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0-semigroups
Published in Takao Nambu, Theory of Stabilization for Linear Boundary Control Systems, 2017
To begin with, let L be a linear closed operator in a Banach space E with the dense domain D(L), and consider an abstract differential equation in E which is described as () dudt+Lu=0,t>0,u(0)=u0∈D(L).
Results on controllability and well-posedness of functional abstract second-order differential equations with state-dependent delay
Published in Applicable Analysis, 2023
Kulandhivel Karthikeyan, Dhatchinamoorthy Tamizharasan, Ozgur Ege
Now, we consider the abstract differential equation of second order of the form: where η is an integrable function defined from to X and . The mild solution of (3)–(4) is and for , the function defined on is We plan to explore further insight into the features of abstract Cauchy problem of second order and cosine functions studied in [33,34]. Let and be the spaces equipped with the norms and , respectively. Notice that , where The space is constructed for all such that , associated with the norm where stands for the first derivative of ϵ.
Exact controllability of a class of nonlinear distributed parameter systems using back-and-forth iterations
Published in International Journal of Control, 2019
Vivek Natarajan, Hua-Cheng Zhou, George Weiss, Emilia Fridman
Let X = H1L(0, 1) × L2(0, 1) be the state space with the following inner product: If we let , then the plant (5.1) can be written as an abstract differential equation on X as follows: where we use the notation and A, B and are defined by Here δ1 is the Dirac pulse at x = 1 and I is the identity operator on L2(0, 1). This follows from the material in Section 10.2.2 of Tucsnak and Weiss (2009), where the linear system corresponding to σ = 0 is shown to be a well-posed boundary control system having the above abstract description. We have added the nonlinear term into the abstract description to obtain (5.2).
On a strongly continuous semigroup for a Black-Scholes integro-differential operator: European options under jump-diffusion dynamics
Published in Applicable Analysis, 2023
An evolution equation such as the PDE (2) or the PIDE (4) can be reformulated as an abstract differential equation where the unknown function belongs to some suitable function space X and A is an appropriately defined linear operator on X. The theory of -semigroups (see [14,15] for instance) is a powerful tool to study such abstract differential equations and problems involving existence, uniqueness and stability (or uniform well-posedness) can be tackled.