Minimax Estimation of Solutions of the First Order Linear Hyperbolic Systems with Uncertain Data

In this paper, we focus on optimal estimation of solutions of the Cauchy problem for the first order linear hyperbolic equation systems (or, more generally, estimation of values of some functionals on their solutions) under incomplete data.


Introduction
In this paper, we will solve the minimax estimation problem for solutions of the first order linear hyperbolic equation system under uncertain data.
Such problems appear in the frequent situation when the right-hand sides of the equations and initial conditions are not known exactly but only satisfy certain restrictions.
To solve the estimation problems we must have supplementary data (observations) where φ is unknown solution, C is an operator that specifies the method of measuring and η is the measurement error. As a rule, this error is not known and belongs to a certain given set and the operator is not invertible. Therefore, in general, from given y, it is not possible to uniquely reconstruct the sought-for solution φ and, consequently, quantity l(φ), where l is a given linear continuous functional. We see that a natural problem arises: to determine an estimate l(φ) which would provide the best (in a certain sense) approximation to the sought-for l(φ). We assume that right-hand sides of equations and initial conditions are unknown and belong to the given bounded subsets of the space of all square integrable functions. It is supposed also that a class of noisy observations distributed on a finite system of bounded domains and observation errors (noises) are realizations of the stochastic processes, with unknown moment functions of the second order also belonging to certain given subsets.
To solve such problems with the lack of reliable information about the distribution of random perturbations, the following approach is used. We are looking for linear with respect to observations optimal estimates of solutions of the considered systems of equations from the condition of minimum of maximal mean square error of estimation taken over the above subsets. Such estimates are called minimax estimates.

Preliminaries and auxiliary results
In this paper we denote matrices by bold capital letters and vectors by bold lower case letters. Denote by H m (R n ) the set of all functions f of L 2 (R n ) whose derivatives up to the order m (in the distribution sense) belong to L 2 (R n ) and by C m ([0, T ]; E) the set of all continuously differentiable functions u(t) (t ∈ [0, T ]) with values in Hilbert space E.
Further, we will use the fact that for any f, and for any functions u, v ∈ C 1 ([0, T ]; E) the following integration by parts formula is valid: where (·, ·) E is the inner product in E (see [2,3] ). Introduce the Hilbert space L 2 (R n × (0, T )) N with the inner product and the Hilbert space L 2 (0, T ; (H s (R n )) N ) consisting of the functions of t with values in the space (H s (R n )) N and norm Consider the first order hyperbolic system ij (x, t)] (k = 1, 2, . . . , n) are symmetric matrices of order N , whose elements b ij (x, t), a (k) ij (x, t) are sufficiently smooth functions of (x, t), bounded as well as their derivatives. By Friedrichs' theorem (see [1]), for any initial data φ 0 ∈ H 1 (R n ) N and for f ∈ L 2 (0, T ; H 1 (R n ) N ) there exsists a unique solution φ of the system (1 It follows immediately from Friedrichs' theorem that for any g ∈ L 2 (0, T ; H 1 (R n ) N ) and ψ 0 (x) ∈ H 1 (R n ) N the following problem formally adjoint to (1.
and the following a priori estimate is valid (1.8)

Statement of the problem
We now assume that in (1.3) and (1.4) the vector-functions f (x, t) and φ 0 (x) are unknown and belong to the set Additionally, we suggest that Q −1 1 and Q −1 2 map the spaces L 2 (0, T ; H 1 (R n ) N ) and H 1 (R n ) N into themselves, respectively.
An estimation problem can be formulated as follows: from observations of the form linear with respect to observations. Here D i , i = 1, . . . , m, are bounded subdomains of R n , D is subdomain which may coincide with R n , ) T are unknown random fields whose choice functions enter in observations (2.2) (hereafter the upper index T denotes the transpose of a vector or matrix).
We suggest that where by G 1 we denote the set of elements R n and zero expectations, satisfying the following conditions: (i) random fieldsη i (x, t) are pairwise uncorrelated, that is, mutually correlative matrices of fieldsη i (x, t) and are assumed to be linear bounded operators acting from . . , m. An example of such an operator is constructed in Appendix. Introduce Definition. The estimate in which vector-functionsû i (x, t) and a numberĉ are determined from the condition will be called the minimax estimate of expression l(φ).
The quantity will be called the error of the minimax estimation of (2.3).
Thus, the minimax estimate is an estimate minimizing the maximal mean-square estimation error calculated for the "worst" implementation of perturbations.
Note that the integrals in (2.8) and (2.10) which are understood as Lebesgue integrals exist with probability 1.

For every fixed
as a solution to the following initial value problem: where χ M (x), is a characteristic function of the set M. The function z(x, t; u) is uniquely determined from equations (3.1) -(3.2). In fact, under our assumptions, the right-hand side of (3.1) equation belongs to Lemma. Finding a minimax estimate of the value of functional l(φ) is equivalent to the problem of optimal control of the system of integro-differential equation (3

.1)-(3.2) with the cost function
By Fubini's theorem we have, recalling thatη i (x, t), i = 1, . . . , m, are vector processes with zero expectations, The latter equality and (3.4) yield From the equalities (3.4) and (3.5) we find, taking into consideration the known relationship Dξ = Eξ 2 − (Eξ) 2 , that relates the variance Dξ = E|ξ − Eξ| 2 of random variable ξ to its expectation Eξ, in which ξ is determined by right-hand side of (3.4), Transform the first term in the right-hand side of (3.6). We get (see the explanations below) The first equality is valid due to the equation (3.1). The second, the third, and the fifth equalities are apparent. The fourth one holds as a result of applying the integration by parts formula (1.2) to the term il (·, t)z l (·, t; u) ∈ H 1 (R n ), g =φ i (·, t) ∈ H 1 (R n ) for any fixed t ∈ (0, T ). Finally, the sixth equality holds sinceφ(x, t) is the solution of Cauchy problem (1.3)- Using relationships (3.6) and (3.7), we have From here, applying the generalized Schwarz's inequality (see, for example, [14]) together with inequalities (2.1) and (2.6), we find inf where functional I(u) is determined according to (3.3) and the infimum with respect to c is attained at The lemma is proved.
Further in the proof of theorem stated below, it will be shown that solving the optimal control problem (3.1)−(3.3) is reduced to solving some system of integro-differential equations.
Theorem 3.1. The minimax estimate of l(φ) can be represented as (3.9) and functionsẑ, p Problem (3.10) -(3.13) is uniquely solvable. The error σ of the minimax estimation of l(φ) is given by the formula Proof Taking into account estimate (1.8) one can easily verify that the funtional I(u) is a strictly convex lower semicontinuous functional on H and since Then, by Remark 1.2 to Theorem 1.1 (see [4]), there exists one and only one elementû = (û 1 , . . . ,û m ) ∈ H such that I(û) = inf u∈H I(u). Hence, for any fixed w ∈ H and τ ∈ R the function s(τ ) := I(û + τ w) reaches its minimum at a unique point τ = 0, so that d dτ Since z(x, t;û + τ w) = z(x, t;û) + τz(x, t; w), wherez(x, t; w) is the unique solution to Cauchy problem (3.1)-(3.2) at u = w and l 0 (x, t) ≡ 0, from (3.3) and (3.15) we obtain 17) † Here we use the following notation. If A(ξ) = [a ij (ξ)] N i,j=1 is a matrix depending on a variable ξ belonging to a measurable set Ω, then we define ∫ Ω A(ξ) dξ by ∫ Then the sum of the first two terms on the right-hand side of (3.16) can be written in the form The last equality can be rewritten as Substituting the latter into (3.1), (3.2) and (3.8) and denotingẑ(x, t) = z(x, t;û), we see thatẑ(x, t) and p(x, t) satisfy system (3.10) -(3.13); the unique solvability of this system follows from the uniqueness of elementû of the functional (3.3). Now let us establish that σ = [l(p)] 1/2 . Substituting expressions (3.9) to (3.3), we obtain Transform the sum of the first and the second summands in the right-hand side of (3.22). Using relationships (3.9) -(3.13), we have Equality (3.14) follows now from two relationships (3.23) and (3.22).
An alternative representation for the minimax estimate of l(φ) in terms of the solution to a system of integrodifferential equations is given in the next theorem. This solution is independent of the specific form of functional (2.3).

Proof
The proof of this theorem is similar to the proof of Theorem 3.1. Remark 2. Using the decoupling technique by J.-L. Lions [4] solving each of the systems of integro-differential equations (3.10) -(3.13) and (3.25)-(3.28) can be reduced to solving a nonlinear operator Riccati equation with the condition at t = T and a first order linear hyperbolic system with the initial conditions at t = 0.

Conclusions
In this paper, the minimax estimation problem for solutions of the first order linear hyperbolic equation system with uncertain data is solved. It was shown that the finding a minimax estimate is equivalent to the corresponding optimal control problem for the system of integro-differential equations. The minimax estimate and the alternative representation for this estimate from the solution to the system of integro-differential equations are obtained. An example of observation operator is constructed in Appendix.