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International Robotics & Automation Journal

Mini Review Volume 3 Issue 3

The case of optimal control with exceptional role of the conditions of transversality

MV Levskii

Khrunichev State scientific & production Space Center, Russia

Correspondence: MV Levskii, Khrunichev State scientific & production Space Center, Russia, Tel (499) 277-37-16

Received: September 25, 2017 | Published: October 17, 2017

Citation: Levskii MV. The case of optimal control with exceptional role of the conditions of transversality. Int Rob Auto J. 2019;3(3):297-298. DOI: 10.15406/iratj.2017.03.00055

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Abstract

This research considered solving the optimal control problem of spacecraft (as solid body) when the conditions of transversality have key significance. It is shown that the assumed criterion of optimality guarantees motion of spacecraft with energy not exceeding the required value. Topicality of article is caused by fact that on concrete example, the conditions of transversality are demonstrated to be very important mathematical instrument (even the only) for finding the main properties, laws and key characteristics (parameters, constants, integrals of motion) of optimal solution of control problem.

Keywords: optimal control, criterion of optimality, maximum principle, conditions of transversality, controlling functions

Introduction

Its investigate motion of spacecraft (as solid body) relative to centre of mass was investigated in this research work. Spacecraft attitude is described by the quaternion Λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amb aa@37EE@  (it give position of the body axes relative to inertial coordinate system) and the vector ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeM8a3b aa@3846@  of absolute angular velocity. Then, equation of motion1

2 Λ ˙ =Λω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdacu qHBoatgaGaaiabg2da9iabfU5amjablIHiVjabeM8a3baa@3E35@             (1)

(It is assumed that Λ( 0 ) =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba Gaeu4MdW0aaeWaaeaacaaIWaaacaGLOaGaayzkaaaacaGLjWUaayPc SdGaeyypa0JaaGymaaaa@3F19@ ). In order to estimate the efficiency of control, the functional to be optimized is introduced as presented in equation 2.

G= 0 T ( a 1 ( J 1 ω 1 2 + J 2 ω 2 2 + J 3 ω 3 2 )+ a 2 ) dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeacq GH9aqpdaWdXbqaamaabmaabaGaamyyamaaBaaajuaibaGaaGymaaqa baqcfa4aaeWaaeaacaWGkbWaaSbaaKqbGeaacaaIXaaabeaajuaGcq aHjpWDdaqhaaqcfasaaiaaigdaaeaacaaIYaaaaKqbakabgUcaRiaa dQeadaWgaaqcfasaaiaaikdaaKqbagqaaiabeM8a3naaDaaajuaiba GaaGOmaaqaaiaaikdaaaqcfaOaey4kaSIaamOsamaaBaaajuaibaGa aG4maaqabaqcfaOaeqyYdC3aa0baaKqbGeaacaaIZaaabaGaaGOmaa aaaKqbakaawIcacaGLPaaacqGHRaWkcaWGHbWaaSbaaKqbGeaacaaI YaaajuaGbeaaaiaawIcacaGLPaaaaeaacaaIWaaabaGaamivaaGaey 4kIipacaWGKbGaamiDaaaa@5C67@    (2)

Where J i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQeada WgaaqcfasaaiaadMgaaeqaaaaa@3885@  are the spacecraft central principal moment of inertia; wi are the components of vector ω( i= 1,3 ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeM8a3n aabmaabaGaamyAaiabg2da9maanaaabaGaaGymaiaacYcacaaMe8Ua aG4maaaaaiaawIcacaGLPaaaaaa@3F89@ ; a 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada Wgaaqcfasaaiaaigdaaeqaaaaa@3869@ =const>0, a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38F8@ =const>0. Let solve following problem of control: take the spacecraft from initial attitude Λ in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aaBaaajuaibaGaamyAaiaad6gaaeqaaaaa@3A1E@ into final attitude Λ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aaBaaajuaibaGaamOzaaqabaaaaa@3928@  obeying Eq. (1) so as to minimize integral (2) (the time T is not given). The taken criterion of optimality combines (in given proportion) the time and integral of energy to be expended for slew maneuver. Aspects of finding economical control are topical now.

For solving the formulated problem, use Pontryagin’s maximum principle2 and the universal variables3 (because the optimized functional does not include positional coordinates). The assumed integral (2) does not include the torques M i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytamaaBaaajuaibaGaamyAaaqabaaaaa@38A8@ ; the sought-for function ω( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeM8a3n aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@3AC8@ is piecewise continuous function of time. For our optimization problem, the Hamiltonian is

H= r 1 ω 1 + r 2 ω 2 + r 3 ω 3 a 1 ( J 1 ω 1 2 + J 2 ω 2 2 + J 3 ω 3 2 ) a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeacq GH9aqpcaWGYbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqaHjpWDdaWg aaqcfasaaiaaigdaaeqaaKqbakabgUcaRiaadkhadaWgaaqcfasaai aaikdaaeqaaKqbakabeM8a3naaBaaajuaibaGaaGOmaaqcfayabaGa ey4kaSIaamOCamaaBaaajuaibaGaaG4maaqabaqcfaOaeqyYdC3aaS baaKqbGeaacaaIZaaajuaGbeaacqGHsislcaWGHbWaaSbaaKqbGeaa caaIXaaabeaajuaGdaqadaqaaiaadQeadaWgaaqcfasaaiaaigdaae qaaKqbakabeM8a3naaDaaajuaibaGaaGymaaqaaiaaikdaaaqcfaOa ey4kaSIaamOsamaaBaaajuaibaGaaGOmaaqcfayabaGaeqyYdC3aa0 baaKqbGeaacaaIYaaabaGaaGOmaaaajuaGcqGHRaWkcaWGkbWaaSba aKqbGeaacaaIZaaabeaajuaGcqaHjpWDdaqhaaqcfasaaiaaiodaae aacaaIYaaaaaqcfaOaayjkaiaawMcaaiabgkHiTiaadggadaWgaaqc fasaaiaaikdaaeqaaaaa@6929@ ,

Where r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOCamaaBaaajuaibaGaamyAaaqabaaaaa@38CD@ are universal variables (as the components of vector r) satisfying equations3

r ˙ 1 =  ω 3   r 2   ω 2  r 3 ,  r ˙ 2 =  ω 1   r 3   ω 3  r 1 ,  r ˙ 3 =  ω 2   r 1   ω 1  r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkhaga GaamaaBaaajuaibaGaaGymaaqabaqcfaOaeyypa0deaaaaaaaaa8qa caGGGcWdaiabeM8a3naaBaaajuaibaGaaG4maaqabaqcfa4dbiaacc kapaGaamOCamaaBaaajuaibaGaaGOmaaqcfayabaGaeyOeI0Ydbiaa cckapaGaeqyYdC3aaSbaaKqbGeaacaaIYaWdbiaacckaaKqba+aabe aacaWGYbWaaSbaaKqbGeaacaaIZaaajuaGbeaacaGGSaWdbiaaccka paGabmOCayaacaWaaSbaaKqbGeaacaaIYaaabeaajuaGcqGH9aqppe GaaiiOa8aacqaHjpWDdaWgaaqcfasaaiaaigdaaeqaaKqba+qacaGG GcWdaiaadkhadaWgaaqcfasaaiaaiodaaKqbagqaaiabgkHiT8qaca GGGcWdaiabeM8a3naaBaaajuaibaWdbiaaiodacaGGGcaajuaGpaqa baGaamOCamaaBaaajuaibaGaaGymaaqcfayabaGaaiila8qacaGGGc WdaiqadkhagaGaamaaBaaajuaibaGaaG4maaqabaqcfaOaeyypa0Zd biaacckapaGaeqyYdC3aaSbaaKqbGeaacaaIYaaabeaajuaGpeGaai iOa8aacaWGYbWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGHsislpeGa aiiOa8aacqaHjpWDdaWgaaqcfasaa8qacaaIXaGaaiiOaaqcfa4dae qaaiaadkhadaWgaaqcfasaaiaaikdaaKqbagqaaaaa@7AFB@             (3)

The Hamiltonian Н is written, ignoring the constraint Λ =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba Gaeu4MdWeacaGLjWUaayPcSdGaeyypa0JaaGymaaaa@3CD6@  since Λ( t ) =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba Gaeu4MdW0aaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLjWUaayPc SdGaeyypa0JaaGymaaaa@3F58@ under any ω( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyYdC3aaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@3AE8@  for Eq. (1) (of course Λ in = Λ f =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba Gaeu4MdW0aaSbaaeaajuaicaWGPbGaamOBaaqcfayabaaacaGLjWUa ayPcSdGaeyypa0ZaauWaaeaacqqHBoatdaWgaaqcfasaaiaadAgaaK qbagqaaaGaayzcSlaawQa7aiabg2da9iaaigdaaaa@46FE@ ). Optimal function r(t) is computed by the quaternion Λ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@3A70@ :1,3

r= Λ ˜ CE Λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhacq GH9aqpcuqHBoatgaacaiablIHiVnaaBaaajuaibaGaam4qaiaadwea aKqbagqaaiablIHiVjabfU5ambaa@4052@ , where CE=Const= Λ in r( 0 ) Λ ˜ in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoeaca WGfbGaeyypa0Jaam4qaiaad+gacaWGUbGaam4CaiaadshacqGH9aqp cqqHBoatdaWgaaqaaKqbGiaadMgacaWGUbaajuaGbeaacqWIyiYBca WGYbWaaeWaaeaacaaIWaaacaGLOaGaayzkaaGaeSigI8Mafu4MdWKb aGaadaWgaaqcfasaaiaadMgacaWGUbaabeaaaaa@4C4C@

For the vector r of universal variables | r | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GaamOCaaGaay5bSlaawIa7aaaa@3A92@  = const ¹ 0. The function Н is maximal if the relations

ω i = r i / 2 a 1 J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeM8a3n aaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0ZaaSGbaeaacaWGYbWa aSbaaKqbGeaacaWGPbaabeaaaKqbagaacaaIYaGaamyyamaaBaaaju aibaGaaGymaaqcfayabaGaamOsaaaaaaa@41F8@            (4)

are satisfied. As is known, the functions r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhada WgaaqcfasaaiaadMgaaeqaaaaa@38AD@  and ω i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeM8a3n aaBaaajuaibaGaamyAaaqcfayabaaaaa@3A11@  should satisfy the conditions of transversality which are r(0) ¹ 0, r(T)¹0 (since left and right endpoints of the trajectory Λ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@3A70@  are fixed) and H=0 because the maneuver end time T is not fixed and the Hamiltonian Н is independent of time in explicit form. After substitution Eq.(4) in expression for H and the requirement H = 0, obtain the equation

( r 1 2 / J 1 + r 2 2 / J 2 + r 3 2 / J 3 )/ 4 a 1 a 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalyaaba WaaeWaaeaadaWcgaqaaiaadkhadaqhaaqcfasaaiaaigdaaeaacaaI YaaaaaqcfayaaiaadQeadaWgaaqcfasaaiaaigdaaeqaaaaajuaGcq GHRaWkdaWcgaqaaiaadkhadaqhaaqcfasaaiaaikdaaeaacaaIYaaa aaqcfayaaiaadQeadaWgaaqcfasaaiaaikdaaeqaaaaajuaGcqGHRa WkdaWcgaqaaiaadkhadaqhaaqcfasaaiaaiodaaeaacaaIYaaaaaqc fayaaiaadQeadaWgaaqcfasaaiaaiodaaeqaaaaaaKqbakaawIcaca GLPaaaaeaacaaI0aGaamyyamaaBaaajuaibaGaaGymaaqabaqcfaOa eyOeI0IaamyyamaaBaaajuaibaGaaGOmaaqabaqcfaOaeyypa0JaaG imaaaaaaa@53A5@

Through which have following key properties of the controlled motion:

r 1 2 / J 1 + r 2 2 / J 2 + r 3 2 / J 3 =const=4 a 1 a 2 ,  J 1 ω 1 2 + J 2 ω 2 2  + J 3 ω 3 2 =const, J 1 2 ω 1 2 + J 2 2 ω 2 2 + J 3 2 ω 3 2 =const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aaS GbaeaacaWGYbWaa0baaKqbGeaacaaIXaaabaGaaGOmaaaaaKqbagaa caWGkbWaaSbaaKqbGeaacaaIXaaabeaaaaqcfaOaey4kaSYaaSGbae aacaWGYbWaa0baaKqbGeaacaaIYaaabaGaaGOmaaaaaKqbagaacaWG kbWaaSbaaKqbGeaacaaIYaaabeaaaaqcfaOaey4kaSYaaSGbaeaaca WGYbWaa0baaKqbGeaacaaIZaaabaGaaGOmaaaaaKqbagaacaWGkbWa aSbaaKqbGeaacaaIZaaabeaaaaqcfaOaeyypa0Jaam4yaiaad+gaca WGUbGaam4CaiaadshacqGH9aqpcaaI0aGaamyyamaaBaaajuaibaGa aGymaaqabaqcfaOaamyyamaaBaaajuaibaGaaGOmaaqabaGaaiilaK qbacbaaaaaaaaapeGaaiiOaaGcbaqcfaOaaiOsamaaBaaajuaibaGa aGymaaqabaqcfaOaeqyYdC3aa0baaKqbGeaacaaIXaaabaGaaGOmaa aajuaGcqGHRaWkcaGGkbWaaSbaaKqbGeaacaaIYaaabeaajuaGcqaH jpWDdaqhaaqcfasaaiaaikdaaeaacaaIYaaaaKqbakaacckacqGHRa WkcaGGkbWaaSbaaKqbGeaacaaIZaaabeaajuaGcqaHjpWDdaqhaaqc fasaaiaaiodaaeaacaaIYaaaaKqbakabg2da9iaadogacaWGVbGaam OBaiaadohacaWG0bGaaiilaaGcbaqcfaOaamOsamaaBaaajuaibaGa aGymaaqabaqcfa4aaWbaaKqbGeqabaGaaGOmaaaajuaGcqaHjpWDda qhaaqcfasaaiaaigdaaeaacaaIYaaaaKqbakabgUcaRiaadQeadaWg aaqcfasaaiaaikdaaeqaaKqbaoaaCaaabeqcfasaaiaaikdaaaqcfa OaeqyYdC3aa0baaKqbGeaacaaIYaaabaGaaGOmaaaajuaGcqGHRaWk caWGkbWaaSbaaKqbGeaacaaIZaaabeaajuaGdaahaaqabKqbGeaaca aIYaaaaKqbakabeM8a3naaDaaajuaibaGaaG4maaqaaiaaikdaaaqc faOaeyypa0Jaam4yaiaad+gacaWGUbGaam4Caiaadshaaaaa@9646@  (5)

Last property follows directly from the demands (4) (they formalize condition of maximum for H). The condition of transversality H=0 takes place at each instant of time.4

The problem of optimal control is reduced to finding the solution to the system of differential equations (1), (3) under the condition that the control ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyYdChaaa@3866@  is chosen based on condition (4) with the simultaneous satisfaction of the condition of transversality H=0 and the boundary conditions Λ( 0 )= Λ in ,Λ( T )= Λ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9iabfU5amnaaBaaa juaibaGaamyAaiaad6gaaeqaaKqbakaacYcacqqHBoatdaqadaqaai aadsfaaiaawIcacaGLPaaacqGH9aqpcqqHBoatdaWgaaqcfasaaiaa dAgaaeqaaaaa@47A6@  (the conditions of transversality r(0) ¹ 0 and r(T)¹0 are satisfied automatically, as it follows from first equality (5) written for optimal motion). The system of differential equations (3) for the variables r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhada WgaaqcfasaaiaadMgaaeqaaaaa@38AD@ , together with the requirement of maximizing the Hamiltonian H and the condition H=0, provides the necessary optimality conditions. Reminding that the coefficients a 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyyamaaBaaajuaibaGaaGymaaqabaqcfaOaeyiyIKRaaGim aaaa@3B98@  and a 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyyamaaBaaajuaibaGaaGOmaaqabaqcfaOaeyiyIKRaaGim aaaa@3B99@ . If we take the ort p= r | r | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadchacq GH9aqpdaWccaqaaiaadkhaaeaadaabdaqaaiaadkhaaiaawEa7caGL iWoaaaaaaa@3D96@ then r 0 = 2 a 1 a 2 /C ;   E k = a 2 /2 a 1 ;  | L |= a 2 / a 1 /C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOCamaaBaaajuaibaGaaGimaaqabaqcfaOaeyypa0ZaaSGb aeaacaaIYaWaaOaaaeaacaWGHbWaaSbaaKqbGeaacaaIXaaabeaaju aGcaWGHbWaaSbaaKqbGeaacaaIYaaabeaaaKqbagqaaaqaaiaadoea aaGaai4oaiaacckacaGGGcGaamyramaaBaaabaGaam4AaaqabaGaey ypa0ZaaSGbaeaacaWGHbWaaSbaaKqbGeaacaaIYaaabeaaaKqbagaa caaIYaaaaiaadggadaWgaaqcfasaaiaaigdaaeqaaKqbakaacUdaca GGGcGaaiiOamaaemaabaGaamitaaGaay5bSlaawIa7aiabg2da9maa lyaabaWaaOaaaeaadaWcgaqaaiaadggadaWgaaqcfasaaiaaikdaae qaaaqcfayaaiaadggadaWgaaqcfasaaiaaigdaaeqaaaaaaKqbagqa aaqaaiaadoeaaaaaaa@5A8C@  where r 0 =| r | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOCamaaBaaajuaibaGaaGimaaqabaqcfaOaeyypa0ZaaqWa aeaacaWGYbaacaGLhWUaayjcSdaaaa@3E46@  ; C= P 10 2 / J 1 + P 20 2 / J 2 + P 30 2 / J 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qaiabg2da9maakaaabaWaaSGbaeaacaWGqbWaa0baaKqb GeaacaaIXaGaaGimaaqaaiaaikdaaaaajuaGbaWdaiaadQeadaWgaa qcfasaaiaaigdaaeqaaaaajuaGpeGaey4kaSYaaSGbaeaacaWGqbWa a0baaKqbGeaacaaIYaGaaGimaaqaaiaaikdaaaaajuaGbaWdaiaadQ eadaWgaaqcfasaaiaaikdaaeqaaaaajuaGpeGaey4kaSYaaSGbaeaa caWGqbWaa0baaKqbGeaacaaIZaGaaGimaaqaaiaaikdaaaaajuaGba WdaiaadQeadaWgaaqcfasaaiaaiodaaeqaaaaaaKqba+qabeaaaaa@4DC1@ ; p i0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCamaaBaaajuaibaGaamyAaiaaicdaaeqaaaaa@3985@  are the components of the vector p 0 =p( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCamaaBaaajuaibaGaaGimaaqabaqcfaOaeyypa0JaamiC amaabmaabaGaaGimaaGaayjkaiaawMcaaaaa@3D63@ ; E k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyramaaBaaabaGaam4Aaaqabaaaaa@3874@  is rotary energy; L is angular momentum;

P 1 2 / J 1 + P 2 2 / J 2 + P 3 2 / J 3 =const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGbaeaacaWGqbWaa0baaKqbGeaacaaIXaaabaGaaGOmaaaa aKqbagaapaGaamOsamaaBaaajuaibaGaaGymaaqabaaaaKqba+qacq GHRaWkdaWcgaqaaiaadcfadaqhaaqcfasaaiaaikdaaeaacaaIYaaa aaqcfayaa8aacaWGkbWaaSbaaKqbGeaacaaIYaaabeaaaaqcfa4dbi abgUcaRmaalyaabaGaamiuamaaDaaajuaibaGaaG4maaqaaiaaikda aaaajuaGbaWdaiaadQeadaWgaaqcfasaaiaaiodaaeqaaaaajuaGpe Gaeyypa0Jaam4yaiaad+gacaWGUbGaam4Caiaadshaaaa@4F7B@ , since | r | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaqWaaeaacaWGYbaacaGLhWUaayjcSdaaaa@3AB2@ =const.

The boundary value problem of the maximum principle is to determine such value of the vector p 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCamaaBaaajuaibaGaaGimaaqabaaaaa@3897@  at which the solution Λ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@3A70@  of the motion equation (1) and differential equations (3) (with the simultaneous satisfying the equalities (4) at each instant of time) satisfies the maneuver conditions Λ( 0 )= Λ in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9iabfU5amnaaBaaa juaibaGaamyAaiaad6gaaeqaaaaa@3EDC@  and Λ( T )= Λ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aabmaabaGaamivaaGaayjkaiaawMcaaiabg2da9iabfU5amnaaBaaa juaibaGaamOzaaqabaaaaa@3E05@  (the quantity r 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOCamaaBaaajuaibaGaaGimaaqabaaaaa@3899@  is calculated unambiguously by p 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCamaaBaaajuaibaGaaGimaaqabaaaaa@3897@  and the coefficients a 1 , a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyyamaaBaaajuaibaGaaGymaaqabaqcfaOaaiilaiaadgga daWgaaqcfasaaiaaikdaaeqaaaaa@3BB8@ ). Optimal vector p0 is determined only by the values Λ in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aaBaaajuaibaGaamyAaiaad6gaaeqaaaaa@3A1E@  , Λ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aaBaaajuaibaGaamOzaaqabaaaaa@3928@  and J 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQeada Wgaaqcfasaaiaaigdaaeqaaaaa@3852@  , J 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQeada Wgaaqcfasaaiaaikdaaeqaaaaa@3853@ , J 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQeada Wgaaqcfasaaiaaiodaaeqaaaaa@3854@  .

Punctual consecutive implementation of procedure of the maximum principle for dynamical problem of optimal slew maneuver (when ω( 0 )=ω( T )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyYdC3aaeWaaeaacaaIWaaacaGLOaGaayzkaaGaeyypa0Ja eqyYdC3aaeWaaeaacaWGubaacaGLOaGaayzkaaGaeyypa0JaaGimaa aa@419E@  and the control torque M is limited) show that maximal rotary energy E k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyramaaBaaajuaibaGaam4Aaaqabaaaaa@38A2@ is no more the ratio a 2 /2 a 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGbaeaacaWGHbWaaSbaaKqbGeaacaaIYaaabeaaaKqbagaa caaIYaaaaiaadggadaWgaaqcfasaaiaaigdaaeqaaaaa@3BDA@  for any instant of time t[ 0,  T ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiDaiabgIGiopaadmaabaGaaGimaiaacYcacaGGGcGaaiiO aiaadsfaaiaawUfacaGLDbaaaaa@3F93@  (independently of duration of acceleration and braking). I.e. always, during optimal rotation from the position Λ in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aaBaaajuaibaGaamyAaiaad6gaaeqaaaaa@3A1E@  into the position Λ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfU5amn aaBaaajuaibaGaamOzaaqabaaaaa@3928@  (in the sense of minimum (2)), rotary energy of spacecraft have restriction by known upper level determined by the coefficients a 1 , a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyyamaaBaaajuaibaGaaGymaaqabaqcfaOaaiilaiaadgga daWgaaqcfasaaiaaikdaaeqaaaaa@3BB8@ of the minimized functional. If assume a 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyyamaaBaaajuaibaGaaGymaaqabaaaaa@3889@ = 0.5 and a 2 = E ad MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyyamaaBaaajuaibaGaaGOmaaqabaqcfaOaeyypa0Jaamyr amaaBaaajuaibaGaamyyaiaadsgaaKqbagqaaaaa@3D94@ then optimization of motion program by criterion (2) give satisfaction of the inequality J 1 ω 1 2 + J 2 ω 2 2 + J 3 ω 3 2 2 E ad MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOsamaaBaaajuaibaGaaGymaaqabaqcfaOaeqyYdC3aa0ba aKqbGeaacaaIXaaabaGaaGOmaaaajuaGcqGHRaWkcaWGkbWaaSbaaK qbGeaacaaIYaaabeaajuaGcqaHjpWDdaqhaaqcfasaaiaaikdaaeaa caaIYaaaaKqbakabgUcaRiaadQeadaWgaaqcfasaaiaaiodaaeqaaK qbakabeM8a3naaDaaajuaibaGaaG4maaqaaiaaikdaaaqcfaOaeyiz ImQaaGOmaiaadweadaWgaaqcfasaaiaadggacaWGKbaabeaaaaa@5157@  for any instant of time, where E ad MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyramaaBaaajuaibaGaamyyaiaadsgaaKqbagqaaaaa@3A0F@ is admissible rotary energy. In our variation problem, find the main properties, laws and key characteristics (parameters, constants, integrals of motion) of optimal solution of control problem using the conditions of transversality as very important and unique mathematical instrument. Chosen criterion of optimality guarantees motion of solid body with rotary energy not exceeding the required value.

Acknowledgments

No financial exists.

Conflict of interest

No conflict of interest exists.

References

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