scholarly journals The Solution Of Nonhomogen Abstract Cauchy Problem by Semigroup Theory of Linear Operator

2017 ◽  
Vol 2 (2) ◽  
pp. 143
Author(s):  
Susilo Hariyanto
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Linear Operator ◽  
Infinitesimal Generator ◽  
Original Problem ◽  
Semigroup Theory ◽  
Abstract Cauchy Problem ◽  
Direct Sum ◽  
Special Operator ◽  
Degenerate Cauchy Problem

<div style="text-align: justify;">In this article we will investigate how to solve nonhomogen degenerate Cauchy problem via theory of semigroup of linear operator. The problem is formulated in Hilbert space which can be written as direct sum of subset Ker M and Ran M*. By certain assumptions the problem can be reduced to nondegenerate Cauchy problem. And then by composition between invers of operator M and the nondegenerate problem we can transform it to canonic problem, which is easier to solve than the original problem. By taking assumption that the operator A is infinitesimal generator of semigroup, the canonic problem has a unique solution. This allow to define special operator which map the solution of canonic problem to original problem. ©2016 JNSMR UIN Walisongo. All rights reserved.</div>

scholarly journals On the Controllability of a Differential Equation with Delayed and Advanced Arguments

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Raúl Manzanilla ◽  
Luis Gerardo Mármol ◽  
Carmen J. Vanegas
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Infinitesimal Generator ◽  
Semigroup Theory ◽  
Exact Controllability

A semigroup theory for a differential equation with delayed and advanced arguments is developed, with a detailed description of the infinitesimal generator. This in turn allows to study the exact controllability of the equation, by rewriting it as a classical Cauchy problem.

A Uniqueness Theorem on the Degenerate Cauchy Problem(1)

1975 ◽  
Vol 18 (3) ◽  
pp. 417-421 ◽  
Author(s):  
Chung-Lie Wang
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Uniqueness Theorem ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Continuous Linear ◽  
Linear Maps ◽  
Image Position ◽  
Degenerate Cauchy Problem ◽  
Densely Defined

In [4] Carroll and the author have treated the following problem(1)where Λ is a closed densely defined self-adjoint operator in a separable Hilbert space H with (Λu, u) ≥ c ‖u‖2, c > 0, Λ-1 ∊ L(H) (L(E, F) is the space of continuous linear maps from E to F; in particular, L(H) = L(H, H)), a(t) > 0 for t > 0 a(0) = 0 and S(t), R(t), B(t) ∈ L(H).

scholarly journals Semigroup theory applied to options

2002 ◽  
Vol 2 (3) ◽  
pp. 131-139 ◽  
Author(s):  
D. I. Cruz-Báez ◽  
J. M. González-Rodríguez
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Infinitesimal Generator ◽  
Semigroup Theory ◽  
European Option ◽  
Market Place ◽  
Underlying Asset ◽  
The Cauchy Problem ◽  
Suitable Space ◽  
Current Value

Black and Scholes (1973) proved that under certain assumptions about the market place, the value of a European option, as a function of the current value of the underlying asset and time, verifies a Cauchy problem. We give new conditions for the existence and uniqueness of the value of a European option by using semigroup theory. For this, we choose a suitable space that verifies some conditions, what allows us that the operator that appears in the Cauchy problem is the infinitesimal generator of aC0-semigroupT(t). Then we are able to guarantee the existence and uniqueness of the value of a European option and we also achieve an explicit expression of that value.

scholarly journals Nonuniqueness and wellposedness of abstract Cauchy problems in a Fréchet space

2001 ◽  
Vol 63 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Peer Christian Kunstmann
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Continuous Selection ◽  
Fréchet Space ◽  
Inhomogeneous Equation ◽  
Frechet Space ◽  
Closed Linear Operator ◽  
Cauchy Problems

Suppose that A is a closed linear operator in a Fréchet space X. We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F, and the part Az of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels.Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

Feedback stabilization for unbounded bilinear systems using bounded control

2018 ◽  
Vol 36 (4) ◽  
pp. 1073-1087
Author(s):  
Rachid El Ayadi ◽  
Mohamed Ouzahra
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Infinitesimal Generator ◽  
Real Hilbert Space ◽  
Sufficient Conditions ◽  
Bilinear System ◽  
Feedback Stabilization ◽  
Bounded Control ◽  
Graph Norm ◽  
Semigroup Of Contractions

Abstract In this paper, we deal with the distributed bilinear system $ \frac{d z(t)}{d t}= A z(t) + v(t)Bz(t), $ where A is the infinitesimal generator of a semigroup of contractions on a real Hilbert space H. The linear operator B is supposed bounded with respect to the graph norm of A. Then we give sufficient conditions for weak and strong stabilizations. Illustrating examples are provided.

scholarly journals A Modified Quasi-Reversibility Method for a Class of Ill-Posed Cauchy Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 627-642 ◽  
Author(s):  
Nadjib Boussetila ◽  
Faouzia Rebbani
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Linear Operator ◽  
Convergence Rate ◽  
A Priori ◽  
Cauchy Problems ◽  
Unbounded Linear Operator ◽  
Ill Posed ◽  
New Perturbation ◽  
Problem Data

Abstract The goal of this paper is to present some extensions of the method of quasi-reversibility applied to an ill-posed Cauchy problem associated with an unbounded linear operator in a Hilbert space. The key point to our proof is the use of a new perturbation to construct a family of regularizing operators for the considered problem. We show the convergence of this method, and we estimate the convergence rate under a priori regularity assumptions on the problem data.

Upper and lower bounds for solutions of linear operator problems with unilateral constraints

1977 ◽  
Vol 76 (2) ◽  
pp. 95-105 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Lower Bounds ◽  
Original Problem ◽  
Adjoint Operator ◽  
Decomposition Theorem ◽  
Positive Definite ◽  
Upper And Lower Bounds ◽  
Unilateral Constraints ◽  
Space Decomposition

SynopsisDual extremum principles characterising the solutions of problems for a positive-definite self-adjoint operator on a Hilbert space which involve unilateral constraints are formulated using a Hilbert space decomposition theorem due to Moreau. Various upper and lower bounds to these solutions are then obtained, these bounds involving the solutions to subsidiary problems with less restrictive conditions than the solution to the original problem.

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