The method presented can be used for the planning of shore protection measures without seawalls or other cliff-protecting structures; it is particularly appropriate in cases where the system is eventually to control itself by adjustment to a dynamic equilibrium regim

Moskovkin, V. M.

Trofimov, A. M.

English

DSpace at Belgorod State University

Z. Geomorph. N. F. 29 3 257-263 Berlin • Stuttgart September 1985Optimal control over geomorphological systemsA . M . T r o f i m o v  and V. M . M o s k o v k i n ,  Kazan, USSR  with 3 figuresZusammenfassung. Ehe man kiistenmorphologische Prozefireaktionssysteme kiinstlich beeinflussen kann, ist es notwendig zu wissen, wieviel Zeit fur den Ubergang eines solchen Systems von einem ,,Anfangsstadium“ zu einem dynamischen Gleichgewichtszustand erforderlich ist. Diese Fage wird hier fur ein aus Kliff und Strand bestehendes System untersucht; das dabei benutzte theoretische Modell beruht auf einer normalen Differentialgleichung, welche die Massenbilanz am Fufi des Kliffs beschreibt. Dabei dient die Zu- oder Wegfiihrung von Schutt als Kontrollfaktor.Die hier vorgestellte Methode laiSt sich in der Planung von Kiistenschutzmafinahmen ohne Schutz- mauern oder andere Kliffschutzkonstruktionen anwenden; sie ist besonders dort geeignet, wo sich das System schliefilich durch das Einspielen auf ein dynamisches Gleichgewichtsregime selbst regulieren soil.A bstract. In order to control coastal geomorphological process-response systems it is necessary to know how much time is required for such a system to change from an „initial state" to a state of dynamic equilibrium. This question is investigated here for a cliff-beach system by means of a theoretical model based on an ordinary differential equation that describes the mass balance at the foot of the cliff. The addition or removal of riprap serves as a control factor.The method presented can be used for the planning of shore protection measures without seawalls or other cliff-protecting structures; it is particularly appropriate in cases where the system is eventually to control itself by adjustment to a dynamic equilibrium regime.Resume. Dans le but de controler des systemes de reponses de processus de geomorphologie cotiere, il est necessaire de savoir combien de temps prend un systeme pour passer d’un etat initial a un etat d’equilibre dynamique. Cette question est etudiee ici sur un systeme plage-falaise au moyen d’un modele theorique, base sur une equation differentielle ordinaire qui decrit le bilan sedimentaire au pied de la falaise. L ’accumulation ou Penlevement de “riprap” sert de facteur de controle. La methode presentee peut etre utilisee pour planifier les mesures de protection de la plage sans construction de digues ou d ’autres structures de protection de falaises; elle est particulierement appropriee dans des cas ou le systeme est eventuellement auto controle par ajustement a un regime d’equilibre dynamique.18 Zeitschrift fur Gcomorphologie N. F. Bd.29, Heft 3 0 372-8854/ 85/ 0029-0257  $ 1.75©  1985 Gebriider Borntraeger, D -1000 Berlin • D -7000 Stuttgart2 58 А. М. T r o f i m o v  and V. М. M o s k o v k inT h e mathematical theory o f optim al control ( P o n t r y a g i n  et al. 1 9 6 1 )  very likely w ill be widely applied to the control o f geom orphological systems in the future. T h e possibility of using this theory for problem s o f the control o f  exogenic proces­ses has been discussed by the authors at the 24 th International G eographical Congress ( T r o f i m o v  &  M o s k o v k in  1 9 8 0 ) . This idea requires a definite level o f  form alization: it is necessary to construct models o f the dynamics systems o f these processes with due regard to a control factor. U nder conditions o f more or less constant inputs, the evolution o f geosystems progresses toward a state of dynamic equilibrium  because o f  negative feedbacks. F or the purpose o f control o f such systems it is important to determine the minimum time they need to change from  an initial state to a state of dynam ic equilibrium ( T r o f i m o v  &  M o s k o v k in  1 9 8 0 , 1 9 8 3 ) . This task is particular­ly relevant for rapidly evolving systems that are frequently subjected to control measures, such as coastal shore processes, stream processes and intensive slope processes.This paper deals with the question o f optim al control in the case o f coastal cliff- beach system s; it has direct relevance to the intensity o f the use o f such coastal zones b y  man.T h e model is based on the balance equation o f the material present at the foot o f the cliff ( E s in  1 9 8 0 )0 )  =  af(\V)H -  kW ,w here W  is the volume o f clastic beach material per unit o f length (m 3/m); f(W ) is the rate o f cliff retreat as the function o f the volume of material (m/year); H  is the height o f the cliff (m ); a is the part of beach-form ing material that is derived from  bedro ck ; к is the abrading coefficient (hardness index for beach pebbles, 1/year); t is tim e (year).N ow  we introduce the control factor ^ (t) in the right part o f equation (1). This factor expresses (according to its sign) the intensity o f addition o f riprap material to the beach or the intensity o f removal o f such material from  the beach. D ifferentia­ting equation (1) on time with the inclusion o f the control factor i; (t) we obtain thefollow ing system o f equations (secondorder dynamic system)- j f = a ^ V H - kW +  u(t).(2) W  = уdtwhere u(t) is the control factor connected with £  (t) by u(t) =  d § (t)/dt, /u/ ^ Р  (som e positive constant, mVvear2). T he condition o f restriction o f the cdntrol factor is due to  physical consideration.T he problem o f optimal time control can now be expressed as a transfer o f the dynam ic system (2) from its initial state (W 0,V 0) to the state (Ws,., O ) in a minimum o f  time. Here Wst. is the stationary point o f equation (1). It can be found from  the solution o f af(W )K-k\V =  O . It is possible to develop some non-linear equation as the function f(W ) for easily-destroyed and for stable rock ( E s in  1 9 8 0 ) . W e use instead the linear function ( T r o f i m o v  &  M o s k o v k in  1 9 8 3 )Optimal control over geomorphological systems 259(3) f(W ) =  y(W max -  W ),where у is some coefficient (1/m year); W m„  is the maximum volume o f material at w hich there is no abrasion (or some other value W  which is needed for the linear approxim ation o f  the non-linear function f(W ).In  order to tranform  this problem  o f optim al control to the classical case, we m ust make a substitution o f the variables in the system equations (2): W ' =  (W -W s, )/(5, V ' =  V/p, u ' =  u/|3. W ith the use o f equation (3) df/dW =  -  y, the classical problem  is then expressed as-  A V ' +  u ',(4) dW ' , , ,s r =  v >/u 1 : i ,w here A =  aH y + k .The problem  of optim al time control is now  reformulated as the transform  of the dynamic system (4) from  the state ((W o-W St.)/p, V 0/P) to the state (0.0). The last one is the coordinate set o f the phass surface (W ', V ').In accordance with P o n t r y a g i n ’ s  maximum principle there exists as the only synthesis o f optimal controls that which can be constructed on the basis of the solution of system equations (4) for u ' =  1 and u ' =  -1  (u' has no more than two intervals of constancy). “ Synthesis o f  optimal co ntrol” refers to the point trajectory on a phase surface w hich leads from  a set state to the origin. It occurs in the minimum o f time. W e give here only final results for the region I (fig. 1), for initial variables when V 0 =  O . Results for regions I and IV  are analogues.Fig. 1. A  com plete calculation synthesis o f  optim al con tro l.2 6 0  А. М. T r o f i m o v  an d  V. М . M o s k o v k in1. The equation o f the point trajectory that leads from  the initial state to the turning point В  (fig. 1) is the solution o f dynam ic system  (4) when u ' =  -1VP )(5) W = W0 + - f r  In A  + -where 1 +  Ф 0.2. The equation o f the point trajectory that leads from  the turning point to the final (stationary) state is the solution o f dynamic system  (4) when u' =  1(6) W ^ - J ^ l n ^ l -  A V ) - _ V ,where 1 -  -Щ - Ф 0, W St =  Wm“3. T he coordinates o f the turning point В (W B, V B) areWB = -  - | r  In (1 -  -^-VA -  - b .  + Wst,(7) Vb =  -  a: (1 ~  exp [ - A2(w ° -  w s,) /p ]) i/2.4. The optimal time o f system transform  from  point (W 0, O ) to the point В along the curve (5) is(8) tWo,B =  -  - L  1„ (1 -  (1 -  e x p [ -A 2(W 0 -  W s,)/|3]),/2).5. The optim al time o f system transform  from  the point В  to the stationary state along the curve (6) is(9) tB,0  =  _  - L  In (1 +  (1 -  exp[—A 2(W 0 -  W s,)/p])1/2).6. The general optimal time of system transform  from  the initial to the final point is(10) T  =  tWo.O =  tWo.B +  1в,о-W ith the help o f Loptile’s law and when W„ >  Ws,. we can find a limit(11) lim T =  2 [(W 0  -  W St,)/(3]l/2.It has been shown that the asym ptotes V  =  ±  (УА are the phase trajectories o f the com plete synthesis of optimal controls, i.e. the dynamic system that is studied here is controlled in the entire rectangle О  <  W  <  W max, /V/ <  const, of phase surface (W ,V ). The direction of m otion by these asym ptotes is shown in fig. 1. For instance, the general optimal time o f m otion from  the inital point (which lies on a straight line V =  -f5/A) to the stationary point can be written as follow s:(12) T  =  ^  (2 In 2 -  1) +  A (W 0  +  W sJ/ p .Optimal control over geomorphological systems 2 6 1Fig. 2. The synthesis o f  optim al co n tro l for different values o f p and W0.Fig. 3. The optima] time dependence o f T  from and W„.2 6 2 А. М. T r o f i m o v  an d  V. М. M o s k o v k inF o r  application to practical problem s, w hich correspond to the geological con ­ditions o f the N ovorosiysk region ( E s i n  1980), experimental calculations according to  expressions (5 -1 0 ) have been made with the help o f a com puter. T he results of one o f  these series are shown in figures 2 and 3. They were obtained with the follow ing param eter values: A =  0.2 year'1 (k  =  0.1 year ) ; a =  0 .3 ; H  =  100m ; у =  1/300 (m  year)’1; W m«  =  50 m2, W St. =  25 m2, W 0 =  30 ; 4 0 ; 50 m2; P =  0 .1 ; 10 m2/ year-2.T h e calculations have shown that the m ost effective control region exists when A  =  0 .2  year"1, Ws,. =  25 m2 <  W D <  50 m2 occur with the range o f the control param eter u =  P between 1 to 10 m2/year‘2. In that case the optimal tim e T  necessary fo r a transform  o f the shore system to a stationary state does not exceed 20 years.F o r  A ,W C =  const., the ratio tWop/tBio  tends towards the value 1 with increa­sing P, and for A ,P =  const., this ratio increases with increasing W 0 (the increase is large if  P is small).T h e  optim al time T  is a linear function o f the initial volume o f beach material W„ and increases w ith decreasing P; the functional relationship T  =  f(P) is well approxim ated by hyperbolas. These relationships are shown in the diagrams of figure 3.Figure 1 demonstrates the com plete synthesis o f optimal controls for A =  0.2 year"1, p =  0.1 m2/year2 and W max =  50 m2. T his synthesis appears to be sensitive to the change o f A.T h e  physical sense o f the initial control parameter | (t) is this: for a given case (Vq =  O , W G >  W St.) in a natural shore system it is necessary to remove the material artificially in the course o f tim e tWo3 (8) and then to slow down this process with the tim e tB о  (9), w hereby the intensity changes in proportion to this time so that J; (t) =  -P t and £  (t) =  P (t-T ).It  is im portant to note that the problem  o f control in a minimum o f time is equivalent to the problem  of optimal control with functional minimization f(W )dt, as the function s(t) =  J '  f(W )dt, s(o) =  О  increases m onotonously (the distance over w hich a cliff retreats, s(t), can only increase with time).T h e  approach considered here can be applied to the planning o f shore protec­tion measures that rely on the form ation o f beach materials rather than on the strengthening o f cliffs by the construction o f protective walls (in this latter case, the negative feedbacks are automatically removed). This approach is especially appro­priate in cases where the control consists o f the adjustment of undisturbed natural system s to a dynamic equilibrium regime.Optimal control over geomorphological systems 2 6 3ReferencesE s i n ,  N . B . (1980): О  roli oblom och n ogo materiala v abrasionnom  processe. — O ceanologia 20 , N  1. [in Russian]P o n t r y a g i n ,  L . S ., V . G . B o l t y a n s k y , P .  V . G a m k r e l i d z e  &  E . P h . M i t c h e n k o  (1 9 6 1 ): M atem ati- cheskaj teoria optim alnich protzessov . M . -  N auka. [in Russian]T r o f im o v , A . M . &  V. M. M o sk o v k in  (1 9 8 0 ): Theoretical Preconditions o f the C o n tro l over Exogenic Processes. — 24th IG C  M ain Session. A bstracts, vol. 1; T ok vo.(1 9 8 3 ): M atem aticheskae m odelirovanie v geom orphologi sklonov. Kazan university, Kazan, [in Russian]T h e  addresses o f  the authors: Prof. D r. A . M . T r o f i m o v , Head o f  the Econom ical -  G eography D epartm ent o f  Kazan State U niversity , Lenin  str 18, K azan, U SS R ; D r. V . M . M o s k o v k i n , H ydraulic Research L aboratory , Institu te o f  W ater C onservation , Bakulina, 6 K harkov, U SSR .

Optimal control over geomorphological systems

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Optimal control over geomorphological systems

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