The existing theory of linear-quadratic problems has been successfully applied to the design of many industrial and military control systems (see, eg, Athans, 1971). A stochastic version of this problem plays today an important role in macroeconomics, where the so-called linear-quadratic economies are considered (see, eg, Ljungqvist and Sargent, 2004; Sent, 1998). These (dynamic stochastic) optimizing models had to have linear constraints with quadratic objective functions to get a linear decision rule (see, eg, Chow, 1976; Kendrick, 1981). However, such stochastic problems are frequently infinite dimensional (see, eg, the work of Federico (2011) and the references cited therein). We will consider infinite dimensional linear control systems which can be represented by two linear continuous operators describing the influence of control, and the constraints imposed on all of the system’s trajectories by given initial and final conditions. The minimum energy and linear-quadratic problems for such systems will be developed. These problems can be studied in an appropriate Hilbert space setting. Then (as is well known) the existence and uniqueness of optimal solutions to the above problems can be easily established, under rather mild assumptions.