Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces

We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2\times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_\Theta$. We suppose that the weight $\Theta$ satisfies some regularity assumption. Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2\left((-T, T); l^2)$ belongs to $C^2\left((-T, T); l^2_\Theta\right)$. And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.

Optimal Control of an abstract Evolution Variational Inequality with Application in Homogenized Plasticity

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not G\^ateaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

Solvability of initial-boundary value problem of a multiple characteristic fifth-order operator-differential equation

In this study, we establish existence-uniqueness of a vector function in appropriate Sobolev-type space for a boundary value problem of a fifth-order operator differential equation. Proper conditions are obtained for the given problem to be well-posed. Much effort is devoted to develop the association between these conditions and the operator coefficients of the investigated equation. In this paper, accurate estimates of the norms of the intermediate derivatives operators are presented and used to determine the solvability conditions.

On Solvability of Third-Order Operator Differential Equation with Parabolic Principal Part in Weighted Space

Sufficient conditions are found for the correct and unique solvability of a class of third-order parabolic operator differential equations, whose principal parts have multiple characteristics, in a Sobolev-type space with exponential weight. The estimates for the norms of intermediate derivative operators are obtained and the relationship between these estimates and solvability conditions is established. Besides, the connection is found between the order of exponential weight and the lower bound for the spectrum of abstract operator appearing in the principal part of the equation.

Mixed Control Problem for the Linearized Quasi-Stationary Phase Field System of Equations

Conditions are obtained for unique solution existence of a mixed control problem without taking in account control expenses for a system that described by an initial-boundary value problem for the linearized quasi-stationary system of phase field equations. The problem is reduced to an optimal control problem for operator differential equation of first order in abstract space with degenerate operator at derivative using start and distributed controls simultaneously. The theorem on the unique solvability of this problem is applied to studying of optimal control problem for the phase field equations system.