# American Institute of Mathematical Sciences

May  2021, 20(5): 1821-1831. doi: 10.3934/cpaa.2021042

## A note on the energy transfer in coupled differential systems

 Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

* Corresponding author

Received  January 2021 Revised  January 2021 Published  May 2021 Early access  March 2021

We study the energy transfer in the linear system
 $\begin{cases} \ddot u+u+\dot u = b\dot v\\ \ddot v+v-\epsilon \dot v = -b\dot u \end{cases}$
made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter
 $b$
.
Citation: Monica Conti, Lorenzo Liverani, Vittorino Pata. A note on the energy transfer in coupled differential systems. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1821-1831. doi: 10.3934/cpaa.2021042
##### References:
 [1] F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.  doi: 10.1051/cocv/2016011. [2] M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019. [3] K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differ. Equ., 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007. [4] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28. [5] G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292.  doi: 10.1007/PL00005977. [6] M. Conti, V. Pata and R. Quintanilla, Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptot. Anal., 120 (2020), 1-21.  doi: 10.3233/asy-191576. [7] F. Dell'Oro and V. Pata, On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differ. Equ., 257 (2014), 523-548.  doi: 10.1016/j.jde.2014.04.009. [8] R. Denk and F. Kammerlander, Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.  doi: 10.1093/imamat/hxy002. [9] H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2. [10] G. R. Goldstein, J. A. Goldstein and G. Perla Menzala, On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199.  doi: 10.1090/S0033-569X-2012-01282-3. [11] J. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159.  doi: 10.1007/s00033-012-0274-0. [12] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974. [13] J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047. [14] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC, Boca Raton, 1999. doi: 0-8493-0615-9. [15] J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058. [16] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [17] L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3. [18] M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012. [19] R. Triggiani, Heat-viscoelastic plate interaction: analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.  doi: 10.3934/eect.2018008.

show all references

##### References:
 [1] F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.  doi: 10.1051/cocv/2016011. [2] M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019. [3] K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differ. Equ., 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007. [4] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28. [5] G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292.  doi: 10.1007/PL00005977. [6] M. Conti, V. Pata and R. Quintanilla, Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptot. Anal., 120 (2020), 1-21.  doi: 10.3233/asy-191576. [7] F. Dell'Oro and V. Pata, On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differ. Equ., 257 (2014), 523-548.  doi: 10.1016/j.jde.2014.04.009. [8] R. Denk and F. Kammerlander, Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.  doi: 10.1093/imamat/hxy002. [9] H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2. [10] G. R. Goldstein, J. A. Goldstein and G. Perla Menzala, On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199.  doi: 10.1090/S0033-569X-2012-01282-3. [11] J. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159.  doi: 10.1007/s00033-012-0274-0. [12] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974. [13] J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047. [14] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC, Boca Raton, 1999. doi: 0-8493-0615-9. [15] J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058. [16] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [17] L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3. [18] M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012. [19] R. Triggiani, Heat-viscoelastic plate interaction: analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.  doi: 10.3934/eect.2018008.
Plot of ${{\mathtt{E}}}$ for $\epsilon = 1$ and $b = 0.99$ (black), $b = 1$ (blue) and $b = 1.01$ (red)
Parametric plot of $t\mapsto (u(t), \dot u(t))$ for $\epsilon = 1$ and $b = \sqrt{\frac{23}{13}+\frac{13}{23}-1}$
${{\mathtt{E}}}$ for $\epsilon = 1$ and $\boldsymbol{z}_0 = (1, 0, 0, 0)$ with different values of $b$
${{\mathtt{E}}}$ for $\epsilon = 1$ and $\boldsymbol{z}_0 = (1, 0.5, 0, 0)$ with different values of $b$
Numerical $u$ (blue) vs asymptotic $u$ (red) for $\epsilon = 1$ with different values of $b$ (and different time-scales)
Numerical $v$ (blue) vs asymptotic $v$ (red) for $\epsilon = 1$ with different values of $b$ (and different time-scales)
Plot of ${{\mathtt{E}}}$ for $\epsilon = 0.5$ and $b = \sqrt{0.5}-0.1$ (black), $b = \sqrt{0.5}$ (blue) and $b = \sqrt{0.5}+0.1$ (red)
Parametric plot of $t\mapsto (u(t), \dot u(t))$ for $\epsilon = \frac12$ and $b = 1$
Parametric plot of $t\mapsto (u(t), \dot u(t))$ for $\epsilon = \frac12$ and $b = 2$
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