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Approximate controllability of a parabolic equation with memory. (English) Zbl 1238.93018
Summary: We study the approximate controllability of a parabolic equation with memory $y_t + y_{xx} + \int^t_0 y(x,s)ds = 0$ by boundary control. The proof relies on the explicit solution of the corresponding homogeneous initial boundary value problem and a duality method.

##### MSC:
 93B05 Controllability 93C20 Control systems governed by PDE 35R09 Integro-partial differential equations
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##### References:
 [1] Volterra, V.: Theory of functionals, (1930) · Zbl 55.0814.01 [2] Yamada, Y.: On a certain class of semilinear Volterra diffusion equations, J. math. Anal. appl. 88, 433-457 (1982) · Zbl 0515.45012 · doi:10.1016/0022-247X(82)90205-0 [3] Yamada, Y.: Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. differ. Equ. 52, 295-326 (1984) · Zbl 0543.35053 · doi:10.1016/0022-0396(84)90165-7 [4] Zhang, N. Y.: On fully discrete Galerkin approximations for partial integrodifferential equations of parabolic type, Math. comp. 60, 133-166 (1993) · Zbl 0795.65098 · doi:10.2307/2153159 [5] Blanchard, D.; Ghidouche, H.: A nonlinear system for irreversible phase changes, European J. Appl. math. 1, 91-100 (1990) · Zbl 0713.35045 · doi:10.1017/S0956792500000073 [6] Barbu, V.; Iannelli, M.: Controllability of the heat equation with memory, Differ. integral equ. 13, 1393-1412 (2000) · Zbl 0990.93008 [7] Fu, X.; Yong, J.; Zhang, X.: Controllability and observability of a heat equation with hyperbolic memory kernel, J. differ. Equ. 247, 2395-2439 (2009) · Zbl 1187.35265 · doi:10.1016/j.jde.2009.07.026 [8] Sakthivel, K.; Balachandran, K.; Nagaraj, B. R.: On a class of non-linear parabolic control systems with memory effects, Internat. J. Control 81, 764-777 (2008) · Zbl 1152.93312 · doi:10.1080/00207170701447114 [9] Lavanya, R.; Balachandran, K.: Null controllability of nonlinear heat equations with memory effects, Nonlinear anal. Hybrid syst. 3, 163-175 (2009) · Zbl 1166.93004 · doi:10.1016/j.nahs.2008.12.003 [10] Prüss, J.: Evolutionary integral equations and applications, (1993) · Zbl 0784.45006 [11] Rosier, L.; Rouchon, P.: On the controllability of a wave equation with structural damping, Int. J. Tomogr. stat. 5, 79-84 (2007)