Approximate observability of abstract evolution equation with unbounded observation operator

On an Abstract Evolution Equation with a Spectral Operator of Scalar Type

It is shown that the weak solutions of the evolution equation y′(t)=Ay(t), t ∈ [0,T ) (0< T ≤∞), where A is a spectral operator of scalar type in a complex Banach space X, defined by Ball (1977), are given by the formula y(t) = etAf , t ∈ [0,T ), with the exponentials understood in the sense of the operational calculus for such operators and the set of the initial values, f ’s, being ⋂ 0≤t<T D(...

Calculating the Best Approximate Solution of an Operator Equation

This paper furnishes two classes of methods for calculating the best approximate solution of an operator equation in Banach spaces, where the operator is bounded, linear and has closed range. The best approximate solution can be calculated by an iterative method in Banach spaces stated in terms of an operator parameter. Specifying the parameter yields some new and some old iterative techniques....

On Attainable Set and Controllability for Abstract Control Problem with Unbounded Input Operator

For linear evolution control system described by ẋ = Ax(t) + Bu(t), x(0) = x0 (A generates a strongly continuous semigroup {S(t)}t≥0 on a Banach space X; B is a linear unbounded operator), the attainable set K (t) set is studied. Conditions of the independence of t for its closure K(t) are established. Controllability conditions for some classes of evolution systems are obtained.

On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

Airy equation with memory involvement via Liouville differential operator

In this work, a non-integer order Airy equation involving Liouville differential operator is considered. Proposing an undetermined integral solution to the left fractional Airy differential equation, we utilize some basic fractional calculus tools to clarify the closed form. A similar suggestion to the right FADE, converts it into an equation in the Laplace domain. An illustration t...