Chapter 2 explores optimal control problems for finite- and infinite-dimensional systems through the lens of the Dubovitskiı̆-Milyutin functional analytical method. After analyzing extremum problems in finite dimensions, the chapter applies this methodology to two distinct optimal control scenarios in infinite dimensions: controlling vibrations of an elastic beam and optimizing sterilization processes for prepackaged food. Key topics covered include necessary optimality conditions, determination of cones, and control strategies tailored to the specific technical requirements of each system. The chapter provides insights into how the Dubovitskiı̆-Milyutin approach can be adapted to solve a wide range of optimal control problems, illustrating its versatility and practical value.