Optimal Filtering Problems for Time-Delay Systems

Let (

Ω

,

F

,

P

) be a complete probability space with an increasing right-continuous family of

σ

-algebras

F

t

,

t

 ≥ 0, and let (

W

1

(

t

),

F

t

,

t

 ≥ 0) and (

W

2

(

t

),

F

t

,

t

 ≥ 0) be independent Wiener processes. The partially observed

F

t

-measurable random process (

x

(

t

),

y

(

t

)) is described by an ordinary differential equation for the dynamic system state

$$ dx(t) = (a_{0}(t)+a(t)x(t)) dt + b(t) dW_{1}(t), ~~ x(t_0)=x_0, ~~ (3.1) $$

and a differential equation with multiple delays for the observation process:

$$ dy(t) = (A_{0}(t)+A(t)x(t)+ \sum \limits _{i=1}^{p} A_i(t)x(t-h_i)) dt + B(t) dW_{2}(t), ~~ (3.2) $$

where

x

(

t

) ∈ 

R

n

is the state vector,

y

(

t

) ∈ 

R

m

is the observation process, the initial condition

$x_0\in R^{n}$

is a Gaussian vector such that

x

0

,

W

1

(

t

),

W

2

(

t

) are independent. The observation process

y

(

t

) depends on delayed states

x

(

t

 − 

h

i

),

i

 = 1,...,

p

, where

h

i

 > 0 are positive delay shifts, as well as non-delayed state

x

(

t

), which assumes that collection of information on the system state for the observation purposes is made not only at the current time but also after certain time lags

h

i

 > 0,

i

 = 1,...,

p

. The vector-valued function

a

0

(

s

) describes the effect of system inputs (controls and disturbances). It is assumed that

A

(

t

) is a nonzero matrix and

B

(

t

)

B

T

(

t

) is a positive definite matrix. All coefficients in (3.1),(3.2) are deterministic functions of appropriate dimensions.