We study the limit behavior of certain classes of dependent random sequences (processes) which do not possess the Markov property. Assuming these processes depend on a control parameter we show that the optimization of the control can be reduced to a problem of nonlinear optimization. Under certain hypotheses we establish the stability of such optimization problems.
Asymptotic properties of the half-linear difference equation (∗) ∆(an|∆xn| α sgn ∆xn) = bn|xn+1| α sgn xn+1 are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to (∗) are considered too. Our approach is based on a classification of solutions of (∗) and on some summation inequalities for double series, which can be used also in other different contexts.
The asymptotic behaviour of the solutions is studied for a real unstable twodimensional system x ' (t) = A(t)x(t) + B(t)x(t − r) + h(t, x(t), x(t − r)), where r > 0 is a constant delay. It is supposed that A, B and h are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as t → ∞ are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wazewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].
The paper discusses the asymptotic properties of solutions of the scalar functional differential equation \[ y^{\prime }(x)=ay(\tau (x))+by(x),\qquad x\in [x_0,\infty ) \] of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution $y(x)$ which behaves in this way.
We study the asymptotic behavior of the solutions of a differential equation with unbounded delay. The results presented are based on the first Lyapunov method, which is often used to construct solutions of ordinary differential equations in the form of power series. This technique cannot be applied to delayed equations and hence we express the solution as an asymptotic expansion. The existence of a solution is proved by the retract method.
Asymptotic properties of solutions of the difference equation of the form ∆ mxn = anϕ(xτ1(n) , . . . , xτk(n) ) + bn are studied. Conditions under which every (every bounded) solution of the equation ∆myn = bn is asymptotically equivalent to some solution of the above equation are obtained.
This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasiiinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as t → ∞.
Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by dX(t) = A(ξ(t))X(t) dt + H(ξ(t))X(t) dw(t), where ξ(t) is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.
As an important artificial neural network, associative memory model can be employed to mimic human thinking and machine intelligence. In this paper, first, a multi-valued many-to-many Gaussian associative memory model (M3GAM) is proposed by introducing the Gaussian unidirectional associative memory model (GUAM) and Gaussian bidirectional associative memory model (GBAM) into Hattori {et al}'s multi-module associative memory model ((MMA)2). Second, the M3GAM's asymptotical stability is proved theoretically in both synchronous and asynchronous update modes, which ensures that the stored patterns become the M3GAM's stable points. Third, by substituting the general similarity metric for the negative squared Euclidean distance in M3GAM, the generalized multi-valued many-to-many Gaussian associative memory model (GM3GAM) is presented, which makes the M3GAM become its special case. Finally, we investigate the M3GAM's application in association-based image retrieval, and the computer simulation results verify the M3GAM's robust performance.