An inverse problem for an abstract evolution equation

An inverse problem for a linear stochastic evolution equation is researched. The stochastic evolution equation contains a parameter with values in a Hilbert space. The solution of the evolution equation depends continuously on the parameter and is Fréchet differentiable with respect to the parameter. An optimization method is provided to estimate the parameter. A sufficient condition to ensure the existence of an optimal parameter is presented, and a necessary condition that the optimal parameter, if it exists, should satisfy is also presented. Finally, two examples are given to show the applications of the above results.

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On an inverse problem for fractional evolution equation

Evolution Equations and Control Theory ◽

10.3934/eect.2017007 ◽

2017 ◽

Vol 6 (1) ◽

pp. 111-134

Author(s):

Nguyen Huy Tuan ◽

◽

Mokhtar Kirane ◽

Long Dinh Le ◽

Van Thinh Nguyen ◽

...

Keyword(s):

Inverse Problem ◽

Evolution Equation ◽

Fractional Evolution Equation

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On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

Open Mathematics ◽

10.1515/math-2019-0083 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1082-1112

Author(s):

Marat V. Markin

Keyword(s):

Banach Space ◽

Evolution Equation ◽

Weak Solutions ◽

A Priori ◽

Complex Banach Space ◽

Spectral Operator ◽

Scalar Type ◽

Improvement Effect ◽

Abstract Evolution Equation ◽

Necessary And Sufficient

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2018/4168609 ◽

2018 ◽

Vol 2018 ◽

pp. 1-14

Author(s):

Marat V. Markin

Keyword(s):

Evolution Equation ◽

Weak Solutions ◽

Normal Operator ◽

A Priori ◽

Complex Banach Space ◽

Complex Hilbert Space ◽

Spectral Operator ◽

Scalar Type ◽

Improvement Effect ◽

Abstract Evolution Equation

Given the abstract evolution equation y′(t)=Ay(t), t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.

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