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Evolution Equations & Control Theory

March 2020 , Volume 9 , Issue 1

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Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints
Abdelmouhcene Sengouga
2020, 9(1): 1-25 doi: 10.3934/eect.2020014 +[Abstract](211) +[HTML](69) +[PDF](935.65KB)

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decays at the rate \begin{document}$ 1/t $\end{document}. We also establish observability results, at one or at both endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.

Almost mixed semi-continuous perturbation of Moreau's sweeping process
Doria Affane, Meriem Aissous and Mustapha Fateh Yarou
2020, 9(1): 27-38 doi: 10.3934/eect.2020015 +[Abstract](170) +[HTML](48) +[PDF](329.4KB)

In this work, we introduce a new concept of semi-continuous set-valued mappings, called almost mixed semi-continuity, by taking maps that are upper semi-continuous with almost convex values in some points and lower semi-continuous in remaining points. We generalize earlier results obtained for both mixed semi-continuous maps and almost convex sets. We discuss the existence of solution for evolution problems driven by the so-called sweeping process subject to external forces, known as perturbation to the system, by this type of set-valued mappings. Finally, we give some topological properties of the attainable and solution sets in order to solve an optimal time problem.

Uniform exponential stability of a fluid-plate interaction model due to thermal effects
Gilbert Peralta
2020, 9(1): 39-60 doi: 10.3934/eect.2020016 +[Abstract](154) +[HTML](96) +[PDF](375.98KB)

We consider a coupled fluid-thermoelastic plate interaction model. The fluid velocity is modeled by the linearized 3D Navier-Stokes equation while the plate dynamics is described by a thermoelastic Kirchoff system. By eliminating the pressure term, the system is reformulated as an abstract evolution problem and its well-posedness is proved by semigroup methods. The dissipation in the system is due to the diffusion of the fluid and heat components. Uniform stability of the coupled system is established through multipliers and the energy method. The multipliers used for thermoelastic plate models in the literature are modified in accordance to the applicability of a certain Stokes map.

On quasilinear parabolic equations and continuous maximal regularity
Jeremy LeCrone and Gieri Simonett
2020, 9(1): 61-86 doi: 10.3934/eect.2020017 +[Abstract](145) +[HTML](48) +[PDF](428.88KB)

We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.

Reachability problems for a wave-wave system with a memory term
Paola Loreti and Daniela Sforza
2020, 9(1): 87-130 doi: 10.3934/eect.2020018 +[Abstract](109) +[HTML](49) +[PDF](587.56KB)

We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theory.

Optimization of the blood pressure with the control in coefficients
Ruxandra Stavre
2020, 9(1): 131-151 doi: 10.3934/eect.2020019 +[Abstract](116) +[HTML](64) +[PDF](432.4KB)

This article is devoted to the study of a distributed control problem, with the control in coefficients, inspired by a disease that can lead to serious health problems: high blood pressure. We are concerned with the determination of a viscosity function that realizes an optimal blood pressure configuration. Using the mathematical model for viscous fluid-elastic structure interaction problems, we present existence, uniqueness, regularity results and estimates for the three unknown functions of the problem: velocity and pressure of the fluid and displacement of the elastic medium. The weak regularity of the state provided by the variational approach of the problem as well as the choice of the control variable induce some difficulties in the proof of the existence of an optimal control. The choice of the cost functional leads to an adjoint system which is not a divergence free one. For analyzing it, we propose a method based on the construction of several functions with suitable properties. Finally, we establish the necessary conditions of optimality.

Initial-boundary value problems for multi-term time-fractional diffusion equations with $ x $-dependent coefficients
Zhiyuan Li, Xinchi Huang and Masahiro Yamamoto
2020, 9(1): 153-179 doi: 10.3934/eect.2020001 +[Abstract](239) +[HTML](112) +[PDF](450.77KB)

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with \begin{document}$ x $\end{document}-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation
Peng Gao
2020, 9(1): 181-191 doi: 10.3934/eect.2020002 +[Abstract](219) +[HTML](115) +[PDF](293.22KB)

In this paper, we establish the global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation by five controls: one control is the right member of the equation and is constant with respect to the space variable, the four others are the boundary controls.

Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach
Andrea Signori
2020, 9(1): 193-217 doi: 10.3934/eect.2020003 +[Abstract](223) +[HTML](110) +[PDF](443.15KB)

In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn–Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into account a non-smooth singular nonlinearity, namely the double obstacle potential. Due to its non-smoothness behavior, the standard procedure to characterize the necessary conditions for the optimality cannot be performed. Therefore, we follow a different strategy which in the literature is known as the "deep quench" approach in order to obtain some optimality conditions that have to be interpreted in a more general framework. We establish the existence of optimal controls and some first-order optimality conditions for the system are derived by employing suitable approximation schemes.

Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators
Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing and M. D. P. Monteiro Marques
2020, 9(1): 219-254 doi: 10.3934/eect.2020004 +[Abstract](219) +[HTML](121) +[PDF](474.55KB)

In this paper, we study the existence of solutions for evolution problems of the form \begin{document}$ -\frac{du}{dr}(t) \in A(t)u(t) + F(t, u(t))+f(t, u(t)) $\end{document}, where, for each \begin{document}$ t $\end{document}, \begin{document}$ A(t) : D(A(t)) \to 2 ^H $\end{document} is a maximal monotone operator in a Hilbert space \begin{document}$ H $\end{document} with continuous, Lipschitz or absolutely continuous variation in time. The perturbation \begin{document}$ f $\end{document} is separately integrable on \begin{document}$ [0, T] $\end{document} and separately Lipschitz on \begin{document}$ H $\end{document}, while \begin{document}$ F $\end{document} is scalarly measurable and separately scalarly upper semicontinuous on \begin{document}$ H $\end{document}, with convex and weakly compact values. Several new applications are provided.

The Kalman condition for the boundary controllability of coupled 1-d wave equations
Sergei Avdonin, Jeff Park and Luz de Teresa
2020, 9(1): 255-273 doi: 10.3934/eect.2020005 +[Abstract](262) +[HTML](116) +[PDF](384.94KB)

The focus of this paper is the exact controllability of a system of \begin{document}$ N $\end{document} one-dimensional coupled wave equations when the control is exerted on a part of the boundary by means of one control. We give a Kalman condition (necessary and sufficient) and give a description of the attainable set. In general, this set is not optimal, but can be refined under certain conditions.

Stability analysis in some strongly prestressed rectangular plates
Jifeng Chu, Maurizio Garrione and Filippo Gazzola
2020, 9(1): 275-299 doi: 10.3934/eect.2020006 +[Abstract](235) +[HTML](121) +[PDF](1083.83KB)

We consider an evolution plate equation aiming to model the motion of the deck of a periodically forced strongly prestressed suspension bridge. Using the prestress assumption, we show the appearance of multiple time-periodic uni-modal longitudinal solutions and we discuss their stability. Then, we investigate how these solutions exchange energy with a torsional mode. Although the problem is forced, we find a portrait where stability and instability regions alternate. The techniques used rely on ODE analysis of stability and are complemented with numerical simulations.

2018  Impact Factor: 1.048



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