On an infinite dimensional linear-quadratic problem with fixed endpoints: The continuity question

The existing theory of linear-quadratic problems has been successfully applied to the design of many industrial and military control systems (see, e.g., 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, e.g., 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, e.g., Chow, 1976; Kendrick, 1981). However, such stochastic problems are frequently infinite dimensional (see, e.g., 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. The purpose of our paper is to explore the conditions under which the solutions to the above-mentioned optimization problems continuously depend on initial and final conditions. Not surprisingly, these continuity (or discontinuity) conditions are strongly related to some concepts of controllability for infinite dimensional (linear) systems. The importance of the continuous dependence of the optimal solution upon the imposed initial and final conditions is obvious, in particular when developing numerical methods for the minimum energy or linear quadratic problem. For infinite dimensional linear control systems, the continuous dependence of optimal solutions on constraints on values of admissible controls has been considered by Przyłuski (1981). A much more general approach to such problems is presented by Kandilakis and Papageorgiou (1992) as well as Papageorgiou (1991).