Regional Controllability of Riemann–Liouville Time-Fractional Semilinear Evolution Equations

In this paper, we discuss the exact regional controllability of fractional evolution equations involving Riemann–Liouville fractional derivative of order q ∈ 0,1 . The result is obtained with the help of the theory of fractional calculus, semigroup theory, and Banach fixed-point theorem under several assumptions on the corresponding linear system and the nonlinear term. Finally, some numerical simulations are given to illustrate the obtained result.

Controllability of Semilinear Evolution Equations with Impulses and Delay

Para mucho sistemas de control en la vida real, los impulsos y los retardos son fenomenos intrinsicos que no modifican su controlabilidad. Por lo tanto, nosotros conjeturamos que, bajo ciertas condiciones, las perturbaciones de un sistemas causadas por cambios abruptos y retardos no afectan ciertas propiedades del mismo como la controlabilidad. En ese sentido, nosotros demostramos que bajo ciertas condiciones los impulsos y los retardos como perturbaciones no destruyen la controlabilidad de un sistema de control gobernado por ecuaciones de evolucion. Como una aplicación consideramos la ecuacion de onda semilineal con impulsos y retardos.

The periodic version of the Da Prato–Grisvard theorem and applications to the bidomain equations with FitzHugh–Nagumo transport

In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal $${{L}}^p$$ L p -regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo, Aliev–Panfilov, or Rogers–McCulloch, it is proved that this set of equations admits a unique, strongT-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T-periodic forces.

Boundedness of Singular Integral Operators with Operator-Valued Kernels and Maximal Regularity of Sectorial Operators in Variable Lebesgue Spaces

This paper is devoted to the maximal regularity of sectorial operators in Lebesgue spaces Lp⋅ with a variable exponent. By extending the boundedness of singular integral operators in variable Lebesgue spaces from scalar type to abstract-valued type, the maximal Lp⋅−regularity of sectorial operators is established. This paper also investigates the trace of the maximal regularity space E01,p⋅I, together with the imbedding property of E01,p⋅I into the range-varying function space C−I,X1−1/p⋅,p⋅. Finally, a type of semilinear evolution equations with domain-varying nonlinearities is taken into account.