Positivity of Quadratic Functionals on Time Scales: Necessity

Hilscher, Roman Šimon

doi:10.1002/1522-2616(200106)226:1<85::aid-mana85>3.0.co;2-h

Cited by 7 publications

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References 11 publications

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“…Recently, a characterization of the positivity of J has been intensively studied in [9,10,14,15,16]. This was expressed among others in terms of the existence of a solution to a certain time scale matrix Riccati equation with appropriate boundary conditions.…”

Section: T X σ (T) X (T) T (P)mentioning

confidence: 99%

Time scale embedding theorem and coercivity of quadratic functionals

Hilscher

Zeidan²

2008

Analysis

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Summary:In this paper we study the relation between the coercivity and positivity of a time scale quadratic functional J , which could be a second variation for a nonlinear time scale calculus of variations problem (P). We prove for the case of general jointly varying endpoints that J is coercive if and only if it is positive definite and the time scale version of the strengthened Legendre condition holds. In order to prove this, we establish a time scale embedding theorem and apply it to the Riccati matrix equation associated with the quadratic functional J . Consequently, we obtain sufficiency criteria for the nonlinear problem (P) in terms of the positivity of J or in terms of the time scale Riccati equation. This result is new even for the continuous time case when the endpoints are jointly varying.

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Section: T X σ (T) X (T) T (P)mentioning

confidence: 99%

Time scale embedding theorem and coercivity of quadratic functionals

Hilscher

Zeidan²

2008

Analysis

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“…Thus, with A k := I + A(k), B k := B(k), C k := C(k), and D k := I + D(k), system (S) and the functional F reduce respectively to (S d ) and F d . Therefore, based on the finding that (S) unifies the Hamiltonian system in the continuous case and the symplectic system in the discrete case the name time scale symplectic system was attributed to (S) in [20,Remark 4.3] and [11].…”

Section: Introductionmentioning

confidence: 99%

Time scale symplectic systems without normality

Hilscher

Zeidan

2006

Journal of Differential Equations

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We present a theory of the definiteness (nonnegativity and positivity) of a quadratic functional F over a bounded time scale. The results are given in terms of a time scale symplectic system (S), which is a time scale linear system that generalizes and unifies the linear Hamiltonian differential system and discrete symplectic system. The novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of F , and in establishing equivalent conditions for the nonnegativity of F without any normality assumption. To reach this goal, a new notion of generalized focal points for conjoined bases (X, U ) of (S) is introduced, results on the piecewise-constant kernel of X(t) are obtained, and various Picone-type identities are derived under the piecewise-constant kernel condition. The results of this paper generalize and unify recent ones in each of the discrete and continuous time setting, and constitute a keystone for further development in this theory.

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