Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique

2016 ◽  
Vol 275 ◽  
pp. 107-120 ◽  
Author(s):  
Fu-Dong Ge ◽  
Hua-Cheng Zhou ◽  
Chun-Hai Kou
Keyword(s):  
Fractional Order ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Impulsive Conditions ◽  
Semilinear Evolution Equations

Finite-approximate controllability of fractional evolution equations: variational approach

2018 ◽  
Vol 21 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Nazim I. Mahmudov
Keyword(s):  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Heat Equations ◽  
Natural Conditions ◽  
Finite Dimensional ◽  
Semilinear Evolution Equations

Abstract In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability (finite-approximate controllability) for the fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linear equation we obtain sufficient conditions for the finite-approximate controllability of the fractional semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated.

scholarly journals Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yongqin Xie ◽  
Zhufang He ◽  
Chen Xi ◽  
Zheng Jun
Keyword(s):  
Evolution Equations ◽  
Global Solutions ◽  
Global Attractors ◽  
Nonlinear Evolution Equations ◽  
Time Behavior ◽  
Asymptotic Regularity ◽  
Compact Attractor ◽  
Long Time ◽  
Critical Exponential Growth ◽  
Semilinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

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