8.6.
BIBLIOGRAPHICAL REMARKS 117
p
k k
G
l k
t p
t l
Figure 8.7: Illustration for Case 2.
8.6 Bibliographical Remarks
The results presented in this chapter appeared in English in full detail for the first time in 1962 in the book by Pontryagin, et al., cited earlier. That book contains many extensions and many examples
and it is still an important source. However, the derivation of the maximum principle given in the book by Lee and Markus is more satisfactory. Several important generalizations of the maximum
principle have appeared. On the one hand these include extensions to infinite-dimensional state spaces and on the other hand they allow for constraints on the state more general than merely initial
and final constraints. For a unified, but mathematically difficult, treatment see Neustadt [1969]. For a less rigorous treatment of state-space constraints see Jacobson, et al, [1971], whereas for a
discussion of the singular case consult Kelley, et al. [1968].
For an applications-oriented treatment of this subject the reader is referred to Athans and Falb [1966] and Bryson and Ho [1969]. For applications of the maximum principle to optimal eco-
nomic growth see Shell [1967]. There is no single source of computational methods for optimal control problems. Among the many useful techniques which have been proposed see Lasdon, et
al. , [1967], Kelley [1962], McReynolds [1966], and Balakrishnan and Neustadt [1964]; also
consult Jacobson and Mayne [1970], and Polak [1971].
118
CHAPTER 8. CONINUOUS-TIME OPTIMAL CONTROL
.
p
k k
G
1 k
t k
G
p
t 1
Figure 8.8: Case 3. The singular case.
8.6.
BIBLIOGRAPHICAL REMARKS 119
. . .
. . .
. .
p
∗
1 t
T s
∗
1 t
T k
∗
t T
p
∗
Case A
t T
t
2
t
1
s
∗
1
µk
G
f k
G
t k
∗
k
G
k t
p
∗
1 t
T t
2
s
∗
1 t
T t
2
k
∗
t T
t
2
p
∗
Case Bi
t T
t
2
t
1
s
∗
t k
∗
t Case
Biii Figure 8.9: The optimal solution of example.
120
CHAPTER 8. CONINUOUS-TIME OPTIMAL CONTROL
Chapter 9
Dynamic programing
SEQUENTIAL DECISION PROBLEMS: DYNAMIC PROGRAMMING FORMULATION The sequential decision problems discussed in the last three Chapters were analyzed by varia-
tional methods, i.e., the necessary conditions for optimality were obtained by comparing the op- timal decision with decisions in a small neighborhood of the optimum. Dynamic programming
DP is a technique which compares the optimal decision with all the other decisions. This global comparison, therefore, leads to optimality conditions which are sufficient. The main advantage of
DP, besides the fact that it give sufficiency conditions, is that DP permits very general problem for- mulations which do not require differentiability or convexity conditions or even the restriction to a
finite-dimensional state space. The only disadvantage which unfortunately often rules out its use of DP is that it can easily give rise to enormous computational requirements.
In the first section we develop the main recursion equation of DP for discrete-time problems. The second section deals with the continuous-time problem. Some general remarks and bibliographical
references are collected in the final section.
9.1 Discrete-time DP
Kategori