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Evolution Equations & Control Theory

June 2018 , Volume 7 , Issue 2

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On state-dependent sweeping process in Banach spaces
Dalila Azzam-Laouir and Fatiha Selamnia
2018, 7(2): 183-196 doi: 10.3934/eect.2018009 +[Abstract](278) +[HTML](146) +[PDF](393.79KB)
Abstract:

In this paper we prove, in a separable reflexive uniformly smooth Banach space, the existence of solutions of a perturbed first order differential inclusion governed by the proximal normal cone to a moving set depending on the time and on the state. The perturbation is assumed to be separately upper semicontinuous.

The recovery of a parabolic equation from measurements at a single point
Amin Boumenir, Vu Kim Tuan and Nguyen Hoang
2018, 7(2): 197-216 doi: 10.3934/eect.2018010 +[Abstract](292) +[HTML](149) +[PDF](5897.52KB)
Abstract:

By measuring the temperature at an arbitrary single point located inside an unknown object or on its boundary, we show how we can uniquely reconstruct all the coefficients appearing in a general parabolic equation which models its cooling. We also reconstruct the shape of the object. The proof hinges on the fact that we can detect infinitely many eigenfunctions whose Wronskian does not vanish. This allows us to evaluate these coefficients by solving a simple linear algebraic system. The geometry of the domain and its boundary are found by reconstructing the first eigenfunction.

Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics
Michele Colturato
2018, 7(2): 217-245 doi: 10.3934/eect.2018011 +[Abstract](206) +[HTML](147) +[PDF](545.67KB)
Abstract:

We consider a singular phase field system located in a smooth bounded domain. In the entropy balance equation appears a logarithmic nonlinearity. The second equation of the system, deduced from a balance law for the microscopic forces that are responsible for the phase transition process, is perturbed by an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. We prove existence and uniqueness of the solution for this resulting highly nonlinear system. Moreover, under further assumptions, the longtime behavior of the solution is investigated.

Robust Stackelberg controllability for linear and semilinear heat equations
Víctor Hernández-Santamaría and Luz de Teresa
2018, 7(2): 247-273 doi: 10.3934/eect.2018012 +[Abstract](225) +[HTML](123) +[PDF](506.51KB)
Abstract:

In this paper, we present a Stackelberg strategy to control a semilinear parabolic equation. We use the concept of hierarchic control to combine the concepts of controllability with robustness. We have a control named the leader which is responsible for a controllability to trajectories objective. Additionally, we have a control named the follower, that solves a robust control problem. That means we solve for the optimal control in the presence of the worst disturbance case. In this way, the follower control is insensitive to a broad class of external disturbances.

On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation
Kazumasa Fujiwara and Tohru Ozawa
2018, 7(2): 275-280 doi: 10.3934/eect.2018013 +[Abstract](187) +[HTML](119) +[PDF](322.42KB)
Abstract:

An explicit lifespan estimate is presented for the derivative Schrödinger equations with periodic boundary condition.

On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions
Tae Gab Ha
2018, 7(2): 281-291 doi: 10.3934/eect.2018014 +[Abstract](214) +[HTML](139) +[PDF](354.74KB)
Abstract:

In this paper, we consider the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. This work is devoted to prove, under suitable conditions on the initial data, the global existence and uniform decay rate of the solutions when the relaxation function is not necessarily of exponential or polynomial type.

Asymptotic behavior of a hierarchical size-structured population model
Dongxue Yan and Xianlong Fu
2018, 7(2): 293-316 doi: 10.3934/eect.2018015 +[Abstract](198) +[HTML](191) +[PDF](730.44KB)
Abstract:

We study in this paper a hierarchical size-structured population dynamics model with environment feedback and delayed birth process. We are concerned with the asymptotic behavior, particularly on the effects of hierarchical structure and time lag on the long-time dynamics of the considered system. We formally linearize the system around a steady state and study the linearized system by \begin{document} $C_0-{\rm{semigroup}}$ \end{document} framework and spectral analysis methods. Then we use the analytical results to establish the linearized stability, instability and asynchronous exponential growth conclusions under some conditions. Finally, some examples are presented and simulated to illustrate the obtained results.

Optimal nonlinearity control of Schrödinger equation
Kai Wang, Dun Zhao and Binhua Feng
2018, 7(2): 317-334 doi: 10.3934/eect.2018016 +[Abstract](218) +[HTML](134) +[PDF](452.32KB)
Abstract:

We study the optimal nonlinearity control problem for the nonlinear Schrödinger equation \begin{document} $iu_{t} = -\triangle u+V(x)u+h(t)|u|^α u$ \end{document}, which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global well-posedness of the equation for \begin{document} $0<α<\frac{4}{N}$ \end{document}, we show the existence of the optimal control. The Fréchet differentiability of the objective functional is proved, and the first order optimality system for \begin{document} $N≤ 3$ \end{document} is presented.

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