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Evolution Equations & Control Theory

September 2019 , Volume 8 , Issue 3

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Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation
Minzilia A. Sagadeeva, Sophiya A. Zagrebina and Natalia A. Manakova
2019, 8(3): 473-488 doi: 10.3934/eect.2019023 +[Abstract](930) +[HTML](287) +[PDF](357.56KB)
Abstract:

The paper presents sufficient conditions for existence of an optimal control of solutions to a non-autonomous degenerate operator-differential evolution equation. We construct families of operators that solve this equation, as well as classical and strong solutions of the multipoint initial-final problem for the equation. We show that there exists a solution of an optimal control problem for a given operator-differential equation with a multipoint initial-final condition. The paper, in addition to the introduction and the bibliography, contains five sections. The first three parts contain information about the solvability of the multipoint initial-final problem for a non-autonomous equation. The fourth section presents the main result of the article; that is, a theorem on existence of optimal control of solutions to a multipoint initial-final problem. In the fifth part, the optimal control problem for the non-autonomous modified Chen – Gurtin model with the multipoint initial-final condition is investigated on the basis of the obtained abstract results.

A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments
Marco Campo, José R. Fernández and Maria Grazia Naso
2019, 8(3): 489-502 doi: 10.3934/eect.2019024 +[Abstract](887) +[HTML](330) +[PDF](406.17KB)
Abstract:

In this paper we study, from the numerical point of view, a dynamic problem which models a suspension bridge system. This problem is written as a nonlinear system of hyperbolic partial differential equations in terms of the displacements of the bridge and of the cable. By using the respective velocities, its variational formulation leads to a coupled system of parabolic nonlinear variational equations. An existence and uniqueness result, and an exponential energy decay property, are recalled. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme. A discrete stability property is shown and a priori error estimates are proved, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some numerical results are shown to demonstrate the accuracy of the approximation and the behaviour of the solution.

A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid
Marcelo M. Disconzi and Igor Kukavica
2019, 8(3): 503-542 doi: 10.3934/eect.2019025 +[Abstract](967) +[HTML](294) +[PDF](489.27KB)
Abstract:

We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to H3, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature and a new compressible Cauchy invariance.

A comparison principle for Hamilton-Jacobi equation with moving in time boundary
Nicolas Forcadel and Mamdouh Zaydan
2019, 8(3): 543-565 doi: 10.3934/eect.2019026 +[Abstract](933) +[HTML](262) +[PDF](378.39KB)
Abstract:

In this paper we consider an Hamilton-Jacobi equation on a moving in time domain. The boundary is described by a \begin{document}$C^{1}$\end{document} function. We show how we derive this equation from the work of [26]. We only prove a comparison principle since the proof of other theoritical results can be found in [20]. At the end of the paper, we consider a short homogenization result in order to reinforce the traffic flow interpretation of the equation.

Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity
Mamoru Okamoto
2019, 8(3): 567-601 doi: 10.3934/eect.2019027 +[Abstract](908) +[HTML](272) +[PDF](518.09KB)
Abstract:

We consider the Cauchy problem of the higher-order KdV-type equation:

where \begin{document}$ {\mathfrak{m}} \ge 4 $\end{document}. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.

Optimal control of evolution differential inclusions with polynomial linear differential operators
Elimhan N. Mahmudov
2019, 8(3): 603-619 doi: 10.3934/eect.2019028 +[Abstract](932) +[HTML](337) +[PDF](323.37KB)
Abstract:

In this paper we have introduced a new class of problems of optimal control theory with differential inclusions described by polynomial linear differential operators. Consequently, there arises a rather complicated problem with simultaneous determination of the polynomial linear differential operators with variable coefficients and a Mayer functional depending on high order derivatives. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and transversality conditions are derived. Formulation of the transversality conditions at the endpoints of the considered time interval plays a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions. The main idea of the proof of optimality conditions of Mayer problem for differential inclusions with polynomial linear differential operators is the use of locally-adjoint mappings. The method is demonstrated in detail as an example for the semilinear optimal control problem and the Weierstrass-Pontryagin maximum principle is obtained. Then the optimality conditions are derived for second order convex differential inclusions with convex endpoint constraints.

Periodic solutions for implicit evolution inclusions
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu and Dušan D. Repovš
2019, 8(3): 621-631 doi: 10.3934/eect.2019029 +[Abstract](850) +[HTML](266) +[PDF](301.48KB)
Abstract:

We consider a nonlinear implicit evolution inclusion driven by a nonlinear, nonmonotone, time-varying set-valued map and defined in the framework of an evolution triple of Hilbert spaces. Using an approximation technique and a surjectivity result for parabolic operators of monotone type, we show the existence of a periodic solution.

Waves and diffusion on metric graphs with general vertex conditions
Klaus-Jochen Engel and Marjeta Kramar Fijavž
2019, 8(3): 633-661 doi: 10.3934/eect.2019030 +[Abstract](752) +[HTML](251) +[PDF](491.73KB)
Abstract:

We prove well-posedness for general linear wave- and diffusion equations on compact or non-compact metric graphs allowing various conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on \begin{document}$ {\mathrm{L}}^p({\mathbb{R}_+}, {\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0, 1], {\mathbb{C}}^m) $\end{document} for \begin{document}$ 1\le p<+\infty $\end{document}.

Two questions arising in the theory of attractors
Vittorino Pata
2019, 8(3): 663-668 doi: 10.3934/eect.2019031 +[Abstract](785) +[HTML](279) +[PDF](288.52KB)
Abstract:

In this note, we dwell on the notions of global and exponential attractors for strongly continuous semigroups acting on a complete metric space. Two natural questions arising in the theory are addressed.

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