Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions

Donchev, Tzanko; Farkhi, Elza

doi:10.1137/s0363012995293694

Cited by 66 publications

(54 citation statements)

References 11 publications

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“…Differential inclusions and their discrete approximations under the more conventional one-sided Lipschitz (OSL) condition have been already studied by the first two authors in papers [7][8][9][10][11] mostly devoted to qualitative theory of OSL differential inclusions and the possibility to uniformly approximate solutions sets to OSL inclusions (1.1) by corresponding solution sets to their discretized counterparts.…”

Section: Introductionmentioning

confidence: 99%

Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces

Donchev

Farkhi

Mordukhovich

2007

Journal of Differential Equations

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We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modified one-sided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity. Our main attention is paid to establishing efficient conditions that ensure the strong approximation (in the W 1,p -norm as p 1) of feasible trajectories for the one-sided Lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations. To proceed with the latter issue, we derive a new extension of the Bogolyubov-type relaxation/density theorem to the case of differential inclusions satisfying the modified one-sided Lipschitzian condition. All the results obtained are new not only in the infinitedimensional Hilbert space framework but also in finite-dimensional spaces.

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Section: Introductionmentioning

confidence: 99%

Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces

Donchev

Farkhi

Mordukhovich

2007

Journal of Differential Equations

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“…In some studies, this condition was weakened allowing to embrace broader classes of differential inclusions. Following the paper by Donchev & Farkhi [11], where the authors presented a generalization of the Filippov theorem for differential inclusions with onesided Lipschitzian r.h.s., several results were obtained extending the averaging method to this case. Recall the definition of one-sided Lipschitzian set-valued map.…”

Section: Averaging On a Finite Time Intervalmentioning

confidence: 99%

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

Gama

Smirnov

2013

Set-Valued Var. Anal

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The averaging method is one of the most used tools to study dynamical systems. With the development of the theory of differential inclusions, a respective generalization of the averaging method followed the steps outlined in the theory of differential equations. Presently, it has been successfully applied to a wide range of problems involving differential inclusions, simplifying the study of the systems under consideration. In this work, the main development trends and methods in the application of the averaging method to the study of stability and optimality of solutions to differential inclusions are surveyed. A detailed list of references is given and some examples of applications are presented.

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“…[10]. On the other hand, after spectral approximation to some partial differential equations, such as parabolic equations, we oftentimes obtain certain systems with the unknown time-dependent coefficients, which possess the property like (3.25).…”

Section: Inserting the Above And (34) Into (33) Gives Thatmentioning

confidence: 99%

Legendre–Gauss collocation methods for ordinary differential equations

Guo

Wang

2008

Adv Comput Math

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In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre-Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre-Gauss Runge-Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.

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Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions

Cited by 66 publications

References 11 publications

Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces

Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

Legendre–Gauss collocation methods for ordinary differential equations

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