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ВЕСТНИК
МГСУ
4/2011
ɂɋɋɅȿȾɈȼȺɇɂȿ ɍɋɌɈɃɑɂȼɈɋɌɂ ɇȿɉɈȾȼɂɀɇɕɏ ɌɈɑȿɄ
ɋɂɋɌȿɆ ɋ ɄȺɋȺɌȿɅɖɇɕɆ ɈȾɇɈɋɌɈɊɈɇɇɂɆ
ɈȽɊȺɇɂɑȿɇɂȿɆ ɉɈ ɉȿɊȼɈɆɍ ɉɊɂȻɅɂɀȿɇɂɘ
LINEAR APPROXIMATION FIXED POINTS STABILITY
INVESTIGATION OF SYSTEMS WITH TANGENT ONE-SIDED
RESTRICTIONS
ȿ.ɂ. Ɉɬɫɬɚɜɧɨɜ
E.I. Otstavnov
ȽɈɍ ȼɉɈ ɆȽɋɍ
ȼɜɨɞɢɬɫɹ ɩɨɧɹɬɢɟ ɫɢɫɬɟɦɵ ɈȾɍ ɫ ɤɚɫɚɬɟɥɶɧɵɦ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ,
ɨɛɨɛɳɚɸɳɟɟ ɩɨɧɹɬɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɫ ɧɟɭɞɟɪɠɢɜɚɸɳɟɣ ɝɨɥɨɧɨɦɧɨɣ ɫɜɹɡɶɸ.
ɍɤɚɡɵɜɚɸɬɫɹɧɟɩɨɞɜɢɠɧɵɟɬɨɱɤɢɢɞɨɤɚɡɵɜɚɸɬɫɹɬɟɨɪɟɦɵɨɛɢɯɭɫɬɨɣɱɢɜɨɫɬɢ.
ODE system with tangent one-sided restriction is introduced as a generalization of mechanical system with one sided restriction. Conditionsforfixedpointsexistenceandstabilityareprovided.
ɋɨ ɜɬɨɪɨɣ ɩɨɥɨɜɢɧɵ XX ɜɟɤɚ ɩɪɨɹɜɥɹɟɬɫɹ ɛɨɥɶɲɨɣ ɢɧɬɟɪɟɫ ɤ ɞɢɧɚɦɢɱɟɫɤɢɦ ɫɢɫɬɟɦɚɦ ɫ ɫɨɭɞɚɪɟɧɢɹɦɢ (ɛɢɥɥɢɚɪɞɚɦ), ɦɧɨɝɢɟ ɢɡ ɤɨɬɨɪɵɯ ɦɨɝɭɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ
ɦɟɯɚɧɢɱɟɫɤɢɟ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɫɬɨɪɨɧɧɢɦɢ ɝɨɥɨɧɨɦɧɵɦɢ ɫɜɹɡɹɦɢ. ȼ ɤɧɢɝɟ [4] ɪɚɫɫɦɨɬɪɟɧ ɜɨɩɪɨɫ ɨ ɪɟɚɥɢɡɚɰɢɢ ɫɜɹɡɟɣ ɩɨɞɨɛɧɨɝɨ ɪɨɞɚ, ɢ ɤɨɧɫɬɪɭɤɰɢɢ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɢɯ
ɩɨɫɬɪɨɟɧɢɹ, ɷɮɮɟɤɬɢɜɧɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɢɡɭɱɟɧɢɹ ɦɨɞɟɥɶɧɵɯ ɫɢɫɬɟɦ, ɧɚɩɪɢɦɟɪ,
ɛɢɥɥɢɚɪɞɚ Ȼɢɪɤɝɨɮɚ. ȼ ɤɧɢɝɟ [3] ɫɢɫɬɟɦɵ ɫ ɫɢɧɝɭɥɹɪɧɵɦɢ (ɭɞɚɪɧɵɦɢ) ɫɢɥɚɦɢ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɫ ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ, ɢ ɩɪɢɜɨɞɹɬɫɹ ɚɜɬɨɪɫɤɢɟ ɨɛɨɛɳɟɧɢɹ ɦɟɬɨɞɚ
ɮɭɧɤɰɢɣ Ʌɹɩɭɧɨɜɚ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɨɥɨɠɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɢ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɞɜɢɠɟɧɢɣ ɜ ɬɚɤɢɯ ɫɢɬɭɚɰɢɹɯ. ȼ ɤɧɢɝɚɯ [3],[4] ɜ ɨɫɧɨɜɧɨɦ ɩɪɢɦɟɧɹɸɬɫɹ ɞɜɚ
ɩɨɞɯɨɞɚ: ɦɟɬɨɞ ɮɭɧɤɰɢɣ Ʌɹɩɭɧɨɜɚ, ɨɛɨɛɳɚɸɳɢɣ ɤɥɚɫɫɢɱɟɫɤɢɟ ɪɟɡɭɥɶɬɚɬɵ, ɢ ɦɟɬɨɞ
ɨɬɨɛɪɚɠɟɧɢɹ ɩɨɫɥɟɞɨɜɚɧɢɹ. ɉɪɹɦɨɣ ɦɟɬɨɞ Ʌɹɩɭɧɨɜɚ ɬɚɤɠɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɫɢɫɬɟɦ ɫ ɢɦɩɭɥɶɫɢɜɧɵɦɢ ɜɨɡɞɟɣɫɬɜɢɹɦɢ [8]-[10].
1. Ɉɩɪɟɞɟɥɟɧɢɟ ɫɢɫɬɟɦɵ ɈȾɍ ɫ ɤɚɫɚɬɟɥɶɧɵɦ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ.
ɋɢɫɬɟɦɨɣ ɈȾɍ ɫ ɤɚɫɚɬɟɥɶɧɵɦ ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɨɝɪɚɧɢɱɟɧɢɟɦ (ɢɥɢ ɩɪɨɫɬɨ ɫɢɫɬɟɦɨɣ) ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɱɟɬɜɺɪɤɭ ሺܺǡ ‫ݎ‬ǡ ࡿǡ ܴሻ, ɫɨɫɬɨɹɳɭɸ ɢɡ:
1) ɋɢɫɬɟɦɵ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ‫ݔ‬ሶ ൌ ܺሺ‫ݔ‬ሻǡ ‫ܴ א ݔ‬௡ , ɧɚɡɵɜɚɟɦɨɣ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɨɣ.
2) ɋɤɚɥɹɪɧɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ ‫ݎ‬ሺ‫ݔ‬ሻ ൒ Ͳ ɧɚɡɵɜɚɟɦɨɝɨ ɫɜɹɡɶɸ,
3) ɨɬɨɛɪɚɠɟɧɢɹ ࡿǣܴ௡ ՜ ܴ௡ , ɧɚɡɵɜɚɟɦɨɝɨ ɨɬɨɛɪɚɠɟɧɢɟɦ ɭɞɚɪɚ,
4) ɜɟɤɬɨɪ-ɮɭɧɤɰɢɢ ܴሺ‫ݔ‬ሻ, ɧɚɡɵɜɚɟɦɨɣ ɪɟɚɤɰɢɟɣ ɫɜɹɡɢ.
Ɉɬɦɟɬɢɦ ɪɚɡɧɢɰɭ ɦɟɠɞɭ ɞɚɧɧɵɦ ɨɩɪɟɞɟɥɟɧɢɟɦ ɢ ɜɜɟɞɺɧɧɵɦɢ ɜ ɞɪɭɝɢɯ ɪɚɛɨɬɚɯ
ɪɚɧɟɟ. Ɇɨɦɟɧɬ ɫɤɚɱɤɚ ɧɟ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɫɜɹɡɚɧɧɵɦ ɡɚɪɚɧɟɟ ɡɚɞɚɧɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɜ
260
4/2011
ВЕСТНИК
МГСУ
ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɢɞɚ ‫ ݐ‬ൌ ߬ሺ‫ݔ‬ሻ ɥɢɛɨ ɮɢɤɫɢɪɨɜɚɧɧɵɦ (ɫɦ. [5]-[7]). Ɋɟɚɤɰɢɹ ܴሺ‫ݔ‬ሻ
ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ ɫɢɫɬɟɦɵ ɩɪɢ ‫ݎ‬ሺ‫ݔ‬ሻ ൌ Ͳ, ɱɬɨ ɨɬɥɢɱɚɟɬ
ɩɪɟɞɥɚɝɚɟɦɵɣ ɩɨɞɯɨɞ ɨɬ ɩɪɢɦɟɧɹɟɦɨɝɨ ɜ [1],[2]. ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɟɞɢɧɨɨɛɪɚɡɧɨ ɨɩɢɫɚɬɶ
ɛɨɥɟɟ ɲɢɪɨɤɢɣ ɫɩɟɤɬɪ ɫɢɬɭɚɰɢɣ ɫ ɪɚɡɧɵɦɢ ɦɨɞɟɥɹɦɢ ɞɜɢɠɟɧɢɹ ɜɞɨɥɶ ɝɪɚɧɢɰɵ ɫɜɹɡɢ.
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ܺሺ‫ݔ‬ሻ, ‫ ݎ‬ൌ ‫ݎ‬ሺ‫ݔ‬ሻ ɢ ܴሺ‫ݔ‬ሻ ɚɧɚɥɢɬɢɱɟɫɤɢɦɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ
‫ ݔ‬ൌ Ͳ. Ɍɨɝɞɚ ɫɩɪɚɜɟɞɥɢɜɵ ɪɚɡɥɨɠɟɧɢɹ:
ଵ
ଵ
(1.1)
‫ݔ‬ሶ ൌ ݂ ൅ ‫ ݔܣ‬൅ ሺ‫ݔܤ‬ǡ ‫ݔ‬ሻ ൅ ‫ڮ‬ǡ ‫ ݎ‬ൌ ‫ݎ‬ሺ‫ݔ‬ሻ ൌ ሺܿǡ ‫ݔ‬ሻ ൅ ሺܷ‫ݔ‬ǡ ‫ݔ‬ሻ ൅ ‫ڮ‬
ଶ
ଶ
డ௥
௜ ௝ ௞
ɝɞɟ ‫ݎ‬ሺͲሻ ൌ Ͳǡ ݂ ൌ ܺሺͲሻ ് Ͳǡ ሺ‫ݔܤ‬ǡ ‫ݔ‬ሻ ؔ ൫ܾ௝௞
‫ ݔ ݔ‬൯ǡ ܿ ൌ ሺͲሻ ് Ͳ.
డ௫
Ʉɪɭɝɥɵɦɢ ɫɤɨɛɤɚɦɢ ɨɛɨɡɧɚɱɟɧɨ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ (ɤɪɨɦɟ ɭɫɥɨɜɧɨɝɨ
ɨɛɨɡɧɚɱɟɧɢɹ ሺ‫ݔܤ‬ǡ ‫ݔ‬ሻ ɞɥɹ ɤɜɚɞɪɚɬɢɱɧɵɯ ɫɥɚɝɚɟɦɵɯ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɈȾɍ). ɇɨɪɦɚ ɜɟɤɬɨɪɚ ɩɨɥɚɝɚɟɬɫɹ ɟɜɤɥɢɞɨɜɨɣ, ɚ ɧɨɪɦɚ ɥɢɧɟɣɧɨɝɨ ɨɩɟɪɚɬɨɪɚ ɟɣ ɩɨɞɱɢɧɟɧɚ. Ɍɚɤɠɟ ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɢɡɦɟɧɟɧɢɟ ɮɭɧɤɰɢɢ ‫ ݎ‬ɧɚ ɪɟɲɟɧɢɢ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ:
‫ݎ‬ሶ ሺ‫ݔ‬ሻ ൌ ሺܿǡ ݂ሻ ൅ ‫ ڮ‬ȿɫɥɢ ሺܿǡ ݂ሻ ൐ Ͳ, ɬɨ ‫ݎ‬ሺ‫ݔ‬ሻ ɛɭɞɟɬ ɜɨɡɪɚɫɬɚɬɶ ɜɞɨɥɶ ɪɟɲɟɧɢɣ ɈȾɍ, ɧɚɱɚɜɲɢɯɫɹ ɞɨɫɬɚɬɨɱɧɨ ɛɥɢɡɤɨ ɤ ɬɨɱɤɟ ‫ ݔ‬ൌ Ͳ ɩɪɢ ‫ݎ‬ሺ‫ݔ‬ሻ ൐ Ͳ. ɗɬɨɬ ɫɥɭɱɚɣ ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ, ɬɚɤ ɤɚɤ ɩɪɢɜɨɞɢɬ ɤ ɭɯɨɞɭ ɪɟɲɟɧɢɹ ɨɬ ɫɜɹɡɢ ɢ ɢɫɤɥɸɱɚɟɬ ɜɨɡɧɢɤɧɨɜɟɧɢɟ ɬɚɦ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɢ. Ɉɫɬɚɸɬɫɹ ɫɥɭɱɚɢ: ሺܿǡ ݂ሻ ൏ Ͳ ɢ ሺܿǡ ݂ሻ ൌ Ͳ. Ɉɫɬɚɧɨɜɢɦɫɹ ɧɚ ɜɬɨɪɨɦ ɢɡ
ɧɢɯ, ɤɨɝɞɚ ɜɟɤɬɨɪɧɨɟ ɩɨɥɟ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɤɚɫɚɟɬɫɹ ɫɜɹɡɢ ɜ ɧɭɥɟ.
2. Ɉɩɪɟɞɟɥɟɧɢɟ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɩɪɢ ሺܿǡ ݂ሻ ൌ Ͳ.
Ʉɨɧɫɬɪɭɤɰɢɹ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ, ɨɬɜɟɱɚɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ‫ݔ‬଴ ,
ɫɥɟɞɭɸɳɚɹ:
a) ȿɫɥɢ (ɩɨɤɚ) ‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ൯ ൐ Ͳ, ɬɨ ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ, ɩɨɤɚ ɧɟ ɜɵɩɨɥɧɟɧɵ b) ɢɥɢ c)
b) ȿɫɥɢ ‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ כ ݐ‬ሻ൯ ൌ Ͳǡ ‫ݎ‬ሶ ൌ Ͳ ɢ ɫɭɳɟɫɬɜɭɟɬ İ ൐ Ͳ, ɱɬɨ ɞɥɹ ɜɫɟɯ ‫ א ݐ‬ሺ‫ כ ݐ‬ǡ ‫ כ ݐ‬൅ İሻ
ɜɞɨɥɶ ɪɟɲɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɫɩɪɚɜɟɞɥɢɜɨ ‫ݎ‬ሷ ൑ Ͳ, ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɨɬ a) ɤ
d), ɥɢɛɨ ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ d) ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ‫ כ ݐ‬ൌ Ͳ. ɗɬɨɬ ɩɟɪɟɯɨɞ ɧɚɡɵɜɚɟɬɫɹ ɜɵɯɨɞɨɦ ɧɚ ɫɜɹɡɶ.
c) ȿɫɥɢ ‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ כ ݐ‬ሻ൯ ൌ Ͳǡ ‫ݎ‬ሶ ൏ Ͳ, ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɭɞɚɪ: ‫ ି ݔ‬ൌ ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ כ ݐ‬ሻ ՜ ‫ ݔ‬ା ൌ
ࡿሺ‫ ି ݔ‬ሻ, ɩɪɢɱɺɦ ‫ݎ‬ሺ‫ ݔ‬ା ሻ ൌ Ͳǡ ‫ݎ‬ሶ ሺ‫ ݔ‬ା ሻ ൒ ͲǤ ȿɫɥɢ ‫ݎ‬ሶ ሺ‫ ݔ‬ା ሻ ൌ Ͳ ɢ ɞɥɹ ɪɟɲɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɫ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ‫ ݔ‬ା ɫɩɪɚɜɟɞɥɢɜɨ b), ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɤ d), ɢɧɚɱɟ
ɪɟɲɟɧɢɟ ɩɪɨɞɨɥɠɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ a).
d) ȿɫɥɢ (ɩɨɤɚ) ‫ݎ‬ሺ‫ݔ‬ሻ ൌ Ͳ, ɬɨ ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ
‫ݔ‬ሶ ൌ ܺሺ‫ݔ‬ሻ ൅ ܴሺ‫ݔ‬ሻ ɩɨɤɚ ɧɟ ɜɵɩɨɥɧɟɧɨ e)
e) ȿɫɥɢ ‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ ככ ݐ‬ሻ൯ ൌ Ͳǡ ‫ݎ‬ሷ ൒ Ͳ ɢ ɫɭɳɟɫɬɜɭɟɬ ɂ ൐ Ͳ, ɱɬɨ ɞɥɹ ɜɫɟɯ ‫ א ݐ‬ሺ‫ ככ ݐ‬ǡ ‫ ככ ݐ‬൅ ɂሻ
ɜɞɨɥɶ ɪɟɲɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɫɩɪɚɜɟɞɥɢɜɨ ‫ݎ‬ሷ ൐ Ͳ, ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɨɬ d) ɤ
a), ɧɚɡɵɜɚɟɦɵɣ ɫɯɨɞɨɦ ɫɨ ɫɜɹɡɢ.
Ɏɭɧɤɰɢɹ ࡿሺ‫ݔ‬ሻ ɨɩɪɟɞɟɥɟɧɚ ɧɚ ɦɧɨɝɨɨɛɪɚɡɢɢ ሼ‫ ܴ א ݔ‬௡ ȁ‫ݎ‬ሺ‫ݔ‬ሻ ൌ Ͳሽ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ ɢ
ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɜ ɬɨɱɤɟ ‫ ݔ‬ൌ Ͳ. ȼ ɫɢɥɭ ɩɪɟɞɩɨɥɨɠɟɧɢɣ ɪɟɲɟɧɢɟ ɡɚɞɚɺɬɫɹ
ɧɚɱɚɥɶɧɵɦɢ ɞɚɧɧɵɦɢ ɟɞɢɧɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ (ɢ ɫɱɢɬɚɟɬɫɹ ɩɪɨɞɨɥɠɢɦɵɦ ɜ ɛɭɞɭɳɟɟ).
3. Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟ ɨɝɪɚɧɢɱɟɧɢɹ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ‫ݎ‬ሺ‫ݔ‬ሻ ɜ ɫɢɥɭ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ‫ݎ‬ሶ ሺ‫ݔ‬ሻ ൌ ሺܿǡ ݂ሻ ൅
ଵ
ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ‫ݔ‬ሻ ൅ ൫ܿǡ ሺ‫ݔܤ‬ǡ ‫ݔ‬ሻ൯ ൅ ሺ‫ݔܣ‬ǡ ܷ‫ݔ‬ሻ ൅ ‫ ڮ‬ȿɫɬɟɫɬɜɟɧɧɨ ɧɚɡɜɚɬɶ ሺܿǡ ݂ሻ ൌ Ͳ ɭɫɥɨɜɢɟɦ
ଶ
261
ВЕСТНИК
4/2011
МГСУ
ɤɚɫɚɧɢɹ. ɍɫɥɨɜɢɟ ɨɛɳɧɨɫɬɢ: ɜɟɤɬɨɪɵ ܿ ɢ ‫ ܿ ்ܣ‬൅ ܷ݂ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ (ɫɥɭɱɚɣ ɨɛɳɟɝɨ ɩɨɥɨɠɟɧɢɹ). ɍɫɥɨɜɢɟ ‫ݎ‬ሷ ȁ௑ሺ௫ሻ ሺͲሻ ൌ ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ݂ሻ ൏ Ͳ ɧɚɡɨɜɺɦ ɭɫɥɨɜɢɟɦ ɜɨɡɜɪɚɳɚɟɦɨɫɬɢ. Ƚɪɭɩɩɭ ɢɡ ɬɪɺɯ ɭɤɚɡɚɧɧɵɯ ɭɫɥɨɜɢɣ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ (Cond).
4. Ɉɬɨɛɪɚɠɟɧɢɟ ɩɨɫɥɟɞɨɜɚɧɢɹ.
ɉɨɥɨɠɢɦ ܴሺͲሻ ൌ െ݂ǡ ࡿሺͲሻ ൌ Ͳ. ɉɭɫɬɶ ɜɵɩɨɥɧɟɧɵ (Cond), ɬɨɝɞɚ ‫ݔ‬ሺͲǡ ‫ݐ‬ሻ ‫ Ͳ ؠ‬ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ (ɫɥɭɱɚɣ d). ɉɨɫɬɪɨɢɦ ɨɬɨɛɪɚɠɟɧɢɟ ɩɨɫɥɟɞɨɜɚɧɢɹ. Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢ ‫ ݐ‬ൌ Ͳ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ‫ݔ‬଴ ɛɥɢɡɤɨɟ ɤ 0 ɢ ɬɚɤɨɟ, ɱɬɨ ‫ݎ‬ሺ‫ݔ‬଴ ሻ ൌ
Ͳǡ ‫ݎ‬ሶ ሺ‫ݔ‬଴ ሻ ൐ Ͳ. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɬɭɬ ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ.
ɉɪɟɞɫɬɚɜɢɦ ɟɝɨ ɜ ɜɢɞɟ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ ɜɪɟɦɟɧɢ:
‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ ൌ ‫ݔ‬଴ ൅ ‫ݔ‬ଵ ‫ ݐ‬൅ ‫ݔ‬ଶ
௧మ
ଶ
൅ ‫ݔ‬ଷ
௧య
ଷ
൅‫ڮ‬
(4.1)
ɗɬɨɬ ɪɹɞ ɫɯɨɞɢɬɫɹ ɧɚ ɦɚɥɨɦ ɩɪɨɦɟɠɭɬɤɟ ɜɪɟɦɟɧɢ. ɉɨɞɫɬɚɜɢɦ (4.1) ɜ (1.1) ɢ ɩɪɢɪɚɜɧɹɟɦ ɫɥɚɝɚɟɦɵɟ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ t. Ɍɨɝɞɚ
ଵ
ଵ
ଵ
ଶ
ଶ
ଶ
‫ݔ‬ଵ ൌ ݂ ൅ ‫ݔܣ‬଴ ൅ ሺ‫ݔܤ‬଴ ǡ ‫ݔ‬଴ ሻǡ ‫ݔ‬ଶ ൌ ‫ݔܣ‬ଵ ൅ ሺ‫ݔܤ‬଴ ǡ ‫ݔ‬ଵ ሻǡ ‫ݔ‬ଷ ൌ ‫ݔܣ‬ଶ ൅ ሺ‫ݔܤ‬ଶ ǡ ‫ݔ‬଴ ሻ ൅
ଵ
൅ ሺ‫ݔܤ‬ଵ ǡ ‫ݔ‬ଵ ሻ
(4.2)
ଶ
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ‫ כ ݐ‬൐ Ͳ ɬɚɤɨɣ, ɱɬɨ
‫ݎ‬൫‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ כ ݐ‬ሻ൯ ൌ Ͳ
(4.3)
ɗɬɭ ɜɟɥɢɱɢɧɭ ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ ɤɨɦɩɨɧɟɧɬ ɦɚɥɨɝɨ ɜɟɤɬɨɪɚ
‫ݔ‬଴ .
‫ כ ݐ‬ൌ ‫ݐ‬଴ ൅ İ‫ݐ‬ଵ ൅ İଶ ‫ݐ‬ଶ ൅ ‫ڮ‬
(4.4)
ȼ ɩɨɫɥɟɞɧɟɣ ɮɨɪɦɭɥɟ ɫɥɚɝɚɟɦɨɟ İ௠ ‫ݐ‬௠ ǡ ݉ ‫ ܰ א‬ɨɛɨɡɧɚɱɚɟɬ ɱɥɟɧɵ ɪɚɡɥɨɠɟɧɢɹ ɫɬɟɩɟɧɢ ݉ ɩɨ ɤɨɦɩɨɧɟɧɬɚɦ ‫ݔ‬଴ ൌ İ݁଴ . ɗɬɨɬ ɪɹɞ ɫɯɨɞɢɬɫɹ. ȼ ɫɢɥɭ ɩɪɟɞɩɨɥɨɠɟɧɢɣ ɜɵɲɟ
ɪɟɲɟɧɢɟ ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ ɹɜɥɹɟɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ ɢ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ (ɜ
ɦɚɥɨɦ), ɤɚɤ ɢ ɮɭɧɤɰɢɹ ‫ݎ‬ሺ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻሻ. Ɍɨɝɞɚ ‫ݎ‬ሷ ሺ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻሻ ɬɚɤɠɟ ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ‫ݔ‬଴ ɢ
ɜɪɟɦɟɧɢ. Ɉɧɚ ɡɧɚɤɨɨɩɪɟɞɟɥɟɧɚ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɭɫɥɨɜɢɹ ɜɨɡɜɪɚɳɚɟɦɨɫɬɢ. ɉɨ ɬɟɨɪɟɦɟ
ɨɛ ɨɛɪɚɬɧɨɣ ɮɭɧɤɰɢɢ ɩɨɥɭɱɢɦ, ɱɬɨ ɥɨɤɚɥɶɧɨ ɫɭɳɟɫɬɜɭɟɬ ɮɭɧɤɰɢɹ ‫ݐ‬ሺ‫ݎ‬ሶ ǡ ‫ݔ‬଴ ሻ, ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɩɨ ɫɨɜɨɤɭɩɧɨɫɬɢ ɫɜɨɢɯ ɚɪɝɭɦɟɧɬɨɜ. ȿɺ ɪɚɡɥɨɠɟɧɢɟ ɩɨ ‫ݔ‬଴ ɢ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ (4.4).
ɉɨɞɫɬɚɜɢɦ (4.4) ɜ (4.3) ɢ ɩɪɢɪɚɜɧɹɟɦ ɫɥɚɝɚɟɦɵɟ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ ߝ. Ƚɥɚɜɧɵɟ
௧మ
ɱɥɟɧɵ ɞɚɞɭɬ బ ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ݂ሻ ൌ െሺܿǡ ݂ሻ ൌ Ͳ (ɜ ɫɢɥɭ ɭɫɥɨɜɢɹ ɤɚɫɚɧɢɹ). ɍɫɥɨɜɢɟ ɜɨɡɜɪɚଶ
ɳɚɟɦɨɫɬɢ ɩɪɢɜɨɞɢɬ ɤ ‫ݐ‬଴ ൌ Ͳ. Ⱦɚɥɟɟ ɢɦɟɟɦ:
İ‫ݐ‬ଵ ൌ െʹ
൅ܷ݂ǡ ‫ݔ‬଴ ሻ െ
൫஺೅ ௖ା௎௙ǡ௫బ ൯
൫஺೅ ௖ା௎௙ǡ௙൯
ǡİଶ ‫ݐ‬ଶ ൌ െ
ଵ ൫௖ǡ஺మ ௙ାሺ஻௙ǡ௙ሻ൯ାଵଶሺ஺௙ǡ௎௙ሻ
ଷ
൫஺೅ ௖ା௎௙ǡ௙൯
య
൫௖ǡሺ஻௫బ ǡ௫బ ሻ൯ାଶሺ஺௫బ ǡ௎௫బ ሻ
൫஺೅ ௖ା௎௙ǡ௙൯
ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ‫ݔ‬଴ ሻଶ
െ
ቀ௖ǡ஺మ ௫బ ାሺ஻௫బ ǡ௙ሻቁାሺ஺௙ǡ௎௫బ ሻ
൫஺೅ ௖ା௎௙ǡ௙൯
మ
(4.5)
ȼɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ ‫ݎ‬ሺ‫ݔ‬ሻ ɜ ɫɢɥɭ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ:
262
ሺ‫ ܿ ்ܣ‬൅
4/2011
ВЕСТНИК
МГСУ
‫ݎ‬ሷ ሺ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻሻ ൌ ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ݂ሻ ൅ ܱሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻ
(4.6)
ɉɪɟɞɥɨɠɟɧɢɟ 1. ȿɫɥɢ ɜɵɩɨɥɧɟɧɵ (Cond), ɬɨ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ, ɧɚɱɚɜɲɟɟɫɹ ɜ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ ‫ ݔ‬ൌ Ͳ, ɨɫɬɚɧɟɬɫɹ ɧɚ ɫɜɹɡɢ ɞɨ ɬɟɯ ɩɨɪ,
ɩɨɤɚ ɧɟ ɩɨɤɢɧɟɬ ɧɟɤɨɬɨɪɭɸ ɤɨɧɟɱɧɭɸ ɨɤɪɟɫɬɧɨɫɬɶ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ.
Ʌɟɦɦɚ 2. ɉɭɫɬɶ ɜɵɩɨɥɧɟɧɵ (Cond). ɋɭɳɟɫɬɜɭɸɬ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɱɢɫɥɚ İǡ ܽǡ ܾ, ɬɚɤɢɟ, ɱɬɨ ɪɟɲɟɧɢɟ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ, ɧɚɱɚɜɲɟɟɫɹ ɜ İ – ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ (ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɧɚ ɫɜɹɡɢ), ɜɟɪɧɺɬɫɹ ɧɚ ɫɜɹɡɶ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ‫ כ ݐ‬ሺ‫ݔ‬଴ ሻ ൒ Ͳ ɢ ɫɩɪɚɜɟɞɥɢɜɵ ɨɰɟɧɤɢ
‫ כ ݐ‬ሺ‫ݔ‬଴ ሻ ൑ ܽξİǡ
ԡ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ כ ݐ‬ሺ‫ݔ‬଴ ሻሻԡ ൑ ܾξİ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɪɚɫɫɦɨɬɪɟɧɢɢ ɡɧɚɤɚ ɪɚɡɥɨɠɟɧɢɹ ‫ ݎ‬ൌ ‫ݎ‬ሺ‫ݔ‬ሺ‫ݔ‬଴ ǡ ‫ݐ‬ሻሻ
ɩɨ ɜɪɟɦɟɧɢ ɫ ɭɱɺɬɨɦ ɭɫɥɨɜɢɹ ɜɨɡɜɪɚɳɚɟɦɨɫɬɢ.
ɉɨɫɬɪɨɢɦ ɨɬɨɛɪɚɠɟɧɢɟ ɩɨɫɥɟɞɨɜɚɧɢɹ. ɉɭɫɬɶ ɜɵɩɨɥɧɟɧɵ (Cond). Ɋɚɫɫɦɨɬɪɢɦ
ɬɨɱɤɭ ‫ݔ‬଴ , ‫ݎ‬ሺ‫ݔ‬଴ ሻ ൌ Ͳǡ ‫ݎ‬ሶ ൫‫ݔ‬ሺ‫ݔ‬଴ ǡ Ͳሻ൯ ൐ Ͳ. ɂɫɩɨɥɶɡɭɹ (4.1), (4.2), (4.4) ɢ (4.5) ɩɨɥɭɱɢɦ ɡɧɚɱɟɧɢɹ ɮɚɡɨɜɵɯ ɩɟɪɟɦɟɧɧɵɯ ɜ ɦɨɦɟɧɬ ɜɨɡɜɪɚɳɟɧɢɹ ɧɚ ɫɜɹɡɶ:
‫ݔ‬෤ ൌ ‫ݔ‬ሺ‫ כ ݐ‬ሺ‫ݔ‬଴ ሻǡ ‫ݔ‬଴ ሻ ൌ ‫ݔ‬଴ െ ʹ
ሺ஺೅ ௖ା௎௙ǡ௫బ ሻ
ሺ஺೅ ௖ା௎௙ǡ௙ሻ
݂ ൅ ܱሺ‫ݔ‬଴ଶ ሻ
(4.7)
Ɇɨɠɧɨ ɜɵɱɢɫɥɢɬɶ, ɱɬɨ ‫ݎ‬ሶ ሺ‫ݔ‬෤ሻ ൌ െሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ‫ݔ‬଴ ሻ ൅ ܱሺ‫ݔ‬଴ଶ ሻ ൌ െ‫ݎ‬ሶ ሺ‫ݔ‬଴ ሻ ൅ ܱሺ‫ݔ‬଴ଶ ሻ ൏ Ͳ ɜ
ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ‫ ݔ‬ൌ Ͳ, ɬɨ ɟɫɬɶ ɫɩɪɚɜɟɞɥɢɜɨ
ɉɪɟɞɥɨɠɟɧɢɟ 2. ȿɫɥɢ ɡɧɚɤ ɫɤɨɪɨɫɬɢ ɢɡɦɟɧɟɧɢɹ ɮɭɧɤɰɢɢ ɫɜɹɡɢ ‫ݎ‬ሺ‫ݔ‬ሻ ɜɞɨɥɶ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɜ ɫɥɭɱɚɟ c) ɩɨɫɥɟ ɭɞɚɪɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɨ ‫ ݔ‬ɫɥɚɝɚɟɦɵɦ, ɬɨ ɩɪɢ
ɫɥɟɞɭɸɳɟɦ (ɫɭɳɟɫɬɜɭɸɳɢɦ ɜ ɫɢɥɭ ɥɟɦɦɵ 2) ɜɵɩɨɥɧɟɧɢɢ ɭɫɥɨɜɢɹ ‫ ݎ‬ൌ Ͳ ɬɚɤɠɟ ɩɪɨɢɡɨɣɞɺɬ ɭɞɚɪ.
Ⱦɚɥɟɟ, ɩɨɫɥɟ ɭɞɚɪɚ ‫ݔ‬෤ ՜ ‫ ݔ‬ା ൌ ࡿሺ‫ݔ‬෤ሻ ൌ ܵ‫ݔ‬෤ ൅ ‫݋‬ሺԡ‫ݔ‬଴ ԡሻǡ ‫ݎ‬ሺ‫ ݔ‬ା ሻ ൌ Ͳǡ ‫ݎ‬ሶ ሺ‫ ݔ‬ା ሻ ൒ Ͳǡ ܵ ൌ
݀ࡿȁ௫ୀ଴ Ɉɩɪɟɞɟɥɢɦ ɨɬɨɛɪɚɠɟɧɢɟ ɩɨɫɥɟɞɨɜɚɧɢɹ ɤɚɤ ࡼǣ‫ݔ‬଴ ՜ ‫ ݔ‬ା , ɝɞɟ ‫ ݔ‬ା ൌ ࡿሺ‫ݔ‬෤ሻ, ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ, ɩɪɨɢɡɨɲɺɥ ɭɞɚɪ ɢɥɢ ɩɟɪɟɯɨɞ ɤ ɫɢɫɬɟɦɟ ɧɚ ɫɜɹɡɢ. ɂɡ (4.7) ɩɨɥɭɱɢɦ:
೅
ࡼሺ‫ݔ‬ሻ ൌ ܲ‫ ݔ‬൅ ‫݋‬ሺԡ‫ݔ‬ԡሻǡ ܲ ൌ ݀ࡼȁ௫ୀ଴ ൌ ܵ ቆ‫ ܧ‬െ ʹ݂
൫஺೅ ௖ା௎௙൯ ீ
ሺ஺೅ ௖ା௎௙ǡ௙ሻ
ቇ
(4.8)
ɝɞɟ ‫ ܩ‬- ɦɚɬɪɢɰɚ Ƚɪɚɦɚ ɫɤɚɥɹɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ ɜ ɮɚɡɨɜɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ.
Ɋɚɫɫɦɨɬɪɢɦ ܲ෠ ൌ ܲȁሺ௖ǡ௫ሻୀ଴ . ɉɨ ɩɨɫɬɪɨɟɧɢɸ ɨɧ ɞɟɣɫɬɜɭɟɬ ɜ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɟ
ሼ‫ ܴ א ݔ‬ȁሺܿǡ ‫ݔ‬ሻ ൌ Ͳǡ ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ‫ݔ‬ሻ ൐ Ͳሽ. Ⱦɨɨɩɪɟɞɟɥɢɦ ɧɚ ɜɫɸ ɝɢɩɟɪɩɥɨɫɤɨɫɬɶ ܲ෠ሺെ‫ݔ‬ሻ ൌ
െܲ෠‫ݔ‬.
௡
Ʌɟɦɦɚ 3 (ɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɹɯ). ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ (Cond) ɨɩɟɪɚɬɨɪ ܲ෠ ɧɟ ɦɨɠɟɬ ɢɦɟɬɶ ɤɨɦɩɥɟɤɫɧɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɫ ɧɟɧɭɥɟɜɵɦ ɚɪɝɭɦɟɧɬɨɦ (݉‫݀݋‬ʌ), ɱɶɢ
ɫɨɛɫɬɜɟɧɧɵɟ ɩɨɞɩɪɨɫɬɪɚɧɫɬɜɚ (ɩɪɢ ɨɜɟɳɟɫɬɜɥɟɧɢɢ) ɧɟ ɫɨɞɟɪɠɚɬɫɹ ɰɟɥɢɤɨɦ ɜ
ሼ‫ܴ א ݔ‬௡ ȁሺܿǡ ‫ݔ‬ሻ ൌ Ͳǡ ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ‫ݔ‬ሻ ൌ Ͳሽ.
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ɂɞɟɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɨɬ ɩɪɨɬɢɜɧɨɝɨ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɧɟɧɭɥɟɜɨɣ ɚɪɝɭɦɟɧɬ ɜɵɡɵɜɚɟɬ ɩɨɜɨɪɨɬ ɜɟɤɬɨɪɚ ɜ ɫɨɛɫɬɜɟɧɧɨɦ ɩɨɞɩɪɨɫɬɪɚɧɫɬɜɟ, ɢ ɧɚɱɚɥɶɧɭɸ ɬɨɱɤɭ ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɬɚɤ, ɱɬɨɛɵ ɨɬɨɛɪɚɠɟɧɢɟ ɭɞɚɪɚ ɫɧɨɜɚ ɩɟɪɟɦɟɫɬɢɥɨ ɟɺ ɜ ɨɛɥɚɫɬɶ ɜɵɯɨɞɚ ɧɚ ɫɜɹɡɶ
ɜɨɩɪɟɤɢ ɨɩɪɟɞɟɥɟɧɢɸ.
5. ɂɫɫɥɟɞɨɜɚɧɢɟ ɭɫɬɨɣɱɢɜɨɫɬɢ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɲɟɧɢɹ ɩɪɢ ሺܿǡ ݂ሻ ൌ Ͳ ɩɨ ɩɟɪɜɨɦɭ
ɩɪɢɛɥɢɠɟɧɢɸ.
Ɍɟɨɪɟɦɚ 2. ɉɭɫɬɶ ɜɵɩɨɥɧɟɧɵ (Cond). ȿɫɥɢ ɜɫɟ ɫ.ɡ. ܲ෠ ɫɬɪɨɝɨ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ ɩɨ
ɦɨɞɭɥɸ ɢ 0 – ɭɫɬɨɣɱɢɜɚɹ ɩɨ Ʌɹɩɭɧɨɜɭ ɧɟɩɨɞɜɢɠɧɚɹ ɬɨɱɤɚ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ, ɬɨ ɪɟɲɟɧɢɟ ‫ݔ‬ሺͲǡ ‫ݐ‬ሻ ‫ Ͳ ؠ‬ɫɢɫɬɟɦɵ ɭɫɬɨɣɱɢɜɨ ɩɨ Ʌɹɩɭɧɨɜɭ. ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɯɨɬɹ ɛɵ ɨɞɧɨ ɫ.ɡ. ܲ෠,
ɛɨɥɶɲɟɟ ɟɞɢɧɢɰɵ ɩɨ ɦɨɞɭɥɸ, ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɫɨɛɫɬɜɟɧɧɨɟ ɩɨɞɩɪɨɫɬɪɚɧɫɬɜɨ ɧɟ
ɫɨɞɟɪɠɢɬɫɹ ɜ ሼ‫ܴ א ݔ‬௡ ȁሺܿǡ ‫ݔ‬ሻ ൌ Ͳǡ ሺ‫ ܿ ்ܣ‬൅ ܷ݂ǡ ‫ݔ‬ሻ ൌ Ͳሽ, ɥɢɛɨ 0 – ɧɟɭɫɬɨɣɱɢɜɚɹ ɩɨ Ʌɹɩɭɧɨɜɭ ɨɫɨɛɚɹ ɬɨɱɤɚ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ, ɬɨ ɪɟɲɟɧɢɟ ‫ݔ‬ሺͲǡ ‫ݐ‬ሻ ‫ Ͳ ؠ‬ɫɢɫɬɟɦɵ ɧɟɭɫɬɨɣɱɢɜɨ ɩɨ
Ʌɹɩɭɧɨɜɭ.
ɂɞɟɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ. ɇɟɨɛɯɨɞɢɦɨɫɬɶ ɭɫɬɨɣɱɢɜɨɫɬɢ, ɥɢɛɨ ɞɨɫɬɚɬɨɱɧɨɫɬɶ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ ɬɨɱɤɢ ‫ ݔ‬ൌ Ͳ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ ɫɥɟɞɭɟɬ ɢɡ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɩɪɟɞɥɨɠɟɧɢɹ 1.
Ɉɫɬɚɺɬɫɹ ɪɚɫɫɦɨɬɪɟɬɶ ɩɨɜɟɞɟɧɢɟ ɪɟɲɟɧɢɣ ɫɢɫɬɟɦɵ, ɩɨɥɭɱɚɟɦɵɯ ɫ ɩɨɦɨɳɶɸ ɫɜɨɛɨɞɧɨɣ ɫɢɫɬɟɦɵ. ȼ ɫɢɥɭ ɥɟɦɦɵ 2 ɞɨɫɬɚɬɨɱɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɪɚɫɫɦɨɬɪɟɧɢɟɦ ɫɥɭɱɚɟɜ, ɤɨɝɞɚ
ɪɟɲɟɧɢɟ, ɜɨ ɩɟɪɜɵɯ, ɩɨɤɢɞɚɟɬ ɫɜɹɡɶ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɨɬɨɛɪɚɠɟɧɢɟ ࡼ, ɜɨɜɬɨɪɵɯ ɧɟ ɩɪɢɜɨɞɢɬ ɤ ɩɟɪɟɯɨɞɭ ɤ ɫɢɫɬɟɦɟ ɧɚ ɫɜɹɡɢ (ɢɧɚɱɟ ɱɟɪɟɡ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɭɞɚɪɨɜ
ɪɚɫɫɦɨɬɪɟɧɢɟ ɫɜɟɞɺɬɫɹ ɤ ɭɠɟ ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɫɢɬɭɚɰɢɢ ɫɢɫɬɟɦɵ ɧɚ ɫɜɹɡɢ ɩɨ ɩɪɟɞɥɨɠɟɧɢɸ 1). ɉɪɢ ɷɬɨɦ ɜ ɩɟɪɜɨɣ ɱɚɫɬɢ ɭɫɥɨɜɢɹ ɨɬɨɛɪɚɠɟɧɢɟ ɩɨɫɥɟɞɨɜɚɧɢɹ ɨɤɚɠɟɬɫɹ ɫɠɢɦɚɸɳɢɦ, ɚ ɜɨ ɜɬɨɪɨɣ ɦɨɠɧɨ ɤɨɧɫɬɪɭɤɬɢɜɧɨ ɩɨɥɭɱɢɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɨɩɪɟɞɟɥɟɧɢɸ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ.
Ⱥɜɬɨɪ ɜɵɪɚɠɚɟɬ ɛɥɚɝɨɞɚɪɧɨɫɬɶ ȼ.ȼ. Ȼɟɥɟɰɤɨɦɭ ɡɚ ɨɛɫɭɠɞɟɧɢɟ ɪɟɡɭɥɶɬɚɬɨɜ ɷɬɨɣ
ɫɬɚɬɶɢ. Ɋɚɛɨɬɚ ɜɵɩɨɥɧɟɧɚ ɩɪɢ ɩɨɞɞɟɪɠɤɟ ɝɪɚɧɬɚ ɊɎɎɂ ʋ 10-01-00406.
Ʌɢɬɟɪɚɬɭɪɚ
1. Ƚɨɪɛɢɤɨɜ ɋ.ɉ. Ʌɨɤɚɥɶɧɵɟ ɨɫɨɛɟɧɧɨɫɬɢ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɫ ɭɞɚɪɧɵɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ. // Ɇɚɬ. Ɂɚɦɟɬɤɢ ɬ.64 ɜɵɩ.4. 1998, ɫɬɪ. 531-542.
2. Ƚɨɪɛɢɤɨɜ ɋ.ɉ. Ɉɫɨɛɟɧɧɨɫɬɢ ɫɬɪɨɟɧɢɹ ɮɚɡɨɜɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɫ
ɭɞɚɪɧɵɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ.// ɂɡɜ. Ⱥɇ ɋɋɋɊ ɆɌɌ 1987, ʋ3, ɫɬɪ. 23-26.
3. ɂɜɚɧɨɜ Ⱥ.ɉ. Ⱦɢɧɚɦɢɤɚ ɫɢɫɬɟɦ ɫ ɦɟɯɚɧɢɱɟɫɤɢɦɢ ɫɨɭɞɚɪɟɧɢɹɦɢ. Ɇ.: Ɇɟɠɞɭɧɚɪɨɞɧɚɹ ɩɪɨɝɪɚɦɦɚ ɨɛɪɚɡɨɜɚɧɢɹ, 1997. 336 ɫ.
4. Ʉɨɡɥɨɜ ȼ.ȼ., Ɍɪɟɳɺɜ Ⱦ.ȼ. Ƚɟɧɟɬɢɱɟɫɤɨɟ ɜɜɟɞɟɧɢɟ ɜ ɬɟɨɪɢɸ ɛɢɥɥɢɚɪɞɨɜ. Ɇ.: ɂɡɞ-ɜɨ ɆȽɍ,
1991. 168 ɫ.
5. Ɇɵɲɤɢɫ Ⱥ.Ⱦ. ɍɫɬɨɣɱɢɜɨɫɬɶ ɪɟɲɟɧɢɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɪɢ ɨɛɨɛɳɺɧɧɵɯ
ɢɦɩɭɥɶɫɧɵɯ ɜɨɡɦɭɳɟɧɢɹɯ.//Ⱥɜɬɨɦɚɬɢɤɚ ɢ ɬɟɥɟɦɟɯɚɧɢɤɚ 2007, ʋ10, ɫɬɪ. 125-133.
6. Ɇɵɲɤɢɫ Ⱥ.Ⱦ., ɋɚɦɨɣɥɟɧɤɨ Ⱥ.Ɇ. ɋɢɫɬɟɦɚ ɫ ɬɨɥɱɤɚɦɢ ɜ ɡɚɞɚɧɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ.//
Ɇɚɬ. ɋɛɨɪɧɢɤ. Ɍ.74 ɜɵɩ 2., 1967, ɫɬɪ. 202-208.
7. ɉɟɪɟɫɬɸɤ ɇ.Ⱥ. ɋɚɦɨɣɥɟɧɤɨ Ⱥ.Ɇ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɢɦɩɭɥɶɫɧɵɦ ɜɨɡɞɟɣɫɬɜɢɟɦ. Ʉɢɟɜ, 1987.
8. Bainov D. Kulev G. Application of Lyapunov’s direct method to the investigation of the global
stability of the solutions of systems with impulse effect.// Applicable Anal. 1988 vol. 26, ʋ4, p.255270.
9. Bainov D. Kulev G.On the asymptotic stability os systems with impulses by the direct method
of Lyapunov.// J.Math.Anal and Appl. 1989 vol. 140, ʋ2, p.324-340.
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10. Bainov D.D., Simeonov P.S. Stability with respect to part of the variables in systems with impulse effect.// J.Math. Anal.AndAppl.1987, vol.124 ʋ2 p.547-560.
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Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɧɟɭɞɟɪɠɢɜɚɸɳɚɹ ɫɜɹɡɶ, ɭɞɚɪ, ɪɟɚɤɰɢɹ ɫɜɹɡɢ, ɭɫɬɨɣɱɢɜɨɫɬɶ ɩɨ Ʌɹɩɭɧɨɜɭ, ɨɬɨɛɪɚɠɟɧɢɟ ɩɨɫɥɟɞɨɜɚɧɢɹ,ɧɟɝɥɚɞɤɚɹ ɦɟɯɚɧɢɤɚ, ɧɟɩɨɞɜɢɠɧɚɹ ɬɨɱɤɚ, ɥɢɧɟɣɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ
Key words: one sided restriction, reaction, impact, Lyapunov stability, first recurrence map, nonsmooth mechanics, fixed point, linear approximation
e-mail: eotstavnov@ya.ru
Ɋ ɟ ɰ ɟ ɧ ɡ ɟ ɧ ɬ : Ɋɨɞɧɢɤɨɜ Ⱥɥɟɤɫɚɧɞɪ ȼɥɚɞɢɦɢɪɨɜɢɱ, ɤ.ɮ.-ɦ.ɧ., ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ «ȼɵɱɢɫɥɢɬɟɥɶɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɢɡɢɤɚ» ɆȽɌɍ ɢɦ. Ȼɚɭɦɚɧɚ
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