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Asymptotic stability of solutions to abstract differential equations
, 2010
"... An evolution problem for abstract differential equations is studied. The typical problem is: ˙u = A(t)u + F(t,u), t ≥ 0; u(0) = u0; ˙u = du dt Here A(t) is a linear bounded operator in a Hilbert space H, and F is a nonlinear operator, ‖F(t,u) ‖ ≤ c0‖u ‖ p, p> 1, c0, p = const> 0. It is assum ..."
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Cited by 16 (14 self)
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An evolution problem for abstract differential equations is studied. The typical problem is: ˙u = A(t)u + F(t,u), t ≥ 0; u(0) = u0; ˙u = du dt Here A(t) is a linear bounded operator in a Hilbert space H, and F is a nonlinear operator, ‖F(t,u) ‖ ≤ c0‖u ‖ p, p> 1, c0, p = const> 0. It is assumed that Re(A(t)u,u) ≤ −γ(t)‖u ‖ 2 ∀u ∈ H, where γ(t)> 0, and the case when limt→ ∞ γ(t) = 0 is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differential inequality.
The Dynamical Systems Method for solving nonlinear equations with monotone operators
"... A review of the authors’s results is given. Several methods are discussed for solving nonlinear equations F(u) = f, where F is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. V ..."
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Cited by 15 (12 self)
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A review of the authors’s results is given. Several methods are discussed for solving nonlinear equations F(u) = f, where F is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM) for solving the equation are formulated. These methods consist of a regularized Newtontype method, a gradienttype method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation F(u) = f is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to equation F(u) = f is justified. New nonlinear differential inequalities are derived and applied to a study of largetime behavior of solutions to evolution equations. Discrete versions of these inequalities are established.
AN ITERATIVE SCHEME FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS
"... An iterative scheme for solving illposed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of illposed operator equations with monotone operators is proposed and its c ..."
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Cited by 11 (6 self)
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An iterative scheme for solving illposed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of illposed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the iterative scheme are formulated and justified.
Solving largescale hybrid circuitantenna problems
 IEEE Transactions on Circuits and Systems I
, 2010
"... Abstract—Motivated by different applications in circuits, electromagnetics, and optics, this paper is concerned with the synthesis of a particular type of linear circuit, where the circuit is associated with a control unit. The objective is to design a controller for this control unit such that cer ..."
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Cited by 3 (3 self)
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Abstract—Motivated by different applications in circuits, electromagnetics, and optics, this paper is concerned with the synthesis of a particular type of linear circuit, where the circuit is associated with a control unit. The objective is to design a controller for this control unit such that certain specifications on the parameters of the circuit are satisfied. It is shown that designing a control unit in the form of a switching network is an NPcomplete problem that can be formulated as a rankminimization problem. It is then proven that the underlying design problem can be cast as a semidefinite optimization if a passive network is designed instead of a switching network. Since the implementation of a passive network may need too many components, the design of a decoupled (sparse) passive network is subsequently studied. This paper introduces a tradeoff between design simplicity and implementation complexity for an important class of linear circuits. The superiority of the developed techniques is demonstrated by different simulations. In particular, for the first time in the literature, a wavelengthsize passive antenna is designed, which has an excellent beamforming capability and which can concurrently make a null in at least eight directions. Index Terms—Antenna radiation pattern, circuit network analysis, circuit optimization, convex optimization, integrated antennas, linear matrix inequalities, reconfigurable antenna. I.
Nonlinear differential inequality
"... A nonlinear differential inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations in Banach spaces. It is applied to a study of global e ..."
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Cited by 2 (2 self)
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A nonlinear differential inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations in Banach spaces. It is applied to a study of global existence of solutions to nonlinear partial differential equations.