42)

where

γ1=−∫01φn(x)dx, γ2=−∫01β(x)φn(x)dx,γ3=∫01(H′′(x−x1)−H′′(x−x2))φn(x)dx.

The general solution of Eq. (33) is given by

Ωn(t)=cnκn(−t)+dnιn(t)+12ϖn∫0t(ιn(t−s)−κn(s−t))(γ1ζ′′(s)+γ2η′′(s))ds+12ϖn∫τt(ιn(t−s)−κn(s−t))(γ3C(s−τ))ds(43)

in which c_n and d_n are constants to be determined by means of Eq. (39). The remaining unknown constants a_n and b_n appearing in Eq. (43) are evaluated by using the terminal conditions given by Eq. (40).

4 Numerical Results and Discussions

In this section, the theoretical results obtained in the previous sections are illustrated to show the effectiveness and capability of the proposed control algorithm for controlling the dynamic response of a smart beam with Kelvin-Voigt damping and a time delay in a minimum level control voltage to be applied to the piezoelectric patch actuator. Due to the selection of the damping coefficient (ξ), three cases: over damping, critical damping and under damping are studied in the literature. Under damping is the most important case due to the insufficient internal damping of the beam. Therefore, in this paper, under damping is studied. In this case, ξ<1λn2 and two complex roots are obtained from Eq. (5). Also, taking the one term solution, when ξ<1λn2, the stability of the system defined by Eqs. (5–7) is guaranteed [9]. The location of the piezoelectric patch actuators and predetermined terminal time are specified as (x₁, x₂) = (0.25,0.75) and t_f = 5, respectively. The time delay in the control voltage function applied to the piezoelectric patch actuator τ is evaluated as 0.002 and the damping coefficient is taken into account as 0.001. The weighting coefficients in the calculations are taken as μ₃ = 10^-2 and μ₁ = μ₂ = 1 for the controlled case. Displacement boundary conditions, moment boundary conditions and initial conditions are evaluated as follows;

For the case a, ζ(t)=0, η(t)=e5−t, u0=2cos(πx) and u1=2sin(πx),For the case b, ζ(t)=5−t, η(t)=0, u0=2sin(πx), and u1=π2cos(πx).

Let us define the dynamic response of the beam and the control voltage as follows, respectively;

𝓓(tf)=∫01[u2(x,tf)+ut2(x,tf)]dx, 𝓒=∫0tfC2(t)dt(44)

The dynamic response of the beam is presented in table 1 for case a and b. Observing table 1, it can be seen that for both cases with/out delay in the control function, the penalty μ₃ on the expenditure of voltage decreases as the dynamic response of the beam decreases corresponding to an increase in the voltage. Also, it can be observed from table 1 that in the cases of τ = 0, the dynamic response of the beam is less than the cases in which τ = 0.002. In case a, the vibrations in the beam are induced by larger external/thermal excitations than case b. Therefore, it seems from table 1 that the value of the dynamic response of the beam corresponding to case b is less than the dynamic response of the beam corresponding to case a. Also, as a parallel result to this, the difference between dynamic responses with/out delay for case a is larger than the corresponding difference for case b. A comparison of the controlled and uncontrolled dynamic responses with/out delay in table 1 demonstrates a substantial decrease as a result of the proposed control method. In table 2, some numerical results are given for the dynamic response of the beam with/out delay at ξ = 0. Due to the absence of internal damping in the system, the dynamic response is much more than the case with internal damping. Also, similarly to table 1, the dynamic response of the beam without delay is less than the case with delay for the undamped case.

Table 1

The values of 𝓓(t_f) at ν = 0.001 for different values of μ₃ in case a and b.

μ₃	𝓓(τ=0.002)a	𝓓(τ=0.000)a	μ₃	𝓓(τ=0.002)b	𝓓(τ=0.000)b
10²	3203.52	3202.10	10²	231.91	231.80
10¹	214.65	212.38	10¹	10.62	10.45
10⁰	29.00	27.90	10⁰	1.82	1.74
10^-1	1.42	1.36	10^-1	0.813	0.808

Table 2

The values of 𝓓(t_f) at ν = 0 for different values of μ₃ in case a and b.

μ₃	𝓓(τ=0.002)a	𝓓(τ=0.000)a	μ₃	𝓓(τ=0.002)b	𝓓(τ=0.000)b
10²	3401.34	3399.82	10²	246.2	246.0
10¹	223.227	220.73	10¹	11.1	10.9
10⁰	30.643	29.462	10⁰	1.92	1.83
10^-1	1.439	1.383	10^-1	0.8141	0.809

In table 3, the dynamic response of the time-delayed controlled beam is given together with the control voltage spent for case a. As stated above, the difference between the dynamic responses in table 3 is an effect of the internal damping. The same observation is valid for the spent voltage to suppress the vibrations in the time-delayed controlled beam. In the case of the system with delay and ξ = 0.001, the control voltage spent in (0, t_f) is less than the case without damping. Parallel results to the ones observed in table 1–3 are obtained from table 4 for case b in the system with delay.

Table 3

The values of 𝓓(t_f) at τ = 0.002 for different values of μ₃ in case a.

μ₃	𝓓(ξ=0.0001)a	𝓒	μ₃	𝓓(ξ=0)a	𝓒
10²	3203.52	29.26	10²	3401	32.55
10¹	214.65	140.2	10¹	223	149
10⁰	29.00	203.2	10⁰	30	215
10^-1	1.42	297.2	10^-1	1.44	314

Table 4

The values of 𝓓(t_f) at τ = 0.002 for different values of μ₃ in case b.

μ₃	𝓓(ξ=0.0001)b	𝓒	μ₃	𝓓(ξ=0)b	𝓒
10²	231.9	2.4	10²	246.1	2.64
10¹	10.6	11.4	10¹	11.08	12.09
10⁰	1.8	16	10⁰	1.91	16.87
10^-1	0.81	21.6	10^-1	0.81	22.8

The un/controlled displacements and velocities are plotted at the midpoint of the beam at x = 0.5 since their maximum occurs at this point at t = 0 owing to the displacement and moment boundary conditions. Therefore, the midpoint gives an idea about the transient behavior of the time-delayed controlled damped beam. By observing Figs. 2–3, it seems that the uncontrolled displacement displays a steady harmonic motion while the controlled displacement gradually decreases in case a. The same observations is valid for the uncontrolled velocity plotted in Figs. 4–5 where the velocity is effectively suppressed because of control. For case b, the displacement and velocity of the beam are plotted in Figs. 6–7 and Figs. 8–9, respectively, illustrating the effect of the control and internal damping are illustrated. It can be concluded from Figs. 6–9 that the uncontrolled displacements and velocities are damped at a given terminal time t_f = 5 as a result of control actuation. Let us focus on the bandwidths in Figs. 2–5 and Figs. 6–9. Because the external excitation/thermal effect is larger in case a than case b, the bandwiths of the Figs. 2–3 and Figs. 4–5 is larger than the bandwidths of Figs. 6–7 and Figs. 8–9. By taking into consideration all tables and figures, it can be concluded that the control method introduced for the beam, with Kelvin-Voigt damping and a time delay in the control voltage, is effective and applicable for the beam with/out damping.

Figure 2

Controlled and uncontrolled displacements with delay at (0.5) for case a.

Figure 3

For 4.5 ≤ t ≤ 5, controlled and uncontrolled displacements with delay at (0.5) for case a.

Figure 4

Controlled and uncontrolled velocities with delay at (0.5) for case a.

Figure 5

For 4.5 ≤ t ≤ 5, controlled and uncontrolled velocities with delay at (0.5) for case a.

Figure 6

Controlled and uncontrolled displacements with delay at (0.5) for case b.

Figure 7

For 4.5 ≤ t ≤ 5, controlled and uncontrolled displacements with delay at (0.5) for case b.

Figure 8

Controlled and uncontrolled velocities with delay at (0.5) for case b.

Figure 9

For 4.5 ≤ t ≤ 5, controlled and uncontrolled velocities with delay at (0.5) for case b.

5 Conclusion

In this paper, the optimal control algorithm to suppress undesirable vibrations in a smart beam, with Kelvin-Voigt damping and a time delay in the control voltage function, subject to the displacement and moment boundary conditions, is investigated by using the maximum principle. The performance index of the control problem consists of a weighted quadratic functional of the dynamic responses of the beam to be minimized at a predetermined terminal time and a penalty term defined as the control voltage spent in the control process. By means of the maximum principle, the optimal control problem is transformed into a coupled system of partial differential equations in terms of state, adjoint and control variables subject to the boundary, initial and terminal conditions. The explicit solution of the problem is sought by the eigenfunction expansion method. By using \verb"MATLAB", numerical results are given to demonstrate the robustness and efficiency of the proposed control algorithm.

References

[1] Ashida F., Sakata S.I., Tauchert T.R., Takahashi Y., Control of Thermally Induced Vibration in a Composite Beam with Damping Effect. Journal of Thermal Stresses, 2006, 29:139-15210.1080/01495730500257458Search in Google Scholar

[2] Ackleh A.S., Banks D.S., Pinter G.A., A nonlinear beam equation. Applied Mathematics Letters, 2002, 15:381-38710.21236/ADA451433Search in Google Scholar

[3] Barnes E.A., Necessary and sufficient optimality conditions for a class of distributed parameter control systems. SIAM Journal on Control, 1971, 9:62-8210.1137/0309006Search in Google Scholar

[4] Cai G.P., Huang J.Z., Yang S.X., An optimal control method for linear systems with time delay. Computers and Structures, 2003, 81:1539-154610.1016/S0045-7949(03)00146-9Search in Google Scholar

[5] Cai G.P., Yang S.X., A discrete optimal control method for a flexible cantilever beam with time delay. Journal of Vibration and Control, 2006, 12:509-52610.1177/1077546306064268Search in Google Scholar

[6] Du H., Zhang N., Active vibration control of structures subject to parameter uncertainties and actuator delay. Journal of Vibration and Control, 2008, 14:689-70910.1177/1077546307083173Search in Google Scholar

[7] Egorov A.I., Necessary optimality conditions for distributed parameter systems. SIAM Journal on Control, 1967, 5:352-40810.1137/0305024Search in Google Scholar

[8] Garg D.P., Gary L., Structural damping and vibration control via smart sensors and actuators. Journal of Vibration and Control, 2003, 9:1421-145210.1177/1077546304031169Search in Google Scholar

[9] Kisacanin B., Agarwal G.C., Linear Control Systems. Kluwer Academic Publishers, Netherlands, 2011Search in Google Scholar

[10] Koshlyakov N.S., Smirnov M.M., Gliner E.B., Differential Equations of Mathematical Physics. North-Holland Publishing Company, Amsterdam, 1964Search in Google Scholar

[11] Kucuk I., Sadek I., Yilmaz Y., Active control of a smart beam with time delay by Legendre wavelets. Applied Mathematics and Computation, 2012, 218:8968-897710.1016/j.amc.2012.02.057Search in Google Scholar

[12] Kucuk I., Yildirim K., Sadek I., Adali S., Optimal control of a beam with Kelvin-Voigt damping subject to forced vibrations using a piezoelectric patch actuator. Journal of Vibration and Control, 2015, 21:701-71310.1177/1077546313488617Search in Google Scholar

[13] Kucuk I., Yildirim K., Necessary and sufficient conditions of optimality for a damped hyperbolic equation in one space dimension. Abstract and Applied Analysis, 2014, 10.1155/2014/49313010.1155/2014/493130Search in Google Scholar

[14] Kucuk I., Yildirim K., Adali S., Optimal piezoelectric control of a plate subject to time-dependent boundary moments and forcing function for vibration damping. Computers and Mathematics with Applications, 2015, 69:291-30310.1016/j.camwa.2014.11.012Search in Google Scholar

[15] Lee E.B., A sufficient condition in the theory of optimal control. SIAM Journal on Control, 1963, 1:241-24510.1137/0301013Search in Google Scholar

[16] Marinaki M., Marinakis Y., Stavroulakis G.E., Vibration control of beams with piezoelectric sensors and actuators using particle swarm optimization. Expert Systems with Applications, 2011, 38:6872-688310.1016/j.eswa.2010.12.037Search in Google Scholar

[17] Sadek I., Necessary and sufficient conditions for the optimal control of distributed parameter systems subject to integral constraints. Journal of the Franklin Institute, 1988, 325:565-58310.1016/0016-0032(88)90033-6Search in Google Scholar

[18] Wang B., Dongbin Z., Alippi C., Liu D., Dual Heuristic dynamic Programming for nonlinear discrete-time uncertain systems with state delay. Neurocomputing, 2014, 134:222-229 Xiang C.L., Cai G.P., Optimal control of a flexible beam with multiple time delays. Journal of Vibration and Control, 2009, 15:1493-151210.1016/j.neucom.2013.06.037Search in Google Scholar

[19] Zachmaonoglou E.C., Thoe D.W., Intoduction to Partial Differential equations with applications, Dover Publ., New York, 1986Search in Google Scholar

[20] Coddington E.A., Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1984Search in Google Scholar

[21] Guliyev H.F., Jabbarova K.S., The exact controllability problem for the second order linear hyperbolic equation, Differential Equations and Control Processes, N3, 2010Search in Google Scholar

[22] Pedersen M., Functional Analysis in Applied Mathematics and Engineering, CRC press, Florida, 1999Search in Google Scholar

Received: 2015-9-27

Accepted: 2016-5-13

Published Online: 2016-7-8

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.