Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data

Abstract

In this paper we construct optimal, in certain sense, estimates of values of linear functionals on solutions to two-point boundary value problems (BVPs) for systems of linear first-order ordinary differential equations from observations which are linear transformations of the same solutions perturbed by additive random noises. It is assumed here that right-hand sides of equations and boundary data as well as statistical characteristics of random noises in observations are not known and belong to certain given sets in corresponding functional spaces. This leads to the necessity of introducing minimax statement of an estimation problem when optimal estimates are defined as linear, with respect to observations, estimates for which the maximum of mean square error of estimation taken over the above-mentioned sets attains minimal value. Such estimates are called minimax mean square or guaranteed estimates. We establish that the minimax mean square estimates are expressed via solutions of some systems of differential equations of special type and determine estimation errors.

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Shestopalov, Y. , Podlipenko, Y. and Nakonechnyi, O. (2014) Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data. Advances in Pure Mathematics, 4, 118-146. doi: 10.4236/apm.2014.44019.

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

Figure 1. Exact solution and its estimate (bold curve) calculated at q = 1, w = 1, N = 5, , and.

Figure 2. Exact solution and its estimate (bold curve) calculated at q = 1, w = 1, N = 5, , and.

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).

Conflicts of Interest

The authors declare no conflicts of interest.

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