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Воронежский государственный технический университет

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. . .

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ISBN 978-5-7731-0689-0 ( . 1)
ISBN 978-5-7731-0688-3
					XV						-		-
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ISBN 978-5-7731-0689-0 ( . 1)					©			«			», 2018
ISBN 978-5-7731-0688-3											», 2018

334.02

. .
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MODELS OF SUSTAINABLE MANAGEMENT IN ACTIVE SYSTEMS

G.A. Ougolnitsky

Southern Federal University

theory of sustainable management in complex dynamical systems with human participation (active systems) is presented. Sustainable development means a simultaneous satisfaction of the requirements of homeostasis and incentivity which are explained in the text. The principal attention is paid to the methodological aspects and the survey of the solved problems.

.	Д1-2Ж,		. .	-
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Ji (u( )) e t gi (x(t),u(t),t)dt max		(1)
0

	ui (t) Ui ,		i N;
	x f (x(t),u(t),t),			x(0) x0 .
	N {1,...,n}-				;	gi , Ji -
					i N ; ui (t) -
i N			t ;	Ui -
u(t) (u1(t),...,un (t))		-				;
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					; [0,1] -
.			T .
			T .
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					-	t
t 0	,				(2).
	x(t) X* Rn, t [0, ).
,		,				(4)
,					(4).
					(4).
					(1)-(3)
NE (				).	, NE ,	Ui
gi (ui ,u i )		ui	Ui		u i U i , i N .
	U* {u U : t x(t) X*}-
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					.

		(2)
		(3)
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		(1)-(3)

NE ,

(1)-(4)

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i N

,
uNE NE \U	DG	u U	DG	: J(u) J(uNE ) ,	(5)
	DG		DG
	(4)				.

uNE NE \U	DG	u U					DG		: J(u) J(uNE ) ,																(6)
																									.
i N uNE NE \U					DG			u U			DG		: J			(u) J					i	(uNE ),			(7)
					DG						DG				i						i				,		-
																									,		-
i N uNE NE \U				DG			u U			DG		: J		i	(u) J					(uNE ),					(8)
				DG						DG				i					i						.		-
, (7) → (5), (6) → (8).																									.		-
, (7) → (5), (6) → (8).
UDG NE U* ,																									(1)-(4)		-
UDG ,									.
UDG ,												(6)										(8),			,		-
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													.
,													.									,
																						,
																						(			)
(3)		.																							(1)-
(3)
J(u()) max																									(9)
(2)-(3).
						Arg max J(u( )) .																UOC U *		,			-
			U			Arg max J(u( )) .																UOC U *	U	,			-
									u( ) U																		.
																											.
,								UOC
																		U		\UOC u UOC : J(u) J(						), . .
																	u	U		\UOC u UOC : J(u) J(					u	), . .
UOC																,									.		-
(4).																,											-
(4).																											-
																											-
																(1)-(4).						,					,
																						(4).					-
(9), (2)-(4).																											-
(9), (2)-(4).
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Д12Ж.

	,									,	,
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			(							).		-

J0 (p( ),q( ),u( )) e t g0 (x(t), p(t),q(t),u(t),t)dt max												(10)
		0
p(t) P,q(t) Q;												(11)

Ji (p( ),q( ),u( )) e t gi (x(t), pi (t),u(t),t)dt max												(12)
		0
ui (t) Ui (qi (t)),		i N;										(13)
	(3)	(4).							g0, J0 -
	;	p(t) (p1(t),..., pn (t)) -										;
q(t) (q1(t),...,qn (t))-										.	,	,
												,
								(4)		.		,
										NE(p,q) ,		-
	.											,
(1)-(3).
Umax	Arg max	max J			0		(p,q,u) -
	p P,q Q u U(q)				0
	p P,q Q u U(q)
	;
Umax	Arg max	max					J	0	(p,q,u) -
NE	p P,q Q u NE( p,q)							0
	p P,q Q u NE( p,q)									;
										;
U*	Arg max	max	J	0		(p,q,u)				-
max	p P,q Q u U*			0
	p P,q Q u U*
,	;

UNE* (p,q) NE(p,q) U * -

;

UHDG(p,q) UNEmax Umax* U

	,
(p,q) -	-

(p,q) P Q:UHDG ,			(10)-(13),
(3)-(4)		.
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. Д15Ж

v(t) (t, x(t)); v(t) (t,u(t)).

), -

v(t) q(t)

SCI J0max J0NE ,

Jmax	max max J	0	(p,q,u),
0	p P,q Q u U(q)	0
	p P,q Q u U(q)

.	,	-
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:v(t) (t)

v(t) (t)

2t 2x v(t) p(t);

J0NE max min

p P,q Q u NE( p,q)

1t	1x (		-
(		) Д17,18Ж;
		(5),	(7)
		(14)
J0 (p,q,u).

			.	SCI 0 .	SCI 0
	.		.			-
	.					-
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.		Д19Ж.
	Д20-21Ж	Д19Ж.				-
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		.	Д22Ж		,	-
.						-
.	Д23Ж					-
	Д23Ж		.	Д24Ж	-	-
			.	Д24Ж		-
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	(	).	Д25Ж			-
-						-
.	Д26Ж	.		-
	Д27Ж	.				-
	Д27Ж					-
	.	Д28Ж				-
	.	Д29-30Ж		-	,	-
	.	Д29-30Ж			,
					(	-
)	.	Д31Ж			-	-
	.	Д32-33]
		8

.
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.

	1.	. .					. -	.:	, 1977. -
255 .
	2.	. .,		. .		:			. -	.:
	, 1999. - 128 .
	3.	. .		, 2007. - 584 .			. -	.:	-	-
-				, 2007. - 584 .
	4.			/	. . .	. - .:	, 2011. - 192 .
	5.	. .,		. .,		. .				-
		. -	.:	, 1991. – 288 .
	6. Laffont J.-J., Martimort D. The Theory of Incentives: The Principal-Agent Model. -
Princeton University Press, 2002. - 421 .
	7.	-		. .,	. .					. -
.:		-	, 2000. - 416 .
	8. Clark W.C. Sustainability Science: A Room of its Own // Proceedings of the National
Academy of Science, 2007 (114), 1737-1738.
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-	:					, 2016. - 940 .
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Long N.V., Sorger G. – Cambridge University Press, 2000. - 382 .
	12.		. .,		. .,	. .		. –	.:	-
	13.	, 2012. - 432 .				/		. .,
	13.					/		. .,
.	.,	.	.,	. . -	.:	, 1983. - 393 .
	14.		. .,	. .		//				-
	, 2014 (6), 86-102.					//				-
	, 2014 (6), 86-102.
	15.		. .,	.	.,	. .
		, 2016, 8(4), 14-29.				//
		, 2016, 8(4), 14-29.

16.Ougolnitsky G.A., Usov A.B. Computer Simulations as a Solution Method for Differential Games // Computer Simulations: Advances in Research and Applications. Eds. M.D. Pfeffer and E. Bachmaier. - N.Y.: Nova Science Publishers, 2018. P.63-106.

17.Olsder G. J. Phenomena in inverse Stackelberg games, part 2: Dynamic problems // Journal of Optimization Theory and Applications, 2009, 143 (3), 601–618.

18.Gorelov M.A., Kononenko A.F. Dynamic models of conflicts. III. Hierarchical games //

Automation and Remote Control, 2015, 76(2), 264-277.
19.	. .				//	-
	. 2018. №1. URL: ivdon.ru/ru/magazine/archive/n1y2018/4670.
20.	. .,	. .,	. .			//
			, 2014, 6(4), 535-542.			//
			, 2014, 6(4), 535-542.
21.	. .,	. .,	. .			-
			, 2015, 7(2), 281-288.			//
			, 2015, 7(2), 281-288.
22.	. .,	. .,	. .		-	-
.	. 55. - ., 2015.	.326-342.			//
.	. 55. - ., 2015.	.326-342.
23.	. .,	. .,	. .,		. .,	. .,
. .,	. .	-				-
	//		.		. 55. - ., 2015. .343-361.
24.	. .,	. .,		. .	.
	, 2016, 28(7), 96-106.				//
	, 2016, 28(7), 96-106.
25.	. .,	. .,	. .	.	-
	, 2017, 6, 122-137.				//	-
	, 2017, 6, 122-137.
26.	. .,		. .,	. .
		(	. 2016. №4.		) //	.
-	.		. 2016. №4.		.25-29.

27.Rokhlin D.B., Ougolnitsky G.A., Usov A.B. A two-level model of optimal harvesting under parameter uncertainty // Far East Journal of Mathematical Sciences, 2017, 102(7), 1365-1380.

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vances in Systems Science and Applications, 2017, 17(2), 52-62.
29.	.	.,	. .,	. .
	. 2017. №6.	.35-47.		//	-
	. 2017. №6.	.35-47.
30.	. .,		. .,	. .	-
. 71. .:	, 2018. .61-97.			//	.
. 71. .:	, 2018. .61-97.
31.		. .,	. .	-
				//	,
2016, 8(4), 673-684.

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vances in Systems Science and Applications, 2015, 15(4), 320-328.
33.	-	. .,	. .,	. .		-
			:		(	-
, 2018, 4, 15-24.				) //		-
, 2018, 4, 15-24.
				10

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