Remarks on Controllability of Second Order Evolution Equations in Hilbert Spaces

1970 ◽  
Vol 8 (1) ◽  
pp. 90-99 ◽  
Author(s):  
Kunio Tsujioka
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Second Order

scholarly journals ON DIFFERENTIAL-NON-AUTONOMOUS REPRESENTATION INTEGRATIVE ACTIVITY OF NEUROPOPULATION BILINEAR SECOND-ORDER MODEL WITH A DELAY

2021 ◽  
Vol 23 (2) ◽  
pp. 115-126
Author(s):  
A.V. Daneev ◽  
◽  
A.V. Lakeev ◽  
V.A. Rusanov ◽  
P.A. Plesnev ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Evolution Equations ◽  
Basic Research ◽  
Second Order ◽  
Infinite Dimensional ◽  
Machine Interface ◽  
Additive Combination ◽  
Basic Research Project ◽  
Principle Of Maximum

For neuromorphic processes specified by the behavior of a local neuropopulation (for example, processes induced by a brain-machine interface platform of the Neuralink type), we study the solvability of the problem of the existence of a differential realization of these processes in the class of bilinear nonstationary ordinary differential equations of the second order (with delay) in separable Hilbert space. This formulation belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The metalanguage of the theory being developed is the constructions of tensor products of Hilbert spaces, lattice structures with orthocompletion, the functional apparatus of the nonlinear Rayleigh-Ritz operator, and the principle of maximum entropy. It is shown that the property of sublinearity of this operator, allows you to obtain conditions for the existence of such differential realizations; along the way, metric conditions for the continuity of the projectivization of this operator are substantiated with the calculation of the fundamental group of its compact image. This work was financially supported by the Russian Foundation for Basic Research (project no. 19-01-00301).

Subharmonic resonance of short internal standing waves by progressive surface waves

1996 ◽  
Vol 321 ◽  
pp. 217-233 ◽  
Author(s):  
D. F. Hill ◽  
M. A. Foda
Keyword(s):  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Internal Waves ◽  
Growth Rates ◽  
Evolution Equations ◽  
Standing Waves ◽  
Second Order ◽  
Inviscid Limit ◽  
Initial Growth

Experimental evidence and a theoretical formulation describing the interaction between a progressive surface wave and a nearly standing subharmonic internal wave in a two-layer system are presented. Laboratory investigations into the dynamics of an interface between water and a fluidized sediment bed reveal that progressive surface waves can excite short standing waves at this interface. The corresponding theoretical analysis is second order and specifically considers the case where the internal wave, composed of two oppositely travelling harmonics, is much shorter than the surface wave. Furthermore, the analysis is limited to the case where the internal waves are small, so that only the initial growth is described. Approximate solution to the nonlinear boundary value problem is facilitated through a perturbation expansion in surface wave steepness. When certain resonance conditions are imposed, quadratic interactions between any two of the harmonics are in phase with the third, yielding a resonant triad. At the second order, evolution equations are derived for the internal wave amplitudes. Solution of these equations in the inviscid limit reveals that, at this order, the growth rates for the internal waves are purely imaginary. The introduction of viscosity into the analysis has the effect of modifying the evolution equations so that the growth rates are complex. As a result, the amplitudes of the internal waves are found to grow exponentially in time. Physically, the viscosity has the effect of adjusting the phase of the pressure so that there is net work done on the internal waves. The growth rates are, in addition, shown to be functions of the density ratio of the two fluids, the fluid layer depths, and the surface wave conditions.

Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor

2014 ◽  
Vol 1651 ◽  
Author(s):  
Thomas Hochrainer
Keyword(s):  
Dislocation Density ◽  
Evolution Equations ◽  
Continuum Theory ◽  
Orientation Distribution ◽  
Line Length ◽  
Second Order ◽  
Dislocation Dynamics ◽  
Alignment Tensor ◽  
Length Changes ◽  
Edge Dislocations

ABSTRACTIn the current paper we present a continuum theory of dislocations based on the second-order alignment tensor in conjunction with the classical dislocation density tensor (Kröner-Nye-tensor) and a scalar dislocation curvature measure. The second-order alignment tensor is a symmetric second order tensor characterizing the orientation distribution of dislocations in elliptic form. It is closely connected to total densities of screw and edge dislocations introduced in the literature. The scalar dislocation curvature density is a conserved quantity the integral of which represents the total number of dislocations in the system. The presented evolution equations of these dislocation density measures partly parallel earlier developed theories based on screw-edge decompositions but handle line length changes and segment reorientation consistently. We demonstrate that the presented equations allow predicting the evolution of a single dislocation loop in a non-trivial velocity field.

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