An Existence Theorem for a Quasilinear Hyperbolic Boundary Value Problem Solved by Semidiscretization in Time

1986 ◽  
Vol 5 (2) ◽  
pp. 157-168 ◽  
Author(s):  
Volker Pluschke
Keyword(s):  
Boundary Value Problem ◽  
Existence Theorem ◽  
Boundary Value ◽  
Hyperbolic Boundary

scholarly journals Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Jamil A. Ali Al-Hawasy ◽  
Lamyaa H. Ali
Keyword(s):  
Optimal Control ◽  
Boundary Value Problem ◽  
Existence Theorem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Continuous Control ◽  
Control Vector ◽  
State Equations ◽  
Vector Solution ◽  
Hyperbolic Boundary

The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.

scholarly journals A Cantilever Beam Problem with Small Deflections and Perturbed Boundary Data

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ammar Khanfer ◽  
Lazhar Bougoffa
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence Theorem ◽  
Cantilever Beam ◽  
Fourth Order ◽  
Boundary Data ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Beam Equation ◽  
Integral Boundary

The boundary value problem of a fourth-order beam equation u 4 = λ f x , u , u ′ , u ″ , u ′ ′ ′ , 0 ≤ x ≤ 1 is investigated. We formulate a nonclassical cantilever beam problem with perturbed ends. By determining appropriate values of λ and estimates for perturbation measurements on the boundary data, we establish an existence theorem for the problem under integral boundary conditions u 0 = u ′ 0 = ∫ 0 1 p x u x d x , u ″ 1 = u ′ ′ ′ 1 = ∫ 0 1 q x u ″ x d x , where p , q ∈ L 1 0 , 1 , and f is continuous on 0 , 1 × 0 , ∞ × 0 , ∞ × − ∞ , 0 × − ∞ , 0 .

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