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International Robotics & Automation Journal

Review Article Volume 4 Issue 1

The dynamic properties in the matrix model of biological system under optimal harvesting

Aleksandr I Abakumov

Institute of Automation and Control Processes, Russia

Correspondence: Aleksandr I Abakumov, Far Eastern Federal University, Russia, Tel +7423 2310439

Received: January 30, 2018 | Published: February 6, 2018

Citation: Abakumova AI. The dynamic properties in the matrix model of biological system under optimal harvesting. Int Rob Auto J. 2018;4(1):47-49. DOI: 10.15406/iratj.2018.04.00091

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Abstract

The problem of optimal harvest in biological system is investigated. The structure of system is described by matrix model. The properties of stabilization indexes of harvest components are shown. The harvesting regime is stabilized on low level for long time interval of harvesting. These properties are analogous for turnpike theorems in models of economic dynamics.

Keywords: matrix model, biological system, optimal harvesting, turnpike phenomenon

Introduction

Matrix models are widely used for describing of ecological, economic and social processes. This article has results for biological applications. The matrices are nonnegative usually by describing of abundance dynamics for biological populations or communities.1 For example the abundance dynamics for a population with age structure may be described by model (1) with matrix A as Leslie matrix L.2 The results of O Perron and of G Frobenius for nonnegative matrices are widely known.3,4 A model of abundance dynamics for biological system has form:

{ x t + 1 = A x t , t = 1 , 2 , ... x 1 = x ¯ 0 ,                               MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabiqaaaqaaiaadIhadaWgaaqcfasaaiaadshacqGHRaWkcaaI XaaabeaajuaGcqGH9aqpcaWGbbGaamiEamaaBaaajuaibaGaamiDaa qabaqcfaOaaiilaiaadshacqGH9aqpcaaIXaGaaiilaiaaikdacaGG SaGaaiOlaiaac6cacaGGUaaabaGaamiEamaaBaaajuaibaGaaGymaa qcfayabaGaeyypa0JabmiEayaaraGaeyyzImRaaGimaiaacYcaqaaa aaaaaaWdbiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiO aaaaa8aacaGL7baaaaa@6176@ (1)

Here А is a nonnegative irreducible matrix of order n[1, 2, 3], parameter t denotes a discrete time, vector x t R + n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadshaaeqaaKqbakabgIGiolaadkfadaqhaaqcfasa aiabgUcaRaqaaiaad6gaaaaaaa@3DCC@ describes an abundance for all groups of the system.

The next result follows for model (1) from Perron-Frobenius Theorem [3,5]: if h is the order of cyclicity of matrix A, then lim t 1 h τ = t t + h 1 X τ + 1 r τ = ( q x ¯ ) p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaxababa GaciiBaiaacMgacaGGTbaabaGaamiDaiabgkziUkabg6HiLcqabaWa aSaaaeaacaaIXaaabaGaamiAaaaadaaeWbqaamaalaaabaGaamiwam aaBaaabaqcfaIaeqiXdqxcfaOaey4kaSscfaIaaGymaaqcfayabaaa baGaamOCamaaCaaajuaibeqaaiabes8a0baaaaaajuaGbaGaeqiXdq Naeyypa0JaamiDaaqaaiaadshacqGHRaWkcaWGObGaeyOeI0IaaGym aaGaeyyeIuoacqGH9aqpdaqadaqaaiaadghacqGHflY1ceWG4bGbae baaiaawIcacaGLPaaacaWGWbaaaa@5B3F@ . The vector p is positive eigenvector for spectral radius ρ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aabmaabaGaamyqaaGaayjkaiaawMcaaaaa@3A88@ of matrix A, the vector q is positive eigenvector for spectral radius ρ ( A * ) = ρ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aabmaabaGaamyqamaaCaaabeqcfasaaiaacQcaaaaajuaGcaGLOaGa ayzkaaGaeyypa0JaeqyWdi3aaeWaaeaacaWGbbaacaGLOaGaayzkaa aaaa@4129@ of transposed matrix A * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeada ahaaqabKqbGeaacaGGQaaaaaaa@383D@ . These vectors are connected by condition q p = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghacq GHflY1caWGWbGaeyypa0JaaGymaaaa@3C6F@ . This result about asymptotic properties in matrix model is very important. We will compare these asymptotical properties of the abundance dynamics of biological systems with the dynamics under optimal harvesting.

We formulate a problem of the optimal harvest in matrix model (1):

{ t = 1 T w U t x t u t [ 0 ; 1 ] sup x t + 1 = A ( I U t ) x t                             x 1 = x ¯                                                                     MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabmqaaaqaamaaqahabaGaam4DaiaadwfadaWgaaqcfasaaiaa dshaaKqbagqaaiaadIhadaWgaaqcfasaaiaadshaaKqbagqaamaaoG cabeqaaiaadwhadaWgaaqcfasaaiaadshaaKqbagqaaiabgIGiopaa dmaabaGaaGimaiaacUdacaaIXaaacaGLBbGaayzxaaaacaGLsgcaci GGZbGaaiyDaiaacchaaeaacaWG0bGaeyypa0JaaGymaaqaaiaadsfa aiabggHiLdaabaGaamiEamaaBaaajuaibaGaamiDaiabgUcaRiaaig daaKqbagqaaiabg2da9iaadgeadaqadaqaaiaadMeacqGHsislcaWG vbWaaSbaaKqbGeaacaWG0baabeaaaKqbakaawIcacaGLPaaacaWG4b WaaSbaaKqbGeaacaWG0baajuaGbeaaqaaaaaaaaaWdbiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcaapaqaaiaadIhadaWgaaqcfasa aiaaigdaaeqaaKqbakabg2da9iqadIhagaqea8qacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcaaaa WdaiaawUhaaaaa@9B86@ (2)

Vector w characterizes the specific income for specific yield. Matrix U t = d i a g ( u t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadshaaeqaaKqbakabg2da9iaadsgacaWGPbGaamyy aiaadEgadaqadaqaaiaadwhadaWgaaqcfasaaiaadshaaeqaaaqcfa OaayjkaiaawMcaaaaa@4231@ contains the harvest vector on main diagonal, 0 U t I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaicdacq GHKjYOcaWGvbWaaSbaaKqbGeaacaWG0baabeaajuaGcqGHKjYOcaWG jbaaaa@3E1B@ . The time is measured in years. The planning time interval is T. All vectors are nonnegative. The abundance dynamics of biological system is researched in this work. The asymptotic properties for the problem of the optimal harvest have other forms than the properties for model (1).

Results

We study the problem (2) for nonnegative irreducible matrix A. The denotation

y t = ( I U t ) x t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMhada WgaaqcfasaaiaadshaaeqaaKqbakabg2da9maabmaabaGaamysaiab gkHiTiaadwfadaWgaaqcfasaaiaadshaaeqaaaqcfaOaayjkaiaawM caaiaadIhadaWgaaqcfasaaiaadshaaKqbagqaaaaa@431A@ (3) will be used. Then next equations are true:

x t + 1 = A y t , U t + 1 x t + 1 = x t + 1 y t + 1 = A y t y t + 1 , U 1 x 1 = x ¯ y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadshacqGHRaWkcaaIXaaajuaGbeaacqGH9aqpcaWG bbGaamyEamaaBaaajuaibaGaamiDaaqcfayabaGaaiilaiaadwfada WgaaqcfasaaiaadshacqGHRaWkcaaIXaaajuaGbeaacaWG4bWaaSba aKqbGeaacaWG0bGaey4kaSIaaGymaaqabaqcfaOaeyypa0JaamiEam aaBaaajuaibaGaamiDaiabgUcaRiaaigdaaeqaaKqbakabgkHiTiaa dMhadaWgaaqcfasaaiaadshacqGHRaWkcaaIXaaabeaajuaGcqGH9a qpcaWGbbGaamyEamaaBaaajuaibaGaamiDaaqcfayabaGaeyOeI0Ia amyEamaaBaaajuaibaGaamiDaiabgUcaRiaaigdaaKqbagqaaiaacY cacaWGvbWaaSbaaKqbGeaacaaIXaaabeaajuaGcaWG4bWaaSbaaKqb GeaacaaIXaaabeaajuaGcqGH9aqpceWG4bGbaebacqGHsislcaWG5b WaaSbaaKqbGeaacaaIXaaabeaaaaa@688F@ Optimization criterion is being transformed to the form: w ( ( x ¯ y 1 ) + t = 1 T 1 ( A y t y t + 1 ) ) sup MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEhada qadaqaamaabmaabaGabmiEayaaraGaeyOeI0IaamyEamaaBaaajuai baGaaGymaaqabaaajuaGcaGLOaGaayzkaaGaey4kaSYaaabCaeaada qadaqaaiaadgeacaWG5bWaaSbaaKqbGeaacaWG0baajuaGbeaacqGH sislcaWG5bWaaSbaaKqbGeaacaWG0bGaey4kaSIaaGymaaqabaaaju aGcaGLOaGaayzkaaaajuaibaGaamiDaiabg2da9iaaigdaaeaacaWG ubGaeyOeI0IaaGymaaqcfaOaeyyeIuoaaiaawIcacaGLPaaacqGHsg IRciGGZbGaaiyDaiaacchaaaa@5776@ . The equations in model (2) are being transformed to form: 0 y t x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaicdacq GHKjYOcaWG5bWaaSbaaKazfa4=baGaamiDaaqabaqcfaOaeyizImQa bmiEayaaraaaaa@4049@ and 0 y t + 1 A y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaicdacq GHKjYOcaWG5bWaaSbaaKazfa4=baGaamiDaiabgUcaRiaaigdaaeqa aKqbakabgsMiJkaadgeacaWG5bWaaSbaaKqbGeaacaWG0baabeaaaa a@43DD@ for t = 1 , ... , T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbGiaadshacq GH9aqpcaaIXaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGubGa eyOeI0IaaGymaaaa@3ECA@ .

Lemma 1: The problem (2) is equivalent to a next problem:

{ Φ = t = 1 T 1 w [ A I ] y t w y T sup 0 y t + 1   A y t   , t = 1 , ... , T 1                   0 y t x ¯                                                                     MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabmqaaaqaaiabfA6agjabg2da9maaqahabaGaam4Damaadmaa baGaamyqaiabgkHiTiaadMeaaiaawUfacaGLDbaacaWG5bWaaSbaaK qbGeaacaWG0baabeaajuaGcqGHsislcaWG3bGaamyEamaaBaaajqwb a9FaaiaadsfaaKqbGeqaaKqbakabgkziUkGacohacaGG1bGaaiiCaa qcfasaaiaadshacqGH9aqpcaaIXaaabaGaamivaiabgkHiTiaaigda aKqbakabggHiLdaabaGaaGimaiabgsMiJkaadMhadaWgaaqcfasaai aadshacqGHRaWkcaaIXaaabeaajuaGqaaaaaaaaaWdbiaacckacqGH KjYOcaGGbbWdaiaadMhadaWgaaqcfasaaiaadshaaeqaaKqba+qaca GGGcGaaiilaiaacshacqGH9aqpcaaIXaGaaiilaiaac6cacaGGUaGa aiOlaiaacYcacaGGubGaeyOeI0IaaGymaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaaWdaeaacaaIWaGa eyizImQaamyEamaaBaaajuaibaGaamiDaaqabaqcfaOaeyizImQabm iEayaaraWdbiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckaaaaapaGaay5Eaaaaaa@A702@ . (4)

Proof: Every possible solution for problem (2) is transformed to possible solution for problem (4) by formula (3). Inverse formulas have forms:

x 1 = x ¯ , x t + 1 = A y t , u t j = { x t j y f o r x t j > 0 e v e r y   v a l u e   f r o m   [ 0 , 1 ] f o r x t j = 0 , t = 1 , ... T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaigdaaeqaaKqbakabg2da9iqadIhagaqeaiaacYca caWG4bWaaSbaaKqbGeaacaWG0bGaey4kaSIaaGymaaqabaqcfaOaey ypa0JaamyqaiaadMhadaWgaaqcfasaaiaadshaaeqaaKqbakaacYca caGG1bWaaSbaaKqbGeaacaWG0bGaamOAaaqabaqcfaOaeyypa0Zaai qaaeaafaqabeGabaaabaGaamiEamaaBaaabaGaamiDaiaadQgaaeqa aiabgkHiTiaadMhadaWgaaqaaiaadAgacaWGVbGaamOCaiaadIhada WgaaqaaiaadshacaWGQbaabeaacqGH+aGpcaaIWaaabeaaaeaacaWG LbGaamODaiaadwgacaWGYbGaamyEaabaaaaaaaaapeGaaiiOa8aaca WG2bGaamyyaiaadYgacaWG1bGaamyza8qacaGGGcWdaiaadAgacaWG YbGaam4Baiaad2gapeGaaiiOa8aadaWadaqaaiaaicdacaGGSaGaaG ymaaGaay5waiaaw2faaiaadAgacaWGVbGaamOCaiaadIhadaWgaaqa aiaadshacaWGQbaabeaaaaaacaGL7baacqGH9aqpcaaIWaGaaiilai aadshacqGH9aqpcaaIXaGaaiilaiaac6cacaGGUaGaaiOlaiaacsfa cqGHsislcaaIXaaaaa@7DB2@ (5)

The optimal solution for problem (2) corresponds to a optimal solution for problem (4).

The following denotation will be used: ψ = w ( A I ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5j abg2da9iaadEhadaqadaqaaiaadgeacqGHsislcaWGjbaacaGLOaGa ayzkaaaaaa@3E53@ .

Lemma 2: The optimal solution for the problem (4) has form:

{ y ^ 1 d i a g ( s g ψ ) x ¯ y ^ t + 1 = d i a g ( s g ψ t + 1 ) A y ^ t     f o r   t = 1 , 2 , .... , T 2 y ^ T = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabmqaaaqaaiqadMhagaqcamaaBaaajqwba+Faaiaaigdaaeqa aKqbakaadsgacaWGPbGaamyyaiaadEgadaqadaqaaiaadohacaWGNb GaeqiYdKhacaGLOaGaayzkaaGabmiEayaaraaabaGabmyEayaajaWa aSbaaKqbGeaacaWG0bGaey4kaSIaaGymaaqcfayabaGaeyypa0Jaam izaiaadMgacaWGHbGaam4zamaabmaabaGaam4CaiaadEgacqaHipqE daWgaaqcfasaaiaadshacqGHRaWkcaaIXaaajuaGbeaaaiaawIcaca GLPaaacaWGbbGabmyEayaajaWaaSbaaKqbGeaacaWG0baabeaajuaG qaaaaaaaaaWdbiaacckacaGGGcGaamOzaiaad+gacaWGYbGaaiiOai aadshacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6ca caGGUaGaaiOlaiaacYcacaWGubGaeyOeI0IaaGOmaaWdaeaaceWG5b GbaKaadaWgaaqcfasaaiaadsfaaKqbagqaaiabg2da9iaaicdaaaaa caGL7baaaaa@7151@ (6)

We use here the next denotations:

{ φ T 1 = ψ φ t = ψ + φ t + 1 *     f o r     t = 1 , 2 , ... , T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabiqaaaqaaiabeA8aQnaaBaaajuaibaGaamivaiabgkHiTiaa igdaaKqbagqaaiabg2da9iabeI8a5bqaaiabeA8aQnaaBaaajuaiba GaamiDaaqabaqcfaOaeyypa0JaeqiYdKNaey4kaSIaeqOXdO2aa0ba aKqbGeaacaWG0bGaey4kaSIaaGymaaqaaiaacQcaaaqcfaieaaaaaa aaa8qacaGGGcGaaiiOaiaadAgacaWGVbGaamOCaiaacckacaGGGcGa amiDaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlai aac6cacaGGSaGaaiivaiabgkHiTiaaikdaaaaapaGaay5Eaaaaaa@5E69@ (7)

a + = max { a , 0 } , s g a = ( 0 f o r   a < 0 1   f o r   a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada ahaaqabKqbGeaacqGHRaWkaaqcfaOaeyypa0JaciyBaiaacggacaGG 4bWaaiWaaeaacaWGHbGaaiilaiaaicdaaiaawUhacaGL9baacaGGSa Gaam4CaiaadEgacaWGHbGaeyypa0deaaaaaaaaa8qacaGGOaqbaeqa biqaaaqaaiaaicdacaWGMbGaam4BaiaadkhacaGGGcGaamyyaiabgY da8iaaicdaaeaacaaIXaGaaiiOaiaadAgacaWGVbGaamOCaiaaccka caWGHbGaeyyzImRaaGimaaaaaaa@576A@ . (8)

Proof: The equation y ^ T = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyEayaajaWaaSbaaKqbGeaacaWGubaajuaGbeaacqGH9aqp caaIWaaaaa@3B1D@ follows from no negativity of vector w (symbol “^” corresponds to some optimal solution). The denotation Φ t = τ = 1 t ψ · y τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agn aaBaaajuaibaGaamiDaaqabaqcfaOaeyypa0ZaaabCaeaacqaHipqE aeaacqaHepaDcqGH9aqpcaaIXaaabaGaamiDaaGaeyyeIuoacqWIpM +zcaWG5bWaaSbaaKqbGeaacqaHepaDaeqaaaaa@48C8@ is being used. The equation Φ   =   Φ T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agb baaaaaaaaapeGaaiiOa8aacqGH9aqppeGaaiiOa8aacqqHMoGrdaWg aaqcfasaaiaadsfacqGHsislcaaIXaaajuaGbeaaaaa@4067@ is true for optimization functional. We will find the optimal value for variable y with help of induction by decrease of variable t.

Step (T-1). We are getting formulas: Φ   =   Φ T 1 = Φ T 2 + ψ y T 1 = Φ T 2 + φ T 1 y T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agb baaaaaaaaapeGaaiiOa8aacqGH9aqppeGaaiiOa8aacqqHMoGrdaWg aaqcfasaaiaadsfacqGHsislcaaIXaaajuaGbeaacqGH9aqpcqqHMo GrdaWgaaqcfasaaiaadsfacqGHsislcaaIYaaajuaGbeaacqGHRaWk cqaHipqEcqGHflY1caWG5bWaaSbaaKqbGeaacaWGubGaeyOeI0IaaG ymaaqcfayabaGaeyypa0JaeuOPdy0aaSbaaKqbGeaacaWGubGaeyOe I0IaaGOmaaqcfayabaGaey4kaSIaeqOXdO2aaSbaaKqbGeaacaWGub GaeyOeI0IaaGymaaqcfayabaGaamyEamaaBaaajuaibaGaamivaiab gkHiTiaaigdaaKqbagqaaaaa@5FD4@ for φ T 1 = ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaaiabg2da9iab eI8a5baa@3E68@ The support task { φ T 1 y T 1 sup 0 y T 1 A y T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabiqaaaqaaiabeA8aQnaaBaaajuaibaGaamivaiabgkHiTiaa igdaaKqbagqaaiaadMhadaWgaaqcfasaaiaadsfacqGHsislcaaIXa aajuaGbeaacqGHsgIRciGGZbGaaiyDaiaacchaaeaacaaIWaGaeyiz ImQaamyEamaaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaai abgsMiJkaadgeacaWG5bWaaSbaaKqbGeaacaWGubGaeyOeI0IaaGOm aaqcfayabaaaaaGaay5Eaaaaaa@538D@ has the solution in form: y ^ T 1 = d i a g ( s g φ T 1 ) A y T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadMhaga qcamaaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaaiabg2da 9iaadsgacaWGPbGaamyyaiaadEgadaqadaqaaiaadohacaWGNbGaeq OXdO2aaSbaaKqbGeaacaWGubGaeyOeI0IaaGymaaqabaaajuaGcaGL OaGaayzkaaGaamyqaiaadMhadaWgaaqcfasaaiaadsfacqGHsislca aIYaaajuaGbeaaaaa@4D3F@ . The general form for this solution is y ^ T 1 = y ^ T 1 ( y T 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadMhaga qcamaaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaaiabg2da 9iqadMhagaqcamaaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbag qaamaabmaabaGaamyEamaaBaaajuaibaGaamivaiabgkHiTiaaikda aKqbagqaaaGaayjkaiaawMcaaaaa@463D@ . Next formula φ T 1 y ^ T 1 = φ T 1 y ^ T 1 ( y T 2 ) = φ T 1 + A y T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaaiqadMhagaqc amaaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaaiabg2da9i abeA8aQnaaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaaiqa dMhagaqcamaaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaam aabmaabaGaamyEamaaBaaajuaibaGaamivaiabgkHiTiaaikdaaKqb agqaaaGaayjkaiaawMcaaiabg2da9iabeA8aQnaaDaaajuaibaGaam ivaiabgkHiTiaaigdaaeaacqGHRaWkaaqcfaOaamyqaiaadMhadaWg aaqcfasaaiaadsfacqGHsislcaaIYaaajuaGbeaaaaa@5C9A@ is being calculated. The formula Φ = Φ T 2 + φ T 1 + A y T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agj abg2da9iabfA6agnaaBaaajuaibaGaamivaiabgkHiTiaaikdaaeqa aiabgUcaRKqbakabeA8aQnaaDaaajuaibaGaamivaiabgkHiTiaaig daaeaacqGHRaWkaaqcfaOaamyqaiaadMhadaWgaaqcfasaaiaadsfa cqGHsislcaaIYaaajuaGbeaaaaa@49D5@ is true.

Step t. Let formula Φ = Φ T + φ t + 1 + A y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agj abg2da9iabfA6agnaaBaaajuaibaGaamivaaqabaGaey4kaSscfaOa eqOXdO2aa0baaKqbGeaacaWG0bGaey4kaSIaaGymaaqaaiabgUcaRa aajuaGcaWGbbGaamyEamaaBaaajuaibaGaamiDaaqcfayabaaaaa@46B8@ be true. Then Φ = Φ T + ψ y t + φ t + 1 + A y t = Φ t 1 + φ t y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agj abg2da9iabfA6agnaaBaaajuaibaGaamivaaqabaqcfaOaey4kaSIa eqiYdKNaamyEamaaBaaajuaibaGaamiDaaqabaGaey4kaSscfaOaeq OXdO2aa0baaKqbGeaacaWG0bGaey4kaSIaaGymaaqaaiabgUcaRaaa juaGcaWGbbGaamyEamaaBaaajuaibaGaamiDaaqcfayabaGaeyypa0 JaeuOPdy0aaSbaaKqbGeaacaWG0bGaeyOeI0IaaGymaaqcfayabaGa ey4kaSIaeqOXdO2aaSbaaKqbGeaacaWG0baabeaajuaGcaWG5bWaaS baaKqbGeaacaWG0baabeaaaaa@58F5@ is true for φ t = ψ + φ t + 1 + A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aaBaaajuaibaGaamiDaaqabaqcfaOaeyypa0JaeqiYdKNaey4kaSIa eqOXdO2aa0baaKqbGeaacaWG0bGaey4kaSIaaGymaaqaaiabgUcaRa aajuaGcaWGbbaaaa@449B@ The solution of the subproblem { φ t y t sup 0 y t A y t 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabiqaaaqaaiabeA8aQnaaBaaajuaibaGaamiDaaqabaqcfaOa amyEamaaBaaajuaibaGaamiDaaqabaqcfaOaeyOKH4Qaci4Caiaacw hacaGGWbaabaGaaGimaiabgsMiJkaadMhadaWgaaqcfasaaiaadsha aKqbagqaaiabgsMiJkaadgeacaWG5bWaaSbaaKqbGeaacaWG0bGaey OeI0IaaGymaaqcfayabaaaaaGaay5Eaaaaaa@4F14@ is forming y ^ t = d i a g ( s g φ t ) A y t 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadMhaga qcamaaBaaajuaibaGaamiDaaqabaqcfaOaeyypa0JaamizaiaadMga caWGHbGaam4zamaabmaabaGaam4CaiaadEgacqaHgpGAdaWgaaqcfa saaiaadshaaeqaaaqcfaOaayjkaiaawMcaaiaadgeacaWG5bWaaSba aKqbGeaacaWG0bGaeyOeI0IaaGymaaqcfayabaaaaa@4A4E@ The formula Φ = Φ t 1 + φ t + A y t 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agj abg2da9iabfA6agnaaBaaajuaibaGaamiDaiabgkHiTiaaigdaaeqa aiabgUcaRKqbakabeA8aQnaaDaaajuaibaGaamiDaaqaaiabgUcaRa aajuaGcaWGbbGaamyEamaaBaaajuaibaGaamiDaiabgkHiTiaaigda aKqbagqaaaaa@488B@ follows from previous formulas.

The lemma 2 is proved by induction by decrease of variable t. Corollary. The optimal solution in problem (2) exists and has form:

{ x ^ 1 = x ¯                                                                                             x ^ t + 1 = A ( d i a g ( s g φ t ) ) x ^ t , t = 1 , 2 , ... , T 1 u ^ T = e                                                                                             u ^ t = s g ¯ φ t                                                                                   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabqqaaaaabaGabmiEayaajaWaaSbaaKazfa0=baGaaGymaaqa baqcfaOaeyypa0JabmiEayaaraaeaaaaaaaaa8qacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcaapaqaaiqadIhagaqcamaaBaaajuaiba GaamiDaiabgUcaRiaaigdaaeqaaiabg2da9iaadgeajuaGdaqadaqa aiaadsgacaWGPbGaamyyaiaadEgadaqadaqaaiaadohacaWGNbGaeq OXdO2aaSbaaKqbGeaacaWG0baabeaaaKqbakaawIcacaGLPaaaaiaa wIcacaGLPaaaceWG4bGbaKaadaWgaaqcfasaaiaadshaaeqaaKqbak aacYcacaWG0bGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6ca caGGUaGaaiOlaiaacYcacaWGubGaeyOeI0IaaGymaaqaaiqadwhaga qcamaaBaaajuaibaGaamivaaqabaqcfaOaeyypa0Jaamyza8qacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcaapaqaaiqadwhagaqcamaa BaaajuaibaGaamiDaaqabaqcfaOaeyypa0Zaa0aaaeaacaWGZbGaam 4zaaaacqaHgpGAdaWgaaqcfasaaiaadshaaeqaaKqba+qacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaaaaa8 aacaGL7baaaaa@0454@ .

The denotations s g ¯ a = { 1 , a < 0 0 , a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaaba Gaam4CaiaadEgaaaGaamyyaiabg2da9maaceaabaqbaeqabiqaaaqa aiaaigdacaGGSaGaamyyaiabgYda8iaaicdaaeaacaaIWaGaaiilai aadggacqGHLjYScaaIWaaaaaGaay5Eaaaaaa@4460@ , e = ( 1 , 1 , ... , 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwgacq GH9aqpdaqadaqaaiaaigdacaGGSaGaaGymaiaacYcacaGGUaGaaiOl aiaac6cacaGGSaGaaGymaaGaayjkaiaawMcaaaaa@4049@ are used. Proof is based on use of formulas (5).

Theorem. The solution of the problem (2) with use formulas (3) has the time points tjand sets I j { 1 , 2 , ... , n } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaadQgaaeqaaiabgkOimNqbaoaacmaabaGaaGymaiaa cYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGUbaaca GL7bGaayzFaaaaaa@43D0@ of indexes for j=1,…,k by some k n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacq GHKjYOcaWGUbaaaa@3A11@ . These points and sets have next properties: 1 = t 1 < t 1 < ... < t k = T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GH9aqpcaWG0bWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGH8aapcaWG 0bWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGH8aapcaGGUaGaaiOlai aac6cacqGH8aapcaWG0bWaaSbaaKqbGeaacaWGRbaajuaGbeaacqGH 9aqpcaWGubaaaa@4723@ and { y ^ t i = 0 f o r   i I j                                                 y ^ t i > 0 f o r   i I j   a n d   t ( t j ; t j + 1 )   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaba qbaeqabiqaaaqaaiqadMhagaqcamaaBaaajuaibaGaamiDaiaadMga aeqaaKqbakabg2da9iaaicdacaWGMbGaam4Baiaadkhaqaaaaaaaaa WdbiaacckapaGaamyAaiabgIGiolaadMeadaWgaaqcfasaaiaadQga aeqaaKqba+qacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckaa8aabaGabmyEayaajaWaaSbaaKqbGeaacaWG0bGaamyA aaqabaqcfaOaeyOpa4JaaGimaiaadAgacaWGVbGaamOCa8qacaGGGc WdaiaadMgacqGHjiYZcaWGjbWaaSbaaKqbGeaacaWGQbaabeaajuaG peGaaiiOa8aacaWGHbGaamOBaiaadsgapeGaaiiOa8aacaWG0bGaey icI48aaeWaaeaacaWG0bWaaSbaaKqbGeaacaWGQbaabeaajuaGcaGG 7aGaamiDamaaBaaajuaibaGaamOAaiabgUcaRiaaigdaaeqaaaqcfa OaayjkaiaawMcaa8qacaGGGcaaaaWdaiaawUhaaaaa@8318@ , I j I j + 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaadQgaaKqbagqaaiabgkOimlaadMeadaWgaaqcfasa aiaadQgacqGHRaWkcaaIXaaabeaaaaa@3EB8@ .

Proof. The property φ T 1 φ T 2 = ψ ( ψ + φ T 1 + A ) = φ T 1 + A 0 φ T 1 φ T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aaBaaajuaibaGaamivaiabgkHiTiaaigdaaeqaaiabgkHiTKqbakab eA8aQnaaBaaajuaibaGaamivaiabgkHiTiaaikdaaeqaaKqbakabg2 da9iabeI8a5jabgkHiTmaabmaabaGaeqiYdKNaey4kaSIaeqOXdO2a a0baaKqbGeaacaWGubGaeyOeI0IaaGymaaqaaiabgUcaRaaajuaGca WGbbaacaGLOaGaayzkaaGaeyypa0JaeyOeI0IaeqOXdO2aa0baaKqb GeaacaWGubGaeyOeI0IaaGymaaqaaiabgUcaRaaajuaGcaWGbbGaey izImQaaGimaiabgkDiElabeA8aQnaaBaaajuaibaGaamivaiabgkHi TiaaigdaaeqaaiabgsMiJMqbakabeA8aQnaaBaaajuaibaGaamivai abgkHiTiaaikdaaeqaaaaa@693C@ follows from formulas (7). We have φ t φ t 1 = ( φ t + 1 + φ t + ) A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aaBaaajuaibaGaamiDaaqabaqcfaOaeyOeI0IaeqOXdO2aaSbaaKqb GeaacaWG0bGaeyOeI0IaaGymaaqabaqcfaOaeyypa0ZaaeWaaeaacq aHgpGAdaqhaaqcfasaaiaadshacqGHRaWkcaaIXaaabaGaey4kaSca aKqbakabgkHiTiabeA8aQnaaDaaajuaibaGaamiDaaqaaiabgUcaRa aaaKqbakaawIcacaGLPaaacaWGbbaaaa@4EFF@ for t = 2 , 3 , ... , T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshacq GH9aqpcaaIYaGaaiilaiaaiodacaGGSaGaaiOlaiaac6cacaGGUaGa aiilaiaadsfacqGHsislcaaIYaaaaa@4099@ . The inequalities

φ T 1 φ T 2 φ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeA8aQn aaBaaajuaibaGaamivaiabgkHiTiaaigdaaKqbagqaaiabgsMiJkab eA8aQnaaBaaajuaibaGaamivaiabgkHiTiaaikdaaKqbagqaaiabgs MiJkabeA8aQnaaBaaajuaibaGaaGymaaqabaaaaa@46E1@ (8)

Follow from previous formula.

The results of the theorem follow from the lemma 2 and formula (8). The meaning of this theorem is in the turnpike phenomenon6 for the solution of problem (2). This phenomenon consists in stabilization of the indexes of the system components which are taken in harvest. This stabilization is being discussed in next part.

Discussion

The variable yti denote the abundance of component i for time moment t after harvesting. The theorem means that the solution for optimal harvesting problem (2) has the index sets I j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaadQgaaeqaaaaa@3885@ of zero components after harvesting. These sets have the inclusion properties I j I j + 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaadQgaaKqbagqaaiabgkOimlaadMeadaWgaaqcfasa aiaadQgacqGHRaWkcaaIXaaabeaaaaa@3EB8@ . This situation is analogous for turnpike phenomenon in models of economic dynamics.6 But the turnpike properties are specific in this case. These turnpike properties are true for indexes of harvesting components: if the component equals zero after harvest in some time point then this component equal zero in any next time points. The inclusion I j I j + 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaadQgaaKqbagqaaiabgkOimlaadMeadaWgaaqcfasa aiaadQgacqGHRaWkcaaIXaaabeaaaaa@3EB8@ means that the new Biosystem components are being included in harvest with increase of time. For example this fact means a decrease of average age of individuals in population with age structure.

The first index set I1 denotes the lightest regime of harvesting. The time moments tj for j 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GHLjYScaaIYaaaaa@39EA@ go to infinity if interval for harvesting went to infinity T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfacq GHsgIRcqGHEisPaaa@3AB0@ . The lightest regime of harvesting dominates in this case. The long time planning interval is preferable to efficient use of biological resources.

Acknowledgments

My research project was partially or fully sponsored by Program of fundamental scientific research “Far Eastern” of Far Eastern Branch of the Russian Academy of Sciences (years 2015-2017) with grant number 15-I-4-006_o.

Conflict of interest

None.

References

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