Applications of Calculus of Variations to Aircraft and Spacecraft Path Planning

Andrea L’Afflitto* and Cornel Sultan† Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University

The problem of finding the trajectory for an aerospace vehicle moving between two fixed points is investigated using Calculus of Variations (CV), which can provide necessary and sufficient conditions for trajectories that optimize a cost index. Within the framework of a systematic study aimed at tackling analytically the trajectory generation problem, this paper presents applications of the CV to find trajectories that optimize kinetic energy, energy consumption, and fuel consumption considering several environmental conditions.

Nomenclature a = acceleration of the vehicle ac = acceleration of the vehicle due to control forces f = generic function fx = derivative of f with respect to the second component fr = derivative of f with respect to the third component g = gravitational acceleration J = optimization/cost/performance index k1, k2… = integration constants r = position vector t = time t1, t2 = initial and final time u = control vector v = velocity vector v1, v2 = initial and final velocity x = state vector x1, x2 = initial and final state y, z = generic piecewise smooth (PWS) functions λ1, λ2… = multipliers μ = gravitational parameter

I. Introduction alculus of variations is the branch of mathematics concerned with finding extrema of functionals. Although the first systematic studies in this field are dated back to the eighteenth century, a new interest raised in the last

forty years leading to great advances in the area of optimization1,2. The problem of finding the optimal trajectory for a spacecraft or aircraft moving between two given fixed positions can be tackled by the theory of the Simplest Problem of Calculus of Variations (SPCV), also known as the Fixed End Problem. With the advent of the computer era several approaches to this problem, which involve selecting arbitrary parameterizations for the trajectories and searching for purely numerical solutions, have been attempted2. The main disadvantage of these techniques is that the selected parameterizations often have no relevance to the performance index of interest. Calculus of Variations

C

* Ph.D. Student, Department of Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall, Blacksburg, VA 24061, andrea.lafflitto@vt.edu, AIAA Space Logistics TC Associate Member, AIAA Senior Member. † Assistant Professor, Department of Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall, Blacksburg, VA 24061, csultan@vt.edu, AIAA Senior Member.

1 American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-6073

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

can be used to construct trajectory parameterizations that optimize a certain performance index eliminating the heuristics associated with selecting arbitrary parameterizations.

In this paper we shall show how, using necessary and sufficient conditions provided by SPCV, we can find the trajectory of a vehicle modeled as a point mass moving between two given fixed positions such that certain performance indices ( )J ⋅ are minimized. In some circumstances it is necessary to directly find the control that achieves an optimal trajectory (the optimal control problem). This paper shows how to use further tools provided by the calculus of variations such as Pontryagin’s minimal principle to address this problem. It is important to notice that, as Pontryagin’s principle generally provides necessary conditions3, a set of candidate optimal trajectories only is found. The actual optimal trajectories may be (in some cases) found by verifying the sufficient conditions provided by the SPCV.

Presenting the mathematical background and results concerning the optimization of the kinetic energy, consumed energy, and fuel consumption of the vehicle, this paper illustrates the rigorous application of the calculus of variations to trajectory optimization problems in the framework of a systematic study using an analytical approach.

II. Historic and Scientific Context It is commonly assumed that the roots of the calculus of variations date back to 1638 when G. Galilei addressed

the problem of finding the trajectory that minimizes the time of descent of a point mass moving between two given points placed at different height under the effect of the gravitational acceleration (brachistochrone problem5). The first systematic solution to this problem was found by Johann and Jakob Bernoulli.

The general approach to this class of problems aimed at minimizing a cost index was established by Euler and Lagrange in the second half of the 18th century and it still constitutes the basis for most of the contemporary optimization problems. Further contributions were then brought by Legendre (1786), Jacobi (1838), Weierstrass (1927) and Caratheodory (1935), which were crucial in assessing the theory of the SPCV and constituted the cornerstones for the general problem of calculus of variations formulated by Bolza3,4 and its solution in presence of constraints known as Pontryagin’s minimal principle5 (1960s). Among the most recent and relevant contributions in finding necessary and sufficient conditions to the constrained problem of calculus of variations it is worth to mention the works of Zeidan6 (1994) and the one of Loewen and Rockafellar7 (1996).

III. Mathematical Background Among the piecewise smooth functions‡ (PWS) ( ) [ ]1 2: , nx t t⋅ ⊂ → , we define the set

( ) ( ) ( ) ( ) 1 2 1 1 2 2, | ,x PWS t t x t x x t xΩ = ⋅ ∈ = = where x1 and x2 are given. The function ( )x ⋅ will be further referred

to as the state vector. We define the cost index§ ( )J ⋅ ,

( )( ) ( ) ( )( ) ( ) ( )(2

1

1 1 2 2, , , , , 't

t

)J x g t x t t x t f t x t x t⋅ = + ∫ dt (1)

where ( ), ,f ⋅ ⋅ ⋅ and ( ), , ,g ⋅ ⋅ ⋅ ⋅ are real valued functions which allow for the ensuing operations to be performed and

( )'x ⋅ is the first derivative of ( )x ⋅ with respect to the independent variable, t in this case.

‡ A function [ ]1 2: , nx t t → is called piecewise smooth if for each component ( )ix ⋅ there exist a piecewise

continuous (PWC) function ( )i ⋅g and a constant ci such that for all [ ]1 2,t t t∈ , ( )i i ( )1

t

it

x t c g s ds= + ∫ . A function

[ ]1 2: ,g t t → is piecewise continuous on [ ]1 2,t t if ( )g ⋅ is bounded on [ ]1 2,t t , the right hand limit ( )g t + exists

and is finite on [ , the left hand limit )1 2,t t ( )g t − exists and is finite on ( ]1 2,t t and there is a finite partition of [ ]1 2,t t ,

such that 1 1 ... mt t= < < 2t t= ( )g ⋅ is continuous on each subinterval ( )1ˆ ˆ,i it t− .

§ Given a function of n variables, it should be reported as ( ,...,q ⋅ ⋅) [ ]( )1...T

nq w w . For the sake of compactness it

will be expressed as . ( )nw1,...,q w

2 American Institute of Aeronautics and Astronautics

The integral in (1) and all the integrals in this paper are meant to be Riemann integrals. A tractation of the calculus of variations with Lebesgue integrals leads to more relaxed results that are beyond the scope of this work.

The scope of the problem of Bolza8 is to find a function ( )*x ⋅ in the set Ω that minimizes ( )( )J x ⋅ : ( )( ) ( )( )*J x J x⋅ ≤ ⋅ (2)

for all . ( )x ⋅ ∈Ω

If we assume that the terminal cost function ( ), , , 0g ⋅ ⋅ ⋅ ⋅ = , finding a function ( )*x ⋅ in the set Ω such that (2) is

satisfied is called the problem of Lagrange. Assuming that ( ), , 0f ⋅ ⋅ ⋅ = , the problem of Bolza reduces to the problem of Mayer. The problem of Lagrange will be dealt hereafter. The problem of Mayer and the problem of Bolza are aimed at penalizing certain terminal states in spite of others, which is quite useful in the presence of conservative vector fields**. Considering that for this paper the initial and final states are always assigned, solving the problem of Lagrange is sufficient.

We introduce a norm on Ω as

( ) ( ) 1 2

2sup

t t tx t x t

≤ ≤= (3)

where 2⋅ denotes the Euclidean norm. Given ( )y ⋅ and ( )z ⋅ in the set Ω, we define two metrics3,9,10

( ) ( )( ) ( ) ( )0 ,d y z y z⋅ ⋅ = ⋅ − ⋅ (4)

( ) ( )( ) ( ) ( )( ) ( ) ( )1 0, , 'd y z d y z y z⋅ ⋅ = ⋅ ⋅ + ⋅ − ⋅' . (5) Given 0δ > , the ( )(0 ,U x )δ⋅ -neighborhood of ( )x ⋅ is the open ball

( )( ) ( ) ( ) ( ) ( )( )0 1 2 0ˆ, , | ,U x x PWS t t d x x ˆδ δ⋅ = ⋅ ∈ ⋅ ⋅ < (6)

while the ( )(1 ,U x )δ⋅ -neighborhood of ( )x ⋅ is the open ball

( )( ) ( ) ( ) ( ) ( )( )1 1 2 1ˆ, , | ,U x x PWS t t d x x ˆδ δ⋅ = ⋅ ∈ ⋅ ⋅ < (7)

If (2) yields for all , then ( )x ⋅ ∈Ω ( )*x ⋅ provides a global minimum. If (2) yields for all

( ) ( )(0 * ,x U x )δ⋅ ∈Ω∩ ⋅

( ) ( )( )1 * ,x U x

, then provides a strong local minimum while if it yields for all ( )*x ⋅

δ⋅ ∈Ω∩ ⋅ , then provides a weak local minimum. ( )*x ⋅

The Euler Necessary Condition (ENC) for a weak, as well as strong local minimum1,4 claims that if ( )*x ⋅ is a

weak/strong local minimum for on Ω, then ( )(J x ⋅ )

E1. There exists a constant nc∈ such that for all [ ]1 2,t t t∈ ⊂ ,

** Given a set , consider a function . This function can be defined as vector field of basis A. If

there exists a scalar function

nA ⊆ : mf A →

( )g ⋅ such that ( )( ) ( )( )g xf x ⋅ = −∇ ⋅ , then the field ( )f ⋅ is conservative. One of the properties of conservative fields is that the path integral along any curve that joins two fixed points P1 and P2 in A does not depend on the path but on P1 and P2 only.

3 American Institute of Aeronautics and Astronautics

( ) ( )( ) ( ) ( )(1

'* * * *, , , ,

t

r xt

)'f t x t x t c f s x s x s ds= + ∫ (8)

E2. ( )* 1 1x t x= and ( )* 2 2x t x= .

E3. Between corners††, the function ( ) ( )( )'* *, ,rf t x t x t is differentiable and

( ) ( )( ) ( ) ( )('* * * *, , , ,r x

d )'f t x t x t f t x t x tdt

= (9)

Any function ( )x ⋅ that verifies eq. (8) is called extremal. Note that condition E2 follows from the fact that we

are solving a fixed end-point problem and it is imposed by all the theorems presented in this paper. The ENC is equivalent to the following theorem1,4: let ( ) ( ) ( ) ( ) 0 1 2 1, | 0V h PWS t t h t h t2= ⋅ ∈ = = be the set of

admissible variations. If is a weak/strong local minimum for ( )*x ⋅ ( )( )J x ⋅ on Ω, then

( ) ( )( ) ( ) ( )( )* *0

, dJ x h J x hd ε

δε =

⋅ ⋅ = ⋅ + ⋅ = 0ε

)

(10)

for all , where is called the first variation of J at ( ) 0h V⋅ ∈ ( ) ( )( * ,J x hδ ⋅ ⋅ ( )*x ⋅ in the direction of . ( )h ⋅

Next, we introduce the Weierstrass Excess Function as ( ) ( ) ( ) ( ) (, , , , , , , , ,T

r )E w y r u f w y u f w y r u r f w y r= − − − (11)

where . ( ), , , n n nw y r u ∈ × × ×

The Weierstrass Necessary Condition1 (WNC) states that if ( )*x ⋅ is a strong local minimum for ( )(J x )⋅ on Ω, then

W1. ( ) ( )( )'

* *, , ,E t x t x t u ≥ 0 for all [ ]1 2,t t t∈ and nu∈ .

The Legendre Necessary Condition1 (LNC) states that if ( )*x ⋅ is a weak local minimum for on Ω, then ( )(J x ⋅ )

L1. ( ) ( )( )'* *, ,rrf t x t x t 0

where the curly inequality sign indicates that the matrix ( ), ,rrf ⋅ ⋅ ⋅ is positive semidefinite.

If we have that (positive definite matrix) this condition becomes the strengthen LNC. ( ) ( )( '* *, ,rrf t x t x t ) 0

If for all , then ( ), , 0rrf w y r ( ), , n nw y r ∈ × × ( ), ,f w y r is called regular. Furthermore if ( ) ( )1, 2x PWS t t⋅ ∈

is an extremal and if 0 at all points where ( ) ( )( ), , 'x t x trrf t ( )'x ⋅ is defined, then ( )x ⋅ is called regular. It can be

proved that if ( ), ,f w y r is regular, all extremals do not have corners3.

Lastly, the Jacobi Necessary Condition1 (JNC) states that if ( )*x ⋅ provides a weak local minimum for ( )( )J x ⋅

on and if is smooth and regular, then Ω ( )*x ⋅

†† A function ( )x ⋅ defined on the interval [t1,t2] has a corner in ( )1 2

ˆ ,t t t∈ if ( ) ( )ˆ ˆt t t t

dx t dx tdt dt

+ −

= =

≠ .

4 American Institute of Aeronautics and Astronautics

J1. There cannot be a value tc conjugate to t1 such that tc<t2.

If condition J1 yields for , then the JNC is called strengthen JNC. 2ct t≤

A value tc is said to be conjugate to t1, if t1 < tc < t2, and there is a nonzero solution ( )cη ⋅ to the Jacobi equation

( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )' ' ' '* * * * * * * *, , , , , , , ,xr c rr c xx c xr c

d ' 'f t x t x t t f t x t x t t f t x t x t t f t x t x t tdt

η η η+ = + η (12)

such that and ( )1 0c tη = ( ) 0c tη ≠ for any t in the interval ( )1, ct t . If tc exists then ( ) 0c ctη = .

Note that ENC and WNC are related to the first variation of ( )( )J x ⋅ while LNC and JNC are related to its second variation.

Two sufficient conditions can be formulated1,3: S1. If ( )*x ⋅ ∈Ω is smooth and satisfies condition ENC, the strengthen LNC, and the strengthen JNC, then

( )*x ⋅ provides a weak local minimum for ( )( )J x ⋅ on Ω. S2. If the function ( ), ,f ⋅ ⋅ ⋅ is regular, the ENC and the strengthen JNC hold, then ( )*x ⋅ provides a strong local

minimum for ( )( )J x ⋅ on Ω. Unfortunately there aren’t efficient necessary and sufficient conditions to identify global minima. The only

known operative condition is the following1: S3. Suppose that the integrand of the cost function is convex. If a function ( )*x ⋅ ∈Ω satisfies conditions S1

and S2, then ( )*x ⋅ provides a global minimum for ( )( )J x ⋅ on Ω. The theory reported in the above applies when ( ), ,f ⋅ ⋅ ⋅ is function of the time t, of ( )x ⋅ , and ( )x′ ⋅ . This can be a

limitation that can be addressed if we extend the theory as follows. Consider the cost index

( )( ) ( ) ( ) ( ) ( )(2

1

, , ' ,...,t

m

t)J x f t x t x t x t⋅ = ∫ dt (13)

where ( ) ( )mx ⋅ represents the m-th derivative of ( )x ⋅ with respect to the independent variable. In these circumstances the Euler Necessary Condition for a weak/strong local minimum, also known as Euler – Poisson Necessary Condition (EPNC), claims that if ( )*x ⋅ is a weak/strong local minimum for on Ω, then ( )(J x ⋅ )

E4. Between corners, functions ( ) ( ) ( ) ( )( ).., m, ,.j

f t x t x tx∂

∂ are differentiable, and

( )( ) ( ) ( )( )

( )0

, ,...,1

mm jj

j jj

f t x t x tddt x=

⎛ ⎞∂⎜− ⎜ ∂⎜ ⎟⎝ ⎠

∑ 0⎟ =⎟ (14)

E5. ( ) ( ) ( )

* 1 1j jx t x= and ( ) ( ) ( )

* 2 2j jx t x= for 0,1,..., 1j m∈ − , where ( )

1jx and ( )

2jx are given.

The Legendre Necessary Condition now states that if ( )*x ⋅ is a weak local minimum for ( )(J x ⋅ ) on Ω, then

5 American Institute of Aeronautics and Astronautics

L2. ( ) ( ) ( )( )

( )( )2

2

, ,...,0.

m

m

f t x t x t

x

∂

∂

As for L1, if the curly inequality sign is strictly positive, then the strengthen LNC is verified.

Again, if ( )( )

( )2

1 12 , ,..., 0mm

f w y yx

+

∂

∂ for all ( )1 1, ,..., ...n n

mw y y + ∈ × × × , then is called

regular. Furthermore if

( )1, ,..., mf w y y +1

( ) ( )1 2,x PWS t t⋅ ∈ is an extremal and if ( )( )

( )(2

2 , ,...m

f t x tx

∂

∂

( ) ( )), 0mx t at all points

where ( )'x ⋅ and the other higher derivatives are defined, then ( )x ⋅ is called regular. It can be proved that if

is regular, all extremals do not have corners. ( 1,...,y )1, mf w y +

If ( )( )

(2

1 12 , ,..., 0mm

f w y yx

+

∂≠

∂) for all ( )1 1, ,..., ...n

mw y y +n∈ × × × , then ( )1, ,..., mf w y y +1 is called non-

singular. Furthermore, if ( ) ( )1 2,x PWS t t⋅ ∈ is an extremal and if ( )( )

( ) ( ) ( )( ,..., mx t x t ≠), 0f t2

2mx

∂

∂ at all points

where ( )'x ⋅ and the other higher derivatives are defined, then ( )x ⋅ is called non-singular Lastly, the Jacobi Necessary Condition1 (JNC) has to be reformulated as follows. If ( )*x ⋅ provides a weak local

minimum for on and if ( )(J x ⋅ ) Ω ( )*x ⋅ is smooth and regular, then the accessory (secondary) minimum problem is: minimize

( )( ) ( ) ( ) ( ) ( )(2

1

', , ,...,t

m

t)J F t t t t dtη η η η⋅ = ∫ (15)

where ( ) ( )21 11 1

1 11 1

, ,...,1, ,...,2

m mm T

m j kj k j k

f tF t

χ χχ χ χ

χ χ

+ ++

+= =

⎛ ⎞∂= ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠∑ ∑ χ . Any ( ) ( ) ( ) ( ) 1 2 1, | 0PWS t t tη η η⋅ ∈ ⋅ ∈ = that

minimizes (15) is called secondary extremal. Assuming that ( )η ⋅ is a non-singular extremal for the accessory equation, it must satisfy the Euler – Poisson Necessary Condition applied to the accessory problem:

( )( ) ( ) ( ) ( )( )

( )

'

0

, , ,...,1

mm jj

j jj

F t t t tddt

η η η

η=

⎛ ⎞∂⎜− ⎜ ∂⎜ ⎟⎝ ⎠

∑ 0.⎟ =⎟ (16)

A value tc is said to be conjugate to t1, if t1 < tc < t2 and if there is a nonzero solution ( )cη ⋅ to (16) such that

( )1 0c tη = and ( ) 0c tη ≠ for any t in the interval ( )1, ct t . If tc exists, then ( ) 0ηc ct = . Furthermore, ( ) ( ) ( ) (j )1t 1,1 *

jc t xη = for . ..., 1j m∈ −

As before, the Jacobi Necessary Condition states that there cannot be a value tc conjugate to t1 such that tc<t2. Again, the Strengthen JNC imposes that there cannot be a value tc conjugate to t1 such that . 2ct t≤ The sufficient conditions (S1-S3) previously presented remain unaltered11.

Further advances in Calculus of Variations lead to more general necessary conditions provided by Pontryagin’s minimal principle. As such, these conditions are not enough to establish whether a trajectory is truly a minimizer for

6 American Institute of Aeronautics and Astronautics

the assigned cost index but they are useful to determine a set of candidate minima; just in some particular cases these conditions become sufficient as well.

Next we describe one version of Pontryagin’s minimal principle3,12-14. Consider the bounded control function and an autonomous system which constrains the state and controls, ( ) : mu A⋅ ⊂ → Γ ⊂

( ) ( ) ( )( )0' ,x t f x t u t= (17)

over a finite, fixed time interval [ ]1 2,t t

( )( )1u t =

. The initial and final states are also given, x1 and x2, respectively. Define Δ as

. The optimal control problem is to find the

optimal (( ) ( ) ( )( ) 1 2 1 1 2 2 2, | , , , ,u PWC t t x t x x t u t x uΔ = ⋅ ∈ = ∈Γ

control )* ⋅ ∈Δ that minimizes the funu ctional

( )( ) ( ) ( )(2

1

,t

t

)J u f x t u t⋅ = ∫ dt (18)

subject to (17). The state ( )x ⋅ associated to ( )*u ⋅ is denoted as ( )*x ⋅ .

It is worth to stress that, in this version of Pontryagin’s principle, the integrand of (18) does not contain the independent variable t explicitly and the cost index is assumed to depend on ( )u ⋅ only. Moreover, ( ) ( )( )0 ,f x u⋅ ⋅

and can be only continuous in ( ) ( )( ,f x u⋅ ⋅ ) ( )u ⋅ . They must, however be differentiable in ( )x ⋅ .

We introduce called the costate vector and the Hamiltonian function ( ) ( ) ( )0 1ˆ ...

Tnλ λ λ λ⋅ = ⋅ ⋅⎡⎣ ⎤⎦

)

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )(0 01

ˆ , , , ,n

j jj

H t x t u t f x t u t t f x t u tλ λ λ=

= +∑ (19)

or equivalently as

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )0 0ˆ , , , ,TH t x t u t f x t u t t f x t u tλ λ λ= + (20)

where . Let’s introduce the system of differential equations ( ) ( ) ( )1 ...

Tnλ λ λ⋅ = ⋅ ⋅⎡⎣ ⎤⎦

( )

( ) ( )( ) ( )( )

'0

0'

0

0

,, 1,...,

nj

i jj i

t

f x t u tt t i

x

λ

λ λ=

=

∂= − ∈

∂∑ n

)

(21)

or equivalently

( )

( )( ) ( )(

'0

'

0ˆ , ,

t

H t x t ut

x

λ

λλ

=

∂= −

∂

(22)

where ( ) ( )(0 ,jf x u⋅ ⋅ ) is the j-th component of the vector ( ) ( )( )0 ,f x u⋅ ⋅ and ( ) ( )( ) ( ) ( )( )00 , ,f x u f x u⋅ ⋅ = ⋅ ⋅ .

We also define the function

( ) ( )( ) ( ) ( )( )ˆ ˆ, min , ,u

.M t x t H t x t uλ λ∈Γ

= (23)

7 American Institute of Aeronautics and Astronautics

Pontryagin's minimal principle states that if ( )*u ⋅ is an optimal control in Δ on [ ]1 2,t t with corresponding

trajectory , then there exists a non-zero costate vector ( )*x ⋅ ( )*λ ⋅ solution to (21) (or equivalently (22)), such that

(17) and the boundary conditions are satisfied, ( ) ( )( )( ) (( ) )( )*ˆ

* *ˆ , , * ,*H t x t M=t uλ t x tλ almost everywhere and

( )( * ( )*, )M t x tλ is constant everywhere. The value of this constant has to be determined according to the problem.

The boundary conditions to (17) are ( )* 1 1x t x= and ( )* 2 2x t x= . For this formulation of Pontryagin’s principle, the

boundary conditions to (21) (or (22)) are such that ( )1* tλ and ( )2* tλ can be arbitrary.

Once a candidate optimal control and the corresponding trajectory ( )*u ⋅ ( )*x ⋅ have been determined, the actual optimal trajectory may, in some situations, be found by verifying the sufficient conditions provided by S1-S3.

It is crucial to make some observations regarding the application of Pontryagin’s principle:

P1. If the Hamiltonian is differentiable with respect to u, (23) requires that ( ) ( )( )

*

*ˆ , ,

0

u u

H t x t u

u

λ

=

∂=

∂.

P2. In addition it must hold that ( ) ( )( )*

2

2ˆ , , 0u . Note that this process only allows the

identification of the minima in the interior of Γ. u u

H t x tu

λ=

∂∂

P3. The optimal control ( )*u ⋅ could be on the boundary of Γ, in which case minimizing ( ) ( )( )ˆ , ,H x uλ ⋅ ⋅ with

respect to u requires additional analysis. This is also the case when there are points/sets interior to Γ where

)( ) ( )( ˆ , ,H x uλ ⋅ ⋅ is not differentiable.

Other situations might also occur but are beyond the scope of this paper. In the examples presented herein we

shall restrict the discussion to situations in which ( ) ( )( )ˆ , ,H x uλ ⋅ ⋅ is differentiable with respect to u and the

controls are restricted to an open and bounded set.

IV. Applications – Cost Indices Optimization In this section we shall illustrate applications of the previous theory. Specifically, we shall examine the optimal

trajectory for the kinetic energy, consumed energy, and fuel consumption of an aerospace vehicle schematized as a point of constant mass. Several environmental conditions will be considered: constant gravity (the vehicle moves in a constant gravitational vector field), constant gravity with aerodynamic forces (the constant gravity model is improved considering also aerodynamic forces acting on the vehicle), radial gravity (the vehicle moves in the gravitational field generated by a central massive point mass under Keplerian assumptions‡‡,16) and radial gravity with aerodynamic forces (the radial gravity environment is improved considering aerodynamic forces).

A. Optimization of the Kinetic Energy For a point mass model, if ( )r ⋅ is the position vector of the vehicle in an inertial reference frame, ( )v ⋅ , ( )a ⋅ and

( )F ⋅ the corresponding velocity, acceleration, and resultant force respectively, the equations of motion (i.e. the

translational equations) are ( ) ( ) ( ) ( )2

2

dv t d r tF t ma t m m

dt dt= = =

where m is the mass of the vehicle.

A measure of the kinetic energy can be defined as15:

‡‡ A central massive body is assumed to be reduced to a point mass and to generate a gravitational field. The mass of the vehicle moving in this field is assumed to be negligible with respect to the mass of the central body.

8 American Institute of Aeronautics and Astronautics

( ) ( )2

1

tT

t

J v t v t dt= ∫ . (24)

We want to find the trajectory that minimizes (24) for given initial and final positions and ( )1r t r= 1 ( )2 2r t r= .

Let ( ) ( )x r⋅ = ⋅ . Then ( ) ( )( ) ( ) ( ) ( ) ( ), , T Tf t x t x t v t v t x t x t′ ′= = ′ and

( ) ( )( ), , 2rrf t x t x t I′ = 0 (25)

where I is the identity matrix. Therefore, if a solution is found, it is regular. From E1 we have

( )'

* 1x t k= (26) which leads to

( ) ( ) ( )* * 1 1 2x t r t k t t k= = − + (27) where k1 and k2 are real constant vectors determined using the given initial and final positions. All the integration constants are reported in the Appendix.

The strengthen LNC condition has already been proved in (25). In conclusion, Jacobi’s equation (12) becomes

( )''2 cI tη 0= (28) Therefore we have that

( ) 3c t k t kη 4= + . (29)

where 3 4 0k k+ ≠ to avoid the trivial solution to (28). Imposing ( )1 0c tη = it results that . The actual

values of k3 and k4 are not of our interest. Because 3 1 4 0k t k+ =

(29) is a hyperline and ( )1 0tcη = , there is no conjugate point to t1 and therefore the strengthen Jacobi necessary condition holds.

It is easy to see now that the S1 condition is verified. Thus we have proved that ( )*x ⋅ in (27) provides a weak local minimum for (24). Moreover, S2 condition also holds: the integrand of (24) is regular while the E1 and the strengthen JNC hold. Hence (27) provides a strong local minimum for (24). Lastly, because the integrand in (24) is convex, according to S3, the unique optimal solution found in (27) is a global minimum for the index considered3.

Note that all of the above apply regardless of the environment considered. However the propulsion system must deliver different forces to ensure that the total acceleration is zero, as required for the globally optimal trajectory. Since for aerospace vehicles the dominant effects are the gravitational and aerodynamic ones and the superposition principle holds, we can write

( ) ( ) ( ) ( )* *c g aa t a t a t a t= + + (30)

where is the optimal control acceleration delivered by the propulsion system, while and ( )*ca ⋅ ( )ga ⋅ ( )aa ⋅ are the accelerations due to gravitational forces and aerodynamic forces respectively. Clearly if the aerodynamic effects are ignored and a constant gravitational acceleration, g, is assumed, ( )ga t g= and ( )*ca t g= − .

For aircraft a simple model for the aerodynamic force, ( )aF ⋅ , can be obtained as

( ) ( ) ( ) ( ) ( ) ( )(1 ˆ ˆ ˆ2

Ta D L )SF t Sv t v t C v t C v t C vρ ⊥ ×= − + + t (31)

9 American Institute of Aeronautics and Astronautics

where ρ is the air density, S is the reference area, CD/L/S are the drag/lift/side force coefficients, ( )v ⋅ , ( )v⊥ ⋅ , and

are the unit vectors in the direction of the velocity, of the lift, and side force, respectively. From ( )v× ⋅ (31) the acceleration due to the aerodynamic force is

( ) ( ) ( ) ( ) ( ) ( )( )ˆ ˆ ˆT

a D La t v t v t k v t k v t k v t⊥ ×= − + + S (32)

where / // / 2

D L SD L S

SCk

mρ

= and

( ) ( ) ( ) ( ) ( ) ( )( )* * * * *ˆ ˆ ˆT

c D La t g v t v t k v t k v t k v t⊥ ×= − + − − *S . (33)

For satellites moving in a radial gravitational field an inverse distance square gravitational field is appropriate. If

we fix the origin of the inertial reference frame at the center of the massive body generating the gravitational field,

( )( )( )

( )3

2

ga r t r tr t

μ= − and the globally optimal trajectory (27) requires a control acceleration

( )( )

( )* 3* 2

ca t r tr t

μ= * (34)

where μ is the gravitational constant.

Satellites, especially those in Low Earth Orbit18 (LEO), are subject to external forces due to the impingement of molecules of air on their surfaces. Commonly this effect is known as aerodynamic drag and is modeled as in (31) with CL/S=0 17. In this case

( )( )

( ) ( ) ( ) ( )* * * *3

* 2

ˆTc Da t r t k v t v t v t

r t

μ= + * . (35)

B. Consumed Energy Optimization First formulation

Another common performance index is the consumed energy that can be expressed as15

( ) ( )2

1

.t

T

t

J a t a t dt= ∫ (36)

We want to minimize (36) for given initial and final positions, ( )1r t r1= and , and velocities, ( )2r t r= 2

( )1v t v= 1 and ( )2v t v= 2 . Let ( ) ( )x r⋅ = ⋅ , then ( ) ( ) ( )( ) ( ) ( ) ( )'' ''T (, , ' , '' )Tf t x t x t x t t= =a t a x t x t , thus enabling direct use of the Euler Poisson Necessary Condition (E4-E5). Candidate minimizers for (36) are solutions of

( ) ( )( ) ( ) ( )( ) ( ) ( )(2

2'' '' '' '' '' '' 0' ''

T T Td dx t x t x t x t x t x tx dt x xdt∂ ∂ ∂

− +∂ ∂ ∂ ) = (37)

which easily leads to

( )* 5a t k t k6= + (38)

10 American Institute of Aeronautics and Astronautics

and therefore

( ) ( ) ( ) ( ) ( )3 3 2 2* * 5 1 6 1 7 1

1 16 2 8x t r t k t t k t t k t t k= = − + − + − + . (39)

where the constants k5, k6, k7 and k8 are determined imposing the boundary conditions.

The strengthen Legendre Necessary Condition holds because

( ) ( )( )2

2 '' '' 2 0''

Tx t x t Ix∂

=∂

(40)

From JNC we have that ( ) ( )23 31 2 3

1 2 31 1

, , ,1, , ,2 j k

j k j k

f tF t

χ χ χχ χ χ χ χ

χ χ= =

⎛ ⎞∂= ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠∑ ∑ , and therefore

( ) ( ) ( )( ) ( ) ( )' '' '' '', , , TF t t t t t tη η η η η= (41)

From the analogy between (41) and the integrand of (36) it follows that any secondary extremal ( )cη ⋅ must be a

cubic polynomial in t. Imposing as boundary conditions that ( )1 0tη = , ( ) ( )'

1 * 1' t x tη = and ( ) 0ctη = , if tc exists, it follows that the position of the point tc conjugate to t1 actually depends on the boundary conditions of the problem and therefore it is impossible to say a priori if the Jacobi Necessary Condition (JNC) is satisfied or not.

Thus (39) is a smooth function and it has been proven to satisfy the Euler Poisson Necessary Condition and the strengthen LNC. If, according to the boundary conditions imposed to the problem the strengthen JNC holds, then the (39) provides a weak local minimum for (36) on [ ]1 2,t t . In addition, it has been proven in (40) that the integrand of

(36) is regular and therefore from S2, if the JNC is verified, (39) provides a strong local minimum for (36) on [ ]1 2,t t . In conclusion, if S1 is verified, because of the convexity of the integrand of (36), according to S3 (39) also provides a global minimum for the consumed energy cost index.

Clearly, the control acceleration that must be delivered by the propulsion system for various environmental conditions can be computed as discussed for kinetic energy optimization (see (30) - (35)).

Second formulation

One can remark that defining the consumed energy as a function of the total acceleration incorporates effects which are not related to the propulsion system such as gravity and aerodynamic forces. Thus, for a different formulation of the consumed energy optimal control problem we introduce another index

( ) ( )2

1

.t

Tc c

t

J a t a t dt= ∫ (42)

It can be assumed that , where ( ) ( )cu a⋅ = ⋅ ( )u ⋅ is the control vector. Then the equations of motion are

( )( )

( ) ( )( ) ( )( )( )

( )( ) ( )( ) ( ) ( ) ( )( )10

1 2

0,

g a g a

v t C x tx t u

Iu t a r t a v t a C x t a C x t

⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ ⎢ ⎥ ⎢ ⎥= = + =⎢ ⎥+ + +⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

t f x t u t (43)

where , ( ) ( ) ( )TT Tx r v⎡ ⎤⋅ = ⋅ ⋅⎣ ⎦ [ ] 3 6

1 0C I ×= ∈ , [ ] 3 62 0C I ×= ∈ , I and 0 are the identity and zero matrices in

respectively, is the acceleration due to gravity and 3 3× ( )ga ⋅ ( )aa ⋅ the one due to the aerodynamic forces.

11 American Institute of Aeronautics and Astronautics

The problem is now to minimize (42) subject to (43) given initial and final positions and velocities. We shall apply Pontryagin’s principle, restricting the controls to an open and bounded set. According to (20) the Hamiltonian can be written as

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( )0 1 2 0ˆ , , ,T T TH t x t u t u t u t t t f x t u tλ λ λ λ⎡ ⎤= + ⎣ ⎦ . (44)

where and . From ( ) ( ) ( )1 2TT Tλ λ λ⎡ ⎤⋅ = ⋅ ⋅⎣ ⎦ ( ) ( )0

ˆ TTλ λ λ⎡⋅ = ⋅⎣⎤⎦ (22), (23) and P1 we obtain that

( ) ( )( )

( )

( ) ( )( )

'0

'

0ˆ , ,

ˆ , ,0

H t x t ut

xH t x t u

u

λ

λλ

λ

=

∂= −

∂

∂=

∂

(45)

from which it follows that the candidate optimal control and the corresponding trajectory satisfy

( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( )

0

'0 * *

0 * 0 * *

constant

,

2 ,

T

T

f x t u t t tx

u t f x t u t tu

λ

λ λ

λ λ

=

∂= −

∂∂

+ =∂

0

(46)

with ( ) ( )( ) ( )( ) ( )( )( )0 * * 2 1 * 2 *,TTT

g af x t u t C a C x t a C x tx x∂ ∂⎡ ⎤

⎥= +⎢∂ ∂⎣ ⎦and ( ) ( )( ) [ ]0 * *, 0 Tf x t u t I

u∂

=∂

.

Consequently we have that

( )( ) ( )( )( ) ( ) ( )

( ) ( )

'1 * 2 *

0 * 2

0

2 .

Tg aa C x t a C x t t t

I x

u t t

λ λ

λ λ

⎡ ⎤∂+ =⎢ ∂⎣

= −

−⎥⎦ (47)

At this point it is necessary to specify an environmental model. For simplicity we assume ( )( )1 *ga C x t g= and

aerodynamic drag only, i.e. . Taking into account the lift and the side forces would lead to a mathematical model whose complexity exceeds the scope of the present paper. In these circumstances

/ 0L Sk =(47) becomes

(48) ( )( ) ( ) ( )

( ) ( )

'

2 *

0 * 2

00

2

t tC x tI

u t t

λ λ

λ λ

⎡ ⎤= −⎢ ⎥

⎢ ⎥⎣ ⎦= −

Ð

where is a symmetric matrix in ( )( 2C x ⋅Ð ) 3 3× whose elements are ( )( )( ) ( )

( )

2 22 32

,2 2

iDi i

C x xk

C x+⋅ − ⋅

⋅ = −⋅

Ð and

( )( )

( )( ) ( )

( )

22 3 32

,2 2

i jDi j

C x x xk

C x+ +⋅ ⋅

⋅ = −⋅

Ð⋅

, where ( ) 2, 1, 2,3 , i ji j ∈ ≠ and ( )kx ⋅ is the k-th component of ( )x ⋅ .

12 American Institute of Aeronautics and Astronautics

We notice that if 0 0λ = , then from (48) it follows that both ( ) ( )1 2 0λ λ⋅ = ⋅ = , which is not an acceptable solution according to Pontryagin’s principle because the costate vector cannot be identically zero; thus, 0 0λ ≠ .

From (48) we obtain that

(49)

( )( ) ( )( ) ( ) ( )

( ) ( )

1 1'

1 2 * 2 2

0 * 2

constant

2 .

t

t C x t t

u t t

λ λ

λ λ

λ λ

= =

+ =

= −

Ð tλ−

In conclusion, according to Pontryagin’s principle the necessary conditions for the minimization of (42) are

( )( )

( )( ) ( )

( ) ( )( ) ( )( ) ( )

2 *'* *

2 *

'* 1 2 * *

* 1 1 * 2 2

0

,

a

C x tx t u

Ig a C x t

u t C x t u t

x t x x t x

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥= + ⎢ ⎥+⎢ ⎥ ⎣ ⎦⎣ ⎦

= −

= =

Ð

t

(50)

where 11

02λ

λλ

= . In addition, according to P2, it must hold that ( ) ( )( )*

2

*2ˆ , ,

u u

H t x t uu

λ=

∂∂

0 .

Eq. (50) appears not to have closed form solutions but assuming that 0Dk = , from (50) we obtain that

( ) ( ) ( )( ) ( )

( )

3 3 2 2* 9 1 10 1 11 1

* 9 10

1 16 2

.

r t k t t k g t t k t t k

u t k t k

= − + + − + − +

= +

12 (51)

The constants k9-12 are reported in the Appendix.

Condition P2 easily leads to ( ) ( )( )2

*2*

ˆ , , 2u u

H x u Iu

λ=

∂⋅ ⋅ =

∂ 0 0λ which is satisfied if 0 0λ > . In addition we

have to check that ( ) ( ) ( )( ) ( ) ( )( )* * * * *ˆ ˆ, , ,H t x t u t M t x tλ λ= . Indeed:

( ) ( )( ) ( ) ( )( )( )

** * * 0 * * 1 2

*

1ˆ , , 22

T T T v tH x t u t u t u t

u t gλ λ λ λ

⎛ ⎞⎡ ⎤⎡ ⎤⎜ ⎟= + ⎢ ⎥⎣ ⎦⎜ ⎟+⎢ ⎥⎣ ⎦⎝ ⎠

(52)

where (from ( )1 *u t kλ ′= = 9 (50) with ) and 0Dk = ( )2 * 9u t k t kλ 10= − = − − . After some algebra we obtain

( ) ( )( ) ( )210 10 9 9* * * 0 1 1 2 1 1 1 1 1 2ˆ , , 2 constan

2 2

T TT T T Tk k k k

H x t u t t k k t k v g k t kλ λ⎛ ⎞

= − − − + − + =⎜ ⎟⎜ ⎟⎝ ⎠

t (53)

and thus everywhere. Lastly, since ( ) ( ) ( )( ) ( ) ( )( )* * * * *

ˆ ˆ, , , constantH t x t u t M t x tλ λ= = (43) is linear and the

integrand of (42) is convex, (51) is guaranteed to be globally optimal5. Consider now a radial gravity environment with aerodynamic drag in eq. (47). In this case, Pontryagin’s

principle imposes that

13 American Institute of Aeronautics and Astronautics

( )( )( )( )

( ) ( )

( ) ( )

1 * '

2 *

0 * 2

0

2

C x tt t

I C x t

u t t

λ λ

λ λ

⎡ ⎤⎢ ⎥ = −⎢ ⎥⎣ ⎦

= −

G

Ð (54)

where is a symmetric matrix in ( )( 1 *C x ⋅G ) 3 3× whose elements are ( )( )( ) ( )

( )

2 21 2

5,1 2

3 ii i

C x x

C xμ

⋅ − ⋅⋅ = −

⋅G and

( ) ( )( )

( )( ) 5,

1 2

3 i ji j

x x

C x

μ ⋅ ⋅⋅ =

⋅G , where ( ) . The similarity between 2, 1,2,3i j ∈ , i j≠ (54) and (48) has to be remarked.

Similarly to the constant gravity case, it can be easily proven that 0 0λ ≠ . Furthermore, operating as before, we arrive to the following set of equations:

( )( )

( )( ) ( )( ) ( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

2 *'* *

1 * 2 *

'* 1 2 * *

'1 1 * *

* 1 1 * 2 2

0

,

g a

C x tx t u

Ia C x t a C x t

u t t C x t u t

t C x t u t

x t x x t x

λ

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥= + ⎢ ⎥+⎢ ⎥ ⎣ ⎦⎣ ⎦

= −

= =

Ð

=G

t

(55)

where ( ) ( )11

02λ

λλ⋅

⋅ = .

Clearly, if 0 0λ > P2 also holds, as required by Pontryagin’s necessary conditions for open and bounded sets. However solving (55) is a difficult task. Moreover, as with all fixed end point control problems (i.e.

( ) ( )* 1 1 2, * 2x t x x= x t = ), controllability of (43) should be guaranteed, which for nonlinear systems is not a trivial task (note that for a constant gravitational field and no aerodynamics controllability of the system of equations of motion, (43), which are linear in this case, is easily verified).

Lastly we remark that the set of state values must exclude a neighborhood of the origin (i.e. ( )1 * 0C x t ≠ ) and

its closure. One can easily remark that actually ( ) ( )1 * 0C x r⋅ = ⋅ = is a physically infeasible situation.

C. Fuel Consumption Optimization Another index of relevance for aerospace applications is the fuel consumption defined as17

( ) ( )2

1

.t

Tc c

t

J a t a t dt= ∫ (56)

Assuming and ( ) ( ) ( )TT Tx r v⎡ ⎤⋅ = ⋅ ⋅⎣ ⎦ ( ) ( )cu a⋅ = ⋅ , the equations of motion are (43) and the problem of interest is

to minimize (56) subject to (43) for given initial and final positions and velocities. Application of Pontryagin’s principle requires construction of the Hamiltonian,

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )(0 1 2 0ˆ , , ,T T T )H t x t u t u t u t t t f x t u tλ λ λ λ⎡ ⎤= + ⎣ ⎦ (57)

14 American Institute of Aeronautics and Astronautics

where and . ( ) ( ) ( )1 2TT Tλ λ λ⎡ ⎤⋅ = ⋅ ⋅⎣ ⎦ ( ) ( )0

ˆ TTλ λ λ⎡ ⎤⋅ = ⋅⎣ ⎦

It is crucial to remark that is and continuous in u but differentiable only for ( ) ( ) ( )( ˆ , ,H x uλ ⋅ ⋅ ⋅ ) 3 \ 0u∈ .

Consequently, we will assume that , where Ψ is an open bounded set and A is a an arbitrary small

neighborhood of the origin and its closure (i.e. ( ) 3 \u ⋅ ∈Ψ ⊂ A

( )u ⋅ can vary in an open bounded set not containing the origin). From (45) it follows that the candidate optimal control and the corresponding trajectory satisfy

( ) ( )( ) ( ) ( )

( )( )

( ) ( )( ) ( )

0

'0 * *

*0 0 * *

* 2

constant

,

, 0

T

T

f x t u t t tx

u tf x t u t t

uu t

λ

λ λ

λ λ

=

∂= −

∂∂

+ =∂

(58)

with ( ) ( )( ) ( )( ) ( )( )( )0 * * 2 1 * 2 *,TTT

g af x t u t C a C x t a C x tx x∂ ∂⎡ ⎤

⎥= +⎢∂ ∂⎣ ⎦and ( ) ( )( ) [ ]0 * *, 0 Tf x t u t I

u∂

=∂

.

From here it follows that

( )( ) ( )( )( ) ( ) ( )

( )( )

( )

'1 * 2 *

*0 2

* 2

0

.

Tg aa C x t a C x t t t

I x

u tt

u t

λ λ

λ λ

⎡ ⎤∂+ =⎢ ⎥∂⎣ ⎦

= −

−

(59)

If we assume that in (59) 0 0λ = then it follows that ( ) ( )1 2 0λ λ⋅ = ⋅ = , which is not an acceptable solution

according to Pontryagin’s principle because the costate vector cannot be identically zero; thus, 0 0λ ≠ . As for the consumed energy, we specialize (59) according to the gravitational model and for simplicity we will

account for aerodynamic drag only. In a constant gravity environment ( )( )1 *ga C x g⋅ = and thus

( )( )

( )( ) ( )

( )( )

( )

( ) ( )( ) ( )( ) ( )

2 *'* *

2 *

*2

* 2'2 1 2 * 2

* 1 1 * 2 2

0

,

a

C x tx t u

Ig a C x t

u tt

u t

t C x t t

x t x x t x

λ

λ λ λ

⎡ ⎤ ⎡ ⎤⎢ ⎥= + ⎢ ⎥+⎢ ⎥ ⎣ ⎦⎣ ⎦

= −

= − −

= =

Ð

t

(60)

where ( ) ( )11

0

λλ

λ⋅

⋅ = and 22

0

λλ

λ= . Substituting ( )2λ ⋅ from the second into the third of (60), we obtain that

( )( )

( )( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )

2 *'* *

2 *

2 3 2' '* * * * * * 1 * 2 * *2 2 2

* 1 1 * 2 2

0

0

, .

a

T

C x tx t u t

Ig a C x t

u t u t u t u t u t u t u t C x t u t

x t x x t x

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥= + ⎢ ⎥+⎢ ⎥ ⎣ ⎦⎣ ⎦

− + +

= =

Ð = (61)

15 American Institute of Aeronautics and Astronautics

Further simplification is possible for , when, from 0Dk = (60), we have that

( ) ( ) ( )

( )( )( ) ( )

2 *'* *

*11 12

* 2

* 1 1 * 2 2

0

, .

C x tx t u

Ig

u tk t k

u t

x t x x t x

⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

= +

= =

t

(62)

Note that if we assume in 0g =

( )

(62), the corresponding equations thus obtained coincide with the ones obtained

optimizing ( )2

1

tT

t

J a t a t dt= ∫ as in the first formulation presented for the energy consumption.

For the case of radial gravitational, from (59) we obtain that

( )( )

( )( ) ( )( ) ( )

( )( )

( )

( )( )

( )( )( )( )

( )( )

( ) ( )

2 *'* *

1 * 2 *

*2

* 2

'1 *1 1

'22 *2

* 1 1 * 2 2

0

0

, .

g a

C x tx t u

Ia C x t a C x t

u tt

u t

C x tt tI tC x tt

x t x x t x

λ

λ λ

λλ

⎡ ⎤ ⎡ ⎤⎢ ⎥= + ⎢ ⎥+⎢ ⎥ ⎣ ⎦⎣ ⎦

= −

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

= =

G

Ð

t

(63)

For , which is a typical assumption for Middle Earth Orbit and High Earth Orbit satellites18,19, we obtain 0Dk =

( )( )( )( ) ( )

( )( )

( )

( ) ( )( ) ( )( ) ( )

2 *'* *

1 *

*2

* 2''2 1 * 2

* 1 1 * 2 2

0

, .

g

C x tx t u

Ia C x t

u tt

u t

t C x t t

x t x x t x

λ

λ λ

⎡ ⎤ ⎡ ⎤⎢ ⎥= + ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

= −

=

= =

G

t

(64)

From (57) it follows that

( ) ( )( )2 22 3 1 2 1 32

2 201 2 1 3 2 32 3

2 221 3 2 3 1 2

ˆ , ,u u u u u u

H x u u u u u u uu u u u u u u u

λλ

⎡ ⎤+ − −∂ ⎢ ⎥⋅ ⋅ = − + −⎢ ⎥∂ ⎢ ⎥− − +⎣ ⎦

(65)

where u1, u2 and u3 are the components of the control vector u.

One can easily check that the eigenvalues of the symmetric matrix that multiplies 03

2uλ

in ( ) ( )( )2

*2ˆ , ,H x u

uλ∂⋅ ⋅

∂

16 American Institute of Aeronautics and Astronautics

are 0 and 22u and the eigenvector associated with the 0 eigenvalue is u. Hence, if 0 0λ > , then

( ) ( )(2

*2ˆ , ,H x u

uλ∂⋅ ⋅

∂ ) 0 for any u in an open bounded set which excludes the origin. This proves that P2 is

verified. Further analysis requires solving the equations obtained for various environmental conditions (i.e. (61), (62), (63) or (64)). Moreover, since ( )* 1 1x t = x and ( )* 2 2x t x= , controllability of the system should be guaranteed,

and for (63) and (64) the set of state values must exclude a neighborhood of the origin (i.e. ( )1 *C x 0t ≠ ) and its closure. Lastly, the solutions (if any) provided by these equations, are candidate minimizers of (56). If they actually minimize that index, it has to be examined through sufficient conditions.

V. Conclusions This work reviews the theory of the Simplest Problem of Calculus of Variations (SPCV), which is a very

powerful tool for finding optimal trajectories for aerospace vehicles because it can provide both necessary and sufficient conditions for optimality. In some relevant situations it can also supply in closed analytical form trajectories which are guaranteed to be optimal.

This study presents the rigorous application of the necessary conditions of Euler, Legendre, Jacobi and Euler-Poisson, as well as of the associated sufficient conditions that guarantee a local and a global minimum. These theorems have been employed to provide closed form analytical solutions that minimize the kinetic energy and the consumed energy - defined as a functional of the total acceleration - for an aerospace vehicle schematized as a point mass moving between two fixed end points.

The application of Pontryagin’s minimum principle is also illustrated on more complex problems involving the minimization of the consumed energy and of the fuel - defined as functionals of the control acceleration. Realistic environments, including a constant or radial gravitational field and aerodynamic forces, have been considered. It is important to stress that Pontryagin’s minimum principle provides not only the optimal trajectory but also the optimal control law and that in general it provides necessary conditions only. A rigorous application of this principle requires analysis of the differentiability properties of the Hamiltonian function and of the topological properties of the sets in which the control and state vectors take value. Only under certain conditions applying Pontryagin’s principle reduces to solving differential equations with assigned boundary values. Such equations are herein provided and for one case (i.e. the minimization of the consumed energy in a constant gravity environment) it is shown how the rigorous application of Pontryagin’s principle leads to a globally optimal closed form solution.

Appendix Hereafter are reported the integration constants used in the present paper.

2 11

2 1

r rk

t t−

=−

, , 2 1k r=

( )

2 21 2 2 1 2 1 1 2

22

2 1 2 1 2

2 2

6 2

r

r t v t

+

+ −

9 3 3 22 1 1 2 1

3 2 2 2 2 2 31 2 1 1 1 1 2 1 1 2 1 2 2 2 2 2 1 2 2

3 3 22 1 2

2

1

11 1 9 1 1

1

0

0 1

2

2 233 3

4 2 3 2 4 6 2 6 32

2

3

1

1

v t v t t t rkt t t t t t

t v t v t t t t v t t v t r t v t r t tt t t

k v k t k g t

k

k

− + − + −= −

− + + −

+ + + − + − − −− +

= − − +

=

= 1.r

21 1 1 2

21 2 1

6 33 3

r t t tt t t

− − ++ −

Note that k5-8 are obtained from k9-12 by setting g=0.

References 1van Brunt, B., “The Calculus of Variations”, Springer Verlag, New York 2004 2Sargent, R.W.H., “Optimal Control”, Journal of Computational and Applied Mathematics, 124 (2000) 361-371 3Ewing, G. M., “Calculus of Variations”, 2nd ed., Dover publications Inc., New York 1985

17 American Institute of Aeronautics and Astronautics

18 American Institute of Aeronautics and Astronautics

4Bliss, G. A., “Lectures in the Calculus of Variations”, The University of Chicago Press, London W.C., 1968 5Lee, E.B., Markus, L. “Foundations of Optimal Control Theory”, John Wiley & sons, New York, 1968 6Zeinden, V., “The Riccati equation for optimal control problems with mixed state-control constraints: necessity and

sufficiency”, SIAM J. Control Optim. 32, 1994 7Loewen, P.D., Rockafellar, R. T., “New necessary conditions for the generalized problem of Bolza”, SIAM J. Control Optim.

36, 1996 8Loewen, P. D., Rockafellar, R. T., “Bolza problems with general time constraints”, SIAM J. Control Optimal, 35 (1997)

2050-2069 9Wan, F. Y. M., “Introduction to the calculus of variations and its applications”, Chapman & Hall Mathematics, New York

1995 10Caratheodory, C., “Calculus of Vatiations and Partial Differential Equations of the First Order”, Holden Day, San Francisco

1967 11Forsyth, A. R., “Calculus of Variations”, Dover Publications, New York, 1960 12Kirk, D. E., “Optimal Control Theory, An introduction”, 3rd ed., Prentice Hall, New York 1970 13Pontryagin, G. M., et al., “The Mathematical of Optimal Processes”, Wiley, New York 1962 14Lee, E.B., Markus, L., “Foundations of Optimal Control Theory“, Wiley & Sons., 1968 15Sultan, C., Seereram, S., Mehra, R. K., “Deep space formation flying spacecraft path planning”, Intl. Journal of Robotics

Research , Vol. 20(4), 405-430, 2007 16Bate, R., Mueller, D., White, J., “Fundamentals of astrodynamics”, 1st ed., Dover Publications, New York 1971 17Beard, R., McLain, T., “Fuel optimization for constrained rotation of spacecraft formations”, AIAA Journal of Guidance,

Control, and Dynamics, 23(2):339-346, 2001 18Larson, W. J., Wertz, J. R. (editors), “Space Missions Analysis and Design”, 3rd ed., Space Technology Library, El Segundo

2005 19Saad, N. A., Ismail M. N., Khalil, K. H. I., “Decay of orbits due to the drag of rotating oblate atmosphere”, Planetary and

Science, November 2007