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Discretization Performance and Accuracy Analysis for the Rocket Powered Descent Guidance Problem

Conference Paper

Discretization Performance and Accuracy Analysis for the Rocket Powered Descent Guidance Problem

Discretization Performance and Accuracy Analysis for
the Powered Descent Guidance Problem
Danylo Malyuta
, Taylor P. Reynolds
, Michael Szmuk
,
Mehran Mesbahi
, Beh¸cet A¸cıkme¸se
,
Dept. of Aeronautics & Astronautics, University of Washington, Seattle, WA 98195, USA
and John M. Carson III
NASA Johnson Space Center, Houston, TX 77058, USA
In this paper we analyze the performance and accuracy properties of several differential
equation discretization methods in the context of powered descent guidance for pinpoint
planetary landing. The guidance problem is formulated as a continuous-time 6-DoF optimal
control problem with nonlinear dynamics and a multitude of state and control constraints.
This problem is to be solved via a direct method whereby it is temporally discretized and
solved iteratively as a sequence of parameter optimization problems. Proper discretization
thus becomes crucial if the resulting thrust commands are to be reproducible by the real
vehicle. Moreover, proper discretization can decrease the overall time required to obtain a
solution, and ultimately to satisfy a real-time computational requirement. We thus carry
out a Monte Carlo performance comparison of the piecewise constant, piecewise linear,
Runga-Kutta and three pseudospectral discretization methods. We study the method’s
performance and accuracy, and discuss how each method may impact the ability to achieve
a real-time solution. These empirical results are backed by a theoretical discussion of how
well each method preserves favorable properties about the discrete optimization problem,
such as sparsity. To the best of our knowledge, these are the first results that provide a
back-to-back and fair test for a large number of discretization methods applied to a the
6-DoF powered descent guidance problem.
I. Introduction
This paper compares the accuracy and performance of several methods for converting a continuous-time
optimal control problem into a finite-dimensional optimization problem. The motivation for this study is to
assess which technique may be suitable for a real-time implementation of a 6 degree-of-freedom (DoF) rocket
powered descent guidance problem for pinpoint planetary landing.
The 6-DoF powered descent guidance problem is a continuous-time free-final time nonlinear optimal control
problem subject to both state and control constraints.1,2 We refer to the process of converting this type of
problem into a finite-dimensional discrete optimization problem as discretization and the methods by which
this is achieved as discretization methods. Discretization consists of expressing the cost function, dynamical
equations and state and control constraints as functions of a finite number of parameters defined on a set of
temporal nodes.
Numerical methods to solve continuous-time optimal control problems can be broadly classified intro indirect
and direct methods.3, 4 Direct methods parameterize either the state or control signal using a set of basis
functions whose coefficients are found via parameter optimization. Indirect methods use techniques from
optimal control theory5–7 to determine the necessary conditions of optimality, which are then solved as a
two-point boundary-value-problem. While indirect methods have been used for powered descent and ascent
problems,8, 9 the necessary conditions for problems with multiple non-trivial state and control constraints
Doctoral Student, W.E. Boeing Department of Aeronautics & Astronautics, AIAA Student Member
{danylo,tpr6,szmuk}@uw.edu
Professor, W.E. Boeing Department of Aeronautics & Astronautics, AIAA Associate Fellow, {mesbahi,behcet}@uw.edu
SPLICE Project Manager, AIAA Associate Fellow.
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can be very difficult to write down, even for problems with simple dynamics.10 As such, our approach is to
use a direct method.
Direct methods can be further classified into techniques that parameterize only the control signal or that
parameterize both the state and control signals. In the context of powered descent guidance, the former
group was the first to appear in literature. For the 3-DoF translational guidance problem, several early works
adopted piecewise constant basis functions to approximate the control function.11–14 This design choice is
valid given that the structure of the optimal solution is known to be piecewise constant. A piecewise linear
approximation, however, can yield a more accurate representation of the optimal control signal over the same
(coarse) temporal grid. In effect, if the density of the temporal discretization (which is fixed) does not yield
a node sufficiently close to an optimal switching time, then sub-optimal results may follow. The piecewise
linear approximation is better suited to handle such cases, and subsequent work on the 3-DoF guidance
problem used this approach.15–19 Recently, these same techniques have been applied to the 6-DoF powered
descent guidance problem, with both piecewise constant20 and piecewise linear approximations.1,2, 21, 22 The
latter group of direct methods that parameterize both the state and control signals are, for our purposes,
global pseudospectral methods.23, 24 These methods have recently been successfully applied to the 3-DoF
translational guidance problem,25 however their application to the 6-DoF powered descent problem has not
yet been reported in literature.
Beyond the specific example of powered descent guidance, there is a significant body of work on both
types of direct methods for aerospace applications such attitude control, proximity operations and formation
reconfiguration.26–32 Historically, it appears that parameterizing the control signal only is the more popular
approach, perhaps due to the fact that common discretization techniques (e.g., forward or backward Euler,
Runga-Kutta, etc.) fall under this category. There is, however, a growing body of work in these areas that
parameterize both the state and control via pseudospectral methods.33, 34
The common theme to all of these applications is that once a discretization method is selected for the dy-
namical equations, the entire continuous-time problem is converted into a discrete parameter optimization
problem. Depending on the intended solver, the optimization problem may be classified as a nonlinear
program (NLP) or as a particular class of convex optimization problems. Our approach is to convert the
non-convex 6-DoF powered descent guidance problem into a convex second order cone programming problem.
These convex problems are then solved iteratively in a process known as successive convexification.1, 19, 21, 35
This choice arises primarily due to the nature of the constraints inherent to this problem, many of which are
expressed as second order cones, and necessitates the linearization of the (nonlinear) dynamical equations.
Convex optimization techniques are desirable for real-time safety-critical applications due to their determin-
istic time complexity and guaranteed convergence to the global optimum. Recent flight heritage of convex
optimization-based powered descent guidance36 has increased the maturity of the technology.
A. Contributions
The primary contribution of this paper is the extensive and fair comparison of the computational performance
and solution accuracy of several popular discretization schemes. In particular, we examing the piecewise
constant and piecewise linear approximations of the control, Runga-Kutta discretization, the Chebyshev-
Gauss-Lobatto pseudospectral method and the differential and integral forms of the Legendre-Gauss-Radau
pseudospectral method. The example that we use to perform this analysis is the 6-DoF powered descent
guidance problem for planetary landing, which is a trajectory optimization problem. To the best of our
knowledge, no analysis as extensive as the one presented herein has been previously published for this
increasingly important spaceflight problem. Moreover, §II serves as a useful and succinct summary of each
discretization method used herein.
B. Outline
This paper is structured as follows. First, §II provides the mathematical details of the six discretization meth-
ods that we consider. Next, §III describes how each discretization method is used to convert a continuous-time
optimal control problem into a discrete finite-dimensional one. The same section describes our Monte Carlo
analysis approach to performance testing. In §IV, Monte Carlo simulation results are presented, and §V
provides a discussion of these results and highlights each discretization method’s merits and drawbacks.
Finally, §VI concludes the paper with an outlook for future research.
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C. Notation
We use the following notation and conventions. Let Rdenote the set of reals, Zthe set of integers and Z++
the set of positive integers. Scalars are lowercase italic, e.g. tR, vectors are lowercase bold, e.g. xRn,
and matrices are uppercase italic, e.g. FRm×n. Exceptions to this rule are Mand Nwhich are scalar
numbers of temporal collocation and discretization nodes respectively. Inline column vectors are written in
parentheses, e.g. v= (1,2,3) R3. For a given matrix F,Fij shall denote the element in row iand column
j,Fishall denote column iand Fi:jshall denote columns ithrough jwhere all indexing is zero-based (i.e.
index 0 is the first element, row or column). An exception to the element indexing rule is the differentiation
matrix D, where Dji is defined Section II. Furthermore, when dealing with block vectors and matrices the
indexing refers to blocks rather than individual elements, e.g. if v= (a,b,c) then v1=b. The notation (k)
shall be shorthand for the evaluation of vector or matrix function at the k-th temporal node. Generally,
shall denote a “some object” placeholder. The symbol Ishall be reserved for the identity matrix, which
is square and whose size shall be clear from context. The number 0 shall denote at once a zero scalar, vector
or matrix of appropriate size depending on context. 1shall denote the vector of all ones whose length shall
be clear from context. We use three different times: tfor real time, τfor scaled time and ηfor pseudo time.
Each of these shall be defined later in the paper. We use the over-dot for differentation with respect to real
time, i.e. ˙
=d
dt , the circle for differentiation with respect to scaled time, i.e.
=d
, and the apostrophe
for differentiation with respect to pseudo time, i.e. 0=d
. Finally, the Kronecker delta function is defined
by δii = 1 and δij = 0 if i6=j.
II. Discretization Methods
In this section we consider the problem of discretizing the following Linear Time Varying (LTV) system:
˙
x(t) = A(t)x(t) + B(t)u(t) + Σ(t)σ+z(t),x(0) = x0, t [0, tf] (1)
where x:RRnis the state, u:RRmis the control input, σRis a parameter, A:RRn×nis the
state zero-input dynamics matrix, B:RRn×mis the control matrix, Σ : RRnis the parameter matrix
and z:RRnis a perturbation that arises if (1) is obtained by linearizing nonlinear dynamics about a
trajectory. Note that the system evolves in real time t, i.e., physical wall clock time. However, we shall see
in §III that by appropriately defining σwe shall be able to equivalently write a scaled time version of (1).
We assume that A,B, Σ, uand zare piecewise continuous functions. Our purpose in the remainder of this
section is to outline several methods for temporally discretizing (1) on a grid of Nnodes tk[0, tf] known
as sampling times. These methods will each be applied to discretize a differential equation constraint of the
type (1) in the powered descent guidance optimal control problem.
Let Φ(·, t0) : RRn×ndenote the state transition matrix for (1). We will make use of the following theorem
from linear systems theory.
Theorem II.1. Φ(t, t0)has the fol lowing properties:
1. Φ(·, t0) : RRn×nis the unique solution of the linear matrix ordinary differential equation
d
dtΦ(t, t0) = A(t)Φ(t, t0),Φ(t0, t0) = I, (2)
and is continuous tRand differentiable everywhere except at the discontinuity points of A.
2. For all t, t0, t1R,Φ(t, t0) = Φ(t, t1)Φ(t1, t0).
3. For all t0, t1R,Φ(t1, t0)is invertible and its inverse is [Φ(t1, t0)]1= Φ(t0, t1).
Proof. See a standard text on linear systems theory.37, 38
A. Zeroth Order Hold
We begin by considering discretization methods that parameterize the control directly. The first and seem-
ingly most popular such method is to keep the input constant between the sampling times, and is known as
zeroth order hold (ZOH). Specifically, the control input is assumed to take the form
u(t) = u(tk)t[tk, tk+1).(3)
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For this purpose we consider a uniformly spaced temporal grid:
tk=k
N1tf, k = 0, . . . , N 1.(4)
Note that (3) amounts to choosing piecewise constant basis functions that render the coefficients of the
resulting control signal independent from each other. The vectors u(tk)Rmdenote the coefficients of the
control input’s representation in this basis. We point out that for a temporal grid with Npoints, this choice
provides N1 degrees of freedom.
Using (2), the state transition matrix can be found between each of the discrete sampling times. Using
an integration routine such as the classical Runge-Kutta method,39, 40 we can compute Φ(tk+1, tk) for each
k= 0, . . . , N 1. It is then a well-known result from linear systems theory that the state at time t[tk, tk+1)
is given by37,38
x(t) = Φ(t, tk)x(tk) + Zt
tk
Φ(t, ξ)B(ξ)u(ξ)dξ+Zt
tk
Φ(t, ξ)Σ(ξ)dξσ +Zt
tk
Φ(t, ξ)z(ξ)dξ. (5)
Using the state transition matrix properties from Theorem II.1 together with (5) and our ZOH assumption
(3) on the control input, we can write the discretized LTV system
x(k+1) =A|kx(k)+B|ku(k)+ Σ|kσ+z|k, k = 0, . . . , N 2,(6)
where we have
A|k= Φ(tk+1, tk),(7a)
B|k=A|kZtk+1
tk
Φ(ξ, tk)1B(ξ)dξ, (7b)
Σ|k=A|kZtk+1
tk
Φ(ξ, tk)1Σ(ξ)dξ, (7c)
z|k=A|kZtk+1
tk
Φ(ξ, tk)1z(ξ)dξ. (7d)
The discretized dynamics (6) can be concatenated for ease of implementation. Define
X:=
x(0)
x(1)
.
.
.
x(N1)
RnN ,U:=
0
u(0)
.
.
.
u(N2)
RmN ,
so that (6) may be expressed as the single constraint
X=
I0 0 · · · 0
A|00 0 · · · 0
0A|10· · · 0
.
.
..
.
........
.
.
0 0 · · · A|N20
X+
0 0 · · · 0
0B|0· · · 0
.
.
..
.
.....
.
.
0 0 · · · B|N2
U+
0
Σ|0
Σ|1
.
.
.
Σ|N2
σ+
0
z|0
z|1
.
.
.
z|N2
.(8)
B. First Order Hold
We now consider a discretization method that paramterizes the control input as a linear function between
sampling times. This method is known as first order hold (FOH), and assumes that the control input takes
the form
u(t) = λ
k(t)u(tk) + λ+
k(t)u(tk+1),t[tk, tk+1 ),(9a)
λ
k(t):=tk+1 t
tk+1 tk
, λ+
k(t):=ttk
tk+1 tk
.
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Note that we maintain the same uniformly spaced temporal grid as in (4). In this case, we are selecting
piecewise linear basis functions that render the coefficients of the control input’s representation dependent on
one another. In particular, both u(tk) and u(tk+1) are used as vector coefficients for the basis function at the
kth sampling time, effectively linking the coefficients of adjacent sampling times. Moreover, for a temporal
grid with Npoints, this choice provides Ndegrees of freedom, since the value of the control at the final
time will now impact the control in the preceding sample interval. This change is reflected by rewriting (5)
and (6) using the new definition (9a) to arrive at
x(k+1) =A|kx(k)+B
|ku(k)+B+
|ku(k+1) + Σ|kσ+z|k, k = 0, . . . , N 2,(10)
where A|k,Σ|kand z|kare the same as in (7) and
B
|k=A|kZtk+1
tk
Φ(ξ, tk)1λ
k(ξ)B(ξ)dξ, (11a)
B+
|k=A|kZtk+1
tk
Φ(ξ, tk)1λ+
k(ξ)B(ξ)dξ. (11b)
Like the ZOH case, the discretized dynamics (10) can be concatenated to a single constraint by defining
X:=
x(0)
x(1)
.
.
.
x(N1)
RnN ,U:=
u(0)
u(1)
.
.
.
u(N1)
RmN ,
and expressing (10) as
X=
I0 0 · · · 0
A|00 0 · · · 0
0A|10· · · 0
.
.
..
.
........
.
.
0 0 · · · A|N20
X+
0 0 0 · · · 0
B
|0B+
|00· · · 0
0B
|1B+
|1· · · 0
.
.
..
.
........
.
.
0 0 · · · B
|N2B+
|N2
U+
0
Σ|0
Σ|1
.
.
.
Σ|N2
σ+
0
z|0
z|1
.
.
.
z|N2
.(12)
C. Classical Runge-Kutta Method
The third method that approximates the control input directly is the classical Runge-Kutta fourth order
integration method (RK41).40 In this case, the state transition over each sample interval is given by
x(k+1) =x(k)+t
6(k|1+ 2k|2+ 2k|3+k|4), k = 0, . . . , N 2,(13)
where,
k|1=A(k)x(k)+B(k)u(k)+ Σ(k)σ+z(k),(14a)
k|2=A(k+1/2) x(k)+t
2k|1+B(k+1/2)u(k+1/2) + Σ(k+1/2) σ+z(k+1/2),(14b)
k|3=A(k+1/2) x(k)+t
2k|2+B(k+1/2)u(k+1/2) + Σ(k+1/2) σ+z(k+1/2),(14c)
k|4=A(k+1)(x(k)+ ∆tk|3) + B(k+1) u(k+1) + Σ(k+1)σ+z(k+1) .(14d)
We use the notation (k+1/2) =(tk+ ∆t/2) and ∆t:=tf/(N1) in accordance with (4).
Remark II.2. If the LTV system (1) arises from a linearization of a nonlinear system, then the quantities
(ξ)correspond to the first-order terms in a Taylor series expansion about some reference {¯
x(ξ),¯
u(ξ),¯σ}. In
particular, they are Jacobian matrices with respect to the state, control or parameters evaluated at the time
ξ. The primary consequence of this is that the quantities in (14b) and (14c) become different. For example,
A(k+1/2) in (14b) becomes the Jacobian with respect to the state evaluated at ¯
x(k)+∆t¯
k|1/2, whereas A(k+1/2)
in (14c) must be evaluated at ¯
x(k)+ ∆t¯
k|2/2. The same holds for each of B(k+1/2),Σ(k+1/2) and z(k+1/2).
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To avoid increasing the input vector dimension to include the intermediate values u(k+1/2), we make a
simplifying assumption that the control may be linearly interpolated to obtain
u(k+1/2) :=u(k)+u(k+1)
2.(15)
This choice retains the property that for a temporal grid of Npoints, this discretization method yields N
degrees of freedom. This simplification of the input signal effectively makes the RK41 method identical
to FOH, except that the discretization is coarsened by removing the integrations (7a), (7c), (7d), (11a)
and (11b). By substituting the k|i,i= 1,...,4, into (13) and simplifying, we obtain the dynamical system
update equation
x(k+1) =A|kx(k)+B0
|ku(k)+B1
|ku(k+1) + Σ|kσ+z|k, k = 0, . . . , N 2,(16)
where A|k,B0
|k,B1
|k, Σ|kand z|kare linear combinations of the elements in (14), and are distinct from
those obtained for ZOH and FOH. The discretized dynamics (16) can be concatenated for implementation
by defining
X:=
x(0)
x(1)
.
.
.
x(N1)
RnN ,U:=
u(0)
u(1)
.
.
.
u(N1)
RmN ,
and expressing (16) as the single constraint
X=
I0 0 · · · 0
A|00 0 · · · 0
0A|10· · · 0
.
.
..
.
........
.
.
0 0 · · · A|N20
X+
0 0 0 · · · 0
B0
|0B1
|00· · · 0
0B0
|1B1
|1· · · 0
.
.
..
.
........
.
.
0 0 · · · B0
|N2B1
|N2
U+
0
Σ|0
Σ|1
.
.
.
Σ|N2
σ+
0
z|0
z|1
.
.
.
z|N2
.(17)
D. Global Pseudospectral Methods
Pseudospectral methods offer an alternative way of discretizing (1) by parameterizing both the state and
control functions. These methods are therefore inherently different from those we have discussed previously.
One of their primary advantages is an exponential convergence rate to the exact solution as the temporal
resolution is increased, assuming that the solution is smooth.24 In this section we describe various pseu-
dospectral flavors, all of which are unified by a common theme of approximating the state function xin a
basis of Lagrange polynomials. In particular, we define D:={ηi[1,1]}N1
i=0 to be the set of discretization
nodes and refer to ηas the pseudo time. Note that it is standard among pseudospectral methods for the
nodes ηi∈ D to be defined on the [1,1] interval.23 The transformation between pseudo time and real time
is given by
t(η) = tf(η+ 1)
2.(18)
Pseudospectral discretization methods use the following approximation for the state and input signals
x(η) =
N1
X
i=0
x(i)φi(η),u(η) =
N1
X
i=0
u(i)φi(η),where φi(η):=
N1
Y
j=0
j6=i
ηηj
ηiηj
.(19)
where we employ the shorthand notation x(η):=x(t(η)) using (18). The function φiis known as a Lagrange
interpolating polynomial of degree N1.
In all pseudospectral methods discussed here, η0=1 and ηN1= 1. Note that |D| =Nwhich stems
from our desire to temporally discretize (1) on a grid of Nnodes to remain consistent with (4). Owing to
the property that φi(ηj) = δij , we see that x(j)=x(ηj) are the coefficients for these basis functions for
j= 0, . . . , N 1. In other words, the coefficients x(j)gives directly the signal approximation at the temporal
node ηj.
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Let C ⊆ D be the set of collocation nodes such that |C| =MN. The collocation nodes are the temporal
values at which the differential equation (1) gets imposed. The core difference between all pseudospectral
methods described below is C, which may or may not contain the terminal node ηN1and which shall differ
in the distribution of the nodes on the [1,1] interval. In each case, Cis chosen so as to avoid the Runge
phenomenon.41
Associated with any given Cand Dis a differentiation matrix DRM×Nsuch that Dji :=φ0
i(ηj). The
matrix Dimposes the differential equation in the finite-dimensional basis of Lagrange polynomials. In other
words
x0(ηj) =
N1
X
i=0
x(i)φ0
i(ηj) =
N1
X
i=0
Dji x(i),ηj∈ C,(20)
where we draw attention to the fact that differentiation is with respect to pseudo time η. Note that when
M < N ,Dis not a square matrix and this has important ramifications for both a method’s formulation
and performance. For example, when Dis square we cannot write an equivalent “integral” version of the
dynamics (see §2). In order to use (20) to approximate (1), the temporal transformation (18) has to be
applied. First, note that
x0=dx
=dx
dt
dt
=tf
2
dx
dt =tf
2˙
x.(21)
Next, substituting (21) into (20) we obtain
˙
x(ηj) = 2
tf
N1
X
i=0
Dji x(i),ηj∈ C,(22)
In the following summary of pseudospectral methods, our main focus is on defining the collocation node
set C. Given C, the corresponding differentiation matrix Dcan be efficiently computed to within machine
rounding error via barycentric Lagrange interpolation.42 Without loss of generality, we assume that the
optimal control problem under consideration has no integral in its cost function – it is of Mayer form.6As
such, we do not discuss quadrature methods associated to each method.
1. Chebyshev-Gauss-Lobatto
Define the Chebyshev polynomial of degree Nas
ςN: [1,1] R
η7→ cos(Narccos(η)).
The Chebyshev-Gauss-Lobatto (CGL) collocation node distribution defines η0, . . . , ηN1as the roots of
(1 η2) ˙ςN1, sorted in increasing order. This admits an explicit formula23
ηj=cos πj
N1, j = 0, . . . , N 1.(23)
The differential equation (1) can now be discretized by imposing that it holds at each collocation node.
Substituting (22) into (1), we obtain
2
tf
N1
X
i=0
Dji x(i)=A(j)x(j)+B(j)u(j)+ Σ(j)σ+z(j), j = 0, . . . , N 1.(24)
The discretized dynamics (24) can be concatenated by defining
X:=
x(0)
x(1)
.
.
.
x(N1)
RnN ,U:=
u(0)
u(1)
.
.
.
u(N1)
RmN ,
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so that (24) can be expressed as a single constraint
2
tf
D00I· · · D0(N1)I
.
.
.....
.
.
D(N1)0I· · · D(N1)(N1)I
X=
A(0) · · · 0
.
.
.....
.
.
0· · · A(N1)
X+
B(0) · · · 0
.
.
.....
.
.
0· · · B(N1)
U
+
Σ(0)
.
.
.
Σ(N1)
σ+
z(0)
.
.
.
z(N1)
.(25)
2. Legendre-Gauss-Radau
Define the Legendre polynomial of degree Nas
`N: [1,1] R
η7→ 1
2NN!
dN
N[(η21)N].
The Legendre-Gauss-Radau (LGR) collocation node distribution defines η0, . . . , ηN2as the roots of `N2+
`N1, sorted in increasing order. Note that this defines a set Cwhich is non-symmetric about the origin
and which does not contain the terminal node ηN1= 1. Nevertheless, the terminal node is included as
a discretization node such that D=C ∪ {ηN1}. In this case, the differentiation matrix DR(N1)×N.
Because ηN1= 1 ∈ D, it is still possible to include terminal constraints via x(N1) in (19). Previous work
on rocket powered descent guidance has explored the flipped LGR distribution, where the negative of the
roots is taken and thus the initial node η0=1 is omitted.43 Since it is more convenient to have u(0)
rather than u(N1) as a primal variable in an optimal control problem, we prefer to use the regular LGR
distribution.44 In our approach we make a simplifying zeroth order hold assumption to recover the terminal
input, i.e., we assume u(N1) =u(N2). The differential equation (1) can now be discretized similarly to
the CGL case by using (22)
2
tf
N1
X
i=0
Dji x(i)=A(j)x(j)+B(j)u(j)+ Σ(j)σ(j)+z(j), j = 0, . . . , N 2,(26)
and we highlight that (1) is not imposed at ηN1, since it is not a collocation node. The discretized
dynamics (26) can be concatenated for implementation by defining
X:=
x(0)
x(1)
.
.
.
x(N1)
RnN ,U:=
u(0)
u(1)
.
.
.
u(N2)
Rm(N1),
so that (26) can be expressed as a single constraint in differential form
2
tf
D00I· · · D0(N1) I
.
.
.....
.
.
D(N2)0I· · · D(N2)(N1)I
X=
A(0) · · · 0
.
.
.....
.
.
0· · · A(N2)
X0:N2+
B(0) · · · 0
.
.
.....
.
.
0· · · B(N2)
U
+
Σ(0)
.
.
.
Σ(N2)
σ+
z(0)
.
.
.
z(N2)
.(27)
Previous research has shown that there may be computational advantages, such as more consistent results,
to an alternative but mathematically equivalent integral form of (27).45 In fact, it has been shown that the
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submatrix D1:N1obtained by removing the first column of Dis full rank and D1
1:ND0=1.24 This allows
us to rewrite (27) in an integral form, which is henceforth refered to as LGR-integral (LGRI), and imposes
the discrete dynamics as
X1:N1=
x(0)
.
.
.
x(0)
+tf
2
D01I· · · D0(N1) I
.
.
.....
.
.
D(N2)0I· · · D(N2)(N1)I
1
A(0) · · · 0
.
.
.....
.
.
0· · · A(N2)
X0:N2+
B(0) · · · 0
.
.
.....
.
.
0· · · B(N2)
U+
z(0)
.
.
.
z(N2)
.
(28)
III. Monte Carlo Analysis
In this section we present the particular optimal control problem and method by which we will compare the
discretization methods presented in §II. This is a 6-DoF powered descent guidance problem and is outlined
in §A. In §B, we describe the Monte Carlo methodology by which the discretization methods are compared.
A. 6-DoF Powered Descent Guidance
The discretization schemes are tested on a 6-DoF powered descent guidance problem. The exact formulation
that we consider is presented as the “baseline problem” in [2, Problem 1]. By expressing the equations of
motion using dual quaternions, this problem models a rigid rocket-powered vehicle with variable mass and
inertia whose objective is to land at a target location using minimum fuel. The sole actuator is a bottom-
mounted rocket engine that can be gimbaled symmetrically about two axes (up to a maximum angle) and
can produce rate-limited thrust with non-zero lower and upper bounds. The following state constraints are
imposed on the vehicle’s trajectories: the vehicle’s vertical axis remains within a maximum tilt angle from
the inertial vertical direction; the position remains inside a glide-slope cone eminating from the landing site;
and the slew rate is bounded. This setup provides a rich and complex optimal control problem to serve as a
reasonable test of the effectiveness of each discretization scheme. While the interested reader may consult2
for the precise problem formulation, we state the problem in a generic form:
min
x(·),u(·),tf
J(x(tf)),(29a)
subject to ˙
x(t) = f(x(t),u(t)),(29b)
gx(x(t)) 0,(29c)
gu(x(t),u(t)) 0,(29d)
b0(x(0),u(0)) = 0,(29e)
bf(x(tf),u(tf)) = 0,(29f)
where J:RnRreturns the final spacecraft mass, gx:RnRnxcaptures the pure state constraints,
gu:RRnucaptures the mixed state and input constraints and b0:RRn0,bf:RRnfhandle the
set of initial and final boundary conditions. In order to handle the fact that [2, Problem 1] is a free final
time problem, we introduce the following variable change
t=στ, (30)
where τ[0,1] is the scaled time and σis a time dilation factor defined so that t[0, tf] under the
mapping (30). An application of the chain rule yields
x=dx
=dx
dt
dt
dτ=σ˙
x,(31)
and as a result the nonlinear dynamics (29b) may be written in scaled time as
x(τ) = σf (x(τ),u(τ)).(32)
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In order to apply the discretization schemes of §II, (32) must be linearized in order to obtain the form of (1).
As explained in,1, 2 an iterative method is used to solve the optimal control problem. Each iteration generates
a state trajectory ¯
x(τ), control trajectory ¯
u(τ) and time dilation factor ¯σby solving an approximation of
(29). These quantities are then used to obtain the first order Taylor series expansion of (32) for any τ[0,1]
as
x(τ)¯σf(¯
x(τ),¯
u(τ)) + ¯σf
x
¯
x,¯
u
(x(τ)¯
x(τ)) + ¯σf
u
¯
x,¯
u
(u(τ)¯
u(τ)) + f(¯
x(τ),¯
u(τ))(σ¯σ),
= ¯σ∂f
x
¯
x,¯
u
x(τ) + ¯σf
u
¯
x,¯
u
u(τ) + f(¯
x(τ),¯
u(τ))σ¯σ∂f
x
¯
x,¯
u
¯
x(τ) + ¯σf
u
¯
x,¯
u
¯
u(τ),
=A(τ)x(τ) + B(τ)u(τ) + Σ(τ)σ+z(τ),(33)
where,
A(τ) = ¯σf
x
¯
x,¯
u
, B(τ) = ¯σ∂f
u
¯
x,¯
u
,Σ(τ) = f(¯
x(τ),¯
u(τ)),
z(τ) = ¯σ∂f
x
¯
x,¯
u
¯
x(τ)¯σ∂f
u
¯
x,¯
u
¯
u(τ).
We note that (33) now has the same form as (1) and can be used directly in the discretization schemes
outlined in §II by simply replacing twith τ.
The remaining state and input constraints of [2, Problem 1] are imposed point-wise at the temporal dis-
cretization nodes. The optimal control problem is thus approximated as a finite-dimensional parameter
optimization problem over x(k),u(k)and σthat can be solved using a numerical optimization algorithm.
For example if ZOH discretization from §II.A is used then (29) becomes:
min
x(k),u(k)J(x(N1)),(34a)
subject to: x(k+1) =A|kx(k)+B|ku(k)+ Σ|kσ+z|k, k = 0, .. . , N 2,(34b)
gx(x(k))0, k = 0, . . . , N 1,(34c)
gu(x(k),u(k))0, k = 0, . . . , N 2,(34d)
b0(x(0),u(0) )=0,(34e)
bf(x(N1),u(N2) )=0.(34f)
Other discretization methods yield similar finite-dimensional optimization problems, with the only difference
being that (34b) is replaced by the appropriate discretized dynamics formulation for the chosen method.
Problem (34) is then solved iteratively until convergence is achieved.1Each iteration solves a convex second
order cone program due to the linear equality constraints and (most generally) second order cone inequality
constraints. As mentioned previously, this strategy is a form of successive convexification (SC), which we
now note uses soft quadratic trust regions and virtual control to guide the convergence process. We use
the SDPT3-4.0 solver46 to solve each of these optimization problems in our implementation. Each solution
produces the discrete state ¯
x(τk), input ¯
u(τk), and time dilation factor ¯σfor each scaled time grid node,
where tk= ¯στk. We can recover the continuous time input trajectory ¯
u(τ) over each sampling interval
τ[τk, τk+1) via (3), (9a) or (19), depending on the chosen discretization method. The continuous time
state trajectory on this interval can then be recovered by integrating the nonlinear dynamics
¯
x(τ) = ¯
x(τk) + Zτ
τk
¯σf(¯
x(ξ),¯
u(ξ))dξ, τ [τk, τk+1 ), k = 0, . . . , N 1,(35)
which we do in practice via RK41 on an Nsub-step uniform grid of the [τk, τk+1] interval, where Nsub = 15.
The continuous time state and control and the time dilation factor are used to compute (33) at the next
iteration, as illustrated in Figure 1. We have found that resetting the initial condition of the integration
in (35) to ¯
x(τk) for each k= 0, . . . , N 1 yields superior convergence properties for the iterative algorithm
described in.1, 2 While this creates a discontinuous state trajectory with which the LTV system (33) is
computed, it can provide a more accurate, robust and faster solution than continuing from final integrated
state of the previous interval. Per [1, Section III.A.3], this strategy is analogous to the well-known multiple
shooting technique. The improved convergence properties can thus be intuitively (though not rigorously)
compared to the improved performace obtained when using multiple shooting versus direct shooting.4
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Figure 1. Illustration of the integration procedure. The right-hand figure shows (35) and uses the states from
the previous successive convexification iteration to initialize the integration over each time interval.
B. Monte Carlo Method
Using each of the discretization methods presented in §II, our goal is to obtain a set of statistics that
measure their relative performance. By controlling for the optimization solver, algorithm weights, constraint
boundaries, and terminal conditions, the performance of each discretization method depends solely on
The resolution of the temporal grid, N;
The initial condition, x(0).
Our approach to Monte Carlo simulation is thus to test each discretization method over a range of initial
conditions and values of N. To be consistent, the same set of initial conditions will be tested for each
discretization method and value of N. To compute the initial condition of the ith Monte Carlo trial, we first
sample the inertial velocity from a uniform distribution according to
vi(0)
U(10,10)
U(10,10)
U(10,0)
m/s,(36)
where the first and second components are in the horizontal plane and the third component is the vertical
direction (planetary curvature is neglected). The inertial position ri(0) is then generated via hit-and-run
sampling47 of the constrained controllability (CC) polytope48–50 generated for the simplified 3-DoF problem13
corresponding to [2, Problem 1]. This random initialization ensures that an initial guess is generated so that
we know the 3-DoF problem is feasible from this initial condition. The initial mass and angular velocity
are fixed, while the initial attitude quaternion is an optimization variable. The optimal 3-DoF trajectory is
used as the initial guess for the algorithm, i.e., to generate the first reference trajectory ¯
x(τ), ¯
u(τ) and ¯σfor
τ[0,1] that are used in (33) to generate the approximate LTV system.1The 6-DoF problem, however, is
not guaranteed to be feasible and the algorithm may fail either due to being fundamentally infeasible for this
initial condition or simply failing to converge. In such cases, the problem is marked as “failed” in our Monte
Carlo data. Note that failure can generally occur for one discretization method but not another for the same
initial condition. This Monte Carlo approach is illustrated in Figure 2 and is formalized as Algorithm 1.
IV. Results
This section analyzes and compares discretization method performance using the Monte Carlo data generated
by Algorithm 1. We use R= 200 initial conditions which are shown in Figure 3. One can see that while
increasing Nfrom 20 to 30 can have a large effect on the trajectory using {ZOH, FOH, RK41}, pseudospectral
trajectories are nearly identically for all N. This echoes the conjecture that pseudospectral methods require
a smaller Nfor a given level of accuracy.25 We also note that FOH and the pseudospectral methods give
perfect success rates while ZOH fails sometimes and RK41 fails many times.
While Figure 3 visually confirms that the trajectories generated by each discretization method appear to be
similar, we proceed to quantify this similarity. Let r(k)and v(k)be the position and velocity at time tkfor
a “baseline” discretization method. Similarly define ˜
r(k)and ˜
v(k)for a “comparison” discretization method.
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Figure 2. Illustration of our Monte Carlo procedure where each combination of discretization method and
temporal resolution Nis tested for a randomly generated initial condition sampled from a CC polytope for a
simplified 3-DoF problem. Two such initial conditions are shown.
Algorithm 1 Monte Carlo data collection for discretization method performance analysis.
1: for i= 1, . . . , R do
2: for discretization method = ZOH,FOH,RK4,CGL,LGR,LGRI do
3: for N= 20,30,40 do
4: Choose vi(0) by sampling (36)
5: Generate CC polytope50 for this v(0)
6: Choose ri(0) via hit-and-run sampling47 of the CC polytope
7: Solve problem 29 for this initial condition, discretization method and N
8: if failed to converge then
9: Save trial as failed
10: else
11: Save trial data
12: end if
13: end for
14: end for
15: end for
Define position and velocity discrepancy measures:
r:= max
k=0,...,N1kr(k)˜
r(k)k2,(37)
v:= max
k=0,...,N1kv(k)˜
v(k)k2.(38)
For a Monte Carlo simulation consisting of Rtrials, Rsuch measures are generated corresponding to a given
initial condition for each trial, excluding initial conditions where either of the discretization methods failed to
converge. Figure 4 plots the median, minimum and maximum ∆rand ∆vusing ZOH and CGL as baseline
methods (using more baseline methods is redundant since the discrepancy relationship is transitive). We
note two important characteristics:
1. The {ZOH, FOH, RK41}and {CGL, LGR, LGRI}method groups converge to distinct solutions of the
problem, since position and velocity discrepancies do not decrease between these groups for larger N;
2. The {ZOH, FOH, RK41}group appears to converge around the same solution as Nincreases, but has
much more variability in the discrepancy measures (up to 60 m and 9.5 m/s) than the {CGL, LGR,
LGRI}group (up to 16 m and 1.5 m/s) for N= 20.
These results suggest that pseudospectral methods provide more consistent trajectories that are less sensitive
to the discretization resolution N.
The next important quantity to consider is the influence of each discretization method on the solution time.
Because SDPT3-4.0 is a relatively slow solver, we look at the solution times as indicative of the relative
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(a) Success rate {99,99,99}%. (b) Success rate {100,100,100}%. (c) Success rate {73,98,95}%.
(d) Success rate {100,100,100}%. (e) Success rate {100,100,100}%. (f ) Success rate {100,100,100}%.
Figure 3. Initial condition cloud. Blue markers show initial conditions where successive convexification failed to
converge. Light, medium and dark gray lines show the generated trajectories for N= 20,30 and 40 respectively
and subfigure captions give the corresponding success rates.
Figure 4. rand vstatistics. Bars show the median and error bars show the minimum and maximum
difference.
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(a) SC global iteration count. (b) Solver iteration count per SC iteration.
(c) Overall problem solution time. (d) Solution time per SC iteration.
Figure 5. Successive convexification iteration and solution time statistics. Bars show the median and error
bars show the minimum and maximum values.
speed of each discretization method for a future real-time implementation. Figure 5 shows the SC iteration
and solution time statistics. From Figure 5a, we see that the group of pseudospectral methods generally
converges in fewer iterations and with less variance than the {ZOH, FOH, RK41}group. We set the SC
method to timeout at 20 iterations and indeed the RK41 method is the worst performing by this measure
as it sometimes requires all 20 iterations. As shown in Figure 5b, however, the median number of iterations
that SDPT3-4.0 takes to solve each SC sub-problem is relatively constant across all methods with a slight
increase for larger N. The most important quantity, however, is the overall time taken to solve the problem.
As shown in Figure 5c, for N= 20 all methods take a median of 12-16s to solve the problem. According to
Figure 5d, for N= 20, SDPT3-4.0 takes a median of 3 s to solve each SC subproblem using {ZOH, FOH,
RK41}and a median of 4-6 s for {CGL, LGR, LGRI}. However, this is offset by the fewer SC iterations
required, yielding an overall similar SC solution time for all methods. Indeed, for N= 20 the LGR method
is the best performing as its SC total solution time’s median and maximum values are both small. The
situation changes for N= 30 and 40, however, as the SDPT3-4.0 solution time for the larger SC sub-
problems increases much more for {CGL, LGR, LGRI}than for {ZOH, FOH, RK41}. This indicates that
the overall solution time is more sensitive to Nfor pseudospectral methods.
The increase in solution time for pseudospectral methods may be explained by a loss of sparsity in the
resulting finite dimensional optimization problem. This is caused by the differentiation matrix Din (20)
which effectively couples the states and controls across all temporal discretization nodes. This is not the case
for the {ZOH, FOH, RK41}methods, where only the neighboring states and controls are explicitly coupled.
Generally speaking, a loss of sparsity will slow down any optimization problem. In the particular case of
SDPT3-4.0 this is manifested through the sparsity of a Schur complement matrix Mwhich is involved in a key
computationally expensive step of finding the search direction in a primal-dual interior-point algorithm [51,
Section 2.1]. The authors note that factorization of Mbecomes very expensive when Mis dense. Figure 6
confirms that the density of Mcan increase up to 3 times for pseudospectral discretization compared to
{ZOH, FOH, RK41}. Furthermore, Figure 7 shows that the Mmatrix can be several orders of magnitude
worse conditioned for pseudospectral methods. We believe that these poor numerical properties of the M
matrix are a key contributor to the observed slowdown in solver time.
Having considered the solution time, it is equally important to consider the accuracy and performance of
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(a) 4.89% non-zero elements. (b) 5.66% non-zero elements. (c) 5.66% non-zero elements.
(d) 14.62% non-zero elements. (e) 14.24% non-zero elements. (f ) 16.89% non-zero elements.
Figure 6. Sparsity patterns of the SDPT3-4.0 Schur complement matrix Musing N= 20. Similar patterns
hold for different Nand initial conditions.
the trajectories generated by each discretization method. We measure accuracy by the final position and
velocity errors obtained by propagating the nonlinear dynamics using the generated thrust profile. That
is, we numerically integrate (29b) using the interpolated continuous thrust commands from the converged
solution. The integration is carried out over the interval [0, tf] in one shot. We stress that only the thrust
is used for this process, not the state vectors obtained as part of the optimization. The performance is
measured by fuel consumption. As illustrated in Figures 8a and 8b, FOH achieves approximately the same
level of accuracy as the pseudospectral methods. The RK41 method is the least accurate, since it mimics
the FOH input profile, but does not include the fine integration steps (11a) and (11b). As a result, RK41
does not sufficiently capture the nonlinear dynamics between sample points. In terms of performance, all
methods achieve a similar fuel consumption.
An ideal discretization method yields both a small SC total solution time and small final position and velocity
errors. The scatter plots in Figure 9 visualize how close each discretization method comes to this ideal, with
better methods located in the left hand corner. Because all methods generate less than about 1 m/s velocity
error (see Figure 8b), we consider only the position error in Figure 9. We see that for N= 20, {FOH, CGL,
LGR}are the best performing, with FOH matching the accuracy of CGL while LGR is the most consistently
faster method (it has less variance in the SC total solution time). The LGRI method is interestingly not
particularly accurate as was originally motivated in §II. Due to the poor Mmatrix conditioning in Figure 7,
this could be a result of solver numerical error (i.e. our implementation) rather than a fault of the LGRI
method itself. As we have already seen, ZOH and RK41 underperform. In particular, it is interesting to
note that the RK41 method is the worst in SC total solution time, this being due to the large number of
SC iterations that it sometimes requires to converge. For N= 30 and 40, the solution time increases for
pseudospectral methods much more than it does for {ZOH, FOH, RK41}. As a result, for higher Nthe
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Figure 7. Condition number of the SDPT3-4.0 Schur complement matrix M. Bars show the median and error
bars show the minimum and maximum values.
(a) Final position error. (b) Final velocity error. (c) Total fuel consumption.
Figure 8. Trajectory accuracy and performance statistics. Bars show the median and error bars show the
minimum and maximum values.
FOH method is the best performing.
An ideal discretization method also ensures that the state and input constraints are not violated between
temporal discretization nodes. In this context, one can check whether the thrust magnitude constraints are
satisfied and is illustrated in Figure 10. One can see that, by design, the {ZOH, FOH, RK41}methods
satisfy the thrust magnitude constraints at all times. The pseudospectral method, however, tends to violate
the constraints whenever the thrust magnitude is near one of the boundaries. This is due to the interpolated
polynomial nature of the input. Despite this fact, pseudospectral methods seem to produce input profiles
that are more similar across Nthan the {ZOH, FOH, RK41}methods, where it is seen that in some cases
the input changes drastically as Nincreases. The ZOH and FOH cases each produce input profiles that are
relatively consistent across N, but perhaps less so than each pseudospectral method. It must also be noted
that each pseudospectral method yields a small number of solutions where the burn time greatly exceeds
the average. Interestingly, since FOH achieved a 100% success rate, we know that it solved these same exact
problems but found a much shorter burn time (we cannot make the same definitive claim for ZOH and RK41
as they did fail some trials). The cause of this phenomenon is difficult to pinpoint, and thus raises some
concern as the difference of roughly 20 s burn time is considerable.
V. Discussion
This section summarizes the findings of Section IV. Table 1 overviews the benefits and drawbacks of each
group of methods (i.e., those that parameterize control only versus those that parameterize state and control).
FOH with N= 20 appears to be the best method that parameterizes the control as it has the lowest overall
solution time in that group, and is accurate for each N. Of the three methods that parameterize state and
control, LGR with N= 20 is preferred as it has the lowest overall solution time and is about as accurate as
CGL and LGRI. We recall that LGR and FOH appear to find slightly different solutions, since increasing N
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Figure 9. Final landing position error versus SC overall solution time. The spread generated by the random
initial conditions for each discretization method is covered with its convex hull polytope for visualization
purposes.
does not bring the trajectories produced by the two methods closer (see Figure 4). Despite LGR matching
the performance of FOH for N= 20, polynomial interpolation of the input induces inter-sample constraint
violation as illustrated in Figure 10. The result is that further constraints would need to be added to
the pseudospectrally-discretized optimal control problem in order to (conservatively) prevent inter-sample
constraint violation. This extra complexity would, however, likely increase the solution time. Our key
conclusion from the present analysis, therefore, is that:
FOH yields an accurate and fast solution with no convex input constraint violation, but has the
disadvantage that the state and input trajectories can vary with N;
LGR performance matches that of FOH for small N, produces state and input trajectories that are
more consistent across N, but has the disadvantage of inter-sample input constraint violation and less
sparse optimization problems.
VI. Conclusions
In this paper, we analyzed and compared the performance and accuracy of six discretization methods in the
context of solving a powered descent guidance problem via direct optimization. These six methods included
three that parameterized the control function only, and three that parameterized both the state and control
functions (using global pseudospectral methods). A Monte Carlo statistical analysis indicated that the
performance and accuracy of both groups matches for discretizations with a low temporal resolution, but the
pseudospectral methods become slower for higher temporal resolutions. It was concluded that the first order
hold discretization and LGR pseudospectral method provide the fastest computational times and achieved
similar performance. While the first order hold method guarantees inter-sample convex input constraint
satisfaction, it showed sensitivity to the discretization’s temporal resolution. Conversely, the LGR method
produces trajectories that are less sensitive to the temporal resolution, however, inter-sample constraint
violation was consistently observed and the optimization problems were noted to be much less sparse. Each
characteristic is important for a real-time implementable algorithm designed to solve 6-DoF powered descent
guidance problems. Our future work will provide discussion of the implementation of these methods in
concert with a real-time capable optimization solver.
Acknowledgements
This research has been supported by NASA grant NNX17AH02A. Government sponsorship acknowledged.
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Figure 10. Thrust profiles with upper and lower bounds shown as dashed black lines. Top to bottom: ZOH,
FOH, RK41, CGL, LGR and LGRI. Left to right: N= 20,30 and 40.
{ZOH, FOH, RK41} {CGL, LGR, LGRI}
+ Higher sparsity in the discretized optimization prob-
lem (Figure 6);
+ Faster solver solution times, in particular for larger
N;
+ FOH provides a terminal position and velocity accu-
racy on par with pseudospectral (Figures 8a and 8b);
Potentially large changes in trajectory across N(Fig-
ure 3);
Potentially large changes in input trajectory across N
(Figure 10);
Generally more SC iterations required (Figure 5a).
+ More consistent trajectories across N(Figure 3);
+ More consistent optimal input trajectories across N
(Figure 10);
+ Fewer SC iterations (Figure 5a);
+ Similar performance to FOH for N= 20;
Longer solver solution time per SC iteration (Fig-
ure 5d);
Higher sensitivity of SC solution time to N(Figures 5c
and 5d);
Lower sparsity in the discretized optimization problem
(Figure 6);
Input constraint violation due to polynomial intepro-
lation (Figure 10).
Table 1. Benefits and drawbacks of the discretization methods.
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Book
I. Examples of Control Problems.- II. Formulation of the Control Problem.- III. Existence Theorems with Convexity Assumptions.- IV. Existence Without Convexity.- V. The Maximum Principle and Some of Its Applications.- VI. Proof of the Maximum Principle.- Bibliographical Notes.