Discrete Quadratic Functionals Research Articles

Multiplicities of focal points for discrete symplectic systems: revisited

In this note, we define a notion of multiplicity of focal points for conjoined bases of discrete symplectic systems. We show that this definition is equivalent to the one given by Kratz in [Discrete oscillation, J. Difference Equ. Appl., 9(1), 135–147 (2003)] and, furthermore, it has a natural connection to the newly developed continuous time theory on linear Hamiltonian differential systems. Many results obtained recently by Bohner, Došlý, and Kratz regarding the non-negativity of the corresponding discrete quadratic functionals, Sturmian separation and comparison theorems, and oscillation theorems relating the number of focal points of a certain special conjoined basis with the number of eigenvalues of the associated discrete symplectic eigenvalue problem, are now formulated in terms of this alternative definition of multiplicities.

Sturmian and spectral theory for discrete symplectic systems

We consider 2 n × 2 n 2n\times 2n symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most n n . In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.

Riccati inequality and other results for discrete symplectic systems

In this paper we establish several new results regarding the positivity and nonnegativity of discrete quadratic functionals F associated with discrete symplectic systems. In particular, we derive (i) the Riccati inequality for the positivity of F with separated endpoints, (ii) a characterization of the nonnegativity of F for the case of general (jointly varying) endpoints, and (iii) several perturbation-type inequalities regarding the nonnegativity of F with zero endpoints. Some of these results are new even for the special case of discrete Hamiltonian systems.

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.

A Remark on Discrete Quadratic Functionals with Separable Endpoints

A characterization of the positivity of a discrete quadratic functional with separable state endpoints constraints is presented in terms of conjugate intervals to 0 , various conjoined bases of the associated linear Hamiltonian difference system, and solutions of the implicit and explicit Riccati difference equations. The boundary conditions are in the form of either equalities or (strict) inequalities. Three sets of results are derived under different underlying assumptions.

POSITIVE SEMIDEFINITENESS OF DISCRETE QUADRATIC FUNCTIONALS

AbstractWe consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

Nonnegativity of discrete quadratic functionals corresponding to symplectic difference systems

We study the nonnegativity of quadratic functionals with separable endpoints which are related to the discrete symplectic system (S). In particular, we characterize the nonnegativity of these functionals in terms of (i) the focal points of the natural conjoined basis of (S) and (ii) the solvability of an implicit Riccati equation associated with (S). This result is closely related to the kernel condition for the natural conjoined basis of (S). We treat the situation when this kernel condition is possibly violated at a certain index. To accomplish this goal, we derive a new characterization of the set of admissible pairs (sequences) that does not require the validity of the above mentioned kernel condition. Finally, we generalize our results to the variable stepsize case.

Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals

Discrete quadratic functionals with variable endpoints for variable stepsize symplectic difference systems are considered. A comprehensive study is presented for characterizing the positivity of such functionals in terms of conjugate intervals, conjoined bases, and implicit and explicit Riccati equations with various forms of boundary conditions. Moreover, necessary conditions for the nonnegativity of these functionals are obtained in terms of the above notions. Furthermore, we show that a variable stepsize discretization of a continuous-time nonlinear control problem leads to a discrete linear quadratic problem and a Hamiltonian difference system, which are special cases of their symplectic counterparts.

Discrete Optimal Control: Second Order Optimality Conditions

In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) . Finally, open problems and perspectives are also discussed.

Positivity of Block Tridiagonal Matrices

This paper relates disconjugacy of linear Hamiltonian difference systems (LHdS) (and hence positive definiteness of certain discrete quadratic functionals) to positive definiteness of some block tridiagonal matrices associated with these systems and functionals. As a special case of a Hamiltonian system, Sturm--Liouville difference equations are considered, and analogous results are obtained for these important objects.