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Evolution Equations & Control Theory

February 2022 , Volume 11 , Issue 1

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Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations
Priscila Santos Ramos, J. Vanterler da C. Sousa and E. Capelas de Oliveira
2022, 11(1): 1-24 doi: 10.3934/eect.2020100 +[Abstract](1366) +[HTML](523) +[PDF](462.89KB)

We discuss the existence and uniqueness of mild solutions for a class of quasi-linear fractional integro-differential equations with impulsive conditions via Hausdorff measures of noncompactness and fixed point theory in Banach space. Mild solution controllability is discussed for two particular cases.

Lifespan of solutions to a damped plate equation with logarithmic nonlinearity
Yuzhu Han and Qi Li
2022, 11(1): 25-40 doi: 10.3934/eect.2020101 +[Abstract](1173) +[HTML](498) +[PDF](374.71KB)

This paper is devoted to the lifespan of solutions to a damped plate equation with logarithmic nonlinearity

Finite time blow-up criteria for solutions at both lower and high initial energy levels are established and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.

Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor
Xueli Song and Jianhua Wu
2022, 11(1): 41-65 doi: 10.3934/eect.2020102 +[Abstract](951) +[HTML](469) +[PDF](449.22KB)

We consider a non-autonomous two-dimensional Newton-Boussinesq equation with singularly oscillating external forces depending on a small parameter \begin{document}$ \varepsilon $\end{document}. We prove the existence of the uniform attractor \begin{document}$ A^\varepsilon $\end{document} when the Prandtl number \begin{document}$ P_r>1 $\end{document}. Furthermore, under suitable translation-compactness and divergence type condition assumptions on the external forces, we obtain the uniform (with respect to \begin{document}$ \varepsilon $\end{document}) boundedness of the related uniform attractors \begin{document}$ A^\varepsilon $\end{document} as well as the convergence of the attractor \begin{document}$ A^\varepsilon $\end{document} to the attractor \begin{document}$ A^0 $\end{document} as \begin{document}$ \varepsilon\rightarrow 0^+ $\end{document}.

Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces
Soniya Singh, Sumit Arora, Manil T. Mohan and Jaydev Dabas
2022, 11(1): 67-93 doi: 10.3934/eect.2020103 +[Abstract](1394) +[HTML](537) +[PDF](410.81KB)

In this paper, we consider the second order semilinear impulsive differential equations with state-dependent delay. First, we consider a linear second order system and establish the approximate controllability result by using a feedback control. Then, we obtain sufficient conditions for the approximate controllability of the considered system in a separable, reflexive Banach space via properties of the resolvent operator and Schauder's fixed point theorem. Finally, we apply our results to investigate the approximate controllability of the impulsive wave equation with state-dependent delay.

Complete controllability for a class of fractional evolution equations with uncertainty
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan and Hoang Viet Long
2022, 11(1): 95-124 doi: 10.3934/eect.2020104 +[Abstract](1325) +[HTML](470) +[PDF](713.78KB)

In this paper, we study the complete controllability for a class of fractional evolution equations with a common type of fuzzy uncertainty. By using Hausdorff measure of noncompactness and Krasnoselskii's fixed point theorem in complete semilinear metric space, we give some sufficient conditions of the controllability for the fuzzy fractional evolution equations without involving the compactness of strongly continuous semigroup and the perturbation function. In addition, the controllable problem is considered in a subspace of fuzzy numbers in which the gH-differences always exist, that guarantees the satisfaction of hypotheses of the problem. An application example related to electrical circuit is given to illustrate the effectiveness of theoretical results.

Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"
Manil T. Mohan
2022, 11(1): 125-167 doi: 10.3934/eect.2020105 +[Abstract](982) +[HTML](433) +[PDF](520.91KB)

The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt (Kelvin-Voight) fluids in bounded domains is considered in this work. We investigate the long-term dynamics of such viscoelastic fluid flow equations with "fading memory" (non-autonomous). We first establish the existence of an absorbing ball in appropriate spaces for the semigroup defined for the Kelvin-Voigt fluid flow equations of order one with "fading memory" (transformed autonomous coupled system). Then, we prove that the semigroup is asymptotically compact, and hence we establish the existence of a global attractor for the semigroup. We provide estimates for the number of determining modes for both asymptotic as well as for trajectories on the global attractor. Once the differentiability of the semigroup with respect to initial data is established, we show that the global attractor has finite Hausdorff as well as fractal dimensions. We also show the existence of an exponential attractor for the semigroup associated with the transformed (equivalent) autonomous Kelvin-Voigt fluid flow equations with "fading memory". Finally, we show that the semigroup has Ladyzhenskaya's squeezing property and hence is quasi-stable, which also implies the existence of global as well as exponential attractor having finite fractal dimension.

Some results on the behaviour of transfer functions at the right half plane
Tahir Aliyev Azeroğlu, Bülent Nafi Örnek and Timur Düzenli
2022, 11(1): 169-175 doi: 10.3934/eect.2020106 +[Abstract](1812) +[HTML](559) +[PDF](284.35KB)

In this paper, an inequality for a transfer function is obtained assuming that its residues at the poles located on the imaginary axis in the right half plane. In addition, the extremal function of the proposed inequality is obtained by performing sharpness analysis. To interpret the results of analyses in terms of control theory, root-locus curves are plotted. According to the results, marginally and asymptotically stable transfer functions can be determined using the obtained extremal function in the proposed theorem.

Optimal control problems for a neutral integro-differential system with infinite delay
Hai Huang and Xianlong Fu
2022, 11(1): 177-197 doi: 10.3934/eect.2020107 +[Abstract](1190) +[HTML](470) +[PDF](382.39KB)

This work devotes to the study on problems of optimal control and time optimal control for a neutral integro-differential evolution system with infinite delay. The main technique is the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature. We first establish the existence and uniqueness of mild solutions and discuss the compactness of the solution operator for the considered control system. Then, we investigate the existence of optimal controls for the both cases of bounded and unbounded admissible control sets under some assumptions. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. An example is given at last to illustrate the applications of the obtained results.

Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems
Gervy Marie Angeles and Gilbert Peralta
2022, 11(1): 199-224 doi: 10.3934/eect.2020108 +[Abstract](1058) +[HTML](515) +[PDF](415.01KB)

We consider a hyperbolic system of partial differential equations on a bounded interval coupled with ordinary differential equations on both ends. The evolution is governed by linear balance laws, which we treat with semigroup and time-space methods. Our goal is to establish the exponential stability in the natural state space by utilizing the stability with respect to the first-order energy of the system. Derivation of a priori estimates plays a crucial role in obtaining energy and dissipation functionals. The theory is then applied to specific physical models.

On time fractional pseudo-parabolic equations with nonlocal integral conditions
Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu and Nguyen H. Tuan
2022, 11(1): 225-238 doi: 10.3934/eect.2020109 +[Abstract](1520) +[HTML](548) +[PDF](356.31KB)

In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order \begin{document}$ \sigma,\; \; 0<\sigma<1 $\end{document} and the space fractional derivative is of order \begin{document}$ \alpha,\beta >0 $\end{document}. In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen \begin{document}$ \alpha, \beta $\end{document}. The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in \begin{document}$ L^p $\end{document} between the regularized solution and the sought solution is obtained.

Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives
Biao Zeng
2022, 11(1): 239-258 doi: 10.3934/eect.2021001 +[Abstract](865) +[HTML](474) +[PDF](361.25KB)

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by fractional impulsive delay evolution equations involving Caputo fractional derivatives in separable reflexive Banach spaces. This work is a continuation of previous work. We firstly give an existence result of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray-Schauder alternative fixed point theorem. Next, by using the Filippove theorem and the Cesari property, we obtain the existence result of feasible pairs for the feedback control system. Finally, some applications are given to illustrate our main results.

Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations
Do Lan
2022, 11(1): 259-282 doi: 10.3934/eect.2021002 +[Abstract](1083) +[HTML](461) +[PDF](350.11KB)

We study the generalized Rayleigh-Stokes problem involving a fractional derivative and nonlinear perturbation. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and asymptotic stability of solutions. In particular, if the nonlinearity is Lipschitzian then the mild solution of the mentioned problem becomes a classical one and its convergence to equilibrium point is proved.

A canonical model of the one-dimensional dynamical Dirac system with boundary control
Mikhail I. Belishev and Sergey A. Simonov
2022, 11(1): 283-300 doi: 10.3934/eect.2021003 +[Abstract](850) +[HTML](401) +[PDF](404.45KB)

The one-dimensional Dirac dynamical system \begin{document}$ \Sigma $\end{document} is

where \begin{document}$ \sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix} $\end{document} is the Pauli matrix; \begin{document}$ V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix} $\end{document} with \begin{document}$ p = p(x) $\end{document} is a potential; \begin{document}$ u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix} $\end{document} is the trajectory in \begin{document}$ \mathscr H = L_2(\mathbb R_+;\mathbb C^2) $\end{document}; \begin{document}$ f\in\mathscr F = L_2([0, \infty);\mathbb C) $\end{document} is a boundary control. System \begin{document}$ \Sigma $\end{document} is not controllable: the total reachable set \begin{document}$ \mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $\end{document} is not dense in \begin{document}$ \mathscr H $\end{document}, but contains a controllable part \begin{document}$ \Sigma_u $\end{document}. We construct a dynamical system \begin{document}$ \Sigma_a $\end{document}, which is controllable in \begin{document}$ L_2(\mathbb R_+;\mathbb C) $\end{document} and connected with \begin{document}$ \Sigma_u $\end{document} via a unitary transform. The construction is based on geometrical optics relations: trajectories of \begin{document}$ \Sigma_a $\end{document} are composed of jump amplitudes that arise as a result of projecting in \begin{document}$ \overline{\mathscr U} $\end{document} onto the reachable sets \begin{document}$ \mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $\end{document}. System \begin{document}$ \Sigma_a $\end{document}, which we call the amplitude model of the original \begin{document}$ \Sigma $\end{document}, has the same input/output correspondence as system \begin{document}$ \Sigma $\end{document}. As such, \begin{document}$ \Sigma_a $\end{document} provides a canonical completely reachable realization of the Dirac system.

Internal control for a non-local Schrödinger equation involving the fractional Laplace operator
Umberto Biccari
2022, 11(1): 301-324 doi: 10.3934/eect.2021014 +[Abstract](715) +[HTML](341) +[PDF](1221.12KB)

We analyze the interior controllability problem for a non-local Schrödinger equation involving the fractional Laplace operator \begin{document}$ (-\Delta)^{\, {s}}{} $\end{document}, \begin{document}$ s\in(0, 1) $\end{document}, on a bounded \begin{document}$ C^{1, 1} $\end{document} domain \begin{document}$ \Omega\subset{\mathbb{R}}^N $\end{document}. We first consider the problem in one space dimension and employ spectral techniques to prove that, for \begin{document}$ s\in[1/2, 1) $\end{document}, null-controllability is achieved through an \begin{document}$ L^2(\omega\times(0, T)) $\end{document} function acting in a subset \begin{document}$ \omega\subset\Omega $\end{document} of the domain. This result is then extended to the multi-dimensional case by applying the classical multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian.

Improved boundary regularity for a Stokes-Lamé system
Francesca Bucci
2022, 11(1): 325-346 doi: 10.3934/eect.2021018 +[Abstract](701) +[HTML](311) +[PDF](517.23KB)

This paper recalls a partial differential equations system, which is the linearization of a recognized fluid-elasticity interaction three-dimensional model. A collection of regularity results for the traces of the fluid variable on the interface between the body and the fluid is established, in the case a suitable boundary dissipation is present. These regularity estimates are geared toward ensuring the well-posedness of the Riccati equations which arise from the associated optimal boundary control problems on a finite as well as infinite time horizon. The theory of operator semigroups and interpolation provide the main tools.

2020 Impact Factor: 1.081
5 Year Impact Factor: 1.269
2020 CiteScore: 1.6



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