Existence of Absolutely Continuous Fundamental Matrix of Linear Fractional System with Distributed Delays

Kiskinov, Hristo; Madamlieva, Ekaterina; Veselinova, Magdalena; Zahariev, Andrey

doi:10.3390/math9020150

Cited by 7 publications

(13 citation statements)

References 19 publications

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“…Proof. According Theorem 9 in [31] the unique solution X(t) of the IP (1), (2) for every t ∈ R + has the representation (30) and hence we obtain that X Φ (t) = T −h (t, −h)Φ(−h) which completes the proof.…”

Section: A Priory Estimates Of the Solutions Obtained Via Their Integral Representationssupporting

confidence: 52%

“…The existence and uniqueness of the fundamental matrix C(t, s) of the system (4) and the matrix T −h (t, s) as well as their properties are proved in [31]. Please note that these matrices are absolutely continuous concerning t and continuous in s on every compact subinterval inR + if s = t and for s = t possess first kind jumps [31].…”

Section: Preliminaries and Problem Statementmentioning

confidence: 97%

“…The next different a priori estimations are obtained using the other most popular approach, which is essentially based on the different kinds integral representations of the solutions of the considered systems obtained in [31,33] and applying the superposition principle. Theorem 4.…”

Section: A Priory Estimates Of the Solutions Obtained Via Their Integral Representationsmentioning

confidence: 99%

“…Proof. According Theorem 9 in [31] the unique solution X Φ (t) of the IP (1), (2) for every t ∈ R + has the following representation:…”

Section: A Priory Estimates Of the Solutions Obtained Via Their Integral Representationsmentioning

confidence: 99%

“…Section 3 is devoted to obtaining a priori estimates of the solutions of nonautonomous fractional differential systems with Caputo type derivatives of incommensurate orders with distributed delays via Gronwall inequality. In Section 4 for the solutions of the same systems we obtain a priori estimates using the approach based on their integral representations obtained in [31]. In Section 5 as application of the proved estimates we obtain sufficient conditions for finite-time stability of the considered systems.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Kiskinov

Veselinova

Madamlieva

et al. 2021

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In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.

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Section: A Priory Estimates Of the Solutions Obtained Via Their Integral Representationssupporting

confidence: 52%

Section: Preliminaries and Problem Statementmentioning

confidence: 97%

Section: A Priory Estimates Of the Solutions Obtained Via Their Integral Representationsmentioning

confidence: 99%

“…Proof. According Theorem 9 in [31] the unique solution X Φ (t) of the IP (1), (2) for every t ∈ R + has the following representation:…”

Section: A Priory Estimates Of the Solutions Obtained Via Their Integral Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Kiskinov

Veselinova

Madamlieva

et al. 2021

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Study on Optimization Method of Visual Odometry Based on Feature Matching

Song

Zhao

Ren

et al. 2022

Mathematical Problems in Engineering

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The mismatching of image features affects the calculation of the fundamental matrix and then leads to poor estimation accuracy of SLAM visual odometry. Aiming at the above problems, a visual odometry optimization method based on feature matching is proposed. Firstly, the initial matching set is roughly filtered by the minimum distance threshold method, and then, the relative transformation relationship between images is calculated by the RANSAC algorithm. If it conforms to the transformation relationship, it is an interior point. The iteration result with most interior points is the correct matching result. Then, the homography transformation between images is calculated, and the fundamental matrix is calculated by it. The interior points are determined by epipolar geometric constraints, and the fundamental matrix with the most interior points is obtained. Finally, the effects of the visual odometry optimization algorithm are verified by the TUM data set from two aspects: feature matching and fundamental matrix calculation. The experimental results show that an improved feature matching algorithm can effectively remove mismatched feature points while improving the operation efficiency. At the same time, the accuracy of feature point matching is increased by 15.8%. The fundamental matrix estimation algorithm not only improves the calculation accuracy of the fundamental matrix but also increases the interior point rate by 11.9%. A theoretical basis for improving the accuracy estimation of visual odometry will be provided.

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Hyers–Ulam and Hyers–Ulam–Rassias Stability for Linear Fractional Systems with Riemann–Liouville Derivatives and Distributed Delays

Kiskinov

Madamlieva

Zahariev

2023

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The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems.

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Existence of Absolutely Continuous Fundamental Matrix of Linear Fractional System with Distributed Delays

Cited by 7 publications

References 19 publications

A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Study on Optimization Method of Visual Odometry Based on Feature Matching

Hyers–Ulam and Hyers–Ulam–Rassias Stability for Linear Fractional Systems with Riemann–Liouville Derivatives and Distributed Delays

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