Further Results: Optimal Identification and Control Problems

Abstract

Let (Ω,F,P) be a complete probability space with an increasing right-continuous family of σ-algebras F _t ,t ≥ t₀, and let (W₁(t), F _t , t ≥ t₀) and (W₂(t), F _t , t ≥ t₀) be independent Wiener processes. The F _t -measurable random process (x(t),y(t)) is described by a linear differential equation with unknown vector parameter θ for the system state

$$ dx(t) = (a_0(\theta, t) + a(\theta , t)x(t))dt + b(t)dW_1(t), ~~ x(t_0)=x_{0}, ~~(2.1) $$

and a linear differential equation for the observation process

$$ dy(t) = (A_{0}(t)+A(t)x(t))dt+ B(t) dW_2(t). ~~ (2.2) $$

Here, x(t) ∈ Rⁿ is the state vector, y(t) ∈ Rⁿ is the linear observation vector, such that the observation matrix A(t) ∈ R^n×n is invertible, and θ(t) ∈ R^p, p ≤ n×n + n, is the vector of unknown entries of matrix a(θ, t) and unknown components of vector a₀(θ, t). The latter means that both structures contain unknown components $a_{0_i}(t)=\theta _k(t)$, k = 1,...,p₁ ≤ n and a _ij (t) = θ _k (t), k = p₁ + 1,...,p ≤ n×n + n, as well as known components $a_{0_i}(t)$ and a _ij (t), whose values are known functions of time. The initial condition $x_0\in R^{n}$ is a Gaussian vector such that x₀, W₁(t), and W₂(t) are independent. It is assumed that B(t)B^T(t) is a positive definite matrix. All coefficients in (2.1)–(2.2) are deterministic functions of time of appropriate dimensions.