Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part II

The paper continues the studies in the equivalence of the measures generated by the solutions of nonlinear evolution differential equations with unbounded linear operators perturbed by random Gaussian processes in a Hilbert space, in particular í. Two different nonlinear evolution differential equations perturbed by the same random Gaussian process in the right-hand side are considered in the space í. The sufficient existence and uniqueness conditions are established for the solutions of these equations, the equivalence of the measures generated by the solutions is proved, and explicit formulas for the Radon–Nikodym density of the respective measures calculated in terms of the coefficients of the considered equations are written.     

Uniform stability with respect to the impulse hypersurfaces of the solutions of differential equations with impulses

Systems of differential equations with impulses are divided into several classes depending on the way in which the moments of the impulses are determined. The initial value problem is considered for systems of differential equations for which the impulses are realized at the moments when the integral curve of the problem meets certain hypersurfaces, called impulse hypersurfaces, of the extended phase space. The notations of strong uniform stability of the solutions of systems without impulses and uniform stability with respect to the impulse hypersurfaces of the solutions with impulses are introduced. The main results are given in two theorems. The first contains sufficient conditions under which strong uniform stability of the zero solution of the respective system without impulses implies uniform stability with respect to the impulse hypersurfaces of the zero solutions of the initial system with impulses. In the second theorem, sufficient conditions are given under which uniform Lipschitz stability of the zero solution of the respective system without impulses implies uniform stability with respect to the impulse hypersurfaces of the zero solution of the initial system with impulses.     

Volterra series and geometric control theory

It is shown here that controlled differential equations which are analytic in the state and linear in the control have solutions which can be expanded in a Volterra series provided there is no finite escape time. The Volterra kernels are computed in terms of the power series expansion of the functions defining the differential equation. We also give necessary and sufficient conditions for a Volterra series to be realizable by a linear-analytic system. These conditions are particularly easy to test if the Volterra series is finite; a complete theory is worked out for this case. In the final section some applications are considered to singular control, multilinear realization theory, etc.     

Optimal Control of a System Described by a Two-Dimensional Quartic Equation with Conjugation Conditions

New problems of optimal control of distributed systems described by partial differential two-dimensional quartic equations with conjugation conditions and quadratic cost functions are considered in the paper. The theorems of existence of unique optimal solutions are proved for every considered case.     

Solving the Wiener-Hopf equation with a probabilistic kernel

Inhomogeneous Wiener-Hopf equations on a halfline with probabilistic kernels are considered. They are analyzed using the theory of isotonic and contraction operators. Necessary and sufficient conditions of existence and general sufficient conditions of uniqueness for the solutions of these equations are established. The conditions for and the rate of convergence of successive approximations of the solution are found. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 39–47, March–April 2006.     

Comparison theorems for set stability of differential equations

Set stability and uniform set stability of differential equations involve known specific bounds on solutions of the differential equations under consideration. Comparison theorems are presented giving sufficient conditions for these forms of stability. The conditions include the existence of a Liapunov-Like function and the set stability of a scalar differential equation. Examples are given to demonstrate the method.     

Theory of fractional hybrid differential equations

In this paper, we develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0<q<1. An existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions.     

On checking existence of infinitely differentiable solutions of partial differential equations with boundary conditions

The problem of checking existence of infinitely differentiable solutions of linear partial differential equations with zero boundary conditions is considered. The coefficients of the equations are assumed to be polynomials over Z in independent variables. It is proved that this problem is algorithmically undecidable. This result extends results of our earlier studies of analytic solutions. The proof relies on the result obtained by Denef and Lipshitz concerning the relationship between a certain subset of nonhomogeneous differential equations of the considered form (but without boundary conditions) and Diophantine equations.     

A Hopfield neural network with multi-compartmental activation

The Hopfield network is a form of recurrent artificial neural network. To satisfy demands of artificial neural networks and brain activity, the networks are needed to be modified in different ways. Accordingly, it is the first time that, in our paper, a Hopfield neural network with piecewise constant argument of generalized type and constant delay is considered. To insert both types of the arguments, a multi-compartmental activation function is utilized. For the analysis of the problem, we have applied the results for newly developed differential equations with piecewise constant argument of generalized type beside methods for differential equations and functional differential equations. In the paper, we obtained sufficient conditions for the existence of an equilibrium as well as its global exponential stability. The main instruments of investigation are Lyapunov functionals and linear matrix inequality method. Two examples with simulations are given to illustrate our solutions as well as global exponential stability.

     

Set stability of differential equations with time delay

Set stability and uniform set stability involve known specific bounds on the solutions of differential equations. These concepts are extended to include the case of differential equations with time delay. Liapunov-like theorems are presented which yield sufficient conditions for the set stability of such systems. Examples are given to demonstrate the method.     

Open-loop Stackelberg strategies in a linear-quadratic differential game with time delay

The open-loop Stackelberg strategies in a linear-quadratic differential game with time delay are constructed in the form of necessary and sufficient conditions. The evolution of a game with time delay is described by coupled differential equations composed of lumped and distributed parameter subsystems. Calculation of solutions is facilitated by transforming two-point boundary-value problems to terminal boundary-value ones by the introduction of Riccati-type differential equations.     

Convolution algebra representation of systems described by linear hyperbolic partial differential equations

The representation of the input-output operator in convolution algebra B(σ₀) is obtained for distributed parameter systems described by linear hyperbolic partial differential equations. Three kinds of system are considered depending on the kind of coefficient matrix corresponding to the space variable derivative, and their properties are studied. Necessary and sufficient conditions for stability are obtained in terms of factorization of the transition matrix. The obtained results allow the use of modern algebraic methods for analysis of such systems.     

The linear periodic output regulation problem

The problem of asymptotic output regulation for linear systems driven by time-varying, T-periodic exosystems is considered in this paper. Necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential Sylvester equations are derived. These conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of Francis. A general algorithm for the synthesis of an error-feedback regulator is given. For the case of minimum-phase systems, it is shown that the regulator design can be carried out without the knowledge of the Floquet decomposition of the exosystem, thus extending significantly the applicability of the general result. The more challenging issue of robust regulation by error feedback is also addressed, and solved under a stronger observability condition.     

L2‐absolute and input‐to‐state stabilities of equations with nonlinear causal mappings

Nonlinear scalar equations with causal mappings are considered. These equations include differential, difference, differential‐delay, integro‐differential and other traditional equations. Estimates for the L²‐norms of solutions are established. These estimates give us explicit conditions for the absolute and input‐to‐state stabilities of the considered equations. The Aizerman‐type problem from the theory of absolute stability is also discussed. The suggested approach enables us to consider various classes of systems from the unified point of view. Copyright © 2008 John Wiley & Sons, Ltd.     

Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order

We investigate the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative. As an application, some examples to illustrate our results are given.     

Analysis of systems of singular ordinary differential equations

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones. Translated from Kibemetika i Sistemnyi Analiz, No. 5, pp. 103–110, September–October, 1999.     

Stability analysis for the solutions of a class of nonlinear nonstationary systems with delay

This paper studies the influence of nonstationary perturbations on the systems of differential equations with delayed argument, whose right-hand sides are representable as the sums of potential and gyroscopic components of vector fields. Finally, the sufficient conditions of asymptotic stability are established for the solutions of the perturbed systems.     

Some results of the degenerate fractional differential system with delay

An analytic study on linear systems of degenerate fractional differential equations with constant coefficients is presented. We discuss the existence and uniqueness of solutions for the initial value problem of linear degenerate fractional differential systems. The exponential estimation of the degenerate fractional differential system with delay and sufficient conditions for the finite time stability for the system are obtained. Finally, an example is provided to illustrate the effectiveness of the presented analytical approaches.     

A note on boundary value problems for a coupled system of fractional differential equations

In this work, we establish sufficient conditions for the existence of solutions to a general class of multi-point boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann–Liouville sense. By means of a fixed point theorem, existence results for the solutions are established. We include an example to show the applicability of our results.     

Nonpathological Lyapunov functions and discontinuous Carathéodory systems

Differential equations with discontinuous right-hand side and solutions intended in Carathéodory sense are considered. For these equations, sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are given. An invariance principle is also proven.