A. Myshkis

Introduction to the Theory and Applications of Functional Differential Equations

Part I: Modelling by Functional Differential Equations. 1. Theoretical Preliminaries. 2. Models. Part II: Theoretical Background of Functional Differential Equations. 3. General Theory. Part III: Stability. 4. Stability of Retarded Differential Equations. 5. Stability of RDEs with Autonomous Linear Part. 6. Liapunov Functionals for Concrete FDEs. 7. Riccati Type Stability Conditions of Some Linear Systems with Delay. 8. Stability of Neutral Type Functional Differential Equations. 9. Applications of the Direct Liapunov Method. 10. Stability of Stochastic Functional Differential Equations. Part IV: Boundary Value Problems and Periodic Solutions of Differential Equations. 11. Boundary Value Problems for Functional Differential Equations. 12. Fredholm Alternative for Periodic Solutions of Linear FDEs. 13. Generalized Periodic Solutions of Functional Differential Equations. Part V: Control and Estimation in Hereditary Systems. 14. Problems of Control for Deterministic FDEs. 15. Optimal Control of Stochastic Delay Systems. 16. State Estimates of Stochastic Systems with Delay. Bibliography. Index.

Applied theory of functional differential equations

Preface. 1. Models. 2. General Theory. 3. Stability of Retarded Differential Equations. 4. Stability of Neutral Type Functional Differential Equations. 5. Stability of Stochastic Functional Differential Equations. 6. Problems of Control for Deterministic FEDs. 7. Optimal Control of Stochastic Delay Systems. 8. State Estimates of Stochastic Systems with Delay. Bibliography. Index.

On some unsolved problems of zero-gravity hydromechanics

IN JANUARY 1978 we gave a number of lectures on zero-gravity hydromechanics at the XII Voronezh School of Mathematics. The lectures were mainly based on our monograph [l]. We tried to present to our listeners the problems already solved in [l] and to formulate problems associated with the material of the book and awaiting closer attention. Thereby we wanted to draw the attention of mathematicians to a large and important group of problems of both purely mathematical and applied interest, which have their own specific features and require the application and development of functional analysis methods as well as various approximate methods. The aim of the present paper is to formulate some of these unresolved problems and in some cases to point out possible ways of investigation.

Fredholm Alternative for Periodic Solutions of Linear FDEs

This chapter is devoted to the conditions of the existence of periodic solutions of linear FDEs and some their properties. Particular case of this problem for RDEs was considered in Subs. 11.1.4 in connection with the Halanay boundary value problem. Here we shall investigate in details the more general case.

Liapunov Functionals for Concrete FDEs

In this chapter the procedure to construct Liapunov functionals for some FDEs is proposed and stability conditions are obtained.

Stability of RDEs with Autonomous Linear Part

General methods for stability analysis, described previously, can be essentially modified for particular clases of FDEs. One of these classes is considered in this chapter. This class consist of nonlinear RDEs with autonomous linear parts. Explicit stability conditions are proposed. They are formulated in terms of the roots of the characteristics polynomials and based on an estimate for the matrix resolvent.

Riccati Type Stability Conditions of Some Linear Systems with Delay

Direct Liapunov method represents a powerful tool for determining systems stability. Along with the general theorems it allows for linear equations to formulate stability conditions in terms of existence of the positive definite solutions of some auxiliary matrix equations.

Boundary Value Problems for Functional Differential Equations

Boundary value problems for FDEs can be roughly subdivided into problems for equations of evolutionary and nonevolutionary types. In the first case the independent variables t is interpreted as a time and the problem keeps features of evolution of some process. In the second case considered in the next section such direct connection is absent. However this subdivision is not precise because there are problems which can be related to the both types.

Problems of control for deterministic FDEs

Many papers have been devoted to control problems for FDEs (see, e.g., [13, 46, 111, 292.8, 345, 395]). Statements of these problems can be obtained from those of for ODEs with natural modifications induced by the infinite dimensional structure of FDEs. We can mention, e.g., optimal control and estimation, controllability, observability, stabilization, control with incomplete state information, large scale systems, adaptive control and identification, etc.

State estimates of stochastic systems with delay

In this chapter we will consider filtering problems for Gaussian unobservable processes and linear observations with delays. We will investigate the dependence of optimal estimates on the delays. We use a least-squares method to estimate the coordinates of the system at each time moment, based on input data and corresponding measurements with noise.