The book is dedicated to functional and convex analysis techniques for the study of recent problems in optimal control and mechanics. It provides a rigorous foundation for the application of variational principles. Through the unifying approach of convex analysis, the text discusses four general types of applications: contact problems in mechanics, control of distributed parameter systems, control of incompressible flow, and identification problems in mechanics. It also considers extensions of convex analysis with application to the homogenization of steady state heat conduction equations, approximation theory in identification problems, and nonconvex variational problems in mechanics. Special emphasis is given to techniques for solving convex optimization problems subject to constraints.