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Особенности планирования и реализации безопасного и оптимального навигационного процесса.

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ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 6, ʋ1, 2003 ɝ.
ɫɬɪ.61-64
Ɉɫɨɛɟɧɧɨɫɬɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ɢ ɪɟɚɥɢɡɚɰɢɢ ɛɟɡɨɩɚɫɧɨɝɨ
ɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ
ȼ.ɂ. Ɇɟɧɶɲɢɤɨɜ, Ʉɭɤɭɢ Ɏɢɪɦɢɧ Ⱦɠɢɜɨ
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɫɭɞɨɜɨɠɞɟɧɢɹ
Ⱥɧɧɨɬɚɰɢɹ. ȼ ɫɬɚɬɶɟ ɢɫɫɥɟɞɭɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɩɟɪɟɧɨɫɚ ɤɚɱɟɫɬɜɚ ɫ ɩɥɚɧɢɪɭɟɦɨɣ ɬɪɚɟɤɬɨɪɢɢ
ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɧɚ ɪɟɚɥɶɧɭɸ ɟɝɨ ɬɪɚɟɤɬɨɪɢɸ. Ⱥɜɬɨɪɚɦɢ ɞɨɤɚɡɚɧɨ, ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ
ɨɝɪɚɧɢɱɟɧɧɵɣ ɤɥɚɫɫ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ, ɜ ɤɨɬɨɪɨɦ ɷɬɢ ɬɪɚɟɤɬɨɪɢɢ ɧɟ ɪɚɡɥɢɱɢɦɵ ɩɨ ɡɚɞɚɧɧɨɦɭ ɤɚɱɟɫɬɜɭ.
Abstract. In the paper the possibility of the quality transfer from the scheduled trajectory of navigational
process to the real one has been researched. The authors have proved that there is a restricted equivalence class
where these trajectories are not distinctive by the given quality.
1. ȼɜɟɞɟɧɢɟ
Ⱦɥɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɤɚɤ ɢ ɛɨɥɶɲɢɧɫɬɜɚ ɭɩɪɚɜɥɹɟɦɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ,
ɧɚɢɛɨɥɟɟ ɫɭɳɟɫɬɜɟɧɧɨɟ ɩɨɜɵɲɟɧɢɟ ɟɝɨ ɷɮɮɟɤɬɢɜɧɨɫɬɢ, ɜ ɫɦɵɫɥɟ ɫɧɢɠɟɧɢɹ ɭɪɨɜɧɹ ɬɟɤɭɳɢɯ ɪɢɫɤɨɜ ɢ
ɩɨɜɵɲɟɧɢɹ ɭɪɨɜɧɹ ɛɟɡɨɩɚɫɧɨɫɬɢ, ɫɥɟɞɭɟɬ ɫɜɹɡɵɜɚɬɶ ɫ ɨɩɬɢɦɢɡɚɰɢɟɣ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɪɟɠɢɦɚ
ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɜ ɪɚɦɤɚɯ ɩɪɢɧɹɬɵɯ ɩɪɚɜɢɥ.
Ⱦɚɥɟɟ ɩɨɞ ɭɫɬɚɧɨɜɢɜɲɢɦɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ ɪɟɠɢɦɨɦ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɛɭɞɟɦ ɩɨɧɢɦɚɬɶ
ɬɚɤɨɣ ɟɝɨ ɪɟɠɢɦ, ɩɪɢ ɤɨɬɨɪɨɦ ɞɥɹ ɤɚɠɞɨɣ ɢɡ N ɫɨɫɬɚɜɥɹɸɳɢɯ yi(x, u, t) ɜɟɤɬɨɪ-ɮɭɧɤɰɢɢ Y(X, U, t), i  N
(Ɇɟɧɶɲɢɤɨɜ, Ʉɭɤɭɢ, 2002) ɫɩɪɚɜɟɞɥɢɜɨ ɭɫɥɨɜɢɟ
t + T/2
(1/T )³ yi(W)dW = B,
(1)
t - T/2
ɝɞɟ B – ɩɨɫɬɨɹɧɧɵɣ ɩɚɪɚɦɟɬɪ ɫɨɫɬɨɹɧɢɹ ɛɟɡɨɩɚɫɧɨɫɬɢ ɦɨɪɟɩɥɚɜɚɧɢɹ ɢɥɢ ɩɨɫɬɨɹɧɫɬɜɚ ɬɪɟɛɨɜɚɧɢɣ,
ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ ɷɬɭ ɛɟɡɨɩɚɫɧɨɫɬɶ, T ɞɥɢɬɟɥɶɧɨɫɬɶ ɜɪɟɦɟɧɧɨɝɨ ɢɧɬɟɪɜɚɥɚ, ɧɚ ɤɨɬɨɪɨɦ ɪɟɲɚɟɬɫɹ
ɛɟɡɨɩɚɫɧɚɹ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɚɹ ɬɪɚɧɫɩɨɪɬɧɚɹ ɡɚɞɚɱɚ ɢɥɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɛɟɡɨɩɚɫɧɚɹ ɜ ɧɚɜɢɝɚɰɢɨɧɧɨɦ
ɫɦɵɫɥɟ ɩɪɨɦɵɫɥɨɜɚɹ ɨɩɟɪɚɰɢɹ, ɩɪɢɱɟɦ 0 < T < f.
ɇɚ ɷɬɚɩɟ ɩɥɚɧɢɪɨɜɚɧɢɹ ɭɩɪɚɜɥɟɧɢɣ ɫɨɫɬɨɹɧɢɟɦ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɛɨɥɶɲɢɧɫɬɜɨ
ɩɚɪɚɦɟɬɪɨɜ ɷɬɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɤɚɤ ɩɪɚɜɢɥɨ, ɡɚɞɚɸɬɫɹ ɬɚɤ, ɱɬɨɛɵ ɭɫɥɨɜɢɟ (1), ɧɚɤɥɚɞɵɜɚɟɦɨɟ ɧɚ ɜɟɤɬɨɪɮɭɧɤɰɢɸ Y(X, U, t), ɜɵɩɨɥɧɹɥɨɫɶ ɞɥɹ ɜɫɟɝɨ ɜɪɟɦɟɧɧɨɝɨ ɢɧɬɟɪɜɚɥɚ T. ɉɨɷɬɨɦɭ ɩɥɚɧɨɜɭɸ ɬɪɚɟɤɬɨɪɢɸ
ɫɨɫɬɨɹɧɢɹ ɩɪɨɰɟɫɫɚ ɏ0(t) ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɬɚɰɢɨɧɚɪɧɭɸ, ɧɚ ɤɨɬɨɪɨɣ ɜɫɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɜɟɤɬɨɪɮɭɧɤɰɢɢ Y(X, U, t), ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɤɚɤ ɫɚɦɨ ɫɨɫɬɨɹɧɢɟ ɩɪɨɰɟɫɫɚ, ɬɚɤ ɢ ɭɩɪɚɜɥɹɸɳɢɟ ɜɨɡɞɟɣɫɬɜɢɹ,
ɩɨɫɬɨɹɧɧɵ.
ɇɚ ɷɬɚɩɟ ɪɟɚɥɢɡɚɰɢɢ ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɢ ɭɫɥɨɜɢɟ (1) ɦɨɠɟɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧɨ ɥɢɲɶ ɞɥɹ
ɧɟɤɨɬɨɪɵɯ ɮɢɤɫɢɪɨɜɚɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ɜɪɟɦɟɧɢ T0, ɩɨɫɤɨɥɶɤɭ ɜɫɟɝɞɚ ɛɭɞɟɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɧɟɨɛɯɨɞɢɦɨɫɬɶ
ɜ ɤɨɪɪɟɤɬɢɪɭɸɳɢɯ ɞɟɣɫɬɜɢɹɯ, ɫɩɨɫɨɛɧɵɯ ɜɨɡɜɪɚɳɚɬɶ ɬɟɤɭɳɟɟ ɫɨɫɬɨɹɧɢɟ ɏ(t) ɜ ɡɚɞɚɧɧɨɟ ɛɟɡɨɩɚɫɧɨɟ
ɫɨɫɬɨɹɧɢɟ ɩɪɨɰɟɫɫɚ ɏ0(t). Ɍɨɝɞɚ ɪɟɚɥɢɡɚɰɢɸ ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɫ ɭɱɟɬɨɦ
ɩɥɚɧɨɜɵɯ ɭɩɪɚɜɥɟɧɢɣ ɢ ɫɨɛɥɸɞɟɧɢɟɦ ɭɫɥɨɜɢɹ (1) ɫɥɟɞɭɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫ ɩɨɡɢɰɢɢ ɰɢɤɥɢɱɟɫɤɨɣ
ɭɩɪɚɜɥɟɧɱɟɫɤɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɫɭɞɨɜɨɝɨ ɩɟɪɫɨɧɚɥɚ ɫ ɩɟɪɢɨɞɨɦ, ɪɚɜɧɵɦ T0  T, ɤɨɬɨɪɵɣ ɞɚɥɟɟ ɞɥɹ
ɭɩɪɨɳɟɧɢɹ ɛɭɞɟɦ ɩɪɢɧɢɦɚɬɶ ɡɚ ɩɨɫɬɨɹɧɧɭɸ ɜɟɥɢɱɢɧɭ.
2. Ɉɫɨɛɟɧɧɨɫɬɢ ɪɚɫɲɢɪɟɧɢɹ ɷɤɫɬɪɟɦɚɥɶɧɨɣ ɡɚɞɚɱɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ
ɋ ɮɨɪɦɚɥɶɧɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ, ɷɬɚɩɵ ɩɥɚɧɢɪɨɜɚɧɢɹ ɭɩɪɚɜɥɟɧɢɹ ɢ ɪɟɚɥɢɡɚɰɢɢ ɡɚɞɚɧɧɨɣ ɬɪɚɟɤɬɨɪɢɢ
ɫɨɫɬɨɹɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɦɨɠɧɨ ɫɜɹɡɚɬɶ, ɟɫɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɭɳɟɫɬɜɭɸɳɟɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ
ɜɨɡɦɨɠɧɨɫɬɢ ɪɚɫɲɢɪɟɧɢɹ ɷɤɫɬɪɟɦɚɥɶɧɵɯ ɡɚɞɚɱ. ɂɞɟɸ ɪɚɫɲɢɪɟɧɢɹ ɷɤɫɬɪɟɦɚɥɶɧɨɣ ɡɚɞɚɱɢ,
ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɩɪɨɛɥɟɦɟ ɭɩɪɚɜɥɟɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɵɦ ɩɪɨɰɟɫɫɨɦ, ɦɨɠɧɨ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɬɶ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.
ɉɭɫɬɶ ɬɟɤɭɳɟɟ ɭɩɪɚɜɥɹɟɦɨɟ ɫɨɫɬɨɹɧɢɟ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɟɧɨ ɧɚ ɦɧɨɠɟɫɬɜɟ D1 ɢ
ɪɟɚɥɢɡɭɟɬɫɹ ɜ ɩɪɨɰɟɫɫɟ ɭɩɪɚɜɥɟɧɢɹ ɬɚɤ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɦɢɧɢɦɭɦ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ R1(z), ɬ.ɟ.
R 1(z) o min.
z  D1
61
(2)
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ., Ʉɭɤɭɢ Ɏɢɪɦɢɧ Ⱦɠɢɜɨ Ɉɫɨɛɟɧɧɨɫɬɢ ɩɥɚɧɢɪɨɜɚɧɢɹ...
ȼ ɬɨ ɠɟ ɜɪɟɦɹ, ɩɪɢ ɩɥɚɧɢɪɨɜɚɧɢɢ ɛɟɡɨɩɚɫɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɷɬɨ
ɫɨɫɬɨɹɧɢɟ ɛɵɥɨ ɡɚɞɚɧɨ ɧɚ ɦɧɨɠɟɫɬɜɟ D2 ɢ ɨɩɪɟɞɟɥɟɧɨ ɦɢɧɢɦɭɦɨɦ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ R2(Z). ɬɨɝɞɚ ɡɚɞɚɱɭ
ɩɥɚɧɢɪɨɜɚɧɢɹ ɫɨɫɬɨɹɧɢɹ ɛɟɡɨɩɚɫɧɨɝɨ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɜ ɨɛɳɟɦ, ɡɚɩɢɲɟɦ
R2(Z) o min.
(3)
z  D2
Ɂɚɞɚɱɢ ɪɟɚɥɢɡɚɰɢɢ (2) ɢ ɩɥɚɧɢɪɨɜɚɧɢɹ (3) ɛɟɡɨɩɚɫɧɨɣ ɧɚɜɢɝɚɰɢɢ ɦɨɠɧɨ ɧɚɡɜɚɬɶ ɢɡɨɦɨɪɮɧɵɦɢ,
ɟɫɥɢ ɤɚɠɞɨɦɭ ɷɥɟɦɟɧɬɭ ɢɥɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɷɥɟɦɟɧɬɨɜ ɢɡ ɦɧɨɠɟɫɬɜɚ D1 ɛɭɞɟɬ ɫɬɚɜɢɬɶɫɹ ɜɨ ɜɡɚɢɦɧɨ
ɨɞɧɨɡɧɚɱɧɨɟ ɫɨɨɬɜɟɬɫɬɜɢɟ ɷɥɟɦɟɧɬ ɢɥɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɷɥɟɦɟɧɬɨɜ ɢɡ ɦɧɨɠɟɫɬɜɚ D2. ɉɪɢɱɟɦ ɷɬɨ
ɫɨɨɬɜɟɬɫɬɜɢɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɚɤ, ɱɬɨ ɡɧɚɱɟɧɢɹ ɰɟɥɟɜɵɯ ɮɭɧɤɰɢɣ R1(z) ɢ R2(Z) ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ
ɷɥɟɦɟɧɬɚɯ ɢɥɢ ɢɯ ɩɪɟɞɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɧɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɹɯ ɪɚɜɧɵ. Ɍɨɝɞɚ ɩɪɢ ɢɡɨɦɨɪɮɧɨɫɬɢ (2) ɢ (3)
ɪɟɲɟɧɢɟɦ ɷɤɫɬɪɟɦɚɥɶɧɨɣ ɡɚɞɚɱɢ (3), ɫ ɩɨɡɢɰɢɣ ɨɛɟɫɩɟɱɟɧɢɹ ɛɟɡɨɩɚɫɧɨɫɬɢ ɦɨɪɟɩɥɚɜɚɧɢɹ, ɫɥɟɞɭɟɬ ɫɱɢɬɚɬɶ
ɬɚɤɭɸ ɡɚɞɚɱɭ (2), ɤɨɬɨɪɚɹ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɫɚɦɨɣ ɡɚɞɚɱɟ (3) ɪɟɲɚɟɬɫɹ ɫ ɫɨɛɥɸɞɟɧɢɟɦ ɞɜɭɯ ɭɫɥɨɜɢɣ:
D1 Ž D2,
R 2(Z) t R 1(z), z  D1.
(4)
Ɂɚɩɢɫɚɧɧɵɟ ɭɫɥɨɜɢɹ (4) ɨɩɪɟɞɟɥɹɸɬ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɭɸ ɫɜɹɡɶ ɦɟɠɞɭ ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɟɣ
ɫɨɫɬɨɹɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢ ɬɪɚɟɤɬɨɪɢɟɣ ɟɝɨ ɫɨɫɬɨɹɧɢɹ ɩɪɢ ɪɟɚɥɶɧɨɦ ɭɩɪɚɜɥɟɧɢɢ. ɂɦɟɧɧɨ
ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɣ (4) ɨɛɟɫɩɟɱɢɜɚɟɬ ɪɟɚɥɢɡɚɰɢɸ ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɫ
ɤɚɱɟɫɬɜɨɦ ɧɟ ɯɭɠɟ ɬɨɝɨ, ɤɨɬɨɪɵɦ ɨɧɚ ɨɛɥɚɞɚɥɚ ɧɚ ɷɬɚɩɟ ɩɥɚɧɢɪɨɜɚɧɢɹ, ɬ.ɟ.
R*2 t R*1,
(5)
ɝɞɟ R*1 = sup R1, R*2 = sup R2, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
ȿɫɥɢ ɡɧɚɱɟɧɢɟ ɤɚɱɟɫɬɜɚ ɜ ɪɚɫɲɢɪɟɧɢɢ (2) ɫɨɜɩɚɞɚɟɬ ɫɨ ɡɧɚɱɟɧɢɟɦ ɤɚɱɟɫɬɜɚ ɜ ɡɚɞɚɱɟ
ɩɥɚɧɢɪɨɜɚɧɢɹ (3), ɬɨ ɬɚɤɨɟ ɪɚɫɲɢɪɟɧɢɟ ɹɜɥɹɟɬɫɹ ɧɟ ɬɨɥɶɤɨ ɢɡɨɦɨɪɮɧɵɦ, ɧɨ ɢ ɷɤɜɢɜɚɥɟɧɬɧɵɦ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ R*2 = R*1, ɩɥɚɧɨɜɚɹ ɬɪɚɟɤɬɨɪɢɹ, ɚ ɬɚɤɠɟ ɟɟ ɩɪɚɤɬɢɱɟɫɤɚɹ ɪɟɚɥɢɡɚɰɢɹ,
ɹɜɥɹɸɬɫɹ ɷɥɟɦɟɧɬɚɦɢ ɨɞɧɨɝɨ ɤɥɚɫɫɚ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ /, ɬ.ɟ. ɧɟ ɪɚɡɥɢɱɢɦɵ ɩɨ ɤɚɱɟɫɬɜɭ.
Ɋɚɫɫɦɨɬɪɢɦ ɦɟɯɚɧɢɡɦ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɵɦ ɪɚɫɲɢɪɟɧɢɟ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɝɨ
ɩɥɚɧɢɪɨɜɚɧɢɹ ɭɩɪɚɜɥɹɟɦɨɝɨ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɛɭɞɟɬ ɜɤɥɸɱɚɬɶ ɜ ɫɟɛɹ ɬɪɚɟɤɬɨɪɢɸ ɫɨɫɬɨɹɧɢɹ,
ɩɨɥɭɱɚɟɦɭɸ ɜ ɩɪɨɰɟɫɫɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɷɬɢɦ ɩɪɨɰɟɫɫɨɦ.
3. Ɉɩɪɟɞɟɥɟɧɢɟ ɛɥɢɡɨɫɬɢ ɩɥɚɧɨɜɨɣ ɢ ɪɟɚɥɶɧɨɣ ɬɪɚɟɤɬɨɪɢɣ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ
ɉɭɫɬɶ ɧɚɜɢɝɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ, ɢɞɭɳɢɣ ɜ ɫɢɫɬɟɦɟ "ɫɭɞɧɨ" ɫ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ,
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɟɪɟɦɟɧɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ x  X ɢ ɭɩɪɚɜɥɟɧɢɹɦɢ u  U. Ɍɨɝɞɚ ɡɚɞɚɱɭ ɩɥɚɧɢɪɨɜɚɧɢɹ
ɭɩɪɚɜɥɟɧɢɹ ɫɨɫɬɨɹɧɢɟɦ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɫ ɩɨɦɨɳɶɸ ɨɩɬɢɦɚɥɶɧɨɣ ɫɬɚɰɢɨɧɚɪɧɨɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ
ɦɨɞɟɥɢ, ɡɚɩɢɫɚɧɧɨɣ, ɧɚɩɪɢɦɟɪ, ɬɚɤ:
f(x, u) = 0,
R2 = f0(x, u) o min,
M(x, u) t 0,
x  X, u  U.
(6)
Ɋɟɚɥɢɡɚɰɢɸ ɠɟ ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ f(x, u) ɩɪɢ ɭɩɪɚɜɥɟɧɢɢ ɢɦ ɫ
ɭɱɟɬɨɦ ɤɨɪɪɟɤɬɢɪɭɸɳɢɯ ɞɟɣɫɬɜɢɣ, ɜɵɩɨɥɧɹɟɦɵɯ ɫ ɩɨɫɬɨɹɧɧɵɦ ɩɟɪɢɨɞɨɦ, ɪɚɜɧɵɦ T0  T, ɨɩɢɲɟɦ
ɨɩɬɢɦɚɥɶɧɨɣ ɞɢɧɚɦɢɱɟɫɤɨɣ ɦɨɞɟɥɶɸ ɜɢɞɚ
dx/dt = f(x(t), u(t)),
T0
R1 = (1/T0) ³ f0(x(t), u(t)) dt o min,
0
T0
0
³ M (x(t), u(t)) dt t 0,
T0
³ f (x(t), u(t)) dt = 0,
0
62
(7)
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 6, ʋ1, 2003 ɝ.
ɫɬɪ.61-64
x X, u U, T0 > 0.
ȼ ɦɨɞɟɥɹɯ (6) ɢ (7) ɩɪɢɧɹɬɵ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ: f0(x(t), u(t)) ɬɟɤɭɳɟɟ ɡɧɚɱɟɧɢɟ ɩɨɤɚɡɚɬɟɥɹ
ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɩɪɨɰɟɫɫɚ, M(x(t), u(t)) – ɬɟɤɭɳɢɣ ɪɚɫɯɨɞ ɦɚɬɟɪɢɚɥɶɧɵɯ ɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɡɚɬɪɚɬ.
Ʉɚɤ ɩɨɤɚɡɚɬɟɥɶ ɤɚɱɟɫɬɜɟɧɧɨɣ ɛɥɢɡɨɫɬɢ ɦɟɠɞɭ ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɟɣ ɩɪɨɰɟɫɫɚ (6) ɢ ɟɟ
ɭɩɪɚɜɥɟɧɱɟɫɤɨɣ ɪɟɚɥɢɡɚɰɢɟɣ (7) ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɟɬɪɢɤɭ ɜɢɞɚ
' = R*2 R*1,
(8)
ɩɪɢɱɟɦ, ɤɚɤ ɷɬɨ ɛɵɥɨ ɫɞɟɥɚɧɨ ɜɵɲɟ, ɜɧɨɜɶ ɩɪɢɦɟɦ, ɱɬɨ R*1 = sup R1 ɢ R*2 = sup R2.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɤɚɡɚɬɟɥɹ ɤɚɱɟɫɬɜɟɧɧɨɣ ɛɥɢɡɨɫɬɢ ' ɜ (8) ɢɫɩɨɥɶɡɭɟɦ ɢɡɜɟɫɬɧɵɟ ɜɨɡɦɨɠɧɨɫɬɢ,
ɡɚɥɨɠɟɧɧɵɟ ɜ ɤɥɚɫɫɢɱɟɫɤɭɸ ɡɚɞɚɱɭ Ʌɚɝɪɚɧɠɚ. Ɍɚɤ, ɩɭɫɬɶ ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ;2(x, u, O, P), ɫɨɫɬɚɜɥɟɧɧɚɹ
ɞɥɹ ɡɚɞɚɱɢ (6), ɨɬɜɟɱɚɟɬ ɭɫɥɨɜɢɸ:
;2(x, u, O, P) = f 0(x, u) + Of (x, u) + PM(x, u) > min,
(9)
uU xX
ɝɞɟ ɜɟɤɬɨɪɵ O ɢ P – ɦɧɨɠɢɬɟɥɢ Ʌɚɝɪɚɧɠɚ, ɚ Of ɢ PM ɫɭɬɶ ɫɤɚɥɹɪɧɵɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɷɬɢɯ ɜɟɤɬɨɪɨɜ.
ɂɫɩɨɥɶɡɭɹ ɜɜɟɞɟɧɧɭɸ ɮɭɧɤɰɢɸ Ʌɚɝɪɚɧɠɚ (9), ɩɨɤɚɠɟɦ, ɱɬɨ ɡɚɞɚɱɚ ɩɨ ɪɟɚɥɢɡɚɰɢɢ ɩɥɚɧɨɜɨɣ
ɬɪɚɟɤɬɨɪɢɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ (7) ɦɨɠɟɬ ɹɜɥɹɬɶɫɹ ɷɥɟɦɟɧɬɨɦ ɪɚɫɲɢɪɟɧɢɹ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɡɚɞɚɱɟ
ɩɥɚɧɢɪɨɜɚɧɢɹ ɭɩɪɚɜɥɟɧɢɹ (6). Ⱦɥɹ ɷɬɨɣ ɰɟɥɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɩɪɢɦɟɦ, ɱɬɨ ɪɚɫɲɢɪɟɧɢɟ ɡɚɞɚɱɢ (6)
ɫɭɳɟɫɬɜɭɟɬ ɩɪɢ ɥɸɛɵɯ ɨɝɪɚɧɢɱɟɧɢɹɯ ɜɢɞɚ O ɢ P t 0, ɟɫɥɢ ɫɨɫɬɚɜɥɹɸɳɚɹ Pk ɜɟɤɬɨɪɚ P ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ
ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ Mk(x, u) > 0.
ɉɭɫɬɶ ɩɪɢ ɨɬɦɟɱɟɧɧɨɦ ɜɵɲɟ ɭɫɥɨɜɢɢ, ɤɨɬɨɪɨɟ ɜ ɩɪɢɧɰɢɩɟ ɦɨɠɧɨ ɨɬɧɟɫɬɢ ɤ ɭɫɥɨɜɢɸ
ɞɨɩɨɥɧɹɸɳɟɣ ɧɟɠɟɫɬɤɨɫɬɢ, ɜɟɤɬɨɪɵ O ɢ P ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ V. Ɍɨɝɞɚ ɞɥɹ ɥɸɛɵɯ ɩɚɪ ɦɧɨɠɢɬɟɥɟɣ
(O, P) Ʌɚɝɪɚɧɠɚ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ V, ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜɨ ɧɟɪɚɜɟɧɫɬɜɨ (5). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫ ɭɱɟɬɨɦ
ɨɫɨɛɟɧɧɨɫɬɟɣ ɩɨɜɟɞɟɧɢɹ ɩɚɪ ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ (O, P), ɞɥɹ ɪɚɫɲɢɪɟɧɧɨɝɨ ɩɨɞɯɨɞɚ ɤ ɡɚɞɚɱɟ
ɩɥɚɧɢɪɨɜɚɧɢɹ ɭɩɪɚɜɥɟɧɢɹ ɛɟɡɨɩɚɫɧɨɣ ɬɪɚɟɤɬɨɪɢɟɣ X0(t) ɧɟɪɚɜɟɧɫɬɜɨ (5) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ:
;2*(O, P) t R*2, (O, P)  V.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɞɥɹ ɧɟɤɨɬɨɪɵɯ ɡɧɚɱɟɧɢɣ ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ O0, P0 ɩɨɫɥɟɞɧɟɟ ɧɟɫɬɪɨɝɨɟ
ɧɟɪɚɜɟɧɫɬɜɨ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɨ ɜ ɫɬɪɨɝɨɟ ɪɚɜɟɧɫɬɜɨ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɮɭɧɤɰɢɹ ;2 ɨɛɹɡɚɧɚ ɢɦɟɬɶ
ɫɟɞɥɨɜɭɸ ɬɨɱɤɭ, ɜ ɤɨɬɨɪɨɣ ɨɧɚ, ɧɚɩɪɢɦɟɪ, ɦɨɠɟɬ ɛɵɬɶ ɦɚɤɫɢɦɚɥɶɧɚ ɩɨ ɫɨɫɬɨɹɧɢɸ ɢ ɭɩɪɚɜɥɟɧɢɸ x, u ɢ
ɦɢɧɢɦɚɥɶɧɚ ɩɨ ɩɚɪɚɦɟɬɪɚɦ O, P, ɩɪɢɱɟɦ ɬɚɤ, ɱɬɨ
min max ;2(x, u, O, P) = R*2.
O, P
(10)
x, u
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɪɚɦɤɚɯ Ʌɚɝɪɚɧɠɟɜɨɝɨ ɩɨɞɯɨɞɚ ɡɚɞɚɱɭ ɬɟɤɭɳɟɝɨ ɭɩɪɚɜɥɟɧɢɹ ɫɨɫɬɨɹɧɢɟɦ
ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ (7), ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɦɨɠɧɨ, ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɪɚɫɲɢɪɟɧɢɟ ɡɚɞɚɱɢ
ɩɥɚɧɢɪɨɜɚɧɢɹ (6), ɩɪɢɱɟɦ ɬɚɤɨɟ ɪɚɫɲɢɪɟɧɢɟ, ɜ ɤɨɬɨɪɨɦ ɩɪɢ ɤɨɧɤɪɟɬɧɨ ɩɪɢɧɹɬɵɯ ɡɧɚɱɟɧɢɹɯ ɩɚɪɚɦɟɬɪɨɜ
O0, P0 ɩɨɪɨɠɞɚɟɬɫɹ ɤɥɚɫɫ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ / ɬɪɚɟɤɬɨɪɢɣ X0(t) ɢ X(t) ɫ ɩɪɢɡɧɚɤɨɦ ɪɚɜɟɧɫɬɜɚ ɢɯ ɰɟɥɟɜɵɯ
ɮɭɧɤɰɢɣ.
Ɉɞɧɚɤɨ ɧɚ ɩɪɚɤɬɢɤɟ ɞɨɛɢɬɶɫɹ ɫɬɪɨɝɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɦɟɠɞɭ ɩɥɚɧɨɜɨɣ
ɬɪɚɟɤɬɨɪɢɟɣ X0(t) ɢ ɟɟ ɪɟɚɥɢɡɚɰɢɟɣ X(t), ɨɪɢɟɧɬɢɪɭɹɫɶ ɬɨɥɶɤɨ ɧɚ ɩɨɞɛɨɪ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɡɧɚɱɟɧɢɣ
ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ O0, P0, ɞɨɫɬɚɬɨɱɧɨ ɫɥɨɠɧɨ. ɉɨɷɬɨɦɭ ɢɦɟɟɬ ɫɦɵɫɥ ɧɚɣɬɢ ɬɚɤɭɸ ɤɨɥɢɱɟɫɬɜɟɧɧɭɸ
ɨɰɟɧɤɭ ɛɥɢɡɨɫɬɢ ɬɪɚɟɤɬɨɪɢɣ, ɤɨɬɨɪɚɹ ɩɨɡɜɨɥɢɥɚ ɛɵ ɨɝɪɚɧɢɱɢɬɶ ɫɜɟɪɯɭ ɝɪɚɧɢɰɵ ɤɥɚɫɫɚ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ.
4. Ɉɰɟɧɤɚ ɛɥɢɡɨɫɬɢ ɩɥɚɧɨɜɨɣ ɢ ɪɟɚɥɶɧɨɣ ɬɪɚɟɤɬɨɪɢɣ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɭɩɪɚɜɥɟɧɢɹ (7) ɢ ɨɬɛɪɨɫɢɦ ɜ ɧɟɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɫɜɹɡɢ. ɉɪɢ
ɨɬɛɪɚɫɵɜɚɧɢɢ ɫɜɹɡɟɣ ɜ (7) ɦɧɨɠɟɫɬɜɨ ɞɨɩɭɫɬɢɦɵɯ ɪɟɲɟɧɢɣ ɪɚɫɲɢɪɹɟɬɫɹ, ɢ ɡɧɚɱɟɧɢɟ ɧɨɜɨɣ ɡɚɞɚɱɢ (ɛɟɡ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɫɜɹɡɟɣ) ɛɭɞɟɬ ɧɟ ɦɟɧɶɲɟ ɡɧɚɱɟɧɢɹ ɫɚɦɨɣ ɡɚɞɚɱɢ (7), ɬ.ɟ. R*0 t R*1, ɝɞɟ R*0 – ɤɪɢɬɟɪɢɣ
ɤɚɱɟɫɬɜɚ ɡɚɞɚɱɢ (7) ɛɟɡ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɫɜɹɡɟɣ.
ȼ ɬɨ ɠɟ ɜɪɟɦɹ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɜ (ɐɢɪɥɢɧ, 1974), ɡɧɚɱɟɧɢɟ R*0 ɦɨɠɟɬ ɫɨɜɩɚɞɚɬɶ ɫɨ ɡɧɚɱɟɧɢɟɦ R*
ɩɪɢ ɪɟɲɟɧɢɢ ɫɥɟɞɭɸɳɟɣ ɡɚɞɚɱɢ
R* = ¦ Jk f0(xk, uk) o min,
k
k
¦ Jk f(xk, uk) = 0,
k
63
¦ Jk M(xk, uk) t 0,
(11)
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ., Ʉɭɤɭɢ Ɏɢɪɦɢɧ Ⱦɠɢɜɨ Ɉɫɨɛɟɧɧɨɫɬɢ ɩɥɚɧɢɪɨɜɚɧɢɹ...
Jk t 0, xk, uk  V, ¦ Jk = 1, k = 1, ..., rɯ + rM + 1,
k
ɩɪɢɱɟɦ ɡɧɚɱɟɧɢɟ R* ɞɥɹ ɡɚɞɚɱɢ (11) ɦɨɠɧɨ ɧɚɣɬɢ ɬɚɤ:
R* = inf sup ;2 (x, u, O, P),
(12)
O, P x, u
ɝɞɟ ;2(x, u, O, P) – ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ ɞɥɹ ɡɚɞɚɱɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ɭɩɪɚɜɥɟɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɵɦ ɩɪɨɰɟɫɫɨɦ (6).
ȿɫɥɢ ɬɟɩɟɪɶ ɜɵɪɚɠɟɧɢɟ (12) ɩɪɢ ɭɫɥɨɜɢɢ R*0 t R*1 ɩɨɫɬɚɜɢɬɶ ɜ (8), ɬɨ ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ
' d inf sup ;2 (x, u, O, P) R*2.
O, P
(13)
x, u  V
ȼɵɪɚɠɟɧɢɟ (13) ɨɩɪɟɞɟɥɹɟɬ ɞɨɩɭɫɬɢɦɵɟ ɝɪɚɧɢɰɵ ɤɥɚɫɫɚ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ /, ɜ ɤɨɬɨɪɨɦ ɤɚɠɞɚɹ
ɩɥɚɧɨɜɚɹ ɬɪɚɟɤɬɨɪɢɹ X0(t) ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɪɚɡɥɢɱɢɦɚ ɩɨ ɤɚɱɟɫɬɜɭ ɫ ɟɟ ɪɟɚɥɢɡɚɰɢɟɣ X(t), ɩɨɥɭɱɟɧɧɨɣ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɭɩɪɚɜɥɟɧɱɟɫɤɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɫɭɞɨɜɨɝɨ ɩɟɪɫɨɧɚɥɚ. Ʉɪɨɦɟ ɬɨɝɨ, ɢɡ (13) ɫ ɭɱɟɬɨɦ (10) ɦɨɠɧɨ
ɞɥɹ ɫɬɪɨɝɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɩɨɥɭɱɢɬɶ ' = 0.
ȿɫɥɢ ɡɚɞɚɱɚ ɩɥɚɧɢɪɨɜɚɧɢɹ ɬɢɩɚ (6) ɨɛɥɚɞɚɟɬ ɪɚɫɲɢɪɟɧɢɟɦ ɢ ɞɥɹ ɷɬɨɝɨ ɪɚɫɲɢɪɟɧɢɹ ɧɚɣɞɟɧɵ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɡɧɚɱɟɧɢɹ O*, P* ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ, ɬɨ ɜɦɟɫɬɨ ɧɚɯɨɠɞɟɧɢɹ ɦɢɧɢɦɭɦɚ ɩɨ ɡɧɚɱɟɧɢɹɦ
O, P ɦɨɠɧɨ ɩɨɞɫɬɚɜɢɬɶ ɨɩɪɟɞɟɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɮɭɧɤɰɢɸ Ʌɚɝɪɚɧɠɚ ɢ ɩɨɥɭɱɢɬɶ ɝɪɭɛɭɸ ɨɰɟɧɤɭ ɞɥɹ
ɝɪɚɧɢɰ ɤɥɚɫɫɚ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ /. ɗɬɭ ɝɪɭɛɭɸ ɨɰɟɧɤɭ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ:
' d sup ;2 (x, u, O, P) R*2.
(14)
x, u  V
5. Ɂɚɤɥɸɱɟɧɢɟ
Ⱦɥɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɤɚɤ ɢ ɛɨɥɶɲɢɧɫɬɜɚ ɭɩɪɚɜɥɹɟɦɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ,
ɧɚɢɛɨɥɟɟ ɫɭɳɟɫɬɜɟɧɧɨɟ ɩɨɜɵɲɟɧɢɟ ɟɝɨ ɷɮɮɟɤɬɢɜɧɨɫɬɢ, ɜ ɫɦɵɫɥɟ ɫɧɢɠɟɧɢɹ ɭɪɨɜɧɹ ɬɟɤɭɳɢɯ ɪɢɫɤɨɜ ɢ
ɩɨɜɵɲɟɧɢɹ ɭɪɨɜɧɹ ɛɟɡɨɩɚɫɧɨɫɬɢ, ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ ɜ ɨɛɥɚɫɬɢ ɨɩɬɢɦɢɡɚɰɢɢ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɪɟɠɢɦɚ
ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɜ ɪɚɦɤɚɯ ɩɪɢɧɹɬɵɯ ɩɪɚɜɢɥ. Ɉɞɧɚɤɨ ɧɚ ɩɪɚɤɬɢɤɟ ɞɨɛɢɬɶɫɹ
ɫɬɪɨɝɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɦɟɠɞɭ ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɟɣ ɩɪɨɰɟɫɫɚ ɢ ɟɟ ɪɟɚɥɢɡɚɰɢɟɣ,
ɨɪɢɟɧɬɢɪɭɹɫɶ ɬɨɥɶɤɨ ɧɚ ɩɨɞɛɨɪ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɡɧɚɱɟɧɢɣ ɦɧɨɠɢɬɟɥɟɣ Ʌɚɝɪɚɧɠɚ O0, P0 ɜ (9),
ɞɨɫɬɚɬɨɱɧɨ ɫɥɨɠɧɨ. ɉɨɷɬɨɦɭ ɢɦɟɟɬ ɫɦɵɫɥ ɧɚɣɬɢ ɬɚɤɨɣ ɜɚɪɢɚɧɬ ɤɨɥɢɱɟɫɬɜɟɧɧɨɣ ɨɰɟɧɤɢ ɛɥɢɡɨɫɬɢ
ɩɥɚɧɨɜɨɣ ɬɪɚɟɤɬɨɪɢɢ ɢ ɬɪɚɟɤɬɨɪɢɢ ɪɟɚɥɢɡɚɰɢɢ, ɤɨɬɨɪɵɣ ɨɛɴɟɞɢɧɹɥ ɛɵ ɷɬɢ ɬɪɚɟɤɬɨɪɢɢ ɜ ɨɞɢɧ ɨɛɳɢɣ
ɤɥɚɫɫ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɢ ɞɟɥɚɥ ɛɵ ɢɯ ɧɟɪɚɡɥɢɱɢɦɵɦɢ ɩɨ ɤɚɱɟɫɬɜɭ.
ȼɵɩɨɥɧɟɧɧɵɟ ɜ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɢɫɫɥɟɞɨɜɚɧɢɹ ɩɨɡɜɨɥɢɥɢ ɨɩɪɟɞɟɥɢɬɶ ɝɪɚɧɢɰɵ ɤɥɚɫɫɚ
ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ, ɜ ɤɨɬɨɪɨɦ ɦɨɠɧɨ ɩɥɚɧɨɜɨɟ ɤɚɱɟɫɬɜɨ ɩɟɪɟɧɨɫɢɬɶ ɧɚ ɬɪɚɟɤɬɨɪɢɸ ɫɨɫɬɨɹɧɢɹ
ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɩɨɥɭɱɟɧɧɭɸ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɭɩɪɚɜɥɟɧɱɟɫɤɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɫɭɞɨɜɨɝɨ
ɩɟɪɫɨɧɚɥɚ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ ɛɥɢɡɨɫɬɶ ɩɨ ɤɚɱɟɫɬɜɭ ɦɟɠɞɭ ɩɥɚɧɨɜɨɣ ɦɨɞɟɥɶɸ ɛɟɡɨɩɚɫɧɨɝɨ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ
ɩɪɨɰɟɫɫɚ ɢ ɟɟ ɪɟɚɥɢɡɚɰɢɟɣ ɧɟ ɢɫɤɥɸɱɚɟɬ ɬɨɝɨ, ɱɬɨ ɮɚɤɬɢɱɟɫɤɚɹ ɬɪɚɟɤɬɨɪɢɹ ɛɭɞɟɬ ɜɫɟ ɠɟ ɨɛɥɚɞɚɬɶ ɫɜɨɢɦɢ
ɢɧɞɢɜɢɞɭɚɥɶɧɵɦɢ ɨɫɨɛɟɧɧɨɫɬɹɦɢ.
Ʌɢɬɟɪɚɬɭɪɚ
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ., Ʉɭɤɭɢ Ɏɢɪɦɢɧ Ⱦɠɢɜɨ, Ɇɨɞɟɥɶ ɢ ɦɟɯɚɧɢɡɦ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ
ɛɟɡɨɩɚɫɧɨɣ ɷɤɫɩɥɭɚɬɚɰɢɟɣ ɫɭɞɨɜ. ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬ.5, ʋ 2, ɫ.171-176, 2002.
ɐɢɪɥɢɧ Ⱥ.Ɇ. Ɉɩɬɢɦɢɡɚɰɢɹ ɜ ɫɪɟɞɧɟɦ ɢ ɫɤɨɥɶɡɹɳɢɟ ɪɟɠɢɦɵ ɜ ɡɚɞɚɱɚɯ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ.
Ɍɟɯɧɢɱɟɫɤɚɹ ɤɢɛɟɪɧɟɬɢɤɚ, ʋ 2, ɫ.143-151, 1974.
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