# American Institute of Mathematical Sciences

eISSN:
2163-2480

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## Evolution Equations & Control Theory

2017 , Volume 6 , Issue 1

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2017, 6(1): 1-13 doi: 10.3934/eect.2017001 +[Abstract](95) +[HTML](0) +[PDF](361.6KB)
Abstract:
In this article, we are concerned with the asymptotic behavior of a class of degenerate parabolic problems involving porous medium type equations, in a bounded domain, when the diffusion coefficient becomes large. We prove the upper semicontinuity of the associated global attractor as the diffusion increases to infinity.
2017, 6(1): 15-34 doi: 10.3934/eect.2017002 +[Abstract](101) +[HTML](1) +[PDF](422.2KB)
Abstract:
We analyze the lumpability of linear systems on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction operator onto a smaller state space in which a self-contained dynamical description exists. We obtain conditions for lumpability of dynamics defined by unbounded operators using the theory of strongly continuous semigroups. We also derive results from the dual space point of view using sun dual theory. Furthermore, we connect the theory of lumping to several results from operator factorization. We indicate several applications to particular systems, including delay differential equations.
2017, 6(1): 35-58 doi: 10.3934/eect.2017003 +[Abstract](62) +[HTML](0) +[PDF](485.2KB)
Abstract:
This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors nonlocal model here studied consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called 'deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.
2017, 6(1): 59-75 doi: 10.3934/eect.2017004 +[Abstract](68) +[HTML](0) +[PDF](416.1KB)
Abstract:
We prove a sharp decay rate for the total energy of two classes of systems of weakly coupled hyperbolic equations. We show that we can stabilize the full system through a single damping term, in feedback form, acting on one component only of the system (\emph{indirect stabilization}). The energy estimate is achieved by means of suitable estimates of the resolvent operator norm. We apply this technique to a wave-wave system and to a wave-Petrovsky system.
2017, 6(1): 77-91 doi: 10.3934/eect.2017005 +[Abstract](50) +[HTML](0) +[PDF](407.0KB)
Abstract:
This paper deals with the identification problem for the $L^1$-valued retarded functional differential equation. The unknowns are parameters and operators appearing in the given systems. In order to identify the parameters, we introduce the solution semigroup and the structural operators in the initial data space, and provide the representations of spectral projections and the completeness of generalized eigenspaces. The sufficient condition for the identification problem is given as the so called rank condition in terms of the initial values and eigenvectors of adjoint operator.
2017, 6(1): 93-109 doi: 10.3934/eect.2017006 +[Abstract](55) +[HTML](0) +[PDF](351.8KB)
Abstract:
In this paper, we focus our study on a multiobjective variational control problem and establish sufficient optimality conditions under the assumptions of α-V-univex function. Furthermore, mixed type duality results are also discussed under the aforesaid assumption in order to relate the primal and dual problems. Examples are given to show the existence of α-V-univex function and to elucidate duality result.
2017, 6(1): 111-134 doi: 10.3934/eect.2017007 +[Abstract](623) +[HTML](0) +[PDF](1770.1KB)
Abstract:
In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time $t = T.$ This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.
2017, 6(1): 135-154 doi: 10.3934/eect.2017008 +[Abstract](70) +[HTML](0) +[PDF](451.8KB)
Abstract:
In this paper, we study a fluid-structure interaction model of Stokes-wave equation coupling system with Kelvin-Voigt type of damping. We show that this damped coupling system generates an analyticity semigroup and thus the semigroup solution, which also satisfies variational framework of weak solution, decays to zero at exponential rate.

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