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Evolution Equations and Control Theory

April 2022 , Volume 11 , Issue 2

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Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints
Stefan Doboszczak, Manil T. Mohan and Sivaguru S. Sritharan
2022, 11(2): 347-371 doi: 10.3934/eect.2020110 +[Abstract](1899) +[HTML](622) +[PDF](522.69KB)
Abstract:

A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows subject to state constraints is established using the Ekeland variational principle. Since the system considered here is of coupled parabolic-hyperbolic type, the well developed control theory literature using abstract semigroup approach to linear and semilinear partial differential equations does not seem to contain problems of the type studied in this paper. The controls are distributed over a bounded domain, while the state variables are subject to a set of constraints and governed by the compressible Navier-Stokes equations linearized around a suitably regular base state. The maximum principle is of integral-type and obtained for minimizers of a tracking-type integral cost functional.

Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains
Mokhtari Yacine
2022, 11(2): 373-397 doi: 10.3934/eect.2021004 +[Abstract](1399) +[HTML](524) +[PDF](670.43KB)
Abstract:

In this paper, we deal with boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domains. By using the characteristics method, we prove under a natural assumption on the boundary functions that the 1D wave equation is controllable and stabilizable from one side of the boundary. Furthermore, the control function and the decay rate of the solution are given explicitly.

An inverse problem for the pseudo-parabolic equation with p-Laplacian
Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov and Guzel Rashitkhuzhakyzy Ashurova
2022, 11(2): 399-414 doi: 10.3934/eect.2021005 +[Abstract](1840) +[HTML](565) +[PDF](533.02KB)
Abstract:

In this article, we study the inverse problem of determining the right side of the pseudo-parabolic equation with a p-Laplacian and nonlocal integral overdetermination condition. The existence of solutions in a local and global time to the inverse problem is proved by using the Galerkin method. Sufficient conditions for blow-up (explosion) of the local solutions in a finite time are derived. The asymptotic behavior of solutions to the inverse problem is studied for large values of time. Sufficient conditions are obtained for the solution to disappear (vanish to identical zero) in a finite time. The limits conditions that which ensure the appropriate behavior of solutions are considered.

$ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations
Michal Fečkan, Kui Liu and JinRong Wang
2022, 11(2): 415-437 doi: 10.3934/eect.2021006 +[Abstract](1196) +[HTML](489) +[PDF](504.52KB)
Abstract:

In this paper, we study \begin{document}$ (\omega,\mathbb{T}) $\end{document}-periodic impulsive evolution equations via the operator semigroups theory in Banach spaces \begin{document}$ X $\end{document}, where \begin{document}$ \mathbb{T}: X\rightarrow X $\end{document} is a linear isomorphism. Existence and uniqueness of \begin{document}$ (\omega,\mathbb{T}) $\end{document}-periodic solutions results for linear and semilinear problems are obtained by Fredholm alternative theorem and fixed point theorems, which extend the related results for periodic impulsive differential equations.

Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative
Tran Bao Ngoc, Nguyen Huy Tuan, R. Sakthivel and Donal O'Regan
2022, 11(2): 439-455 doi: 10.3934/eect.2021007 +[Abstract](1189) +[HTML](528) +[PDF](519.4KB)
Abstract:

In this paper, we consider a nonlinear fractional diffusion equations with a Riemann-Liouville derivative. First, we establish the global existence and uniqueness of mild solutions under some assumptions on the input data. Some regularity results for the mild solution and its derivatives of fractional orders are also derived. Our key idea is to combine the theories of Mittag-Leffler functions, Banach fixed point theorem and some Sobolev embeddings.

Almost periodic type functions and densities
Marko Kostić
2022, 11(2): 457-486 doi: 10.3934/eect.2021008 +[Abstract](1154) +[HTML](474) +[PDF](595.18KB)
Abstract:

In this paper, we introduce and analyze the notions of \begin{document}$ \odot_{g} $\end{document}-almost periodicity and Stepanov \begin{document}$ \odot_{g} $\end{document}-almost periodicity for functions with values in complex Banach spaces. In order to do that, we use the recently introduced notions of lower and upper (Banach) \begin{document}$ g $\end{document}-densities. We also analyze uniformly recurrent functions, generalized almost automorphic functions and apply our results in the qualitative analysis of solutions of inhomogeneous abstract integro-differential inclusions. We present plenty of illustrative examples, results of independent interest, questions and unsolved problems.

Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling
Hedy Attouch, Aïcha Balhag, Zaki Chbani and Hassan Riahi
2022, 11(2): 487-514 doi: 10.3934/eect.2021010 +[Abstract](1560) +[HTML](527) +[PDF](1180.6KB)
Abstract:

In a Hilbert setting, we develop fast methods for convex unconstrained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three coefficients varying with time, one is a viscous damping coefficient, the second is attached to the Hessian-driven damping, the third is a time scaling coefficient. We study the convergence rate of the values under general conditions involving the damping and the time scale coefficients. The obtained results are based on a new Lyapunov analysis and they encompass known results on the subject. We pay particular attention to the case of an asymptotically vanishing viscous damping, which is directly related to the accelerated gradient method of Nesterov. The Hessian-driven damping significantly reduces the oscillatory aspects. We obtain an exponential rate of convergence of values without assuming the strong convexity of the objective function. The temporal discretization of these dynamics opens the gate to a large class of inertial optimization algorithms.

Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass
Masahiro Ikeda, Ziheng Tu and Kyouhei Wakasa
2022, 11(2): 515-536 doi: 10.3934/eect.2021011 +[Abstract](1302) +[HTML](418) +[PDF](494.36KB)
Abstract:

In the present paper, we study small data blow-up of the semi-linear wave equation with a scattering dissipation term and a time-dependent mass term from the aspect of wave-like behavior. The Strauss type critical exponent is determined and blow-up results are obtained to both sub-critical and critical cases with corresponding upper bound lifespan estimates. For the sub-critical case, our argument does not rely on the sign condition of dissipation and mass, which gives the extension of the result in [14]. Moreover, we show the blow-up result for the critical case which is a new result.

BV solutions of a convex sweeping process with a composed perturbation
Alexander Tolstonogov
2022, 11(2): 537-557 doi: 10.3934/eect.2021012 +[Abstract](1062) +[HTML](440) +[PDF](474.25KB)
Abstract:

A measurable sweeping process with a composed perturbation is considered in a separable Hilbert space. The values of the moving set generating the sweeping process are closed, convex sets. The retraction of the sweeping process is bounded by a positive Radon measure. The perturbation is the sum of two multivalued mappings. The values of the first one are closed, bounded, not necessarily convex sets. It is measurable in the time variable, is Lipschitz continuous in the phase variable, and satisfies a conventional growth condition. The values of the second one are closed, convex, not necessarily bounded sets. We assume that this mapping has a closed with respect to the phase variable graph.

The remaining assumptions concern the intersection of the second mapping and the multivalued mapping defined by the growth conditions. We suppose that this intersection has a measurable selector and has certain compactness properties.

We prove the existence of solutions for our inclusion. The proof is based on the author's theorem on continuous with respect to a parameter selectors passing through fixed points of contraction multivalued maps with closed, nonconvex, decomposable values depending on the parameter, and the classical Ky Fan fixed point theorem. The results which we obtain are new.

Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations
Brahim Alouini
2022, 11(2): 559-581 doi: 10.3934/eect.2021013 +[Abstract](1076) +[HTML](418) +[PDF](529.71KB)
Abstract:

In the current issue, we consider a general class of two coupled weakly dissipative fractional Schrödinger-type equations. We will prove that the asymptotic dynamics of the solutions for such NLS system will be described by the existence of a regular compact global attractor in the phase space that has finite fractal dimension.

Dynamics of piezoelectric beams with magnetic effects and delay term
Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos and Jamille L.L. Almeida
2022, 11(2): 583-603 doi: 10.3934/eect.2021015 +[Abstract](1277) +[HTML](401) +[PDF](528.99KB)
Abstract:

In this paper, we consider a piezoelectric beams system with magnetic effects and delay term. We study its long-time behavior through the associated dynamical system. We prove that the system is gradient and asymptotically smooth, which as a consequence, implies the existence of a global attractor, which is characterized as unstable manifold of the set of stationary solutions. We also get the quasi-stability of the system by establishing a stabilizability estimate and therefore obtain the finite fractal dimension of the global attractor.

Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space
Ankit Kumar, Kamal Jeet and Ramesh Kumar Vats
2022, 11(2): 605-619 doi: 10.3934/eect.2021016 +[Abstract](1261) +[HTML](411) +[PDF](441.3KB)
Abstract:

This paper aims to establish sufficient conditions for the exact controllability of the nonlocal Hilfer fractional integro-differential system of Sobolev-type using the theory of propagation family \begin{document}$ \{P(t), \; t\geq0\} $\end{document} generated by the operators \begin{document}$ A $\end{document} and \begin{document}$ R $\end{document}. For proving the main result we do not impose any condition on the relation between the domain of the operators \begin{document}$ A $\end{document} and \begin{document}$ R $\end{document}. We also do not assume that the operator \begin{document}$ R $\end{document} has necessarily a bounded inverse. The main tools applied in our analysis are the theory of measure of noncompactness, fractional calculus, and Sadovskii's fixed point theorem. Finally, we provide an example to show the application of our main result.

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

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