Objectives of Meeting Movements - Application for Ship in Maneuvering

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1 Iteatoal Joual of Mechacal Egeeg ad Applcatos 25; 3(3-): Publshed ole May 6, 25 ( do:.648/.mea.s ISSN: X (Pt); ISSN: (Ole) Obectves of Meetg Movemets - Applcato fo Shp Maeuveg Nguye Xua Phuog, Vu Ngoc Bch 2 Faculty of Navgato, Ho Ch Mh Cty Uvesty of Taspot, Ho Ch Mh cty, Vetam 2 Depatmet of Scece Techology Reseach ad Developmet, Ho Ch Mh Cty Uvesty of Taspot, Ho Ch Mh cty, Vetam Emal addess: Phuogx968@gmal.com (N. X. Phuog), vubch@hcmutas.edu.v (Vu N. B.) To cte ths atcle: Nguye Xua Phuog, Vu Ngoc Bch. Obectves of Meetg Movemets - Applcato fo Shp Maeuveg. Iteatoal Joual of Mechacal Egeeg ad Applcatos. Specal Issue: Taspotato Egeeg Techology pat II. Vol. 3, No. 3-, 25, pp do:.648/.mea.s Abstact: The pape devotes the fomulato of the poblem of optmzg the ocomg taffc ad gves a descpto of the cocept ad cotol system that mplemets the avgato of shps maeuves. I autcal pactce, the shp has bee ecouteed the specal stuatos, such as: avodg collso, matag the tme avg the plot stato, pckg up plot, bethg as schedules, salg cofed wate aea... I ode to solve ths ssue, the authos peset the eseaches about the task of tecepto optmal tme ad the omal ad degeeate poblem; also they gve the emaks about globally-optmal cotol ad optmal cotol. Accodgly, the esult s appled fo shp cotol maeuveg. Keywods: Itecepto Optmal Tme, the Nomal ad Degeeate Poblem, Shp Maeuveg. Itoducto I autcal pactce, the shp has bee ecouteed the specal stuatos, such as: avodg collso, matag the tme avg the plot stato, pckg up plot, bethg as schedules, salg cofed wate aea... ode to solve these ssues, we wll fomulate the poblem of optmzg the ocomg taffc ad gve a descpto of the cocept ad cotol system that mplemets the avgato of shps maeuves. The optmzato poblems ca be classfed fo whch you ae to mmze the tasto tme fom the tal state to the fal aea elates to the tasks of the optmal tme. I ths secto we fomulate the poblem pecsely cotol the optmal tme to be cosdeed at a patcula physcal example. Most of ths secto s devoted to a dscusso of the poblem fom a geometc pot of vew. We show that the tme-optmal poblem essetally educes to fdg [, 4,, 2]: ) The fst tme at whch the aea of eachable states meets the aea S; 2) Cotol, whch t caes out. 2. The Task of Itecepto Optmal Tme The vessel wll be cosdeed as a dyamc system wth state x, a ext y( t ) ad the cotol u( t ), defed by the equatos [2, 7,, 2]: Let's assume that Also that f [ x, t + B[ x, t]u(t) (2.) y(t)h[ x( t )] (2.2) x dmesoal vecto y m dmesoal vecto (2.3) u - dmesoal vecto m > (2.4) Thus, f - a -dmesoal vecto fucto; B[x,t] - the matx-fucto of the sze ad h s a m-dmesoal vecto fucto. We wll cosde that compoets of a vecto of cotol u(t) ae lmted o sze by equaltes [] u m,,2,..., (2.5) Let z( t ) - a vecto wth m compoets. We wll agee to ame a z( t ) desable ext. Let e y z - Eo

2 5 Nguye Xua Phuog ad Vu Ngoc Bch: Obectves of Meetg Movemets - Applcato fo Shp Maeuveg vecto. Let t - tal tme ad x(t ) - statgg state of dyamc system. It s equed to fd cotol, whch: ) Satsfes to estctos (2.5); 2) Opeates system such a mae that dug the fal momet of tme e( T ) E Whee E - some set subset fom ; 3) Mmses tasto tme T t. If the dyamc system [3, 7] descbed by (2.) ad (2.2), s completely obsevable, to eveyoe y(t) thee coespods a uque status x(t). Hece, aea S spacee of statuses ca be defed paty: S { x( T) : y( T) h[ x( T)];y(T) Y} (2.7) We use Potyag s mmum pcplee [4, 3] to eceve the systematzed appoach to the decso of poblems o optmum speed. Receved esults the aalytcal fom ca be used fo umecal epesetato of decsos. We wll cosde cotol, optmum o speed, fo moble aea S t. The system s gve. f[ x, t] + b[ x(t),t] u ;,2,... o f [ x, t] + B[ x, t] u Set smooth aea S s defed by pates: Compoets paty: gα [ x, t], α,2,..., β; o g[ x, t] R m β vecto wth compoett g α u, u2( t),..., u ; β (2.6) (2.8) (2.9) ae lmted o sze by - Taslated x(t ) systems (2.8) aea S; - Mmsed fuctoal J(u). O the bass of a mmum pcple [4, 3] t s possble to asset that thee s (optmum) addtoal vecto p(t) coespodg to optmum cotol u(t) ad a optmum taectoy x(t). Exstece p(t) ) s a ecessay codto. It s ecessay, those compoets x k ad pk, k,2,..., satsfed to the tal equatos: 3. Nomal ad Degeeate Poblem 3.. Nomal Task Suppose [, 6, 5, 6] that the teval coutable set of pots t, t 2, t 3,, Such that H[ x, p, u, t] k ; pk ( t) H[ x, p, u, t] pɺ k xk γ t T γ t [, ],,2,3,...;,2,..., (3.) q b [ x, t] p ( t ) f t tγ; othe case [ t, T (2.2) ] has a (3.2) I ths case, the poblem of optmal speed wll be called omal. Fg. 3. shows the fucto q ad the coespodg u. Fucto q vashes oly solated momets tme, ad theefoe cotol, tme-optmal, a pecewse costat fucto wth smple umps. If all fuctos q have the same popetes, the task s a omal cotol. It s usually sad that the cotol u wll swtch whe t tᵞ ad that whe the umbe of swtches u s equal to the geatest umbe (o ). Cotol u show Fg. 3. wll swtch 4 tmes. Cosequetly, the umbe of swtches s fou. u,,2,..., wth all t o u Ω (2.) Fuctoal t s defed a kd: T J ( u) dt T t t (2.) Whee T - t s fee. To fd such cotol u(t), that t: - Satsfed to estctos (2.); Fg. 3.. A fucto q ( t ) that gv ves a well-defed cotol u ( t ).

3 Iteatoal Joual of Mechacal Egeeg ad Applcatos 25; 3(3-): Degeeate Poblem Assume [, 6, 5, 6] thee s a teval (o moe) [ T, T2 ] of sub-slot [ t, T], such that [ t, T] ( ) [ ( ), ] ( ) w t [, 2] of oe q t b x t t p t th T T (3.3) Ths poblem s called degeeate, ad the teval [T,T 2 ] (o tevals) - teval degeeacy. Fucto q (t) show Fg. 3., s equal to zeo fo all t of [T,T 2 ], ad theefoe coespods to a degeeate poblem. Thus, the degeeate case the poblem exsts at least oe tme sub-slot u sg b[ x, t] p fo whch the ato does ot deteme the optmum cotol, ad as a fucto of x(t) ad p(t). { } { } u SIGN q SIGN B[ x, t] p ) (3.5) Thus, f a omal task, the compoets of the cotol-optmal ae a pecewse-costat (o elay) fuctos of tme. The followg theoem ca be poved by dect substtuto. Theoem 2. Peequstes [,, 2]. Let u(t) optmal cotol fo the poblem, x(t) state at tme-optmal taectoy ad p(t) coespodg to a addtoal vecto. Let T mmum tme. If a omal task, t s ecessay to: A) Satsfes the degeeate poblem (3.2); B) The codto x(t) ad a addtoal vecto p(t) comply wth the smplfed caocal equatos: f [ x, t] k k bk[ x, t] sg b[ x, t] p (3.6) f[ x, t] k xk pɺ p + (3.7) [ ( ), ] b x t t + sg b [ x, t] p p ( ) t xk fo the k, 2,, ad t ϵ [t,t]; C) Hamltoa alog the optmal taectoy s detemed by the equato Fg Show the fgue coespods to the fucto q (t) of a degeeate poblem of optmal cotol. The last statemet does ot mea that the optmal cotol does ot exst o caot be detemed. It oly meas that a ecessay codto does ot gve a defte elato betwee x(t), p(t), u(t), t. Degeeate poblems ae typcal fo shp addessg the meetg of movemets. We cosde the poblem of optmal omal speed. I ths case, thus excluded u(t) fom all the ecessay codtos. Theefoe, all the codtos ae lad dow by u(t), step wll be educed to the ecessay codtos beyod the cotol of u(t). As we wll see step 3, ths fact wll allow us to fd the cotol-optmal. State two theoems that summaze these deas. Theoem. Relay Pcple [,, 2]. Let u(t) - optmal cotol fo the poblem, but also x(t) ad p(t) - ts coespodg phase taectoy ad a addtoal vecto. If the task s omal, compoets u (t), u 2 (t),, u (t) of cotol u(t) shall be detemed by the elatos: u sg b[ x, t] p,2,..., (3.4) fo the t ϵ [t,t] Equato (3.4) ca be wtte moe compactly: + H[ x, p, u, t] f [ x, t] p - - b [ x (t),t] p t [ t, T ]; D) The fal tme T the elato + f [ x ( T ), T] p ( T) b [ x (T),T] p ( T) β e α α g [ x ( T), T ] α T E) At the tal tme the fal tme T (3.8) (3.9) x ( t ) x( t ) (3.) gα [ x ( T), T ], α,2,..., - β; β ; (3.) p ( T ) t β [ x ( T ), T ] kα (3.2) α x ( T ) We gve a geometc tepetato of Theoem 2 -

4 52 Nguye Xua Phuog ad Vu Ngoc Bch: Obectves of Meetg Movemets - Applcato fo Shp Maeuveg Peequstes Assume that 3 ad 2. As show Fg. 3.3, the matx sze B [x(t),t] assocated wth the coveso 2 3, dsplayg 3-dmesoal vecto p(t) a 2-dmesoal vecto q(t) B [x(t),t]p(t). Fg Geometc tepetato of the fact that the cotol of u(t) should mmze the scala poduct [u(t), q(t)]. I ode to mmze the scala poduct [u(t), q(t)], vecto cotol u(t) must have a maxmum value ad be dected opposte to the vecto q(t). So f, q(t) s the fst quadat, the vecto u(t) should be "estg" o the agle A squae estctos. If q(t) the secod quadat, the u(t) should be set to agle B, ad so o. Peequstes lead to a symmetc method fo fdg optmal cotol. Ths wll be dscussed detal the steps below. Results of degeeate poblem ad assocated optmal values ae ecessay codtos. If ths cotol u(t) ad the coespodg taectoy s ot satsfed ay of the ecessay codtos, t follows that u(t) s ot optmal cotol. Steps ae set ato that must be met fo optmal cotol u(t), states x(t), coespodg p(t), ad a mmum tme T. The essece of the challege s to fd the optmal cotol, ad so the questo ases: how ca usg all of these theoems to fd the optmal cotol poblem. The aswe to ths questo wll be gve below. I addto, each step of ou agumet wll be ettled, whch wll allow to tace the logcal coecto betwee them. Step. Fomato of the Hamltoa [6, 6]. We fom the Hamltoa H[x(t),p(t),u(t),t] system t f [ x, t] + B[ x, t] u ad fuctoal J( u) dt. Hamltoa usg expessos ca be wtte as H[ x, p, u, t] + f [ x, t], p + + u, B [ x, t] p (3.3) whch emphaszes that x(t), p(t), u(t) Vectos epesetg a fucto of tme. At ths pot, we do ot mpose estctos o ay vecto values x(t), p(t), u(t), o by t. Step 2. Mmzg the Hamltoa [6, 6]. Hamltoa H[x(t), p(t), u(t), t] depeds o vaables. Let us assume that we have fxed x(t), p(t), u(t) ad t ad cosde the behavo of the Hamltoa (whch ow s oly a fucto of u, as x(t), p(t), ad t ae costat) whe chagg u(t) lmtatos Ω. I patcula, we wat to fd a cotol whch the Hamltoa has the absolute mmum. Theefoe, we defe H-mmal cotol as follows. Defto. H-mmal cotol [6]. Admssble cotol u (t), H-called mmal f t satsfes H[ x, p, u, t] H[ x, p, u, t] (3.4) fo all u(t) ϵ Ω ad all x(t), p(t) ad t. Pevously, t was foud that the mmum cotol H - u (t), fo the Hamltoa of the type (3.3) s gve by equato: o vecto fom, u sg b [ x, t] p,2,..., (3.5) { } u ( t ) SIGN B [ x ( t ), t ] p ( t ) (3.6) Substtute the H-mmal cotol u (t), expesso (3.3): H[ x, p, u, t] + f [ x, t], p { } SIGN B [ x, t] p, B [ x, t] p Cosequetly, H[ x, p, u, t]+ f [ x, t] p - - b [ x(t),t] p (3.7) (3.8) The ght sde of (3.8) s a fucto oly of the x(t) ad p(t). We defe the fucto H [x(t), p(t), t] by the elato

5 Iteatoal Joual of Mechacal Egeeg ad Applcatos 25; 3(3-): H [ x, p, t] m H[ x, p, u] u Ω (3.9) These deftos ad equatos ae ot explctly lked wth the taectoes ad optmal values. Step 3. Restcto x(t), ad p(t). We eque that the (as yet udetemed) vectos x(t) ad p(t) satsfes the dffeetal equato [4]: o, equvaletly, dffeetal equatos H [ x, p, t] (3.2) p H [ x, p, t] pɺ (3.2) x x ɺ k fk[ x, t] bk [ x, t] sg b [ x, t] p (3.22) f[ x, t] pɺ k p + x ( ) k t (3.33) b [ x, t] + sg b[ x, t] p p x ( ) k t fo k,2,...,. Note that ad H [ x, p, t] H[ x, p, u, t] p t p t ( ) ( ) u t u t ( ) ( ) H [ x, p, t] H[ x, p, u, t] x t x t ( ) ( ) u t u t ( ) ( ) (3.24) (3.25) Step 4. The pupose of ths secto s to fd the optmal cotol u(t), tasfes the system f [ x, t] + B[ x, t] u fom a gve tal state x(t ) to S. We assume that ths poblem s omal. Model the equato (3.22) ad (3.23) o a compute. At a ceta tal tme t use we have take the tal values of the phase coodates as the tal codtos of the system (3.22). As tal values of the fuctos p (t ), p 2 (t ),...,p (t ) wll use some of the expected values [6, 6]. Let q (t),, 2,..., fuctos defed by the elatos Assume that q b [ x(t),t] p (3.26) q ( t ) fo,2,..., (3.27) Equatos (3.27), (3.26) ad (3.5) mply that the umbe of u sg{ q } equal o -. Thus, the soluto of the equatos (3.22) ad (3.23) t s detemed, at least fo t, close to t. We deote the solutos of equatos (3.22) ad (3.23) though x x[ t, t, x( t ), p( t )] p p[ t, t, x( t ), p( t )] (3.28) to emphasze the depedece o a kow tal state x(t ) ad the teded tal value p(t ). Smulato s as follows. Measug sgals x(t ) ad p(t ), at each tme we get ad egste sgals: (3.29) q b [ x, t] p,,2,..., qɺ, qɺɺ, ɺɺɺ q,,2,..., (3.3) (3.3) H [ x, p, t] + f [ x, t] p q gα [ x, t], α,2,..., β (3.32) g α [ x, t],,2,..., t α β (3.33) [ x, t] hα [ x, t], α,2,..., β (3.34) x Usg cocete (adomly selected) value p(t ), sequetally fo each tme t some teval [t, T], ask ouselves the followg questos: Questo. If q (t), the q (t)? If qɺɺ ( t ), the qɺɺ? (Ad so o). If the aswe to the fst questo s postve (.e. Yes ), the we ask the secod questo. If the aswe s egatve (.e. No ), the we chage the value p(t ) ad epeat aga the fst questo. Questo 2. If the aswe to the fst questo s Yes, s thee a tme T, fo whch satsfes gα [ x( T), T], fo allα,2,..., β? (3.35) If the aswe to the secod questo s No, we chage p(t ) ad stat all ove aga. If the aswe s Yes, the ask a thd questo. Questo 3. If the aswe to the secod questo s Yes, ae thee pemaet thd questo. e, e 2,, e -β such that the elato of: β [ x( T), T] H [ x( T), p( T), t] eα? (3.36) T α If the aswe s No, the we must chage p(t ) ad stat all ove aga. If the aswe s Yes, the go to questo 4.

6 54 Nguye Xua Phuog ad Vu Ngoc Bch: Obectves of Meetg Movemets - Applcato fo Shp Maeuveg Questo 4: If the aswe to the thd questo s Yes, ae thee pemaet k, k 2,, k -β such that the elato of: β [ x( T), T ] p( T) kα? (3.37) x( T) α If the aswe s No, the we must chagee p(t ) ad stat all ove aga. If the aswe s Yes, t meass that we foud p(t ) oe whch the aswes to all questos - 4 ae postve. I ths case, we emembe accepted p(t ) ad beg to expemet at fst, utl we fd all the vectos p(t ), fo whch the aswes to questos - 4 ae postve. The logcal sequece of questos s show Fg Step 5. Possble cotol-optmal. Fomalze the esults of the modelg doe step 4. We have detfed the set I, whch s a set of tal values p, coespodg to a gve x(t ) ad havg the popety that the aswes to all questos - 4 wll be postve (.e. Yes I s ). It s clea that s a subspace of the -dmesoal space R. You ca mage I as a way out of the logcall pocess show Fg. 3.4 moe pecsely I s defed as follows [, 4, 6]. Fg Logc dagam modelg that ca be used fo fdg the optmal cotol. Defto 2. Let I aea of tal states a addtoal vaable p, wth the followg popet ) Fo each p I es [4,, 6]: coespodg solutos of (3.22) ad (3.23), deoted by satsfy the elato x x[ t, t, x, p] p p[ t, t, x, p] q b [ x, t] p, (3.38),2,..., (3.39) oly o a coutable set of pots t; 2) Thee s a tme T (depedg o x(t ) ad p ), such that t s possble to fd the costats e, e 2,, e -β ad k, k 2,, k -β, that espects the followg elatoshps: H [ x( T ), p( T ), T ] f [ x ( T ), T ] p ( T) g [ x( T), T], α,2,..., β ; (3.4) α β [ x( T ), T ] p( T ) kα? ɺɺ x( T ) α You ca etu to Theoemm 2 ad compae the elato (3.4) ad (3.9), (3.4) wth (3.) ad (3.42) ad (3.2). By vtue of the fact that the fuctos q (t) zeo oly o a coutable set t, ad also smla to the equatos (3.22) ad (3.23) wth (3.6) ad (3.7) we obta the followg lemma. Lemma. Each soluto x( t ) ad p, t [ t, T], poduced by the elemet of the set I, satsfes all the ecessay codtos fo smplfed Theoem 2 [6, 6]. We have show that H-mmal cotol u (t) (see. Defto. H-mmal cotol) s gve by [see. ato (3.6)] u ( t ) SIGN { B [ x ( t ), t ] p ( t )} f the x x, p p fo ay x(t), p(t) ad t. As fo ( t ) ad t [ t, T], fd u ( t ) 垐 x x u ( t ) SIGN { B [ x, t] p}, t [ t, T] (3.43) p p β (3.4) b x( T ), T [ x( T ), T ] p ( T ) eα? T Compag the expesso (3.43) wth (3.6) ad takg to accout Lemma 2, we obta the followg lemma. Lemma 2. Each cotol u( t ), poduct of the elemets of I, satsfes the ecessay codtos of theoem - Relay pcple. Note that [6, 6] fo all u Ωadt [ t, T]. Now to clafy the meag of Lemmas 2 ad 3, ad the usefuless of the ecessay codtos fo fdg the cotol-optmal. To be specfc, let us assume that thee ae thee dffeet cotol-optmal, tasfomg the system fom a gve tal state x(t ) to S. All thee cotols, by defto, eque the same mmum tme T. We deote these (tme-optmal) cotols so ( ), 2( ), 3 u t u t u If you daw a 4 modelg step, we defe the set I. Suppose that we ca fd [the expesso (3.43)] fve dffeet depatmets, coespodg to the elemets I. These cotols wll be u, u, u, u, u, 2 α H[ x, p, u, t] H[ x, p, u, t] (3.42) (3.44), t [ t, T ] (3.45) (3.46) ad the coespodg slots whch they ae defed s

7 Iteatoal Joual of Mechacal Egeeg ad Applcatos 25; 3(3-): deoted [ t, T ],[ t, T ],[ t, T ],[ t, T ],[ t, T ], (3.47) espectvely. It ca be agued that thee of the fve depatmets (3.46) wll be detcal to the thee offces, the optmal tme [see. (3.45)]. Fo defteess, we assume: u u, T T, t [ t, T ]; u2 u2, T2 T, t [ t, T ]; u3 u3, T3 T, t [ t, T ]. (3.48) The questo ases: what s the sgfcace of cotols u 4 ad u 5? These two cotols must be locally-optmal. Sce thee s the pcple of mmum codtos fo a local, t caot dstgush local fom global optmal cotols. The oly way to deteme whch depatmets u,..., u5 ae globally optmal - s to measue ad compae the tmes T,..., T5 ad, thus, foud that T T2 T3 T ; T4 > T ; T5 > T. (3.49) Fo ths easo, we emphasze that the ecessay codtos gve oly cotols that ca be optmal. I the ext secto we dscuss the esults obtaed above. I the pevous sectos wee obtaed ecessay codtos fo optmal cotol ad developed a systematc method fo detemg the dealzed offces, oe of whch may be the best pefomace, but also establshed (Theoem ) that f the poblem s omal, the the compoets of the cotol-optmal, ae pecewse costat fuctos of tme [, 4,, 6]. As fo the omal compoets of the poblem optmal cotol must be pecewse costat fuctos of tme, oe of the ecessay codtos, amely: H x t p t u t t [ ( ), ( ), ( ), ] H[ x, p, u, t]; u Ω allow you to estct the seach fo optmal class cotol u,,2,...,. Ths s pehaps the most useful esult obtaed fom the mmum pcple, whle the est of the ecessay codtos gve moe appopate bouday codtos ad tasvesely codtos. It should be oted that the Hamltoa [6, 6] H[ x, p, u, t] + f [ x, t], p + + u, B [ x, t] p ad dffeetal equatos (3.5) H[ x, p, u, t] p H[ x, p, u, t] pɺ x (3.5) System s fully defed ad fuctoal ad thus depedet of the bouday codtos ad at the ego S. I addto, the mmum cotol H u (t) (cm. Defto. H-mmal cotol), defed by the equato [6] { } u ( t ) SIGN B [ x ( t ), t ] p ( t ) (3.52) depedetly (fuctoal) of the bouday codtos mposed. Thus, steps - 3, ae exactly the same fo ay poblem about the optmal speed. Necessay codtos fo the Hamltoa ad a addtoal vaable the fal tme T togethe wth a gve tal state ad equatos ego S povde eough bouday codtos fo the soluto of the system 2 dffeetal equatos. We showed step by step pocess used to deteme the cotols u, the esultg taectoes x( t ) ad appopate addtoal vaables p, meet all the ecessay codtos. I ode to hghlght these values, we make the followg behavo. Defto 3. Exteme vaables. The cotol u called exteme f u ad the coespodg taectoy x( t ) ad a addtoal vaable p meet all the codtos [.e. Equato (3.38) ad (3.4) - (3.44)]. It wll also be called x( t ) ad p extemely taectoes state ad a addtoal vaable, espectvely [, 9, 3]. 4. Remaks I geeal, ca be a lot of exteme cotol. Each exteme cotol gves a taectoy that may be optmal ethe locally o globally. Sce exteme cotol satsfes all the ecessay codtos, we ca ote the followg [, 4, 4]. Remak. If the optmal cotol u(t) exsts ad s uque ad thee s o othe local optmal cotols, thee s oly oe extemely cotol u, whch s the optmal tme,.e. e. u u. It s clea that the assumpto of the absece of othe locally-optmal cotols made Remak makes the pcple of mmum of ecessay ad suffcet codto. Remak 2. If thee s oly a vaety of optmal cotols ad f thee m 2 cotol, optmal locally, but ae ot optmal globally, the all wll be m + m 2 exteme cotol. Remak 3. If a globally-optmal cotol does ot exst ad thee m 2 dffeet locally optmal cotols, thee s a m 2 exteme cotol. Theefoe, the exstece of exteme cotol does ot mply the eed fo a globally-optmal cotol. Remak 4. If the optmal cotol exsts, t ca be foud by calculatg the tme T equed by each of the exteme cotol ad cotol by mmzg T. These emaks lead to the cocluso that dealg wth the

8 56 Nguye Xua Phuog ad Vu Ngoc Bch: Obectves of Meetg Movemets - Applcato fo Shp Maeuveg poblem of optmal cotol, we eed to kow the aswes to the followg questos: ) Whethe thee s a cotol-optmal? 2) Oly f the optmal cotol? 3) Whethe a task s omal? 4) Does ot cota addtoal fomato that s ecessay codtos fo the data system ad the aea S? Ufotuately, fo abtay olea systems ad aeas of S aswes to these questos have ot yet bee eceved. Thee ae, howeve, a umbe of esults fo a class of lea systems. Sce ths class of systems s extemely mpotat, we wll devote a few paagaphs to t to get addtoal esults that ae mpotat, both fom theoetcal ad pactcal pots of vew. 5. Cocluso Accodgly, the eseach devotes the fomulato of the poblem of optmzg the ocomg taffc ad gves a descpto of the cocept ad cotol system that mplemets the avgato of shps maeuves. I sum, we ca coclude as followg [5,, 4]: - The substatato of statemet of poblems of cotol s made by a meetg of movemets ad geometcal tepetato of a poblem of a fdg of shp cotol, optmum o tme, the fom of movg aeas space of statuses s offeed due couse. - Possbltes of a pcple of a mmum fo a fdg of optmum cotols ae cosdeed ad ways of ecepto of umecal decsos ae offeed. - The easos of occuece omal ad degeeate cotol poblems of shp cotol ae establshed by a meetg of movemets [8]. Refeeces [] Athas M, Falb P. Optmal cotol. - M.: Egeeg, p. [2] Bas A.M, Moskv G.I. Coastal Vessel Taffc Cotol System. M.: Taspot, p. Bogaphy Nguye Xua Phuog, (967, Hao); Mae Maste; PhD Systems Aalyss, Cotol ad Ifomato Pocessg, (2, Russa). He cuetly s a lectue of Navgato faculty, Ho Ch Mh Cty Uvesty of Taspot (Vetam). Hs eseach teests ae wth geeal lea/olea cotol theoy fo maeuveg systems wth applcatos towad gudace, avgato, ad cotol of ocea vehcles. [3] Blekhma I.I. Sychozato of dyamc systems. - M.: Scece, p. [4] I. M. Ross A. Pme o Potyag's Pcple Optmal Cotol, Collegate Publshes, 29. [5] Clake, D. The foudatos of steeg ad maeuveg. Poceedgs of the IFAC cofeece o maeuveg ad cotollg mae cafts, IFAC, Goa, Spa, 23. [6] Lopaev VK, Makov AV, Maslov Y, Stuctue V. Appled mathematcs egeeg ad ecoomc calculatos/ Collecto of scetfc papes. St. Petesbug, 2, pp [7] Kulbaov YM. Dyamc model vese poblems of taffc cotol. Collecto of scetfc papes "Maagg taspot systems SPb.: SPGUVK, 995, pp [8] Iose H., T. Hama; Taffc Cotol. - M.: Taspot, p. [9] Leve, Wllam S., ed.. The Cotol Hadbook. New Yok: CRC Pess, 996. (ISBN ) [] Zemlyaovsky DK. Calculato Elemets Maeuveg fo Pevetg Collsos. Poc. Ist / Novosbsk Isttute of Wate Taspot Egees p. [] Coft E.A, Feto R.G, Behabb B. Tme-optmal tecepto of obects movg alog pedctable paths. Assembly ad Task Plag Poceedgs IEEE Iteatoal Symposum o, 995, pp (ISBN ) [2] Ik Sag Sh; Sag-Hyu Nam; Robets, R.G.; Moo, S.B. "Mmum-Tme Algothm Fo Iteceptg A Obect O The Coveyo Belt By Robot", Computatoal Itellgece Robotcs ad Automato, 27. CIRA 27. Iteatoal Symposum o, pp [3] Seth, S. P.; Thompso, G. L. Optmal Cotol Theoy: Applcatos to Maagemet Scece ad Ecoomcs (2d ed.). Spge, 2. (ISBN ) [4] Geeg, H. P. Optmal Cotol wth Egeeg Applcatos. Spge. 27 (ISBN ) [5] Johstoe, Pete, Notes o Logc ad Set Theoy, Cambdge Uvesty Pess, 987 (ISBN ) [6] Afke, G. Mathematcal Methods fo Physcsts, 3d ed. Olado, FL: Academc Pess, 985 Vu Ngoc BICH (96, Haphog), PhD Automated desg system (27, Russa), Assocate Pofesso (23). He s Decto of Scece Techology Reseach ad Developmet Depatmet at the Ho Ch Mh cty Uvesty of Taspot (HCMUTRANs). Ma eseach aeas ae desg ad costucto the shp, R&D, educato. Fome Dea of Naval Achtectue ad Offshoe Egeeg at HCMUTRANs. He has authoed 6 books ad 2 publcatos scetfc papes ad pesetatos o atoal cofeeces.