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**Boundary controllability of differential equations with nonlocal condition.**
*(English)*
Zbl 0917.93009

Global exact boundary controllability in a given time-interval of a semilinear abstract dynamical system is considered. It is generally assumed that the dynamical system is defined in an infinite-dimensional Banach space, that it contains both a linear and a nonlinear part, and that, moreover, it satisfies the so-called nonlocal conditions. Using a Banach fixed-point theorem, a sufficient condition for exact boundary global controllability is formulated and proved. As an application, boundary exact controllability of certain semilinear distributed parameter dynamical systems with Dirichlet boundary conditions is investigated. Similar controllability problems for semilinear abstract dynamical systems have been recently considered in the paper of the authors and Y.-C. Kwun [Indian. J. Pure Appl. Math. 29, No. 9, 941-950 (1998; Zbl 0917.93008)].

Reviewer: J.Klamka (Katowice)

### MSC:

93B05 | Controllability |

93C25 | Control/observation systems in abstract spaces |

93C10 | Nonlinear systems in control theory |

### Keywords:

global exact boundary controllability; semilinear abstract dynamical systems; nonlocal conditions; Banach fixed-point theorem### Citations:

Zbl 0917.93008
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\textit{H.-K. Han} and \textit{J.-Y. Park}, J. Math. Anal. Appl. 230, No. 1, 242--250 (1999; Zbl 0917.93009)

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### References:

[1] | Balachandran, K.; Balasubramaniam, P.; Dauer, J. P., Controllability of nonlinear integrodifferential systems in Banach space, J. Optim. Theory Appl., 84, 83-91 (1995) · Zbl 0821.93010 |

[2] | Barbu, V.; Precupanu, T., Convexity and Optimization in Banach Spaces (1986), Reidel: Reidel New York · Zbl 0594.49001 |

[3] | Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162, 494-505 (1991) · Zbl 0748.34040 |

[4] | Park, J. Y.; Han, H. K., Controllability for some second order differential equations, Bull. Korean Math. Soc., 34, 411-419 (1997) · Zbl 0889.93008 |

[5] | Quinn, M. D.; Carmichael, N., An approach to non-linear control problems using fixed-point methods, degree theory and pseudo-inverses, Numer. Funct. Anal. Optim., 7, 197-219 (1984-1985) · Zbl 0563.93013 |

[6] | Washburn, D., A bound on the boundary input map for parabolic equations with application to time optimal control, SIAM J. Contr. Optim., 17, 652-671 (1979) · Zbl 0439.93035 |

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