towards the end. Thus, on the one hand, it is by no means necessar

for a reader of this book to have been exposed to real analysis; an

yet, on the other hand, such prior exposure cannot help but increas

the book’s accessibility.

Students taking a ﬁrst course on this topic typically have divers

backgrounds among engineering, mathematics and the natural or so

cial sciences. The range of potential applications is correspondingl

broad: the calculus of variations and optimal control theory hav

been widely used in numerous ways in, e.g., biology [27, 35, 58],

criminology [18], economics [10, 26], engineering [3, 49], ﬁnance [9

management science [12, 57], and physics [45, 63] from a variet

of perspectives, so that the needs of students are too extensive t

be universally accommodated. Yet one can still identify a solid cor

of material to serve as a foundation for future graduate studies, re

gardless of academic discipline, or whether those studies are applie

or theoretical. It is this core of material that I seek to expoun

as lucidly as possible, and in such a way that the book is suitabl

not only as an undergraduate text, but also for self-study. In othe

words, this book is primarily a mathematics text, albeit one aime

across disciplines. Nevertheless, I incorporate applications—cance

chemotherapy in Lecture 20, navigational control in Lecture 22 an

renewable resource harvesting in Lecture 24—to round out the theme

developed in the earlier lectures.

Arnold Arthurs introduced me to the calculus of variations i

1973-74, and these lectures are based on numerous sources consulte

at various times over the 35 years that have since elapsed; sometime

with regard to teaching at FSU; sometimes with regard to my own re

search contributions to the literature on optimal control theory; an

only recently with regard to this book. It is hard now to judge th

relative extents to which I have relied on various authors. Neverthe

less, I have relied most heavily on—in alphabetical order—Akhieze

[1], Bryson & Ho [8], Clark [10], Clegg [11], Gelfand & Fomin [16

Hadley & Kemp [19], Hestenes [20], Hocking [22], Lee & Marku

[33], Leitmann [34], Pars [47], Pinch [50] and Pontryagin et al. [51

and other authors are cited in the bibliography. I am grateful to a

of them, and to each in a measure proportional to my indebtedness

1Bold

numbers in square brackets denote references in the bibliography (p. 245

for a reader of this book to have been exposed to real analysis; an

yet, on the other hand, such prior exposure cannot help but increas

the book’s accessibility.

Students taking a ﬁrst course on this topic typically have divers

backgrounds among engineering, mathematics and the natural or so

cial sciences. The range of potential applications is correspondingl

broad: the calculus of variations and optimal control theory hav

been widely used in numerous ways in, e.g., biology [27, 35, 58],

criminology [18], economics [10, 26], engineering [3, 49], ﬁnance [9

management science [12, 57], and physics [45, 63] from a variet

of perspectives, so that the needs of students are too extensive t

be universally accommodated. Yet one can still identify a solid cor

of material to serve as a foundation for future graduate studies, re

gardless of academic discipline, or whether those studies are applie

or theoretical. It is this core of material that I seek to expoun

as lucidly as possible, and in such a way that the book is suitabl

not only as an undergraduate text, but also for self-study. In othe

words, this book is primarily a mathematics text, albeit one aime

across disciplines. Nevertheless, I incorporate applications—cance

chemotherapy in Lecture 20, navigational control in Lecture 22 an

renewable resource harvesting in Lecture 24—to round out the theme

developed in the earlier lectures.

Arnold Arthurs introduced me to the calculus of variations i

1973-74, and these lectures are based on numerous sources consulte

at various times over the 35 years that have since elapsed; sometime

with regard to teaching at FSU; sometimes with regard to my own re

search contributions to the literature on optimal control theory; an

only recently with regard to this book. It is hard now to judge th

relative extents to which I have relied on various authors. Neverthe

less, I have relied most heavily on—in alphabetical order—Akhieze

[1], Bryson & Ho [8], Clark [10], Clegg [11], Gelfand & Fomin [16

Hadley & Kemp [19], Hestenes [20], Hocking [22], Lee & Marku

[33], Leitmann [34], Pars [47], Pinch [50] and Pontryagin et al. [51

and other authors are cited in the bibliography. I am grateful to a

of them, and to each in a measure proportional to my indebtedness

1Bold

numbers in square brackets denote references in the bibliography (p. 245