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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 6(366), Pages 89–114 (Mi umn1678)  

This article is cited in 66 scientific papers (total in 66 papers)

Optimization of boundary controls of string vibrations

V. A. Il'inab, E. I. Moiseevb

a Steklov Mathematical Institute, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: For a large time interval $T$ boundary controls of string vibrations are optimized in the following seven boundary-control problems: displacement control at one end (with the other end fixed or free); displacement control at both ends; elastic force control at one end (with the other end fixed or free); elastic force control at both ends; combined control (displacement control at one end and elastic force control at the other). Optimal boundary controls in each of these seven problems are sought as functions minimizing the corresponding boundary-energy integral under the constraints following from the initial and terminal conditions for the string at $t=0$ and $t=T$, respectively. For all seven problems, the optimal boundary controls are written out in closed analytic form.

DOI: https://doi.org/10.4213/rm1678

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English version:
Russian Mathematical Surveys, 2005, 60:6, 1093–1119

Bibliographic databases:

UDC: 517.984.5
MSC: Primary 35L05, 49J20, 35B37; Secondary 74K05, 74H45, 93C20
Received: 06.08.2005

Citation: V. A. Il'in, E. I. Moiseev, “Optimization of boundary controls of string vibrations”, Uspekhi Mat. Nauk, 60:6(366) (2005), 89–114; Russian Math. Surveys, 60:6 (2005), 1093–1119

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    This publication is cited in the following articles:
    1. V. A. Il'in, E. I. Moiseev, “Optimization of a boundary control by a displacement at one end of a string based on minimization of the integral of the modulus of the derivative of the displacement raised to an arbitrary power $p\ge 1$”, Dokl. Math., 74:3 (2006), 878–882  mathnet  crossref  mathscinet  zmath  isi  elib  elib
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    3. V. A. Il'in, E. I. Moiseev, “Optimization of the boundary control of string vibrations by an elastic force on an arbitrary sufficiently large time interval”, Differ. Equ., 42:12 (2006), 1775–1786  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. V. A. Il'in, E. I. Moiseev, “Minimization of the $L_p$-norm for $p\ge 1$ of the derivative of a boundary displacement control on an arbitrary sufficiently large time interval $[0,T]$”, Differ. Equ., 42:11 (2006), 1633–1644  mathnet  crossref  mathscinet  zmath  isi  elib
    5. A. Yu. Popov, “Minimization of the variation in the derivative of a boundary control by damping the vibrations of a string with a fixed end”, Dokl. Math., 74:1 (2006), 487–490  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. V. A. Il'in, E. I. Moiseev, “Optimization of a boundary control of an elastic force at one end of a string based on minimization of the integral of the modulus of the elastic force raised to an arbitrary power $p\ge 1$”, Dokl. Math., 75:1 (2007), 123–126  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    7. M. M. Potapov, “Observability of irregular solutions to the Neumann problem for the wave equation with variable coefficients”, Dokl. Math., 75:1 (2007), 138–142  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    8. V. A. Il'in, E. I. Moiseev, “Boundary control of string vibrations that minimizes the integral of power $p\ge1$ of the module of control or its derivative”, Autom. Remote Control, 68:2 (2007), 313–319  mathnet  crossref  mathscinet  zmath
    9. A. Yu. Popov, “Integral minimization of the second derivative module of the boundary control of string vibrations with the fixed end”, Autom. Remote Control, 68:2 (2007), 327–336  mathnet  crossref  mathscinet  zmath  elib  elib
    10. V. A. Il'in, E. I. Moiseev, “Optimal boundary control by the displacement of one endpoint of the string with the other endpoint being free in an arbitrarily large time interval”, Differ. Equ., 43:10 (2007), 1403–1414  crossref  mathscinet  zmath  isi  elib
    11. V. A. Il'in, E. I. Moiseev, “Optimization of a boundary control by a displacement at one end of a string with second end free during an arbitrary sufficiently large time interval”, Dokl. Math., 76:3 (2007), 806–811  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    12. V. A. Il'in, E. I. Moiseev, “Optimization of boundary controls by displacements at two ends of a string during an arbitrary sufficiently large time interval”, Dokl. Math., 76:3 (2007), 828–834  mathnet  crossref  mathscinet  zmath  isi  elib  elib
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    38. Moiseev E.I., Kholomeeva A.A., “Optimal boundary displacement control of string vibrations with a nonlocal evenness condition of the second kind”, Differ. Equ., 47:1 (2011), 126–133  crossref  mathscinet  zmath  isi  elib  elib
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