MATHEMATICAL TREATMENT
MATHEMATICAL TREATMENT
OF OPTIMAL OCEAN SHIP ROUTEING
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HOGESCHOOL DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C.J.D.M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT
TE VERDEDIGEN OP WOENSDAG 20 NOVEMBER 1968 TE 14 UUR
DOOR
CORNEUSDEWIT
DOCTORANDUS WIS- EN NATUURKUNDE
GEBOREN TE ROTTERDAM
Dit proefschrift is goedgekeurd door de promotor PROF. DR. R. TIMMAN.
C O N T E N T S
page
1. INTRODUCTION 9 1.1 The stage preceding ship routeing 9
1. 2 The K. N. M. I. ship routeing department 10 1. 3 Description of the dynamical system 11
2. COORDINATE SYSTEM 14
3. ANALYSIS OF THE CONSTRUCTION OF AN OPTIMAL TRACK 19
3.1 Controlled dynamical systems 19 3. 2 Restricted coordinates 20 3. 3 The unrestricted problem 20 3.4 E x t r e m a l s , optimal controls, timefronts 24
3. 5 Pontryagin's maximum principle 28 3. 6 Behaviour of the tangent and the normal to a timefront
along an e x t r e m a l 31 3. 7 Construction of the solution. Examples 39
3. 8 Remarks on the occurrence of more than one solution 46 3. 9 Modifications in c a s e of coordinate r e s t r i c t i o n s 50
4. PRACTICAL DATA 57 4 . 1 The wave prediction problem 57
4. 2 Some basic elements of ocean wave theory 57 4. 3 The Sverdrup-Munk wave forecasting method 59
4. 4 The P i e r s on-Neumann theory 62 4. 5 The performance of a ship in a given wave field 70
page
4. 5.1 P r a c t i c a l approach 70 4. 5. 2 Scientific approach 71 5. EVALUATION 0 F THE LEAST TIME TRACK 75
5.1 Oceanographic and meteorological data 75
5. 2 Evaluating the timefronts 76 5. 3 Revision of the timefronts 78 5. 4 Determination of the trajectory and the time gain 79
5. 5 The Algol program to compute the least time track 80 5. 6 Considerations regarding data incertainties and
M A T H E M A T I C A L T R E A T M E N T O F O P T I M A L O C E A N S H I P R O U T E I N G
1. I N T R O D U C T I O N
This chapter contains a description of the practical data, that formed the b a s i s of ocean navigation for merchant ships in the past.
The foundation of a Ship Routeing Department at the Royal Dutch Weather
Institute (K. N. M . I . ) in 1960 can be seen as an end of this period for two r e a s o n s . In the first place the issue to a ship of navigational advices by a shore station was an important renewal in navigation history, while secondly this was a s e r i o u s attempt to furnish ships with information regarding the sea conditions to be expected, which was never but occasionally done before.
This chapter concludes with the description of a suitable mathematical model of the ship's movement with r e s p e c t to the e a r t h ' s surface, suitable in the sense that it can be used for solving the problem of constructing the shortest -i . e . least t-ime - track between two f-ixed po-ints.
1.1. T h e s t a g e p r e c e d i n g s h i p r o u t e i n g .
The entirely autonomous navigation of merchant ships a c r o s s the North Atlantic Ocean in either East or West direction was committed on the following b a s i s :
a. Statistical information regarding the occurrence of bad weather and rough sea furnished by books, called "Ocean P i l o t s " and by Pilot C h a r t s , issued by the U . S . Hydrographic Office.
b . Information regarding a r e a s to be avoided on account of iceberg danger. c. Determination of a great c i r c l e track and computation of the intersections
of this track with meridians of 10 W, 20 W etc.
d. Meteorological information from a weather forecast for a period of one to five days ahead.
This meteo information contained little or nothing regarding the sea conditions to be expected. The ship's m a s t e r was compelled to draw his conclusions from these weather forecasts all by himself, led by his experience and by his common s e n s e . Therefore, the deviations from the predetermined track seldomly w e r e
m o r e than occasional and for low powered ships, this procedure not r a r e l y
resulted in a considerable delay of the time of a r r i v a l .
1.2. T h e K . N . M . I . S h i p R o u t e i n g D e p a r t m e n t .
Seeking to avoid these delays and the high costs they implied, two Dutch shipping companies studied the possibility of furnishing their ships with more detailed information on the sea to be expected, such as height and direction of sea waves, direction and period of swell.
For this purpose, the K. N. M. I. established a special "Ship Routeing Department" with the assignment of predicting sea conditions for and giving navigational a d -vices to ships "under treatment".
This routeing office was thus confronted with various problems of which no
systematical study had been made so far.
To s t a r t with, a reliable picture of the weather situation was needed, not only for the present and the past, but also for the next (at most) eight days to come. In the case of an obviously stable weather situation, this stability was e x t r a -polated over the period the crossing was expected to take and a least time track was constructed in accordance with it.
In the m o r e frequently occurring case of a rapidly changing weather picture, no assumptions were made at all for m o r e than two days ahead. Here the routeing department confined itself to pilot the ship through the dangers of the next two days to come. Working this way, disappointments still occur, although with considerably less frequency than in the preceding period with no shore based navigating control.
The second difficulty was the prediction of the sea conditions as a r e s u l t of a given weather situation. Although many aspects of this problem a r e still subject to serious studies, the routeing department has by now attained a satisfying degree of reliability regarding these forecasts.
Thirdly the ship's response to the sea and swell waves had to be considered. As this response depended upon various c h a r a c t e r i s t i c properties of the routed ship, such as its draught, shape, length, longships and thwartships stability, the routeing was started on a strictly voluntary basis with two ships of the Holland America Line. After some experience, obtained with these t r i a l models, the s e r v i c e was extended to other ships of the same general cargo type, while nowadays also tankers and container c a r r i e r s a r e frequently accompanied by
the K.N. M.I. routeing office on their way a c r o s s the North Atlantic.
1.3. D e s c r i p t i o n of t h e d y n a m i c a l s y s t e m .
It is the aim of this thesis to indicate a construction of the least time track on a mathematical b a s i s , mainly formed by Pontryagin's theory of optimally controlled p r o c e s s e s .
To s t a r t with, the weather and sea situation will be assumed to be known in the entire navigating a r e a and for a time last, that covers the average t r i p ' s duration. F r o m a mathematical point of view it s e e m s perhaps most appropriate to
describe the ship's movement in a twodimensional manyfold the e a r t h ' s s u r -face - by means of complete equations of motion. However, practical navigation methods as well as some n u m e r i c a l facts regarding the motion of an ocean v e s s e l necessitate some simplifying assumptions.
In the first place ocean navigation is c a r r i e d out with the aid of a conformal ocean chart of a small scale with such a slight scale alteration, that a straight line between two points on this chart can be taken as a good enough approximation of the shortest distance between these points. A chart with these properties would be Kahn's oblique cilindrical projection. This chart will be treated more closely in the second chapter.
For the time being, it will be assumed that the ship's position and movement can be accurately enough described with the aid of an orthonormal 2-dimensional coordinate s y s t e m .
The dynamical system, describing the horizontal movement of the vessel, calls for the following notations:
x , y : Ship's position coordinates. (It is customary to neglect the ship's dimensions in sea navigation)
u, V : Components of the ship's speed in the X and Y directions, relative to w a t e r .
n ' : Number of ship's engine rotations per time unit.
•y : Steering angle, i . e . the s h a r p angle between the rudder plane and the plane of keel and s t e m s .
(The numbers n and y could be regarded as control parameters) l,d : Longships and thwartships directions of the driving power per m a s s
unit, excited by propeller and/or rudder.
w , , w , : Resistance force components per m a s s unit in 1- and d-directions as a r e s u l t of ship's speed and situation of sea and swell.
t : Time.
t , t, : Times of departure and a r r i v a l . 1 2 + +
c , c : X and Y components of the sea c u r r e n t , assumed to be stationary. x , y , ü , v : Time derivatives of x , y , u , v.
§ . , i = l ( l ) m : Sea r e s i s t a n c e p a r a m e t e r s .
With these notations, the equations of motion a r e : Ü = (1 - w.)cos a - (d - w ,)sin a
( 1 3 1) V = (1 - Wj)sin a + {d - w^)cos a X = u + c
2 y = V + c
The driving force component 1 is a function of n and y , while d is dependent of n, y, u and v, so
(1.3.2) l = l(n,'Y); d = d ( n , Y , u , v )
The controls n and y a r e limited: \n\4 N ; \y\ 4:
F-2 F-2 "^ The r e s i s t a n c e components w, and w , depend on the longships speed (u + v )^ and the rudder angle y as well as on the p a r a m e t e r s 5., which a r e functions of X, y and t.
The problem to be solved would now be to find a trajectory satisfying (1. 3.1) and begin and end conditions like x(t )=x , y(t )=y , x(t, )=x, , y(t, )=y, and to choose n and y as measurable functions of t in such a way as to minimize the time t, for a given value of t .
This system, however, is unfit for practical use for two reasons:
a. On account of safety considerations regarding the ship's oscillations the necessity to reduce n in heavy weather frequently o c c u r s . Thus n no longer satisfies the requirements of a freely choosable control. b . The practical r e s u l t of (1. 3.1) always consists of long periods (12 hours)
of ultimately slightly changing u and v, interrupted by short time i n t e r -vals of a few minutes, during which u and v change to another value as a r e s u l t of a pulse type alteration of y and possibly of n.
On account of the fact, that the influence of the sea p a r a m e t e r s ?• ~ such as height and direction of the sea waves> direction and period of swell - on the ship's r e s i s t a n c e is by now far from exactly known for one thing, while on the other hand one has good reasons to expect e r r o r s in these p a r a m e t e r s , the values of u and v, obtained from integrating (1. 3.1) can hardly be expected to have any pr-ictical use.
It is for these r e a s o n s , that another model will be adopted.
Given certain sea conditions in a point (x,y) at a time t, the maximum attainable speed relative to water can be r e g a r d e d as a function of the course angle a.
2 2
-Denoting s = (u + v )^, the equations of motion a r e : ( 1 3 3) ^ " s ( x , y , a , t ) c o s a + c (x,y)
y = s ( x , y , a , t ) s i n a + c (x,y)
The problem is now to find a trajectory satisfying (1. 3. 3) and the starting condition:
(1. 3. 4) (Given t^) : (x(t^),y(t^)) = (x^,y^)
while for some, yet undetermined time t, the condition (1.3.5) (x(tj^),y(t^^)) = (Xj^,y^^)
must hold. F u r t h e r m o r e a as a function of time is to be chosen so that it minimizes t, .
This system is fairly in accordance with the practical c a s e .
The way s depends on a can in some occasions be very peculiar. In heavy s e a , the highest attainable speed can for c e r t a i n value intervals of a be extremely s m a l l or even z e r o , meaning that a value of a in that interval is highly unad-visable or even forbidden.
This is the main r e a s o n , why an analogy with the air navigation problem, studied and treated by H. M. de Jong (K. N. M . l . publ. 64) was not possible. In that case s was entirely independent of a, which made the problem solvable as a direct application of the c l a s s i c a l calculus of variations.
It must be noticed that the course angle a is to be considered as a "course made good". It differs from the steering course a by the drift angle d, as showm in figure 1. 3. a.
ship's velocity relative to water
2. C O O R D I N A T E S Y S T E M
System (1. 3. 3) supposes a plane orthonormal coordinate system. Crossing the North Atlantic involves an a r e a of the geoid of 4000 * 1200 square nautical m i l e s , that must be scanned to find the best ship's route.
As it is a practical habit of navigators to work with a conformal sea chart - to avoid tedious corrections of measured or plotted angles, like c o u r s e s , sight and radio bearings - it is most convenient to construct a plane conformal mapping of this a r e a with very slight scale alteration.
Considering the length/width r a t i o of the a r e a , amounting to about 3. 3, it s e e m s most appropriate to make an oblique Mercator projection.
In a Mercator projection the equator is mapped as the X-axis and the mapping of some meridian is the Y-axis. The isometric grid of p a r a m e t e r lines, that is bjeing mapped as the net of lines parallel to the X - and Y - a x i s , is formed by parallel c i r c l e s of equal geographic latitude and meridians. It would imply quite a few computational difficulties to construct such an isometric system on an a r b i t r a r y part of the geoid.
In order to avoid these difficulties, one can make a conformal mapping of the e a r t h ' s surface onto a sphere by way of intermediate step. Beside being con-formal, this mapping must have a practically constant scale.
Denoting the spherical mappings of A and B - the vertices of the ship's ocean route - as A' and B ' , we can consider the great c i r c l e over A' and B' as an oblique equator, while the collection of great c i r c l e s through the poles of this great c i r c l e over A' and B' serve as skew meridians. The conformal mapping with this basis is called a Kahn projection.
Notations:
cp ,X : geographical latitude and longitude. a : radius of the equator.
6 : excentricity of an elliptic meridian. p : curvature radius of a meridian. N : curvature radius of the prime vertical.
^ , L : latitude and longitude of the spherical mapping of a point (9 , X ). R : radius of the sphere.
o : index, r e f e r r i n g to the projection c e n t r e , when mapping the (Hayford) 14
ellipsoid onto a s p h e r e . n : longitude scale -rr- .
k : distance scale sphere/ellipsoid. a-pci '• ë^^^^ c i r c l e distance from P to Q.
V „ : maximal latitude of the great c i r c l e over P and Q.
6 : spherical distance from a point P ' of the sphere to the oblique equator. y : angle between the oblique meridians of P ' and S, the intersection of the
oblique and normal equator closest to the projection c e n t r e . m : angle between P'S and the n o r m a l equator.
s : scale Kahn c h a r t / s p h e r e . s : value of s for 6 = 0 .
o
Adopting the Hayford ellipsoid as a suitable approximation of the e a r t h ' s surface, we work with: a = 6378388 m e t e r s e = 0.08226889 „ = a g - e^) P / , , 2 . 2 , 3 V (1 - e sin cp) N = / 2 2 V (1 - e sin 9 )
The transformation of (9 ,\) to {if , L) will now be described.
Taking >|i = \li (9) and L = 1,{\), a conformal mapping r e q u i r e s the equality: , _ Rdt|) R cos ill dL
pd9 N cos 9 dX Taking n = -rr- constant and L = X , this gives:
L = X + n(X - X ) and o o r , r (1 - e
)sec
f J sec p dp = n J '- J ~ ^ • o 9o 'P^ „ 2, df 1 - e''sin''fThe apparent freedom to select the constants iji and n is now used to make
9 ° dk d k
-T— and —p equal to z e r o for 9 = 9 . The r e s u l t s a r e : ^ d9 °
n = sin 9 / s i n \li ^o o
R = V(p^N^)
The coordinates (i|i, L) of the spherical mapping of a point on the e a r t h (9,X) follow from:
a =ln{tg(TT/4 + 9 / 2 ) ( f ^ ^ ^ ^ ) l ]
9 ^ " ^ " 1 + e sin 9 ' •' 1(1 = 2 arctg(exp(hi(tg(TT/4 + i/ /2)) + n(a - a ))) - TT/2 L = X + n(X - X ) o ^ o'In order to demonstrate the slight scale alteration,, k was computed, with 9 = 4 6 ° , for 9 = 15°(1°)60°:
(9)
15°
20°
25°
30°
40°
50°
55°
60°
(k)
1.00029868 1.00017889 1.00009431 1.00004036 1.00000025 1.00000086 0.99999470 0.99997084Taking k=l implies an e r r o r over a 2000 miles s t r e t c h from the edge to the c e n t r e of less than 0. 5 m i l e . For navigational practice this is negligible. Taking the great circle on the sphere as an approximation of the geodesic on the ellipsoid introduces another e r r o r of a still s m a l l e r magnitude. The
dk -7 s m a l l value of -j— keeps this e r r o r s m a l l e r than 2*10 miles over a 100 miles
s t r e t c h .
P '
S L - L equator
Figure 2. a.
Concentrating now on the transformation of (iji, L) to ( 6 , 7 ) , the quantities v and L a r e computed by:
—^— = arccos(cos ili^.cos •^^, + sin \|t^,sin if g , c o s ( L ^ , - L^,)),
*^A'B' V = arccos(cos if. ,cos i|(_,sin(L., - L_,)cosec(—-—)) ,
Lg = L ^ , - arcsin(cotg v tg \|i^,)
Then the angle m follows from m = arctg(tg <|i cosec(L - L )). Finally, 6 and y can be computed:
dgpt
—5— = arccos(cos(L - L )cos i|i) , d g p ,
6 = arcsin(sin(—5—) sin(m - v)) ,
XV
d g p ,
y = arctg(tg(-^—) cos(m - v))
The X and ycomponents of the mapping P " of P ' into the obUque Mercator p r o -jection can be computed by means of:
•Y_,= arcsin(sin i|i , cosec v), X = s^R('Y - Yg,) .
y = s R hi(tg(n/4 + 6/2) , s = s sec 5 .
o
One can take account of the scale alteration by multiplying a line element 2 2
-ds = (dx + dy )^ with the factor sec 6, where 6 follows from y according to 5 = 2 arctg(exp(y/s R)) - TT/2.
Another possible e r r o r source is the fact, that a straight line segment is taken as the s h o r t e s t track between two points.
For a distance d between two points P and Q, measured along a straight line in the Kahn chart, the difference with the great c i r c l e distance between P and Q is maximal if 5-r, = 6 „ , namely:
j r . . / r • A sec 5,, d - 2 arcsin(cos 6 sin (—g-g—)) .
With 15k 10 and working with distance steps of at most 200 m i l e s , the e r r o r per step never exceeds 0. 06 m i l e s , so the relative e r r o r is at most 0. 03%. On a 4000 miles s t r e t c h , this could accumulate to 1. 2 m i l e s , which is negligible for navigational practice.
3. A N A L Y S I S O F T H E C O N S T R U C T I O N O F A N O P T I M A L T R A C K
Optimal ship routeing can be incorporated into the c l a s s of optimal control p r o b l e m s . These problems were investigated and analyzed by a group of Russian mathematicians, directed by Pontryagin, while Halkin later published a m o r e generalized theory. It was Pontryagin's m e r i t to effectuate something much like a breakthrough from the c l a s s i c a l variation calculus into modern applied mathematics by announcing his maximum principle. Halkin gave the problem statement and the adjacent theory in a somewhat different and m o r e general form. His concept of the "set of reachable events" can be seen a s a continuation of Hamilton's wave front theory. It also opens practical p o s s i -bilities for an actual solution of the problem.
With this b a s i s , the third chapter has no other pretentions than to be an e x -planation of this application of the general theory.
3 . 1 . C o n t r o l l e d d y n a m i c a l s y s t e m s . 1 2
Consider a ship, starting in A:(x , x ) at a time t , following a course ot, which a a a .. g
is a given function of the time t, and arriving in B:(x, , x , ) at a time t, . This sequence of events can be conceived as a dynamical s y s t e m . The events y = (x,t) a r e elements of the event space, which is the Cartesian product of the two
1 2
dimensional space X - with points x = (x , x ) - and the r e a l time axis T = [ t | t < t ] . S o y = (x,t) 6 X * T .
3.
A dynamical system is specified by the fact that the various elements (x,t) a r e mutually connected by a binary relation R on X * T:
(a) yRy for all y in X * T.
(b) if yj^Ryg and y^y^ ^^^^ ^ l ^ ^ S ' (c) if y^Ryg and ygRy]^ then y^^ = y^.
(d) if y-^Ryg and Y-^Y^ then either yg^yg or
y3Ï^y2-This relation R is caused by a system of differential equations (3.1.1) X = f(x,ci'(t), t) almost everywhere on T. As a(t) is a given function of t, this equation can be written as
X = f*(x,t).
However, by varying a (t), the solution of (3.1.1) will in general vary as well. 19
System (3.1.1) can therefore be seen as a controlled dynamical system - or "control s y s t e m " for short - and the, yet to be chosen, function ^(t) is called a control or steering function for the obvious r e a s o n that it can vary the solution of (3.1.1) to a certain extent.
The optimal control problem now consists of choosing a (t) so that the a r r i v a l time at B i s not g r e a t e r than any other t, , generated by some a ( t ) .
3.2. R e s t r i c t e d c o o r d i n a t e s .
Before studying this control system more precisely, the fact that the coordinates 1 2
X and X of the ship's position a r e limited, cannot remain unmentioned, because it might play a part when determining the least time track.
In p r a c t i c e , the only r e s t r i c t i o n s of importance a r e the 10, 20 or 30 fathoms depth lines - dependent of the ship's draught - and the boundaries of iceberg danger a r e a s .
These r e s t r i c t i o n s can be expressed mathematically by the requirement, that the coordinate vector x is an element of G, a bounded subset of X, determined by the condition G = [ x | 9(x) < 0 j , where 9(x) is a s c a l a r function of x. As the coordinate r e s t r i c t i o n s appear to be working as slight modifications of the general maximum principle, this problem will be taken into consideration later in this chapter.
3 . 3 . T h e u n r e s t r i c t e d p r o b l e m .
Consider the following data:
1 2
(1) The initial point A : x = (x , x ), from where the ship s t a r t s at a time t .
a a a a (2) The time half line T : { 111 < t ] .
1 2 (3) The point of destination B : x, = (x, , x , ) .
(4) A collection C of available c o u r s e s a. What this collection consists of, may depend on the ship's position (x) as well as on the time (t) the ship is at that position, so C = C(x,t).
In a field of low waves C is the entire interval [ 0,2Tr).
However, if the wave heights exceed 4 m e t e r s , some course intervals may become prohibited.
For a wave direction p , C(x, t) can thus be the union of the intervals [ 0 , 0 + a ] , [TT + S-b,TT + e + b ] and [ 2 T T + B - a , 2TT).
Figure 3. 3. a.
1 2
(5) Thinking of a certain trajectory with points x(t) = (x .(t),x (t)), the values of a, chosen in these points, must be elements of C(x(t),t). Following some track, a can thus be seen as a function of t.
F o r practical r e a s o n s , we confine ourselves to control functions a(t) that a r e piecewise differentiable, piecewise continuous with a finite number of discontinuous jumps.
(6) The actual motor of the control system is the vector valued function f (x, a, t) = (f (x , X , a , t), f (x , x , a , t)) . i af^
f a n d — r - with i, j = 1, 2 - a r e assumed to be bounded, continuously diffe-1 2
rentiable with r e s p e c t to x , x and t and differentiable with r e s p e c t to a. (7) A special study will be made of functions a (t), as mentioned in (5), with the
property that the differential system
(3. 3.1) X = f(x,a (t), t) with the initial condition (3. 3. 2) x(t ) = X and the end condition
(3. 3. 3) '^^t,) = X, for some t, > t has a unique and continuous solution.
The a r r i v a l time t, obviously depends on the choice of the function a(t). The problem can now be formulated as to find a function a(t), defined in (7) s o that the a r r i v a l time t, , determined by a(i), is not g r e a t e r than any other tf, generated by some a(t).
Some additional r e m a r k s have to be made concerning the components of f(x,Q',t). According to (1. 3. 3) we have
i l 2 1 2 i l 2 (3. 3.4) f (x , x ,Q',t) = s(x ,x , a , t ) s i n ( n i / 2 - cc) + c (x ,x ).
1 2 1 Considering a fixed point P with coordinates (x , x ) at a time t, the values of f
2 1 2
and f can be plotted out into the X - and X -direction respectively for all 1 2 possible values of a in C(x,t), using a r b i t r a r y , but equal units for x and x . The c u r v e , that is thus obtained, will be called the "original velocity indicatrix". In a field of low to moderate waves, a can vary continuously from 0 to 2n . The s c a l a r function s ( x , a , t ) , indicating the maximum attainable speed for a true course a, is in that case differentiably dependent of a • The indicatrix is now a closed curve with a continuous tangent, but not necessarily convex, as it may be n e c e s s a r y to reduce speed in certain c o u r s e s to avoid too heavy rolling or pitching.
Figure 3. 3. b.
As the s c a l a r value of the sea c u r r e n t c never exceeds 1. 5 miles per hour, while s is not less than 8 miles per hour, the centre P , from where the vectors f were plotted out, always lies well inside the original velocity in-dicatrix. It is c l e a r that in this c a s e , the ship is able to make headway from P into all directions.
Considering the c a s e , that there a r e prohibited s e c t o r s for a, the original indicatrix consists of separate a r c s (AB and CD in fig. 3. 3.c).
At first hand the possibilities for a ship to proceed from P seem to be limited to the s e c t o r s APB and CPD. However, it is still possible to make headway into a direction, that lies within one of the gaps BPC or DPA. To explain this, let us replace the velocity indicatrix by a position indicatrix,
1 2
meaning that from P , the values of f (x,a , t)6t and f (x,a , t)6t a r e plotted out
B (1-X) K
Figure 3. 3.c.
1 2 1
into the X - and X - d i r e c t i o n s , with 6t > O and using the adopted units for x and X .
1 2 1 2
As f and f a r e assumed to be partially differentiable with respect to x , x and t, we have for some fixed a:
X^t + 6t) = x\t) + J f^X,Q',T)dT t+6t
I
t= x'(t) + f^x,Q',t)5t + o(6t), where o(6t) has the property: lim
6t-0 o(6t)
6t 0 . The end points of the vectors
x(t) + f(x,a,t)5t
can thus be considered as approximations of the furthest attainable points from P at time t after a small time interval 5t. The e r r o r s in the coordinates of these points can be made a r b i t r a r i l y small by taking 5t small enough. If one thinks of figure 3. 3. c to be constructed this way, it is clear that all points of the straight line segment BC a r e now attainable by taking linear combinations of the c o u r s e s a. and a during the 6t-time interval. The point K for instance is attainable by taking course a, during a time X6t and course
a during the interval (t+X6t, t+6t), where X follows from KC = X* BC. Points of AD can be reached by tacking between the c o u r s e s a and a , in a
a ci proper time ratio.
The objection to this idea could be made, that it is practically impossible for
a ship to tack on two considerably different courses during a s m a l l time interval. However, practical data r e v e a l , that in wave fields on the North Atlantic circumstances a r e changing very slightly, so that one hour is a
small enough choice of fit to obtain a practically negligible e r r o r in the sense
mentioned above.
In 3. 5 it will be shown, that it is the (smallest possible) convex envelope of the original indicatrix, that plays an important part in the constructional procedure of the least time track. This envelope will be called the "effective velocity indicatrix".
As a result of practical experience it can be a s s e r t e d , that for the merchant
ships that were routed on the North Atlantic, every point P lies inside its own effective indicatrix, so that the foregoing considerations justify the conclusion,
that it is always and everywhere possible to make headway from P into all horizontal directions from 0 to 2TI .
3.4. E x t r e m a l s , o p t i m a l c o n t r o l s , t i m e f r o n t s .
If a trajectory from A to B, satisfying (3. 3.1) for some control function «(t), generates an a r r i v a l time t, not greater than any other t, , caused by some a ( t ) , the control a (t) is called "optimal" and the corresponding trajectory is called an " e x t r e m a l " .
Given the initial event (x , t ) and the system (3. 3.1), the set of reachable
+ a a
points H (t;x , t ) can be defined as the collection of points x(t) that can be
a. 3.
reached at the instant t, by starting from A at a time t and then following a
a trajectory, governed by (3. 3.1) for all possible control functions a ( t ) . So H"'(t;x . t ) = [x(t)|x(t ) = x k = f(x,a(T),T),T€(t t ] } .
a. ci ci OL o. This point collection, shortly denoted as H (t), is everywhere dense, closed
and bounded.
1 2 The boundedness is an evident consequence of the fact that f and f a r e bounded. The density of H (t) means to express that, given a trajectory x = x(T;a(T)), we can always find a control function 0 ( T ) , different from « ( T ) ,
with the property that, given an a r b i t r a r y c > 0, every point X*(T) of the 24
trajectory, produced by this control function 3 (T), has a distance to X(T) l e s s than e . (t < T ^ t).
^ 1 This property, proved extensively by Halkin ), is guaranteed by the boundedness
1 2 1 2
of f and f and of their partial derivatives to x and x . It i s , for instance, possible to choose B(T) a r b i t r a r i l y different from aif) during the interval (t , t + 6t), while taking B(T) = a (T) for t + 5 t < T < t. By taking 5t sufficiently
a a a s m a l l , we can always find X*(T) in an e-vicinity of X(T).
A consequence of these two properties is the possibility of covering H (t) by a finite number of these e - c i r c l e s , which means that H (t) is compact and - by virtue of this - closed.
I now wish to consider the boundary of H (t), called the timefront S (t).
-'•^^ e x t r e m a l s
Figure 3.4. a.
In what follows I a s s u m e , a s i s plausible from topological considerations, that S (t) is a continuous closed curve with an almost everjrwhere continuous tangent. As this boundary of H (t) can be seen a s the collection of ultimately reachable points at a time t, every trajectory from A to a point of S (t) is an e x t r e m a l .
An important property of these boundaries follows from the considerations and practical data regarding f ( x , a , t ) , mentioned in 3. 3.
Consider the points of S (t) for some t ^ t . As all these points lie inside their
3.
own effective indicatrix, a ship is able to move away from such a point into all directions. This implies that for every 6t > 0, the boundary S (t + 5t) of
+ + +
H (t + 6t) lies wholly "outside" S (t), i . e . two S -timefronts never have points in common.
) Journal d'Analyse Mathênialique, Vol. XII, Jerusalem Acad. Press.
Another useful concept is the set of initial points H ( t ; x ^ , t , ) . This set can be defined as the collection of points, from where, starting at a time t, B can be reached at a given time t, , following (3. 3.1) for some proper a(T). So we can write
H'(t;xjj,tj^) = [x(t)lx(tj^) =xj^, t < t j ^ , x = f(x,a(T),T), T6(t,tj^)] .
This s e t , shortly denoted as H (t), has the same properties as H (t), while its boundary S (t) has the same properties as those, that were mentioned for S''(t).
I now wish to state and prove a fimdamental property of e x t r e m a l s :
Lemma 3. 4.o': If a trajectory from A to B is an extremal and P is an a r b i t r a r y point of this trajectory, then the a r c s AP and PB of this extremal a r e " e x t r e -m a l s " as well.
The e x t r e m a l a r c AP is an extremal in the sense that, starting from A at a time t and following the e x t r e m a l AB, one a r r i v e s at P at a time t < t*, where t* is the a r r i v a l time at P for an a r b i t r a r y trajectory from A to P and governed by X = f(x,a'''(T),T).
The e x t r e m a l a r c PB is to be considered an e x t r e m a l in such a way, that every other trajectory from P to B with a r r i v a l time t, , following (3. 3.1), demands a starting time r < t.
Figure 3 . 4 . b.
To prove this lemma, let us a s s u m e that - fig. 3 . 4 . b - the fully drawn curve AB is an extremal from A (at t ) to B. (Whenever a part of this curve, say
a
from C to P , is considered, it will be indicated as CP ). e'
Taking an a r b i t r a r y point of this curve, let us first suppose that PB is not an e x t r e m a l . This would mean, that a moving point M', starting from A at 26
time t , following ACP , arriving at P at a time t, could then reach B in a a e
s h o r t e r last of time than t, -t by following some other track - the dotted line for instance - and would thus a r r i v e in B at a time t/ < t, . This would contradict the assumption that t, was optimal, so PB must be an e x t r e m a l .
As for the proof, that AP is an e x t r e m a l , suppose it is not and a s s u m e , that the dashed a r c ADP leads from A (at time t ) to P at a time t* < t. Consider
^ a' now two points M and M*, moving from A to B.
M takes track ACPB , a r r i v e s at P at time t and is in B at time t, , not later
e b than any other point, going from A to B.
M* s t a r t s from A at the same moment t , but follows the dashed curve ADP, a
which brings it in P at an e a r l i e r moment t*. Then M* goes to B along track P B .
e
For the a r r i v a l time t* of M* there a r e now two possibilities, merely thinking of the fact that M* leaves P e a r l i e r than M:
(1) t* < t, . This is in contradiction with the optimality assumption regarding t, and therefore impossible.
(2) M* is being overhauled by M somewhere on the way between P and B, say in Q. Let this overhauling take place at a time t'. Q may even coincide with B, so for t' the inequality t < t' < t, holds.
F r o m that moment on, the positions of M and M* coincide permanently until t, , so they a r r i v e in B at the s a m e moment.
Consider now the sets H (t*; x. , t^) and H (t; x, , t^) with S (t*) and S (t) as their respective boimdaries.
As t* was assumed to be smaller than t, the boundary S (t*) must be wholly outside S (t) and these two curves have no common points. Now, as M, starting in P at time t, follows an extremal to B, this starting point P is situated on S (t), the boundary of H (t). This implies that P is an interior point of H (t*). Now the conclusion is justified, that PB is not an e x t r e m a l for M* to reach B. In other w o r d s , M* could reach B at an instant t* < t, , if some other track would be taken from P to B. This however contradicts the optimality assumption of t, once m o r e .
So the supposition, that P could be reached at an e a r l i e r time than t, is e r r o -neous. This completes the proof of lemma 3. 4. a .
The main r e a s o n for giving this lemma some more attention i s , that it indicates a possibility of constructing the least time track from A to B by means of the S -timefronts.
S ( t , ) , on which B is located. The other points x(t) of the extremal from
+
A to B a r e subsequently located on the S (t)-fronts. Working backwards from B, all these points can be traced and thus the track can be determined.
The way, this "working backwards" as well as finding S (t,) takes place, will be indicated after having treated some p r o p e r t i e s regarding the relation between timefronts and e x t r e m a l s . These properties will be discussed in the following
paragraphs.
3 . 5 . P o n t r y a g i n ' s M a x i m u m P r i n c i p l e .
Reconsidering the r e s u l t of the preceding paragraph, it can be stated that a (total) e x t r e m a l from A to B is an e x t r e m a l all the way through. In other words: Every line element of an extremal between the times t and t + e 6 t - with e and 6t positive - i s optimal when it comes to c r o s s i n g the lane between the two consecutive timefronts S (t) and S (t + e fit).
Figure 3. 5. a.
This optimal crossing c h a r a c t e r i z e s the e x t r e m a l ' s direction.
Let r be an e x t r e m a l , intersecting the fronts S (t) and S (t + e 5t) in P and Q respectively. The width PR of the lane between these two fronts, measured perpendicular to S (t) in P , is proportional to e fit + o(^, with lim — ^ = 0.
e - o ^ As the vector PQ is equal to
t+efit
x(t + e fit) - x(t) = J f(x(T),a^(T),T) dT = t
= f(x(t),a (t),t)e5t + o(e) ,
we see that, with the e x t r e m a l ' s velocity f(x(t),a (t),t) at P , this width is covered faster than with any other velocity f ( x ( t ) , a , t ) .
Neglecting t e r m s of o(e), it turns out that the projection of this velocity vector onto the vector •^ , drawn in P perpendicular to S (t), has to be g r e a t e r than or at least equal to the projection of any other vector f(x(t),a,t) onto \|ip. Maximum principle. Analytically, this means that in every point of an e x t r e m a l the control p a r a m e t e r a must be so chosen, that it maximizes the inner product
{if , f(x,Q', t)) = H ( a ; x , * , t ) , for given values of the vectors x and >(i and the p a r a m e t e r t.
So for all t > t the control function a (t) that belongs to an e x t r e m a l , must satisfy the requirement
(t(t), f(x(t), ag(t), t)) ^ (^, (t), f(x(t), a, t))
for all possible values of a •
This requirement is the formulation of Pontryagin's maximum principle in this c a s e .
TransversaUty. The special crossing of the time fronts by the extremals is called " t r a n s v e r s a l intersection".
In this c a s e it may be useful to point out the possibility of a discontinuous jump of the velocity x along an extremal as a r e s u l t of a continuous change of i)i . This can occur, whenever the maximum value of H((y,x.rji, t), denoted as M(i^,x, t) is attained for two different values of a , say a^ and a„, with a, < cy^' while for all a between these values, H is s m a l l e r . In such a case a jumps instantaneously from a^ to a_ or r e v e r s e l y . This means that the e x t r e m a l trajectory makes an abrupt bend.
j . ^ / 1 ^ polar curve of f(x,a, t)
Figure 3.5.b.
These bending points a r e allowed for in Pontryagin's theory, as (3. 3.1) was to hold almost everywhere on T, which practically means, that a finite number of discontinuous jumps of a - and therewith of x - is permitted. Let us once m o r e consider the original velocity indicatrix, as described in 3. 3.
Figure 3 . 5 . C .
The true course a , for which the inner product e
2
(ili,f) = (\li,c) + E i|i. s sin(n i/2 - a)
is maximized, can be graphically determined with the aid of the original indi-c a t r i x :
Draw a line perpendicular to the given i|(-direction, denoted by 9 , so that at least one point of the indicatrix lies on this line, while all other points a r e located on the same side of it as the c e n t r e P .
Letting 9 vary from 0 to 2n , we find the optimal c o u r s e a as a function of 9. This function a (9) turns out to be piecewise continuously differentiable. Discontinuous jumps can be expected in two c a s e s :
1. iji is perpendicular to a tangent of the indicatrix with two or more touching points.
2. i|i is perpendicular to a straight Une, that covers a gap in the original indicatrix.
As for the determination of o , it is now clear that the original indicatrix
e " can be replaced by its convex c l o s u r e , for which the name "effective indicatrix" s e e m s suitable.
Figure 3. 5.d shows the graph of a (9), following from the indicatrix of figure 3. 5.C. 2TT 3 n / 2 n / 2 a e n / 2 • 1* ^ 1 1 , 4 - - ^ 1 \ 1 1 1 1 1 1 1 1 1 ' / 1 1 1 1 1 1
1 1
1 1 1 ji !
1 ' 1 ' 1 '! 1
3 TT/2 - * V Figure 3 . 5 . d . 3.6. B e h a v i o u r of t h e t a n g e n t a n d t h e n o r m a l t o a t i m e f r o n t a l o n g a n e x t r e m a l .Let [r] be the collection of all extremals from A to a point of S (T), with T > t . Stated m o r e precisely, [ r j is the collection of solutions of x = f ( x , a , t )
^ +
with initial value x(t ) = x and X(T) on S (T). a a
Figure 3. 6. a.
A subcollection C of { r ] is formed by a field of extremals - i. e. a family of e x t r e m a l s with no mutual intersection points but A - with the additional property that the timefronts S (t) a r e differentiable curves as long as they a r e drawn in relation with elements of C.
The e x t r e m a l s of C, covering a subregion G of H (T), can be characterized by attaching a r e a l number s to each specimen.
Thus the coordinates of a point of G a r e functions of t and s: x = X (t, s) with i = 1,2, or in vector notation x = x(t, s).
In view of the assumptions regarding C, these functions a r e partially differen-tiable with r e s p e c t to s everywhere in G, except of course in the edge points, where there is only left or right differentiability.
Let r be an extremal of C with s = s , generated by the control function a (t).
o o o Take t, and t so that t < t < t ^ T .
r i n t e r s e c t s the timefronts S (t-) and S (t^) in P- and P^ respectively. o • • 1. di \. dt
So Xp = x ( t . , s ) with i , j = l , 2 . j •' °
Analogously Q- and Q„ a r e points of another e x t r e m a l I' of C, generated by a (t), so that x„ = x (t,, s +e). This situation is exposed in figure 3. 6.b.
6 "^j J o ^ The components of the tangent vector 6x(t), touching S (t) in a point of r , a r e defined by:
Ö X ( t , s )
(3.6.1) 6 x ' ( t ) = g ^ . a = 1 . 2 ) . so that
(3.6.2) x \ t , s + e) = x ^ ( t , s ) + e 6 x \ t ) + o(e) ,
where o(e) has again the property: lim - ^ = 0. e - o ®
For what follows a lemma is needed regarding the optimal steering functions a (t) and a (t). I first want to prove:
Lemma 3. 6. a: If f(x) is a Lebesque-integrable function, defined on [ a , b ] with x
the property f(t)dt = 0 for x ê [ a , b ] then f(x) = 0 almost everywhere on [ a , b ] . a
To prove this, I introduce the collections B, B and B»: B = [ x | f(x) ?^0 , X e [ a , b ] i , B^ = {x|f(x) > 0 , x € [ a , b ] ] , B2 = [ x | f ( x ) < 0 , X € C a , b ] ] .
Then B = B + B , B and B^ a r e disjoint, so n (B) = p,(Bj^) + ^^(B^). Now B^^ and B„ can be sequentially covered by a finite or countably infinite number of measurable s e t s :
B. C £ B.. so u,(B.) = inf Z u(B..) 1 (j) 13 ^^ i ' (j) i j '
The sets B.. a r e so defined, that, with a.. = lim inf(x I x e B..) and 1] ' ' ij ^ 1 i j '
b . . = lim sup ( x l x £ B..), it follows from a..'^ x ^ b . . that x € B . . , in other iJ ' 1] 1] i j 1] words: B.. is a closed interval.
X
F r o m the assumption that f(t)dt = 0 it follows that a
^ij
hj
fa
[
f(t)
dt = [f(t)
dt - [f(t)
dt = 0Hi
As f(t) > 0 for i = 1 and f(t) < 0 for i = 2, it is c l e a r , that p, (B..) = 0.
Now from the choice of B.. we see that B.. and B., a r e disjoint for j 7^ k, so 1] 1) i k •• •" that
p,(B ) = E p,(B ) = 0 while M,(B) = ti (B ) + p,(B ) , 1 (J) ij 1 2 s o p,(B) = 0, which means that f(x) = 0 almost everywhere on [ a , b ] . We a r e now in a position to state and prove:
Lemma 3.6.B: lim f(x(t,s + e ) , Q ' (t), t) = f(x(t, s ), a (t),t) K. ^ ^ ^ Q e ^ ' ' ^^ o' o ' ' almost everywhere on [t , T ] .
As we a r e dealing with a field of e x t r e m a l s , it is obvious, that for t € [t , T ] a we have:
(3.6.3) lim x ( t , s + e ) = x ( t , s ) . e - 0 o ' ^ o'
(If r is a boundary curve of G, then this "e - 0" should be replaced by "e i 0" or "e T 0") Comparing now: t x(t, s^ + e) = x^ + J [ f ( x ( § , s ^ + e ) , a g ( 5 ) , § ) d | and ta t x ( t , s ) = x + f ( x ( | , s ) , a ( 5 ) , l ) d 5 , it is now c l e a r that O a J o o ta t Um f { f ( x ( | , s + e ) , o (?),?) - f ( x ( § , s ) , « ( ? ) , § ) } d§ = 0 ta f o r t € [t , T ] , or written componentwise: [ [ Um f^(x(§,s + e ) , a (§),?) - f'(x(|, s ) , a ( ? ) , ? ) } d§ = 0 . ' e_.0 u e u o ta
According to lemma 3. 6. a we may now conclude: Um f^(x(t, s + e ), a (t), t) = f^x(t, s ), a (t), t) Ê —^ 0 o o almost everywhere on [ t , T ] .
Corollary:
1 2 In r e g a r d of the assumption (6) in paragraph (3. 3), that f and f were
1 2
continuously differentiable with r e s p e c t to x and x and as a r e s u l t of this last lemma we may state, that
Um e - O Sf (x(t,s + e),a (t),t) o e 3xJ af^(x(t,s^),a^(t),t) (i,j = 1.2) almost everywhere on [ t , T ] a Figure 3 . 6 . b .
Seeking now a relation between 6 x ( t J and 6x(t ), we subtract:
(3. 6. 5^)
(3. 6. 5 ^
t)dt and x ( t 2 , s ^ + e) = x ( t ^ , s ^ + e) + j f(x(t, s^ + e ) , a ^ ( t ) ,
t l
x(t2,s^) = x ( t ^ , s ^ ) + J f(x(t,s^),a^(t),t)dt and find,
in view of ( 3 . 6 . 2 ) :
t2
(3.6.6) e 5x(t2) = e 6x(tj) + j [f(x(t, s^+e),Q'^(t), t)-f(x(t,s^),tï^(t), t)} dt t l
where o„(e) denotes a vector in R with components of the type o(e) We can spUt up (3. 6. 6) in two ways:
(3. 6. 7) 1 : e 5x(t2) = e fi x(t ) + I + J + 02(e) with
(3. 6. 7^) 1= j {f(x(t,s^+e),a (t),t) - f ( x ( t , s ^ + e ) , a ^ ( t ) , t ) } d t and
(3.6. 7 ^ J = J {f(x(t,s^+e),c.^(t),t) - f ( x ( t , s ^ ) , a ^ ( t ) , t ) } d t
(3. 6. 8) 2°: e 5x(t2) = e 6x(t ) + 1+ + J* + o (e) with
I*= f[
(3.6.8^ I* = r[f(x(t,s ),a (t),t) -f(x(t, s ),a (t),t)}dt and
t l t2
( 3 . 6 . 8 ^ J*= f [f(x(t,s + e ) , a (t),t) - f ( x ( t . s ) , a (t),t)} d t . t l
The expressions for J and J* can be written respectively a s : r^ 2 öf(x(t,s ) , a (t),t) . J = { E 2 ^ - ^ (xJ(t,s + e ) - x ' ( t , s )) + o (e)3 dt , t\ J=l Sv" o o 2 *2 2 af(x(t,s + e ) , a (t),t) J * = \ [ T. 2 _ ^ — ^ - ( x ^ t . s + e ) - x J ( t , s )) + o„(e)]dt . t J ^ j = i ^^j ^ - o - - o " 2^ In view of the corollary to lemma (3. 6. p) we may conclude now, that
*2 2 a f ( x ( t , s J , o J t ) , t )
ti'-'
(3.6.9) lim J / e = lim J * / e = I S ^ ^-^-^ - 5 x ^ ( t ) d t e - 0 e - O J 3=1 gx^
Concentrating on I and I*, we can r e m a r k two things: a. F r o m (3. 6. 7) and (3. 6. 8) it follows that
-,c
( 3 . 6 . 7 ) 5 x ( t J = 5x(tJ + lim l/e + lim j /
2 1 c - 0 e-O € - 0 e- ~ e and
(3.6.8^^) fix(t,) = fix(t,)+ Um I*/e + Um J * / e . ^ 1 e - 0 e - 0
Considering (3. 6. 9) it can be a s s e r t e d now that
lim I/e = lim I*/e = L e - 0 e - 0
'2 f(x(t, s J , ajt), t) - f(x(t, s J , a„(t), t) b . Writing L as lim
.•^ e-.( e - O
2.—! "^—"^—_dt
we may conclude that L is almost everywhere differentiable to its upper bound, 36
so that--r— exists almost everjrwhere on [t , T ] . F r o m (3. 6. 7 ^ we see that
(3.6.10) - ^ = f ( x ( t , s ^ + e),cy^(t),t) - f(x(t, s^ + e ) , a ^ ( t ) , t) .
Remembering the maximum principle (-• 3. 5) and the assumption that a (t) was an optimal control function - as F is an e x t r e m a l - it is c l e a r that
(^[^,f(x(t,s^+ e ) , a ^ ( t ) , t ) ) 5 ( ^ , ^ , f ( x ( t , s ^ + e),a^(t),t))
where i|i is a n o r m a l vector to S (t) in the point x ( t , s + e) . So according to (3. 6.10) we have:
(3. 6.11) ^'l' • dt"^ - ° ^^ obviously
(3. 6.12 & 13) for e > 0 : W g . - ^ ^ ) = 0 and for e < 0 : (^ •'^t ^ " ° ' a
Regarding I*, we find from (3. 6. 8 ):
(3. 6.14) ^ = f(x(t, s^), a^ (t), t) - f (x(t, s^), a^(t), t) .
As a (t) is an optimal control function, generating the e x t r e m a l r , the maximum principle r e v e a l s :
{^, f (x(t, s), a^ (t), t)) S (,1,^, f(x(t, s ), o (t), t)) ,
so from (3. 6.14) it follows that for e > 0 :
dT*/f ^ dl*/e (3. 6.15 & 16) (* ,^=^) = 0 and for e < 0: {^1 ^,^( ) 1 0 .
Letting e descend to z e r o , ^ tends to •^Q on every timefront S (t), t 6 [t , T ]
Ê J J a
F r o m (3. 6.12) it follows now, that {if ,-j^) ^ 0 almost everywhere on [t , T ] ,
J T O Q t a
while (3.6.15) leads to (>li ,-j—) i 0 a. e. on [ t , T ] . So for e i 0 we see that
J -r O d t a
(>li ,-7-) = 0 almost everywhere on [t , T ] . For e T 0 the arguments a r e analogous:
(3. 6.16) gives ( * o . ^ ) § 0 a . e . o n [ t ^ , T ]
So the conclusion is that
(3.6.17) ( l i ^ . ^ ) = 0 a . e . on [ I ^ . T ]
Differentiating (3. 6. 7^^) to t and omitting the index (t2:=t) vve find in view of ( 3 . 6 . 9 ) :
j / t 1 9 9 f ( x ( t , s ),a ( t ) , t ) . ,.,
(3.6.18) ^ = I o ^ ^ 6 x J ( t ) . f i a . e . o n [ t , T ] . dt j = i g^j dt a
Investigating the behaviour of a vector if (t) along an e x t r e m a l , for which the relation (^i (t),5x(t)) = 0 holds for all t € [t , t ^ ] , we s e e that it has to satisfy the requirement
~{if{t),bx{t)) = 0 .
In view of ( 3 . 6 . 1 8 ) w e s e e t h a t
i - l j - 1 OX''
Because of (3.6.17) and arranging this relation somewhat differently, the components if^ and \|i„ of the covariant vector \|i a r e found to vary with t according to:
(3.6.19) ^. = - Z af^(x(t),a(t),t)^ (i = l , 2 ) a . e . on[t^,t^^] J-1 3x •'
A vector i(t with this property and the condition, that
(Ht^),6x(t^)) = 0
for some t e [t , t ^ ] is called an adjoint vector to the trajectory x(t;a'(t)) 38
3.7. C o n s t r u c t i o n of t h e s o l u t i o n . E x a m p l e s .
Summarizing the foregoing p a r a g r a p h s , a solution x = x(t) was sought of the controlled dynamical system x = f(x,Q', t) with the initial condition x(t ) = x and
a a for some - yet undetermined - value t, the requirement x(t ) = x, . Among all the r e a l valued functions a ( t ) , almost everywhere continuous, that generate a solution of this problem, an optimal control function a (t) was sought, generating a minimal a r r i v a l time t, .
The trajectory that satisfies all these r e q u i r e m e n t s , could be traced by c a r r y -ing out the follow-ing procedure:
Select an a r b i t r a r y vector i)i = (1(1., *„). Select a(t ) so that it maximizes the 1 z a
inner product H(0) = (i)r ,f(x , a (t ) , t )). o a a a
With \li (t ) = i|i , let the components of ijj (t) vary according to (3. 6.18), while a
x(t) changes as indicated by (3. 3.1) and for all t è t , let a (t) maximize the a
inner product
H(t) = (^(t),f(x(t),a(t),t) .
Considering the maximum principle and the fact, that (3. 6.18) is homogeneous in i^^ and \|(„, it can be seen, that the initial choices if and p\|i - for some r e a l p > 0 - generate the same trajectory, so if can be taken with unit length, say
ili° = (cos 9^, sin 9^)
The coordinates of points of the timefront S (t) a r e consequently functions of t and CD , so X = X (t,9 ) .
If analytical expressions for these functions can be deduced, t, and 9 can be solved from the equations
. X (t, , 9 ) = X,
(3.7.1) ( 2 2
It should be r e m a r k e d h e r e , that these equations may very well have more than one set of solutions (t*,9*). With the object of finding the least time track it is obvious, that of all solutions for t, , we a r e only interested in the smallest value of it, providing it is g r e a t e r than t .
If 9* is the solution of 9 , corresponding to this minimal a r r i v a l time t*, then 39
the e x t r e m a l trajectory from A to B can be constructed by solving (3. 3.1) and (3. 6.18), using the maximum principle for the determination of a and taking X (t ) = X and \j((t ) = (cos 9*, sin 9*) as initial conditions.
a a a 0 0
R e m a r k 1:
Considering the effective indicatrix of fig. 3. 7. a, drawn with r e s p e c t to the starting point A, it appears that all initial values 9 with
cp' = cp — cp"
^ o ^ o ^ o
generate the s a m e initial c o u r s e a
Figure 3 . 7. a.
F r o m this We can conclude, that the initial a r c s of e x t r e m a l s , issuing from A, may partly coincide. This happens when C(x , t ), the collection of possible
a a
c o u r s e s in A, does not consist of the entire interval [0,2Tr] , so that the effective indicatrix has points where the tangent is not continuous. Let, in figure 3. 7.b, F be such an a r c of partly coinciding e x t r e m a l s . Ill a point K of that curve, the value of i|i is not quite determined, because it depends on which neighbouring extremal F is compared with.
Let a, correspond with a bending point like Q in figure 3. 7. a and let i|i' and \|i"
Figure 3. 7.b.
be the vectors perpendicular to the left and right tangents to the indicatrix of K in such a bending point.
If 9 ' and 9 " a r e the directions of i)i' and i|i'' respectively, it is clear that the tangent to the timefront S (t) in K bends abruptly from 9 ' +,rr/2 to 9 " + 11/2 .
Remark 2:
Along every e x t r e m a l the maximum value of H(a,x,ili, t), equal to M(\li,x,t) satisfies the relation
(3.7.2) dM 5M r,
To prove this, consider the three possibilities for a in a point of an e x t r e m a l trajectory:
a. a is continuously variable and f is partially differentiable with r e s p e c t to a. We then know, that for H = M:
ÖH _ 2 5fi
ha i=i 1 So; and so:
dM dt
2 <**i 4 2 5fi dxJ ^ SM E -TT- t + E 11. r —TT + i=l dt i , j = l ' i g ^ dt St With (3. 3.1) and (3. 6.17) we find that
dM dt - V / 9f , fi ^ , Sf^ J , ^ aM 1,1=1 Av' 3 1 ^J ax-'
at
so dM ^ aM dt atb . If a is constant - compare r e m a r k 1 - then there simply is no change in a as a r e s u l t of a change of t, so the same relation holds.
c. If a jumps discontinuously from a^ to a , we have the same maximum of H for a^ and a„, so in this case t h e r e is no change in H as a r e s u l t of a
1 z
continuous change in \|( and x.
For an autonomous system: x = f(x,a) it now follows that the maximum value of H is constant along an e x t r e m a l .
Remark 3:
The Hamilton function H enables us to write (3. 3.1) and (3. 6. 2) in a canonical form:
.i an
'^ - a*
(3.7.3) \ (i = l , 2 ) •, _ 3H^ i - - - T
ax
I now wish to enUghten this procedure by means of some elementary examples:
Example 1:
On the X-axis a moving point has at a time t = t a position x with r e s p e c t to the origin and a velocity y along the X - a x i s . The movement of this point along
o the X-axis can be controlled by:
X = a with the limitations - 1 = a ^ 1 .
The assignment is to choose a as a function of t, so that the moving point a r r i v e s at the origin with velocity z e r o in the least possible time.
Introducing the velocity as a new variable y = x, the equations of motion a r e :
( 3 . 7 . 1 . 1 ) | _ x = y
y = a
AppUcation of (3. 3.1) and (3. 6. 2) gives 42
r *i
(3.7.1.2) I /
= O
*2 " ~ *1 ' '^^^^^ H = 1)1^ y + 1II2 a
H is maximized by taking a = sign {•ij.) .
Taking the a r r i v a l time t^ = 0 and putting \)i(0) = (cos 9 , sin 9 ), the solution of (3. 7 . 1 . 2) becomes
if At) = c o s 9 ,
(3.7.1.3) I ^ °
iji (t) = sin 9 - t cos 9 (t = 0)
Concentrating on the S -timefronts, various possibiUties a r e considered:
a. 0 ^ 9 ^ Ti/2 .
^o
For t < 0 we see that i|;_ = sin 9 - t cos 9 > 0, so o(t) = +1. F o r this 9 - s e c t o r , the S -timefront consists of merely one point:
(t2/2, t), solution of (3. 7.1.1) for a = 1 . b . TT/2 < 9 < TT .
^o
Now i)(„ = sin 9 - t cos 9 is positive for tg 9 < t ^ 0 and negative for t < tg 9 . So, proceeding in t i m e , we have:
a(t) = - 1 for t < tg 9 and G'(t) = +1 for tg 9 < t < 0 . Solving (3. 7 . 1 . 1 ) , we find:
x = t ^ / 2 , y = t for 1 6 [tg 9^ , 0 ] ,
X = - t g 9^ + 2 t t g 9 ^ - t / 2 , y = 2 t g 9 ^ - t for t e [ t ^ , t g 9 ^ ] . Eliminating 9 , we find the equation for S (t) : 4 x + (y - t) = 2 t . At the time t* = tg 9 , the switching from 01 = -1 to a = + 1 takes place.
C. TT ^ 9 ^ 3 T T / 2 .
\|i„ = sin 9 - t cos 9 is negative for t < 0. So a(t) = - 1 maximizes H(a,x, •>;). CI O O
The trajectory is (-t2/2, -t) and the S -timefront is just this one point. d.i 3rr/2 < 9 < 2 n .
^o
As \(i„(t) = sin 9 - t cos 9 is negative for tg 9 < t ^ 0 and (li„(t) > 0 for
z o o o z
t < tg 9 , we find:
x(t) = tg2;p^ - 2 t tg 9^ + t^/2 , y(t) = - 2 tg 9^ + t for t^ < t < tg 9 ^ . The switching point is (-g tg 9 , - tg 9 ) , the switching time is t* = tg 9 and a jumps from +1 to - 1 .
2
F o r t e [tg 9 , 0] we have x = - j t , y = - t .
- " 2 2 The S -timefront a r c has the equation 4 x - ( y + t) = 2 t .
find t = - 6 , tg 9 o ' ^ ^o - 6 i t ^ - 2:
- 2 ^ t ^ 0:
- 2 , and the optimal track i s :
s = -t
.x = - 5 t ^ - 4 t - 4 4 ,{
X = è t ^ y = t Figure 3. 7.c. Example 2:In a 2-dimensional plane with an orthonormal coordinate system, the velocity field is defined as follows:
In a direction a with r e s p e c t to the X -direction the velocity is v I sin a I . (v > 0). The problem is to find the least time track from a given point A to another given point B.
The velocity components in the X - and Y -directions a r e : . X = V I sin a\ cos a , f ol ' 1 y = V I sin a \ sin a
The adjoint system i s awfully simple: ill.. = ili„ = 0 with solutions \|i = cos 9 \|i„ = sin 9 , where 9 is a constant angle.
In this c a s e we have to maximize
H = V I sin a I cos a cos 9 + v I sin a I sin a sin 9
ol I ^o o 1 I ^ o = V I sin a I cos (a - 9 ) .
o I I o F o r various values of 9 we find:
^o
a. 9 = O : H ^o max for a = TI/4 or 7Tr/4 b . 0 < 9 < T i : H = 2 V ( 1 + sin 9 ) for a =.v/4 + 5 9 . ^o max o' ^o " ^o c . 9 = T T : H = è v for a = 3Tr/4 or 5 n / 4 . ^o max ^ o d. n<cp < 2 T I : H = 2 V ( 1 - sin 9 ) for a = 3Tr/4 + 5 9 . ^o max o ^ o ' •* ^o Taking the origin in A, the S (t)-timefronts can be described by: a. 9^ = 0 : X = i v^t , - z v^t g y < i v^t . b. 0 < 9 < T T : x = è v t cos 9 , y = 5 v t ( l + sin 9 ) . ^ o •^ o ^ o -^ o ^ o c. 9^ = n : X = - è v^t , - è v^t ^ y = è v^t . d. TT<9 < 2 r r : x = 5 v t cos 9 , y = 5 v t (sin 9 - 1) . Figure 3. 7 . d .
Figure 3. 7. d shows the extremal tracks for various c a s e s . Denoting the direction of AB as 0 , we find:
1. If 7 n / 4 ^ B < 2 T r or 0 ^ p $ n / 4 , then the extremal exists of a number of Une segments with direction TI/4 and 7TI/4. The tracks A P . B and AQ. B .
give the same a r r i v a l time at B. and so does any track along the periphery or inside the rectangle AP B^Q^, as long as it leads from A to B. in courses of 45 and 315 alternatively.
2. If Ti/4 < g < 3TI/4 , a = B and the track is one straight Une.
3. For 3Tr/4 i B = 5 n / 4 , the extremal track consists of line segments with directions 135 and 225 . The number of 90 -left or -right turns can be finite (one a t least) or even countably infinite. This h a s the practical sense that this number of turns can be as large as one could possibly Uke.
4. If 5 n / 4 < B < 7 n / 4 then a = g . The e x t r e m a l track is again a straight Une.
This example can be appUed with sUght modifications to a head - to - wind saiUng ship. It shows that a windward position can sometimes better be reached by tacking up against the wind than by steering a constant c o u r s e .
3 . 8 . R e m a r k s on t h e o c c u r r e n c e of m o r e t h a n o n e s o l u t i o n .
In regions of small values of | f | the speed d e c r e a s e causes a d e c r e a s e of the distance between two consecutive timefronts S (t) and S (t+fit). As these regions, in the c a s e of ship routeing, very frequently have a m o r e or less elUptic s t r u c -t u r e , -they also cause a convergence of -the e x -t r e m a l s . The r e s u l -t of -these -two phenomena often consists of a s e c t o r of ambiguity, i. e. m o r e than one solution of the system of equations (3. 3.1) and (3. 6.17) combined with the maximum principle, with A as starting point and a point in this sector as point of a r r i v a l . This phenomenon will now be explained at the hand of an elementary example. Let the starting point A have coordinates (- 3 , 0) and let the velocity compo-nents be given by:
(3. 8.1) X = (2 - exp(- èx^ - iy^))cos a , 2 2
y = (2 - exp(- 5X - Jy ))sin a , a € [ 0 , 2 TT] .
This impUes that in every point of the X-Y-plane the velocity indicatrix is a 2 2
c i r c l e with radius r(x,y) = 2 - exp( - gx - hy )• This radius has a minimal length in the origin,
Min r(x,y) = r ( 0 , 0 ) = 1 , (x,y)
and r is obviously constant on a c i r c l e with (0,0) as c e n t r e . Applying the maximum principle, the form
I
H(\lij^,\(i2.x,y,a) = {if^ cos a + 1II2 sin a) r(x,y) (3.8.2)
h a s to be minimized for given values of \|i^, if„, x and y. The adjoint variables i|i^ and (|i change with time according to:
^ = - x(2 - r)(i(i cos a + 'I'o ^^^ °') >
(3.
8. 3) {
• 2 " " y(2 - r){ij^ cos a + ii^ sin a)
Writing i|i. = m sin(Tii/2 - 9) with 1 = 1,2, it follows that H is maximal for 0 = 9. Thus the optimal dynamical system is described by:
2 2 (1) X = (2 - exp(- i x - | y ))cos 9 ,
2 2 (2) y = (2 - exp(- Jx - èy ))sin 9 ,
2 2
(3) 9 = exp(- 5X - zy )(x sin 9 - y cos 9) , 2 2
(4) m = - m exp(- Jx - gy )(x cos 9 + y sin 9) (3.8.5)
with starting conditions x(0) = - 3, y(0) = 0, 9(0) = 9^ and m(0) = l / ( 2 - exp(- 4i)) The fact that H = (2
field, implies that
2 2
The fact that H = (2 - exp(- gx - gy ))m is constant in this stationary velocity
(3.8.6) m = - ^
2 2 2 2 2 - exp(- gx - èy ) 2 - exp(- Jx - gy )
By taking various initial values, like 9 : = 0(0. 01)1. 20 the equations (3. 8. 4) can be integrated numerically by means of a Runge-Kutta method with fit = 0 . 1 as a sufficiently s m a l l step width. This was c a r r i e d out with the aid of a TR-4 computer and for t = 1(1)12 the values of x, y, 9 and m were printed out. The result is exposed in figure 3. 8. a.
As for a quaUtative explanation, it can be r e m a r k e d that m = I \(i| is constantly piroportional to | grad S|, where S(x, y, t) = 0 is the equation of a timefront S (t). According to ( 3 . 8 . 6 ) , m has a maximum value 1 for (x,y) = (0,0). This means
+ + that the distance between two subsequent timefronts S (t) and S (t + fit) is minimal in the origin. This explains that the timefronts become less convex n e a r the X-axis as t i n c r e a s e s .
The overlapping of the timefronts can be explained by thinking of a timefront S (t + fit) to originate from S (t) as the envelope of the fit-position-indicatrices, drawn for all points of S (t).
Building up the subsequent timefronts this way and reminding that the radius of
Figure 3 . 8 . a.
Figure 3 . 8 . b .
the indicatrix, r(x,y)6t for a fixed value of x is minimal for y = 0, there is a fair possibiUty that the timefront through 0 has a z e r o curvature in 0 or even a concave part for - y ^ y ^ y . In that c a s e the next timefront has a deeper dent, which is symmetric with r e s p e c t to the X - a x i s .
Let us consider a small line element p * p p * * of S (t) near the X - a x i s , with P on the X-axis and the distance P P * = P P * * small enough to be able to a s s u m e r(x,y) equal in P , P* and P**.
Now if the curvature radius of S (t) in P is equal to r(Xp,yp)6t, the trajectories through P , P* and P** coincide in Q, the curvature c e n t r e of S (t) for P . Q is known to be called the conjugate point to A for the central extremal, with 9 = 0 .
^o
•p**
Figure 3. 8. c .
The solutions of (3. 8. 4) with fixed x , y and variable 9 as starting values for x(0), y(0) and 9(0) a r e functions of t and 9 . In Q the partial derivatives of x and y with r e s p e c t to 9 a r e z e r o , while for all points on the Une through Q parallel
° av
to the Y-axis the derivative = ^ is positive. 390
Turning back to the approximative construction, it is now obvious that the two parts of S (t + fit) on each side of the X-axis a r e transformed, by a next time i n c r e a s e fit, to the intersecting fronts R*S* and R**S**; The frontal a r c b e tween R* and R** is formed by the "t + 2 fitpoints" of e x t r e m a l s that i n t e r -sected the timefront through P between P* and P**.
If the curvature radius of S (t + fit) in Q is equal to r(x„,yQ)fit, then R* and R** a r e the conjugate points to A for the extremals through P* and P** r e s -pectively.
The further propagation of the timefronts i s analogous. For t* > t + fit they apparently have a double point on the central " e x t r e m a l " through Q and turning
