Generalized solvability and optimal control for an integro-differential equation of a hyperbolic type

2021 ◽  
pp. 8-9
Author(s):  
Andrii Anikushyn ◽  
Oleksandra Zhyvolovych
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Differential Operator ◽  
Existence And Uniqueness ◽  
Generalized Solution ◽  
Hyperbolic Type ◽  
Well Posedness ◽  
Volterra Type ◽  
Integral Term ◽  
Type Integral

We consider an integro-differential operator with Volterra type integral term. We provide a priory inequalities in negative norms for certain spaces. Further, using obtained inequalities we prove well-posedness (existence and uniqueness of the (weak) generalized solution) of the corresponding boundary value problem as well as a theorem on optimal control existence.

scholarly journals Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu Song ◽  
Lingling Zhang ◽  
Bibo Zhou ◽  
Nan Zhang
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Mixed Monotone Operator ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential

Abstract In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of φ-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.

scholarly journals On one problem with dynamic nonlocal condition for ahyperbolic equation

2017 ◽  
Vol 21 (3) ◽  
pp. 44-52
Author(s):  
A.E. Savenkova
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Input Data ◽  
Nonlocal Condition ◽  
Generalized Solution ◽  
Embedding Theorems ◽  
Hyperbolic Partial Differential Equation ◽  
Apriori Estimates

In this article, boundary value problem for hyperbolic partial differential equation with nonlocal data in an integral of the second kind form is considered. The emergence of dynamic conditions may be due to the presence of a damping device. Existence and uniqueness of generalized solution is proved in a given cylindrical field. There is some limitation on the input data. The uniqueness of generalized solution is proved by apriori estimates. The existence is proved by Galerkin’s method and embedding theorems.

scholarly journals New Fixed Point Results via (q,y)R-Weak Contractions with an Application

2020 ◽  
Author(s):  
Mohammad Imded ◽  
Based Ali ◽  
Waleed M. Alfaqih ◽  
Salvatore Sessa
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Volterra Type ◽  
Auxiliary Functions ◽  
Type Integral

In this paper, inspired by Jleli and Samet [journal of inequalities and applications 38 (2014) 2 1–8] we introduce two new classes of auxiliary functions and utilize the same to define (q, y)R-weak 3 contractions. Utilizing (q, y)R-weak contractions, we prove some fixed point theorems in the setting 4 of relational metric spaces. We employ some examples to substantiate the utility of our newly proved 5 results. Finally, we apply one of our newly proved results to ensure the existence and uniqueness of 6 solution of a Volterra-type integral equation.

scholarly journals BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients

2019 ◽  
Vol 4 (1) ◽  
pp. 139-150 ◽  
Author(s):  
Sadibou Aidara ◽  
Yaya Sagna
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Backward Stochastic Differential Equation ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Type Integral ◽  
Divergence Type ◽  
Mutually Independent ◽  
Uniqueness Of A Solution

AbstractThis paper deals with a class of backward stochastic differential equation driven by two mutually independent fractional Brownian motions. We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.

scholarly journals New Fixed Point Results via (θ,ψ)R-Weak Contractions with an Application

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 887
Author(s):  
Mohammad Imdad ◽  
Based Ali ◽  
Waleed M. Alfaqih ◽  
Salvatore Sessa ◽  
Abdullah Aldurayhim
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Volterra Type ◽  
Auxiliary Functions ◽  
Type Integral ◽  
Uniqueness Of The Solution

In this paper, inspired by Jleli and Samet (Journal of Inequalities and Applications 38 (2014) 1–8), we introduce two new classes of auxiliary functions and utilize the same to define ( θ , ψ ) R -weak contractions. Utilizing ( θ , ψ ) R -weak contractions, we prove some fixed point theorems in the setting of relational metric spaces. We employ some examples to substantiate the utility of our newly proven results. Finally, we apply one of our newly proven results to ensure the existence and uniqueness of the solution of a Volterra-type integral equation.

scholarly journals Nonlinear Fuzzy Differential Equation with Time Delay and Optimal Control Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Wichai Witayakiattilerd
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Time Delay ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Existence And Uniqueness ◽  
Fuzzy Differential Equation ◽  
Initial Value ◽  
Existence Of A Solution ◽  
Delay Function

The existence and uniqueness of a mild solution to nonlinear fuzzy differential equation constrained by initial value were proven. Initial value constraint was then replaced by delay function constraint and the existence of a solution to this type of problem was also proven. Furthermore, the existence of a solution to optimal control problem of the latter type of equation was proven.

scholarly journals Existence of Optimal Control for a Nonlinear Partial Differential Equation of Hyperbolic Type

2019 ◽  
Vol 12 (4) ◽  
pp. 1595-1601
Author(s):  
Dieudonne Ampini ◽  
Mabonzo Vital Delmas
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Nonlinear Partial Differential Equation ◽  
Hyperbolic Type ◽  
Partial Differential ◽  
Integral Sign ◽  
Hyperbolic Problem ◽  
Existence Of Optimal Control ◽  
Passage To The Limit

In this paper, we prove the existence of an optimal control for a nonlinear hyperbolic problem, examined in [3]. An estimation is used which makes it possible to extract from a minimizable sequence of controls and from the sequence of corresponding solutions weakly convergent sub sequences. To prove the passage to the limit in a true equality for every element of the minimizable sequence, Lebesgue’s theorem on the passage to the limit under the integral sign and the theorem of immersion have been used.

scholarly journals A nonlinear control system with a Hilfer derivative and its optimization

2019 ◽  
Vol 24 (2) ◽  
pp. 279-296
Author(s):  
Rafał Kamocki
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Control System ◽  
Optimal Control Problem ◽  
Nonlinear Control ◽  
Existence And Uniqueness ◽  
Nonlinear Control System ◽  
Sufficient Optimality Conditions ◽  
Necessary And Sufficient ◽  
Uniqueness Of A Solution

In this work, we consider a fractional optimal control problem (FOCP) containing a nonlinear control system, described by a differential equation involving a Hilfer derivative, and an integral cost functional. We study the existence and uniqueness of a solution of the control system as well as the necessary and sufficient optimality conditions of FOCP.

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