In this work we present necessary and sufficient conditions for the nonnegativity and positivity of discrete quadratic functionals for the optimal control setting and generalizations to quadratic functionals related to symplectic difference and dynamic systems. These results have been obtained by the author and his scientific collaborators O.Dosly and V.Zeidan during the years 1999-2002, and have been published or submitted for publication in the papers [ 31 ], [ 46 ], and [ 51 ], [ 52 ], [ 53 ], [ 54 ], [ 55 ], [ 56 ].

The great importance of discrete quadratic functionals resides in the fact that they arise as the second variation of nonlinear problems in the discrete calculus of variations and optimal control theory. Such discrete optimization problems are obtained by discretizing an optimal control (or calculus of variations) problem. Since the nonnegativity and positivity of the second variation are necessary and sufficient conditions, respectively, for optimality in the original discrete-time problem, the conditions presented in this work are, in this sense, second order necessary and sufficient optimality conditions for nonlinear discrete problems. Most of the presented results on discrete quadratic functionals can be regarded as the discrete counterparts of the corresponding continuous-time statements. Thus, the discrete-time control theory is developed in a parallel way to the continuous one, although many essential discrepancies occur and have to be understood. For the classical theories of (continuous) calculus of variations and optimal control, which have been intensively studied in the literature, we recommend to see e.g. [ 7 ], [ 37 ], [ 39 ], [ 62 ].

The discrete problem has one great advantage over the continuous one, namely, it is a finite dimensional problem. Therefore, our investigation of discrete quadratic functionals uses the methods of mathematical programming, sensitivity technique, transformation of separable endpoints to fixed endpoints, and transformation of joint endpoints to separable endpoints via augmenting the problem into the double dimension. Some of these methods are known in the continuous-time control setting, but are much more complex. On the other hand, most of the discrete-time results require ``only'' a suitable and clever applications of the matrix analysis and linear algebra.

A key role in the investigation of the nonnegativity and positivity of discrete quadratic functionals is played by the notions of focal points for conjoined bases of the associated linear Hamiltonian difference system , conjugate intervals (as opposed to conjugate points in the continuous-time theory), partial quadratic functionals, and implicit and explicit Riccati difference equations . Focal points, in their full generality, have been introduced in 1996 by M.Bohner in his landmark work [ 12 ], where the positivity of a discrete quadratic functional was characterized in terms of the nonexistence of such focal points, for the so-called singular case , i.e., for the case that would include quadratic functionals for the higher-order Sturm-Liouville difference equations. Therefore, Bohner's paper [ 12 ] can be regarded as a starting point for a new discrete-time theory, a theory that is studied, and in several aspects completed, in this work.

The text is divided into five chapters. Each chapter contains author's original results and their relation to the known ones from the literature. In Chapter 1, we introduce a nonlinear discrete optimal control problem with equality control constraints and equality state endpoint constraints as a discretization of the corresponding continuous-time problem. Then, we present first and second order necessary and sufficient optimality conditions in terms of the second variation, which is a discrete quadratic functional with appropriate boundary conditions. The discrete analog of the Legendre-Clebsch transformation is then performed on the second variation to obtain a transformed second variation, which is the discrete quadratic functional studied further in the text.

Next, we proceed in Chapter 2 by characterizing the positivity and nonnegativity of such discrete quadratic functionals in terms of the previously mentioned conjugate intervals, focal points of conjoined bases, partial quadratic functionals, and implicit and explicit Riccati equations, each of them having equality or inequality endpoint constraints. The results on the positivity precede the ones on the nonnegativity only from historical reasons, since the characterization of the nonnegativity of discrete quadratic functionals, proved recently by the author and V.Zeidan in [ 56 ], has been an open problem long after the corresponding result on the positivity was established.

In Chapter 3, we investigate the nonnegativity and positivity of quadratic functionals in the discrete calculus of variations. In this case, our previous results from Chapter 2 can be applied, if a certain invertibility condition is satisfied. Under this condition, it is well-known that any calculus of variations problem can be written as a special control problem. However, if the invertibility condition is not satisfied (the so-called singular case in the calculus of variations), we can still prove characterizations of the nonnegativity and positivity of the corresponding discrete quadratic functionals by invoking the theory of three term recurrence equations .

In Chapter 4, we offer a generalization to variable stepsize discrete symplectic systems and the corresponding discrete quadratic functionals, which include the Hamiltonian systems, considered in Chapter 2, as a special case.

Further generalization and unification with the known continuous-time results is presented in Chapter 5, where symplectic dynamic systems are introduced on the so-called time scales (or measure chains ). Such unification of continuous and discrete theories provides also a clear and quantitative explanation of the discrepancies between these two theories in terms of the graininess function of the time scale. Finally, each chapter contains a list of relevant open problems and perpectives for further research.

The author would like to thank cordially to all colleagues and friends that supported him in the research and shared with him their professional and personal experience. My greatest acknowledgements belong to Calvin Ahlbrandt, Martin Bohner, Zuzana Dosla, Ondrej Dosly, Stefan Hilger, Werner Kratz, Allan Peterson, Pavel Rehak, and Vera Zeidan.

Last change: October 9, 2003. (c) Roman Hilscher