On Optimization Techniques for the Construction of an Exponential Estimate for Delayed Recurrent Neural Networks

This work is devoted to the modeling and investigation of the architecture design for the delayed recurrent neural network, based on the delayed differential equations. The usage of discrete and distributed delays makes it possible to model the calculation of the next states using internal memory, which corresponds to the artificial recurrent neural network architecture used in the field of deep learning. The problem of exponential stability of the models of recurrent neural networks with multiple discrete and distributed delays is considered. For this purpose, the direct method of stability research and the gradient descent method is used. The methods are used consequentially. Firstly we use the direct method in order to construct stability conditions (resulting in an exponential estimate), which include the tuple of positive definite matrices. Then we apply the optimization technique for these stability conditions (or of exponential estimate) with the help of a generalized gradient method with respect to this tuple of matrices. The exponential estimates are constructed on the basis of the Lyapunov–Krasovskii functional. An optimization method of improving estimates is offered, which is based on the notion of the generalized gradient of the convex function of the tuple of positive definite matrices. The search for the optimal exponential estimate is reduced to finding the saddle point of the Lagrange function.

Construction of the fundamental solution of a class of degenerate parabolic equations of high order

In the article, using the modified Levy method, a Green's function for a class of ultraparabolic equations of high order with an arbitrary number of parabolic degeneration groups is constructed. The modified Levy method is developed for high-order Kolmogorov equations with coefficients depending on all variables, while the frozen point, which is a parametrix, is chosen so that an exponential estimate of the fundamental solution and its derivatives is conveniently used.

Exponential Stability of Linear Discrete Systems with Multiple Delays

The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with multiple delays xk+1=Axk+∑i=1sBixk-mi, k=0,1,…, where s∈N, A and Bi are square matrices, and mi∈N. New criterion for exponential stability is proved by the Lyapunov method. An estimate of the norm of solutions is given as well and relations to the well-known results are discussed.