Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data ()
1. Introduction
The theory of finding estimates of solutions to stochastic differential equations has been intensively developing since the classical works of Kalman and Bucy [1] [2] . This theory found numerous applications in the treatment of the results of experiments in physics, biology, medicine, and many other areas of science and technology. Such successful and broad applications are explained by the fact that Kalman-Bucy methods provide differential equations for optimal mean square estimates which can be solved numerically in the real-time mode. At the same time, it should be noted that Krasovskii and Kurzhanskii proposed in [3] [4] an alternative approach to estimating the solutions of differential equations where perturbations and inaccuracies of additional data about solution were not known and the only thing given was that they belong to a certain domain.
Let us formulate a general approach to the problem of minimax estimation. If a state of a system is described by a linear ordinary differential equation
and a function 
is observed in a time interval
, where 
, 
and 
are known matrices, the minimax estimation problem consists in the most accurate determination of a function 
at the “worst” realization of unknown quantities 
taken from a certain set. N.N. Krasovskii was the first who stated this problem in [3] . Under different constraints imposed on function 
and for known function 
he proposed various methods of estimating inner products
, where 
in the class of operations linear with respect to observations that minimize the maximal error. Later these estimates were called minimax a priori or guaranteed estimates (see [3] [4] ).
This theory was further developed in the works of Chernous’ko, Pshenichnyi, Kuntzevich, Nakonechnyi, Kirichenko, Podilipenko, and their disciples; one may refer e.g. to [4] -[10] and the bibliography therein.
We note that the duality principle elaborated in [3] [4] , and [5] proved its efficiency for the determination of minimax estimates [5] . According to this principle, finding minimax a priori estimates can be reduced to a certain problem of optimal control of a system; this approach enabled one to obtain, under certain restrictions, recurrent equations, namely, the minimax Kalman-Bucy filter (see [5] ).
The essential results within the frames of 
-theory are obtained in [11] .
In this paper, we study estimation of solutions of boundary value problems (BVPs) for ordinary differential equations at fixed points of interval from additional data about their solutions. Such settings may be considered as inverse problems when additional data are given with errors. We assume that these errors are random with unknown correlation functions. Similar problems arise in data processing of observations of the objects or processes described by BVPs for ordinary differential equations with unknown perturbations of the right-hand sides or boundary conditions. We solve the estimation problems using guaranteed linear estimates that minimize maximal mean square estimation errors. It is shown that optimal guaranteed estimates are expressed via solutions to special BVPs for ordinary differential equations.
2. Preliminaries and Auxiliary Results
Assume that it is given a vector-function 
with the components belonging to space 
and vectors 
and
. Consider the following BVP: find a vector-function 
that satisfies a system of linear first-order ordinary differential equations
(0.1)
almost everywhere on an interval 
and the boundary conditions
(0.2)
at the points 0 and
. Here 
is an 
matrix with the entries 
continuous on
;
and 
are 
and 
matrices of rank 
and
, respectively; upper index 
denotes transposition of a matrix or a vector and the upper bar throughout the whole text of the paper that e.g. index 
takes all values from 1 to
; 
is a space of functions absolutely continuous on 
for which the derivative that exists almost everywhere on 
belongs to space 
and
The problem of finding a function 
that satisfies on 
the equation
(0.3)
and the boundary conditions
(0.4)
will be called the homogeneous BVP corresponding to BVP (0.1), (0.2).
The solution 
to homogeneous BVP (0.3), (0.4) is called the trivial solution.
BVP (0.1), (0.2) can be written in a scalar form:
(0.5)
(0.6)
Let
(0.7)
be a fundamental system of solutions to (0.3) (for the definition, see e.g. [12] p. 179). Then the solutions to (0.3), (0.4) have the form
where, by virtue of (0.4), constants 
must be such that
(0.8)
Thus, if the matrix
(0.9)
has rank
, the homogeneous BVP has only the trivial solution. The inverse statement is also valid: if the homogeneous BVP has only the trivial solution then the rank of matrix (0.9) equals 
(see, for example [13] ).
Assume in what follows that homogeneous BVP (0.3), (0.4) corresponding to BVP (0.1), (0.2) has only the trivial solution or what is the same that matrix (0.9) has rank 
As is known [13] , under this assumption, initial BVP (0.1), (0.2) is uniquely solvable at any right-hand sides 
, and 
Formulate the notion of a BVP conjugate to (0.1), (0.2). To this end, introduce the following designations: 
is the 
unit matrix; 
is the 
null matrix; 
is a square nondegenerate 
submatrix of the matrix 
; 
is an 
submatrix of 
obtained as a result of deleting in 
all columns of matrix 
(so that
); 
is an 
matrix such that its 
th column equals 
th column of matrix 
(its size is
), 
and 
th column equals lth column of matrix 
is an 
matrix such that its 
th column equals k-th column of matrix 
and 
th column equals lth column of matrix 
is an 
matrix such that its 
th column equals kth column of matrix 
and 
th column equals lth column of matrix 
Introduce more similar notations: 
is a square nondegenerate 
submatrix of the matrix 
is a 
submatrix of the matrix 
obtained as a result of deleting in 
all columns of matrix 
(so that
); 
is an 
matrix such that its 
th column equals kth column of matrix 
(the size of the latter is
), 
and 
th column equals lth column of matrix 
is an 
matrix such that its 
th column equals kth column of matrix 
and 
th column equals lth column of matrix 
is an 
matrix such that its 
th column equals kth column of matrix 
and 
th column equals lth column of matrix 
By
we will denote the inner product of vectors 
and 
in the Euclidean space 
Then, if 
we have
(0.10)
where the differential operator
will be called formally conjugate to operator 
Let us show that the term 
in (0.10) can be represented as
(0.11)
Note first that
where
Then 
and
where 
and 
are vectors composed of components of vector 
with the numbers equal to the numbers of components of vectors 
and
, respectively. Taking into account that
we have
Analogously
These two equalities yield representation (0.11); using the latter and (0.10), we obtain
(0.12)
In order to write the sum of the first four terms on the right-hand side of (0.12) in a scalar form, introduce the following notations:
(0.13)
(0.14)
(0.15)
(0.16)
(0.17)
(0.18)
Then the Equality (0.12) can be written as
(0.19)
Now we can introduce the notion of the adjoint BVP.
Definition 2.1 The homogeneous BVP
(0.20)
(0.21)
is called adjoint to homogeneous BVP (0.3), (0.4).
Definition 2.2 The inhomogeneous BVP
(0.22)
(0.23)
where 
is called adjoint to inhomogeneous BVP (0.1), (0.2).
The results contained in [13] [14] imply that the following statement is valid (for more detailed explanations see [9] , pp. 9-11).
Theorem 2.1 If homogeneous BVP (0.3), (0.4) has only the trivial solution, then the corresponding adjoint BVP (0.20), (0.21) also has only the trivial solution and inhomogeneous BVP (0.22), (0.23) has one and only one solution 
at any 
.
3. Statement of the Minimax Estimation Problem and Its Reduction to an Optimal Control Problem
Let a vector-function
(0.24)
with the values from the space 
be observed on an interval
; here 
is an 
matrix with the entries that are continuous functions on 
and 
is an unknown random vector process whose realizations enter observations (0.24).
Denote by 
the set of random vector processes 
with zero expectation 
and second moments 
integrable on 
such that their correlation matrices 
satisfy the inequality
(0.25)
where 
is a positive definite matrix of dimension 
the entries of 
and 
are continuous on 
is a given positive number, 
denotes the trace of the matrix
.
Set
(0.26)
where 
are given vectors; 
is a given vector-function;
, and 
are positive definite matrices of dimensions 
, and
, respectively, the entries of 
and 
are continuous on
, 
is a given positive number.
Assume that the right-hand sides 
and 
of Equation (0.1) and boundary conditions (0.2) are not known exactly and it is known only that the element 
belongs to a set 
and, additionally, 
Further we also will assume, without loss of generality, that in (0.25) and (0.26) 
Let a vector-function 
be a solution to BVP (0.1), (0.2).
We will look for an estimation of the inner product
(0.27)
in the class of estimates linear with respect to observations that have the form
(0.28)
where 
1 and 
is a vector belonging to 
, 
, 
, and 
is certain constant. Then 
Definition 3.1 An estimate
for which vector-function 
and constant 
are determined from the condition
where
is a solution to BVP (0.1), (0.2) at 
and
will be called a minimax mean square estimate of inner product 
The quantity
will be called an error of the minimax estimation.
We see that the minimax mean square estimate of inner product 
is an estimate at which the maximum mean square estimation error calculated for the worst realization of perturbations attains its minimum.
We will show that solution to the minimax estimation problem is reduced to the solution of a certain optimal control problem.
For every fixed 
introduce vector-functions 
and 
as a solution to the following BVP:
(0.29)
Lemma 3.1 Determination of the minimax mean square estimate of inner product 
is equivalent to the problem of optimal control of the system described by BVP (0.29) with the cost function
(0.30)
Proof. Show first that BVP (0.29) is uniquely solvable under the condition that functions 
and 
belong, respectively, to the spaces 
and 
Since homogeneous BVP (0.3), (0.4) has only the trivial solution, the BVP
(0.31)
has, in line with Theorem 2.1, the unique solution for any right-hand side, in particular, at
(0.32)
Denote this solution by 
and its restrictions on intervals 
, and 
by 
and
, respectively. Note that function 
is absolutely continuous on 
(see [15] ).
Let us show that the problem
(0.33)
has one and only one solution at any vector 
Denote by 
the coordinates of vector-function 
Let 
be the fundamental system of solutions of the equation system 
on 
Then we can represent functions 
in the form
where 
are constants. Taking into account the boundary conditions at the points 
and transmission conditions at 
in (0.33), we see that the solution to BVP (0.33) is equivalent to the solution of the following linear equation system with 
unknowns 
(0.34)
(0.35)
(0.36)
(0.37)
(0.38)
where
denote the coordinates of vector 
and 
and 
denote the entries of matrices 
and
, respectively.
Show that system (0.34)-(0.38) is uniquely solvable at any vector 
To this end, note that homogeneous system (0.34)-(0.38) (at
) has only the trivial solution.
Indeed, setting 
in Equations (0.35) and (0.36), taking into account (0.37) and the fact that
, and 
because 
is the fundamental system of solutions of the equation system 
on 
we obtain
Coefficients 
satisfy Equations (0.34) and (0.38); therefore vector-function 
with the components 
is a solution to conjugate BVP (0.20), (0.21) which has only the trivial solution 
on 
by Theorem 2.1. This implies
, so the homogeneous linear equation system (0.34)- (0.38) (at
) has only the trivial solution. Consequently, system (0.34)-(0.38) and therefore BVP (0.33) which is equivalent to this system are uniquely solvable at any vector 
Then vector-functions 
form the unique solution to BVP (0.29).
Show next that the determination of the minimax estimate of inner product 
is equivalent to the problem of optimal control of the system described by BVP (0.29) with the cost function (0.30).
Using the second and third equations in (0.29) and the fact that 
is a solution to BVP (0.1), (0.2) at 
and 
we easily obtain the relationships
Taking into account the equalities
and that
(we refer to the reasoning on p. 4) we obtain
Taking into notice that
and therefore
we use the last equality to obtain
(0.39)
where
Recalling that 
is a vector process with zero expectation, we use condition (0.25) and the known relationship 
that couples the dispersion 
of random variable 
with its expectation 
to obtain
which yields
(0.40)
In order to calculate the supremum on the right-hand side of (0.40) we apply the generalized CauchyBunyakovsky inequality [16] and write this inequality in the form convenient for further analysis: for any
, the generalized Cauchy-Bunyakovsky inequality holds
in which the equality is attained at
Setting in the generalized Cauchy-Bunyakovsky inequality
and denoting
we obtain, in line with (2.7), the inequality
where the equality is attained at
Thus,
(0.41)
at 
Calculate the second term on the right-hand side of (0.40). Applying the generalized Cauchy-Bunyakovsky inequality, we have
(0.42)
Here 
can be placed under the integral sign according to the Fubini theorem because we assume that 
is a random process of the integrable second moment. Transform the last factor on the right-hand side of (0.42):
Taking into account that (0.25) holds, we see that (0.42) yields
(0.43)
It is not difficult to check that here, the equality sign is attained at the element
where 
is a random variable such that 
and 
We conclude that statement of the lemma follows now from (0.40), (0.41), and (0.43).
4. Representations for Minimax Mean Square Estimates of Functionals from Solutions to Two-Point Boundary Value Problems and Estimation Errors
In this section we prove the theorem concerning general form of minimax mean square estimates. Solving optimal control problem (0.29), (0.30), we arrive at the following result.
Theorem 4.1 The minimax mean square estimate of expression 
has the form
where
(0.44)
(0.45)
and vector-functions 
and
, 
are determined from the solution to the problem
(0.46)
Here 
and 
The minimax estimation error
(0.47)
Problem (0.46) is uniquely solvable.
Proof. We will solve optimal control problem (0.29), (0.30). Represent solutions 
of problem (0.29) as 
where 
and 
denote the solutions to this problem at 
and 
respectively. Then function (0.30) can be represented in the form
where
Since solution 
of BVP (0.31) is continuous2 with respect to right-hand side 
defined by
(0.32), the function 
is a linear bounded operator mapping the space 
to
Thus, 
is a continuous quadratic form corresponding to a symmetric continuous bilinear form
is a linear continuous functional defined on 
and 
is a constant independent of
. We have
Using Theorem 1.1 from [17] , we conclude that there is one and only one element 
such that
Therefore
Taking into consideration the latter equality, (0.30), and designations on p. 11, we obtain
(0.48)
Introduce functions 
and 
as a unique solution to the following problem3:
(0.49)
Now transform the sum of the last for terms on the right-hand side of (0.48) taking into notice that 
and 
We have
(0.50)
From Equalities (0.48)-(0.50) it follows that
so that
(0.51)
Functions 
and 
are absolutely continuous on segments 
, and
, respectively, as solutions to BVP (0.49); therefore, functions 
and 
that perform optimal control are continuous on 
and 
Replacing in (0.29) functions 
and 
by 
and 
defined by formulas (0.51) and denoting 
we arrive at problem
(0.46) and equalities (0.44).
Taking into consideration the way this problem was formulated we can state that its unique solvability follows from the fact that functional (0.30) has one minimum point
.
Now let us prove representation (0.47). Substituting into formula 
expressions (0.44) for 
and 
we have
(0.52)
Next, we can apply the reasoning similar to that on p. 4 and use (0.46) to obtain
which yields
(0.53)
In a similar manner, using the equality
we obtain
(0.54)
Relationships (0.52)-(0.54) yield
which is to be proved.
It is easy to see that function 
defined by (0.45) and the function
(0.55)
satisfy the following uniquely solvable BVP
(0.56)
where
and 
is the characteristic function of interval
.
Now Theorem 4.1 can be restated as follows:
Theorem 4.1' The minimax estimate of expression 
has the form
where
(0.57)
and vector-functions 
and 
are determined from the solution to problem 0.56.
Obtain now another representation for the minimax mean square estimate of quantity 
which is independent of 
and
. To this end, introduce vector-functions 
and 
as solutions to the problem
(0.58)
at realizations 
that belong with probability 1 to space 
Note that unique solvability of problem (0.58) at every realization can be proved similarly to the case of (0.46). Namely, one can show that solutions to the problem of optimal control of the system
with the cost function
can be reduced to the solution of problem (0.58) where the optimal control 
is expressed in terms of the solution to this problem as 
; the unique solvability of the problem follows from the existence of the unique minimum point 
of functional
.
Considering system (0.58) at realizations 
it is easy to see that its solution is continuous with respect to the right-hand side. This property enables us to conclude, using the general theory of linear continuous transformations of random processes, that the functions 
considered as random fields have finite second moments.
Theorem 4.2 The following representation is valid
Proof. By virtue of (0.44) and (0.58),
(0.59)
Next,
(0.60)
Similarly,
(0.61)
From (0.59), (0.60), and (0.61) it follows
(0.62)
However,
(0.63)
(0.64)
Subtracting from (0.63) equality (0.64), we obtain
or
(0.65)
Since
we can use the latter equalities, (0.65), and the fact that 
is a symmetric matrix to obtain
(0.66)
Performing a similar analysis, one can prove that
(0.67)
From (0.66), (0.67), and (0.62) and the expression for 
it follows
The theorem is proved.
As is easily seen from (0.58), the functions 
and 
defined by
and
satisfy the following uniquely solvable BVP:
(0.68)
at realizations 
that belong with probability 1 to space
.
Thus, Theorem 4.2 can be restated in the following form.
Theorem 4.2' The minimax mean square estimate of expression 
has the form
where vector-function 
is determined from the solution to problem (0.68).
Remark. Function 
can be taken as a good, in certain sense, estimate of solution 
of initial BVP (0.1), (0.2) on 
As an example, consider the case when a vector-function 
is observed on an interval
, where a vector-function 
with values in 
is a solution to the BVP
(0.69)
and operator 
is defined by the relation
where 
is a positive definite 
-matrix whose entries are continuous functions on 
Note that this problem has the unique classical solution if 
is continuous on 
and the unique generalized solution if 
Assume that, as well as in the previous case, 
is an 
matrix with the entries that are continuous functions on 
and 
is a random vector process with zero expectation 
and unknown 
correlation matrix
. Assume also that domains 
and 
are given in the form (0.25) and (0.26) where matrices 
, and 
entering (0.26) have dimensions 
, and 
Write Equation (0.69) as a first-order system by setting 
and introducing a vector-function
with 
components, a vector 
with 
components, a 
-matrix
matrices 
and
. Then system (0.69) can be written as
(0.70)
(0.71)
Applying Theorems 1 and 2 and performing necessary transformations in the resulting equations that are similar to (0.46) and (0.58) (in terms of the designations introduced above) we prove the following Theorem 4.3 The minimax mean square estimate of expression 
has the form
where
and vector-functions
, 
and 
are determined from the solutions to the problems
and
respectively.
5. Minimax Mean Square Estimates of Solutions Subject to Incomplete Restrictions on Unknown Parameters
Assume again that observations have form (0.24) and unknown parameters
, 
, and 
belong to the domain
(0.72)
where 
is given in (0.26). The correlation function of process 
belongs to domain (0.25). Introduce the set
(0.73)
here 
where 
is the solution to BVP (0.29).
Lemma 5.1
where
(0.74)
This lemma can be proved using formula (0.39).
Lemma 5.2 
is a convex closed set in the space
.
Proof. The convexity of set 
is obvious. Let us prove that this set is closed.
Note that functions 
and 
can be represented as
(0.75)
where 
and 
are known matrix functions with the elements from 
and 
and 
are vectors. Expression (0.75) can be obtained if we introduce a vector 
such that 
Then 
where 
is a solution to the equation
and 
is the unit matrix. Next,
Since BVP (0.29) is uniquely solvable, there exists one and only one vector 
satisfying the system of algebraic equations
Solving this system we determine 
in the form
where 
is a known matrix function continuous on 
and 
is a known vector. Taking into account this equality, we obtain expressions (0.75). From these relationships, it follows that if a sequence 
converges in 
to a function 
then
which proves that 
is a closed set.
Assume now that 
is nonempty. Then the following statement is valid.
Theorem 5.1 There exists the unique minimax mean square estimate of expression 
which can be represented in the form (0.44) at 
where vector-functions 
and 
solve the equations
(0.76)
Proof. Similarly to Theorem 4.1 one can show that for 
the following equality holds
where
and 
are solutions to Equations (0.29) at 
and 
is a strictly convex lower semicontinuous functional on a closed convex set 
and 
Therefore there exists one and only one vector 
such that 
This vector can be determined from the relationship
where
and 
are Lagrange multipliers.
Further analysis is similar to the proof of Theorem 4.1. Let vector-functions 
and
, 
be solutions to the system
(0.77)
Theorem 5.2 Assume that for any vector 
set 
is nonempty. Then system (0.77) is uniquely solvable and the equality
holds Proof. Introduce functions
, 
as a solution to the BVP
where 
Define a set
Since 
is nonempty, the same is valid for 
for any vector
. Similarly to the case of 
one can show that 
is a convex closed set. Denote by 
the functional of the form
One can show, following Theorem 5.1, that on set 
there is one and only one point of minimum of functional 
namely,
where functions 
and 
are determined from system (0.77). The proof of the equality
is similar to that in Theorem 4.2.
6. Conclusions
For a system described by a one-dimensional two-point BVP with decoupling boundary conditions at the endpoints of the interval and quadratic restrictions imposed on the unknown deterministic data and the second moments of observation noise, we have obtained guaranteed mean square estimates of inner product 
where 
is the unknown solution of the BVP at a point 
and
. Guaranteed estimates are obtained using the duality of the problems of estimation and optimal control. We have shown that guaranteed mean square estimates and estimation errors are expressed via solutions to special optimal control problems for conjugate BVPs. The solutions to these optimal control problems enable one to find explicit expressions for estimates and estimation errors both for distributed and point observations.
The obtained results are applied to minimax estimation of solutions of two-point BVPs for linear ordinary second-order differential equations.
Methods and results of the paper may be used for estimation under uncertainties of the states of the systems described by more general linear BVPs for different classes of functional--differential equations; in particular, for systems of differential equations with impulse perturbations, differential equations with multipoint conditions, and in several other cases.
7. Results of Numerical Experiments
Let realizations of the random variables
(0.78)
be observed. Here 
are independent random variables for which
; 
is a solution of the BVP
(0.79)
(0.80)
and
are eigenfunctions of the operator
, where 
.
We assume that function 
is not known exactly and is chosen arbitrarily from the set
, where 
is a certain constant.
Applying the technique similar to the estimation methods developed in Section 4, it is possible to obtain expressions for the minimax mean square estimates in the case when observations have the form (0.78). In particular, in this case the function 
which approximates the solution 
of BVP (0.79)-(0.80) on the interval 
is determined from the system
and can be represented in the form
The exact solution 
and its estimate 
(bold curves) calculated on the basis of the modelled observations are presented in Figure 1 and Figure 2. Calculations are performed at
, 
, and 
for 
using the parameters 
and 
(Figure 1) and 
and 
(Figure 2).
As can be seen from these figures, parameter 
plays the crucial role as far as the estimation quality is concerned. In fact, this parameter directly influences the signal-to-noise ratio.
The calculations were performed using Wolfram Mathematica.
NOTES
*Corresponding author.
1If 
then the minimax estimation problem can be solved in a similar manner but somewhat simpler.
2This continuous dependence follows from the representation of function 
in terms of Green's matrix 
of BVP (0.31) (see , p. 115):
3The unique solvability of problem (0.49) can be proved similarly to problem (0.29).