V. Kolmanovskii

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.

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.

Optimal Control of Stochastic Delay Systems

In practical applications the behaviour of many dynamical systems depends not only on their previous history, but also on unknown disturbances.

Stability of stochastic functional differential equations

Here we will consider the Ito type SRDE