Existence Theorem

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Shuqin Zhang - One of the best experts on this subject based on the ideXlab platform.

J Dai - One of the best experts on this subject based on the ideXlab platform.

Roger C E Tan - One of the best experts on this subject based on the ideXlab platform.

Frank Osterbrink - One of the best experts on this subject based on the ideXlab platform.

  • Existence Theorem for Geometrically Nonlinear Cosserat Micropolar Model Under Uniform Convexity Requirements
    Journal of Elasticity, 2015
    Co-Authors: Patrizio Neff, Mircea Bîrsan, Frank Osterbrink
    Abstract:

    We reconsider the geometrically nonlinear Cosserat model for a uniformly convex elastic energy and write the equilibrium system as a minimization problem. Applying the direct methods of the calculus of variations we show the Existence of minimizers. We present a clear proof based on the coercivity of the elastically stored energy density and on the weak lower semi-continuity of the total energy functional. Use is made of the dislocation density tensor $\overline{\boldsymbol{K}}= \overline{\boldsymbol{R}}^{T}\operatorname{Curl}\overline{\boldsymbol{R}}$ as a suitable Cosserat curvature measure.

  • Existence Theorem for geometrically nonlinear cosserat micropolar model under uniform convexity requirements
    arXiv: Analysis of PDEs, 2014
    Co-Authors: Patrizio Neff, Mircea Bîrsan, Frank Osterbrink
    Abstract:

    We reconsider the geometrically nonlinear Cosserat model for a uniformly convex elastic energy and write the equilibrium problem as a minimization problem. Applying the direct methods of the calculus of variations we show the Existence of minimizers. We present a clear proof based on the coercivity of the elastically stored energy density and on the weak lower semi-continuity of the total energy functional. Use is made of the dislocation density tensor $\bar{\boldsymbol{K}}=\bar{\boldsymbol{R}}^T\,\mathrm{Curl}\,\bar{\boldsymbol{R}}$ as a suitable Cosserat curvature measure.

K O Besov - One of the best experts on this subject based on the ideXlab platform.

  • on balder s Existence Theorem for infinite horizon optimal control problems
    Mathematical Notes, 2018
    Co-Authors: K O Besov
    Abstract:

    Balder’s well-known Existence Theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f0, 0} of the utility function (integrand) f0 is relaxed to the requirement that the integrals of f0 over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.

  • on balder s Existence Theorem for infinite horizon optimal control problems
    arXiv: Optimization and Control, 2017
    Co-Authors: K O Besov
    Abstract:

    Balder's well-known Existence Theorem (1983) for infinite-horizon optimal control problems is extended to the case when the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part $\max\{f_0,0\}$ of the utility function (integrand) $f_0$ is relaxed to the requirement that the integrals of $f_0$ over intervals $[T,T']$ be uniformly bounded from above by a function $\omega(T,T')$ such that $\omega(T,T')\to 0$ as $T,T'\to\infty$. This requirement was proposed by A.V. Dmitruk and N.V. Kuz'kina (2005); however, the proof in the present paper does not follow their scheme but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.