ABSTRACT
The purpose of this paper is to study the Timoshenko system with the nonlocal time-delayed condition. The well-posedness is proved by Hille-Yosida theorem. Exploring the dissipative properties of the linear operator associated with the full damped model, we obtain the exponential stability by using Gearhart-Huang-Prüss theorem.
Keywords:
Timoshenko system; nonlocal time-delayed condition; exponential stability
RESUMO
O objetivo deste artigo é estudar o sistema de Timoshenko com uma condição de retardo de tempo não local. A boa colocação é provada através do teorema de Hille- Yosida. Explorando as propriedades dissipativas do operador linear associado ao modelo totalmente amortecido, obtemos a estabilidade exponencial usando o Teorema de Gearhart- Huang-Prüss.
Palavras-chave:
sistema de Timoshenko; condição de retardo não local; estabilidade exponencial
1 INTRODUCTION
The history of nonlocal problems with integral conditions for partial differential equations is re- cent and goes back to 44 J. R. Cannon. The solution of heat equation subject to the specification of energy. Quart. Appl. Math., 21 (1963), 155-160.. In particular, a review of the progress related to the nonlocal models with integral type was given in 33 Z. P. Bažant & M. Jirásek. Nonlocal Integral Formulation of Plasticity And Damage: Survey of Progress. J. Eng. Mech., 128 (2002), 1119-1149. with many discussions about physical justifications, advan- tages, and numerical applications. For a nonlocal hyperbolic equation with integral conditions of the 1st kind, we cite 2020 L. S. Pul’kina. A nonlocal problem for a hyperbolic equation with integral conditions of the 1st kind with time-dependent kernels. Izv. Vyssh. Uchebn. Zaved. Mat., 10 (2012), 32-44.. Dissipative properties associated with the Timoshenko system have been studied by several authors by considering the dissipative mechanism of frictional or vis- coelastic type. An interesting problem was brought out when the dissipation acts in different ways on the domain. For the case of terms of time-varying delay in the internal feedbacks, the stability result of the Timoshenko system can be found in 1010 M. Kirane, B. Said-Houari & M. N. Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pur. Appl. Anal ., 10 (2011), 667-686.. On the other hand, for the case of delay and boundary feedback, we can see in 2424 B. Said-Houari & A. Soufyane. Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inf., 6 (2012), 1-16.. The Timoshenko beam system with delay in the boundary control was studied in 2525 G. Xu & H. Wang. Stabilisation of Timoshenko beam system with delay in the boundary control. Int. J. Control., 86 (2013), 1165-1178. where the exponential stabilization result is proved via a test of exact observability of the system. Distributed delay in the boundary control was considered in 1313 X. F. Liu & G. Q. Xu. Exponential Stabilization for Timoshenko Beam with Distributed Delay in the Boundary Control. Abstr. Appl. Anal . 2013 (2013), 726794.. Distributive delay in a Timoshenko-type system of thermoelasticity of type III was considered in [8] and then, in 99 M. Kafini , S. A. Messaoudi & M. I. Mustafa. Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay. Appl. Anal., 93 (2014), 1201- 1216., the same problem was dealt with constant delay. The Timoshenko system with second sound and the internal distributed delay was investigated in 22 T. A. Apalara. Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay. Electron. J. Differential Equations, 254 (2014), 1-15..
The transmission problem with delay in porous-elasticity was considered in 2121 C. A. Raposo, T. A. Apalara & J. O. Ribeiro. Analyticity to transmission problem with delay in porous-elasticity. J. Math. Anal. Appl., 46 (2018), 819-834.. For a nonlinear Timoshenko system with delay, we cite 55 B. Feng & M. L. Pelicer. Global existence and exponential stability for a nonlinear Timo- shenko system with delay. Bound. Value Probl., 26 (2015), https://doi.org/10.1186/s13661-015-0468-4.
https://doi.org/10.1186/s13661-015-0468-...
and reference therein. As far as we know, there is no result for the Timoshenko system with nonlocal delay.
Let be an interval on ℝ. In this paper, describe the small transverse displacement of the beam and the rotation angle of a filament of the beam in Ω, respectively at the time t. For a constant and
bounded functions, we define the nonlocal time delayed integral of the 1st kind condition by
These conditions (1.1) are called nonlocal because the integral is not a pointwise relation, so it provides a problem with distributed delay. Well-posedness and exponential stability for this kind of nonlocal time-delayed for a wave equation were studied in 1717 S. Nicaise & C. Pignotti. Stabilization of the wave equation with boundary or internal distributed delay. Differential Integral Equations, 21 (2008), 935-958.), (2222 C. A. Raposo, H. Nguyen, J. O. Ribeiro & V. B. Oliveira. Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition. Electron. J. Differential Equations, 279 (2017), 1-11. by different techniques.
Let b, k, α, β be positive constants. The Timoshenko system with frictional damping and nonlocal time-delayed condition is given by
We consider the Dirichlet boundary conditions as follows
Here the initial data
belong to suitable spaces and
We use the Sobolev spaces with its properties as in Adams 11 R. A. Adams. Sobolev Spaces. Academic Press, New York 1975. and the semigroup theory ( see Pazy 1818 A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer- Verlag, New York 1983.). In this paper, we apply the semigroup technique for dissipative systems (see Liu and Zheng 1414 Z. Liu & S. Zheng. Semigroups Associated with Dissipative Systems. Chapman & Hall 1999.), that is different from some others in the literature, for example, like as the energy method (see Rivera 2323 J.E.M. Rivera. Energy decay rates in linear thermoelasticity. Funkcial EKVAC, 35 (1992), 9-30.), the direct method (see Kormonik 1111 V. Komornik. Exact controllability and stabilization. The multiplier method. RAM: Research in Applied Mathematics, John Wiley & Sons, Paris, 1994.), (1212 V. Kormonik & E. Zuazua. A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl., 69 (1990), 33-54.) and the Nakao’s method (see 1515 M. Nakao. On the decay of solutions of some nonlinear dissipative wave equations. Math.Z. Berlin., 193 (1986), 227-234.). This manuscript is organized as follows. In Section 2, we deal with the semigroup setting where we prove the well-posedness of the system. In section 3, we show the exponential stability by using the Gearhart-Huang-Prüss theorem, 66 L. Gearhart. Spectral theory for contraction semigroups on Hilbert spaces. Trans. Amer.Math. Soc., 236 (1978), 385-394.), (77 F. Huang. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Diff. Eqns., 1 (1985), 45-53.), (1919 J. Pruss. On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. 284 (2) (1984), 847-857..
2 SEMIGROUP SETUP
As in Nicaise and Pignotti 1717 S. Nicaise & C. Pignotti. Stabilization of the wave equation with boundary or internal distributed delay. Differential Integral Equations, 21 (2008), 935-958. we introduce the new variables
where
The new variables z, y satisfy
Moreover, using the approach as in 1616 S. Nicaise & C. Pignotti. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim., 45 (2006), 1561-1585., the equations
has unique solution
respectively. The problem (1.2)-(1.7) is equivalent to
with the Dirichlet boundary condition (1.8) and on the boundary .
Defining and , we formally get that U satisfies the Cauchy problem
where the operator 𝒜 is defined by
We introduce the energy space
equipped with the inner product
for and .
The domain of 𝒜 is defined by
Clearly, D(𝒜) is dense in ℋ and independent of time . Next, we will prove that the operator 𝒜 is dissipative.
Lemma 2.1.For. Assume that
then we have
Proof.
Integrating by parts on Ω,
Taking into account we have
and
Inserting (2.17) and (2.18) into (2.16), apply Young’s inequality and simplifying the terms, we obtain
Finally, by the assumption (2.14) we conclude the proof. □
The well-posedness of (2.5)-(2.12) is ensured by the following theorem.
Teorema 2.1.For, there exists a unique weak solution U of (2.13) satisfying
Moreover, if, then
Proof. We will use the Hille-Yosida theorem. Since 𝒜 is dissipative and D(𝒜 ) is dense in ℋ , it is sufficient to show that 𝒜 is maximal; that is, is surjective. Given , we must show that there exists satisfying which is equivalent to
Suppose that we have found φ and ψ with the appropriated regularity. Therefore, (2.21) and (2.22) give
It is clear that u,.
From (2.1),(2.3) it follows that equation (2.25) has a unique solution given by
and from (2.2),(2.4) it follows that equation (2.26) has a unique solution given by
So, from (2.27) and (2.28),
and, in particular,
where defined by
By (2.23), (2.24), (2.27) and (2.28), we see that the functions φ and ψ satisfy the following system
where
Solving the system (2.29) is equivalent to finding such that
for all .
Now, we observe that solving the system (2.30) is equivalent to solve the problem
where the bilinear form
and the linear form
are defined by
and
It is easy to verify that ϒ is continuous and coercive, and L is continuous. So applying the Lax-Milgram theorem, we deduce that for all the problem (2.31) admits a unique solution
Applying the classical elliptic regularity, it follows from (2.30) that
Therefore, the operator is surjective. As consequence of the Hille-Yosida theorem [1414 Z. Liu & S. Zheng. Semigroups Associated with Dissipative Systems. Chapman & Hall 1999., Theorem 1.2.2, page 3], we have that 𝒜 generates a C 0-semigroup of contractions on ℋ. From semigroup theory, is the unique solution of (2.13) satisfying (2.19) and (2.20). The proof is complete. □
3 EXPONENTIAL STABILITY
The necessary and sufficient conditions for the exponential stability of the C 0-semigroup of con- tractions on a Hilbert space were obtained by Gearhart 66 L. Gearhart. Spectral theory for contraction semigroups on Hilbert spaces. Trans. Amer.Math. Soc., 236 (1978), 385-394. and Huang 77 F. Huang. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Diff. Eqns., 1 (1985), 45-53. independently, see also Pruss 1919 J. Pruss. On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. 284 (2) (1984), 847-857.. We will use the following result due to Gearhart.
Teorema 3.2.Let ρ(𝒜) be the resolvent set of the operator 𝒜 andbe the C 0-semigroup of contractions generated by 𝒜. Then, S(t) is exponentially stable if and only if
The main result of this manuscript is the following theorem.
Teorema 3.3. The semigroup generated by 𝒜 is exponentially stable.
Proof. It is sufficient to verify (3.1) and (3.2). If (3.1) is not true, it means that there is a such that , iζ is in the spectrum de 𝒜. From the compact immersion of D(𝒜 ) in ℋ, by spectral theory, there is a vector function
such that , which is equivalent to
Using (3.3) we obtain . Multiplying by u x , integrating on Ω and using Young’s inequality we have
from where it follows that
Applying Poincaré’s inequality in (3.9) we obtain a.e. in L 2(Ω). Note that (2.3) gives us as the unique solution of (3.7), which implies a.e. in . Similarly, it is proved that a.e. in L 2(Ω) and a.e. in . But is a contradiction with and then (3.1) holds.
To prove (3.2) we use contradiction argument again. If (3.2) is not true, there exists a real sequence ζ n , with and a sequence of vector functions that satisfies
Hence
Since it follows that there exists a unique sequence
with unit norm in ℋ such that
Denoting we have from (3.10) that
and then strongly in ℋ as .
Taking the inner product of ξ n with U n we have
Using Lemma 2.1, follows that
and taking the real part we have
As U n is bounded and we obtain
Now, for is equivalent to
where
for .
From (3.11), (3.12), (3.13) and (3.18), we obtain that
By (2.1) and (2.3), we have that
is the unique solution of (3.16). Using the Euler formula for complex numbers in (3.20) we obtain
Since and limited and by (3.11), (3.18) deducing that
Analogously, we conclude that
Finally, (3.11), (3.19), (3.21) and (3.22) give us a contradiction with . The proof is complete. □
REFERENCES
-
1R. A. Adams. Sobolev Spaces Academic Press, New York 1975.
-
2T. A. Apalara. Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay. Electron. J. Differential Equations, 254 (2014), 1-15.
-
3Z. P. Bažant & M. Jirásek. Nonlocal Integral Formulation of Plasticity And Damage: Survey of Progress. J. Eng. Mech., 128 (2002), 1119-1149.
-
4J. R. Cannon. The solution of heat equation subject to the specification of energy. Quart. Appl. Math., 21 (1963), 155-160.
-
5B. Feng & M. L. Pelicer. Global existence and exponential stability for a nonlinear Timo- shenko system with delay. Bound. Value Probl, 26 (2015), https://doi.org/10.1186/s13661-015-0468-4
» https://doi.org/10.1186/s13661-015-0468-4 -
6L. Gearhart. Spectral theory for contraction semigroups on Hilbert spaces. Trans. Amer.Math. Soc, 236 (1978), 385-394.
-
7F. Huang. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Diff. Eqns, 1 (1985), 45-53.
-
8M. Kafini, S. A. Messaoudi & M. I. Mustafa. Energy decay result in a Timoshenko-type system of thermoelasticity of type III with distributive delay. J. Math. Phys, 54 (2013), 101503.
-
9M. Kafini , S. A. Messaoudi & M. I. Mustafa. Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay. Appl. Anal, 93 (2014), 1201- 1216.
-
10M. Kirane, B. Said-Houari & M. N. Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pur. Appl. Anal ., 10 (2011), 667-686.
-
11V. Komornik. Exact controllability and stabilization. The multiplier method RAM: Research in Applied Mathematics, John Wiley & Sons, Paris, 1994.
-
12V. Kormonik & E. Zuazua. A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl, 69 (1990), 33-54.
-
13X. F. Liu & G. Q. Xu. Exponential Stabilization for Timoshenko Beam with Distributed Delay in the Boundary Control. Abstr. Appl. Anal . 2013 (2013), 726794.
-
14Z. Liu & S. Zheng. Semigroups Associated with Dissipative Systems Chapman & Hall 1999.
-
15M. Nakao. On the decay of solutions of some nonlinear dissipative wave equations. Math.Z. Berlin, 193 (1986), 227-234.
-
16S. Nicaise & C. Pignotti. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim, 45 (2006), 1561-1585.
-
17S. Nicaise & C. Pignotti. Stabilization of the wave equation with boundary or internal distributed delay. Differential Integral Equations, 21 (2008), 935-958.
-
18A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations Springer- Verlag, New York 1983.
-
19J. Pruss. On the spectrum of C0-semigroups. Trans. Amer. Math. Soc 284 (2) (1984), 847-857.
-
20L. S. Pul’kina. A nonlocal problem for a hyperbolic equation with integral conditions of the 1st kind with time-dependent kernels. Izv. Vyssh. Uchebn. Zaved. Mat, 10 (2012), 32-44.
-
21C. A. Raposo, T. A. Apalara & J. O. Ribeiro. Analyticity to transmission problem with delay in porous-elasticity. J. Math. Anal. Appl, 46 (2018), 819-834.
-
22C. A. Raposo, H. Nguyen, J. O. Ribeiro & V. B. Oliveira. Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition. Electron. J. Differential Equations, 279 (2017), 1-11.
-
23J.E.M. Rivera. Energy decay rates in linear thermoelasticity. Funkcial EKVAC, 35 (1992), 9-30.
-
24B. Said-Houari & A. Soufyane. Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inf, 6 (2012), 1-16.
-
25G. Xu & H. Wang. Stabilisation of Timoshenko beam system with delay in the boundary control. Int. J. Control, 86 (2013), 1165-1178.
Publication Dates
-
Publication in this collection
30 Nov 2020 -
Date of issue
Sep-Dec 2020
History
-
Received
16 Jan 2020 -
Accepted
27 May 2020