## On the Cauchy problem for a linear harmonic oscillator with pure delay.(English)Zbl 1422.34188

Summary: In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. Finally, the solution asymptotics as the delay parameter goes to zero is studied. In contrast to earlier works, no positivity conditions are imposed.

### MSC:

 34K06 Linear functional-differential equations 39A06 Linear difference equations 39B42 Matrix and operator functional equations 34K26 Singular perturbations of functional-differential equations
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### References:

 [1] Khusainov, D, Pokojovy, M, Azizbayov, E: Representation of classical solutions to a linear wave equation with pure delay. Bull. Kyiv Natl. Univ., Ser. Cybern. 13, 5-12 (2013) [2] Els’gol’ts, LE, Norkin, S: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Mathematics in Science and Engineering, vol. 105. Elsevier, Burlington (1973) [3] Hale, J, Lunel, S: Introduction to Functional Differential Equations. Springer, New York (1993) · Zbl 0787.34002 [4] Bátkai, A, Piazzera, S: Semigroups for Delay Equations. Research Notes in Mathematics, vol. 10. AK Peters, Wellesley (2005) · Zbl 1089.35001 [5] Khusainov, D, Agarwal, R, Davidov, V: Stability and estimates for the convergence of solutions for systems involving quadratic terms with constant deviating arguments. Comput. Math. Appl. 38, 141-149 (1999) · Zbl 0981.34069 [6] Khusainov, D, Agarwal, R, Kosarevskaya, N, Kojametov, A: Spectrum control in linear stationary systems with delay. Comput. Math. Appl. 39, 39-55 (2000) · Zbl 0976.93029 [7] Travies, C, Webb, G: Partial differential equations with deviating arguments in the time variable. J. Math. Anal. Appl. 56(2), 397-409 (1976) · Zbl 0349.35071 [8] Di Blasio, G, Kunisch, K, Sinestari, E: The solution operator for a partial differential equation with delay. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 74(4), 228-233 (1983) · Zbl 0553.35082 [9] Di Blasio, G, Kunisch, K, Sinestari, E: L $$2{L}^2$$-Regularity for parabolic partial integro-differential equations with delay in the highest-order derivatives. J. Math. Anal. Appl. 102(1), 38-57 (1984) · Zbl 0538.45007 [10] Di Blasio, G: Delay differential equations with unbounded operators acting on delay terms. Nonlinear Anal. 52(1), 1-18 (2003) · Zbl 1034.34095 [11] Diblík, J, Khusainov, D, Kukharenko, O, Svoboda, Z: Solution of the first boundary-value problem for a system of autonomous second-order linear partial differential equations of parabolic type with a single delay. Abstr. Appl. Anal. 2012, Article ID 219040 (2012) · Zbl 1250.35117 [12] Khusainov, D, Pokojovy, M, Racke, R: Strong and mild extrapolated L $$2{L}^2$$-solutions to the heat equation with constant delay. SIAM J. Math. Anal. 47(1), 427-454 (2015) · Zbl 1331.35357 [13] Nicaise, S, Pignotti, C: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5), 1561-1585 (2006) · Zbl 1180.35095 [14] Nicaise, S, Pignotti, C: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9-10), 935-958 (2008) · Zbl 1224.35247 [15] Nicaise, S, Pignotti, C, Valein, J: Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst., Ser. S 4, 693-722 (2011) · Zbl 1215.35030 [16] Khusainov, D, Pokojovy, M: Solving the linear 1D thermoelasticity equations with pure delay. Int. J. Math. Math. Sci. 2015, Article ID 479267 (2015) · Zbl 1476.35264 [17] Khusainov, D, Diblík, J, Růz̆ic̆ková, M, Lukác̆ová, J: Representation of a solution of the Cauchy problem for an oscillating system with pure delay. Nonlinear Oscil. 11(2), 276-285 (2008) · Zbl 1276.34055 [18] Khusainov, D, Ivanov, A, Kovarzh, I: The solution of wave equation with delay. Bull. Taras Shevchenko Natl. Univ. Kyiv., Ser. Phys. Math. 4, 243-248 (2006) (in Ukrainian) · Zbl 1142.35621 [19] Diblík, J, Fečkan, M, Pospıšil, M: Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices. Ukr. Math. J. 65, 58-69 (2013) · Zbl 1283.34057 [20] Diblík, J, Fečkan, M, Pospıšil, M: Representation of a solution of the Cauchy problem for an oscillating system with multiple delays and pairwise permutable matrices. Abstr. Appl. Anal. 2013, Article ID 931493 (2013) · Zbl 1283.34057 [21] Khusainov, D, Shuklin, G: On relative controllability in systems with pure delay. Prikl. Mekh. 41(2), 118-130 (2005) · Zbl 1100.34062 [22] Arendt, W, Batty, C, Hieber, M, Neubrander, F: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001) · Zbl 0978.34001 [23] Dreher, M, Quintanilla, R, Racke, R: Ill-posed problems in thermomechanics. Appl. Math. Lett. 22(9), 1374-1379 (2009) · Zbl 1173.80301 [24] Rodrigues, H, Ou, C, Wu, J: A partial differential equation with delayed diffusion. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 14, 731-737 (2007) · Zbl 1142.35097
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