Success stories of optimal control theory in various applications are too numerous to be listed here; see [CEHS87,ST05,Swa84] for some examples from engineering and well beyond. The reader interested in applications will easily find many other references.
The material in Section 1.2 can be found in standard texts
on optimization, such as [Lue84] or [Ber99].
See also Sections 5.2-5.4 of the book [AF66], which
will be one of our main references for the optimal control chapters. Complete
proofs of
the results presented in Section 1.2.2, including the fact that the condition (1.20) is sufficient for 
to be a tangent vector, are given
in [Lue84, Chapter 10]. The alternative argument based on the
inverse function theorem is adopted from [Mac05, Section 1.4]. The necessary condition in terms of Lagrange multipliers can also be derived from a cone separation argument (Farkas' lemma) as shown, e.g., in [Ber99, Section 3.3.6]; we will see this type of reasoning in the proof of the maximum principle in Chapter 4.
Section 1.3 is largely based on [GF63], which will be our main reference for calculus of variations; function spaces, functionals, and the first variation are introduced in the first several sections of that book, while the second variation is discussed later in Chapter 5. Essentially the same material can be found in [AF66, Section 5.5]. In [GF63] as well as in [Mac05] the first and second variations are defined via (1.38) and (1.43), while other sources such as [You80] follow the approach based on (1.33) and (1.39).
For further background on function spaces and relevant topological concepts, the reader can consult [Rud76] or [Sut75] (the latter text is somewhat more advanced). Another recommended reference on these topics is [Lue69], where Gateaux and Fréchet derivatives and their role in functional minimization are also thoroughly discussed. A general treatment of convex functionals and their minimization is given in [Lue69, Chapter 7]; for convexity-based results more specific to calculus of variations and optimal control, see the monograph [Roc74], the more recent papers [RW00] and [GS07], and the references therein.