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Evolution Equations & Control Theory

December 2021 , Volume 10 , Issue 4

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A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case
Wenhui Chen and Alessandro Palmieri
2021, 10(4): 673-687 doi: 10.3934/eect.2020085 +[Abstract](1480) +[HTML](565) +[PDF](360.77KB)
Abstract:

In this paper, we study the blow – up of solutions to the semilinear Moore – Gibson – Thompson (MGT) equation with nonlinearity of derivative type \begin{document}$ |u_t|^p $\end{document} in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow – up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent \begin{document}$ p $\end{document} for the nonlinear term satisfies \begin{document}$ 1<p\leqslant (n+1)/(n-1) $\end{document} for \begin{document}$ n\geqslant2 $\end{document} and \begin{document}$ p>1 $\end{document} for \begin{document}$ n = 1 $\end{document}. In particular, we find the same blow – up range for \begin{document}$ p $\end{document} as in the corresponding semilinear wave equation with nonlinearity of derivative type.

Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation
Tomomi Yokota and Kentarou Yoshii
2021, 10(4): 689-699 doi: 10.3934/eect.2020086 +[Abstract](1091) +[HTML](554) +[PDF](331.61KB)
Abstract:

This paper deals with the solvability in the semilinear abstract evolution equation with countable time delays,

in a Banach space \begin{document}$ X $\end{document}, where \begin{document}$ -A $\end{document} is a generator of a \begin{document}$ C_0 $\end{document}-semigroup with exponential decay and \begin{document}$ F: X \times X^\mathbb{N} \to X $\end{document} is Lipschitz continuous. Nicaise and Pignotti (J. Evol. Equ.; 2018;18;947–971) established global existence and exponential decay in time for solutions of the above equation with finite time delays in Hilbert spaces under global or local Lipschitz conditions. The purpose of the present paper is to generalize the result to the case of countable time delays in Banach spaces under a global Lipschitz condition.

Deterministic control of stochastic reaction-diffusion equations
Wilhelm Stannat and Lukas Wessels
2021, 10(4): 701-722 doi: 10.3934/eect.2020087 +[Abstract](1317) +[HTML](566) +[PDF](889.44KB)
Abstract:

We consider the control of semilinear stochastic partial differential equations (SPDEs) via deterministic controls. In the case of multiplicative noise, existence of optimal controls and necessary conditions for optimality are derived. In the case of additive noise, we obtain a representation for the gradient of the cost functional via adjoint calculus. The restriction to deterministic controls and additive noise avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schlögl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.

Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity
Haixia Li
2021, 10(4): 723-732 doi: 10.3934/eect.2020088 +[Abstract](1101) +[HTML](557) +[PDF](307.69KB)
Abstract:

In this paper, an initial boundary value problem for a parabolic type Kirchhoff equation with time-dependent nonlinearity is considered. A new blow-up criterion for nonnegative initial energy is given and upper and lower bounds for the blow-up time are also derived. These results partially generalize some recent ones obtained by Han and Li in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(2018), 3283-3297].

S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations
Pallavi Bedi, Anoop Kumar, Thabet Abdeljawad and Aziz Khan
2021, 10(4): 733-748 doi: 10.3934/eect.2020089 +[Abstract](1262) +[HTML](565) +[PDF](550.75KB)
Abstract:

In this article, we deal with the existence of S-asymptotically \begin{document}$ \omega $\end{document}-periodic mild solutions of Hilfer fractional evolution equations. We also investigate the Ulam-Hyers and Ulam-Hyers-Rassias stability of similar solutions. These results are established in Banach space with the help of resolvent operator functions and fixed point technique on an unbounded interval. An example is also presented for the illustration of obtained results.

Approximate controllability of network systems
Yassine El Gantouh, Said Hadd and Abdelaziz Rhandi
2021, 10(4): 749-766 doi: 10.3934/eect.2020091 +[Abstract](1128) +[HTML](535) +[PDF](431.47KB)
Abstract:

In this paper, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory are used to formulate and solve control problems for network systems. In fact, we derive necessary and sufficient conditions for approximate controllability of such systems. These criteria, in some particular cases, are given by the well-known Kalman's controllability rank condition.

Uniform stability in a vectorial full Von Kármán thermoelastic system with solenoidal dissipation and free boundary conditions
Catherine Lebiedzik
2021, 10(4): 767-796 doi: 10.3934/eect.2020092 +[Abstract](841) +[HTML](483) +[PDF](399.43KB)
Abstract:

We will consider the full von Kármán thermoelastic system with free boundary conditions and dissipation imposed only on the in-plane displacement. It will be shown that the corresponding solutions are exponentially stable, though there is no mechanical dissipation on the vertical displacements. The main tools used are: (i) partial analyticity of the linearized semigroup and (ii) trace estimates which exploit the hidden regularity harvested from partial analyticity.

Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms
Andrew R. Becklin and Mohammad A. Rammaha
2021, 10(4): 797-836 doi: 10.3934/eect.2020093 +[Abstract](951) +[HTML](510) +[PDF](513.6KB)
Abstract:

This article is concerned with Hadamard's well posedness of a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain \begin{document}$ \Omega\subset\mathbb{R}^3 $\end{document} which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of \begin{document}$ \Omega $\end{document}. The system is influenced by several competing forces. In particular, the source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from \begin{document}$ H^1_{\Gamma_0}(\Omega) $\end{document} into \begin{document}$ L^2(\Omega) $\end{document}. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions. Moreover, we prove that such solutions depend continuously on the initial data, and uniqueness is obtained in two different scenarios.

Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions
El Mustapha Ait Ben Hassi, Salah-Eddine Chorfi, Lahcen Maniar and Omar Oukdach
2021, 10(4): 837-859 doi: 10.3934/eect.2020094 +[Abstract](1453) +[HTML](518) +[PDF](435.98KB)
Abstract:

In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.

Stabilization of higher order Schrödinger equations on a finite interval: Part I
Ahmet Batal, Türker Özsarı and Kemal Cem Yılmaz
2021, 10(4): 861-919 doi: 10.3934/eect.2020095 +[Abstract](1158) +[HTML](515) +[PDF](2705.14KB)
Abstract:

We study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation.

Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $
Brahim Boufoussi and Soufiane Mouchtabih
2021, 10(4): 921-935 doi: 10.3934/eect.2020096 +[Abstract](1035) +[HTML](462) +[PDF](381.72KB)
Abstract:

This article investigates the controllability for neutral stochastic delay functional integro-differential equations driven by a fractional Brownian motion, with Hurst parameter lesser than \begin{document}$ 1/2 $\end{document}. We employ the theory of resolvent operators developed by [10] combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result.

Optimal distributed control of the three dimensional primitive equations of large-scale ocean and atmosphere dynamics
Bo You
2021, 10(4): 937-963 doi: 10.3934/eect.2020097 +[Abstract](892) +[HTML](460) +[PDF](437.97KB)
Abstract:

The main objective of this paper is to study the optimal distributed control of the three dimensional non-autonomous primitive equations of large-scale ocean and atmosphere dynamics. We apply the well-posedness and regularity results of solutions for this system as well as some abstract results from the nonlinear functional analysis to establish the existence of optimal controls as well as the first-order necessary optimality condition for an associated optimal control problem in which a distributed control is applied to the temperature.

Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains
Nathanael Skrepek
2021, 10(4): 965-1006 doi: 10.3934/eect.2020098 +[Abstract](995) +[HTML](593) +[PDF](587.15KB)
Abstract:

We consider a port-Hamiltonian system on an open spatial domain \begin{document}$ \Omega \subseteq \mathbb{R}^n $\end{document} with bounded Lipschitz boundary. We show that there is a boundary triple associated to this system. Hence, we can characterize all boundary conditions that provide unique solutions that are non-increasing in the Hamiltonian. As a by-product we develop the theory of quasi Gelfand triples. Adding "natural" boundary controls and boundary observations yields scattering/impedance passive boundary control systems. This framework will be applied to the wave equation, Maxwell's equations and Mindlin plate model. Probably, there are even more applications.

Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay
Vu Manh Toi
2021, 10(4): 1007-1023 doi: 10.3934/eect.2020099 +[Abstract](912) +[HTML](463) +[PDF](391.03KB)
Abstract:

We consider the 3D Navier-Stokes-Voigt equations in a bounded domain with unbounded variable delay. We study the stability of stationary solutions by the classical direct method, and by an appropriate Lyapunov functional. We also give a sufficient condition of parameters for the polynomial stability of the stationary solution in a special case of unbounded variable delay. Finally, when the condition for polynomial stability is not satisfied, we stabilize the stationary by using the finite Fourier modes and by internal feedback control with a support large enough.

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

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