ASYMPTOTIC EXPANSION OF A SOLUTION FOR ONE SINGULARLY PERTURBED OPTIMAL CONTROL PROBLEM IN R n WITH A CONVEX INTEGRAL QUALITY INDEX

The paper deals with the problem of optimal control with a convex integral quality index for a linear steady-state control system in the class of piecewise continuous controls with a smooth control constraints. In a general case, for solving such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. In this work, we deduce an equation to which an initial vector of the conjugate system satisfies. Then, this equation is extended to the optimal control problem with the convex integral quality index for a linear system with a fast and slow variables. It is shown that the solution of the corresponding equation as ε → 0 tends to the solution of an equation corresponding to the limit problem. The results received are applied to study of the problem which describes the motion of a material point in Rn for a fixed period of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion.


Introduction
The paper is devoted to studying the asymptotics of the initial vector of a conjugated state and an optimal value of the quality index in the optimal control problem [1]- [3] for a linear system with a fast and slow variables (see review [4]), convex integral quality index [3,Chapter 3], and smooth geometrical constraints for control.
Singularly perturbed problems of optimal control have been considered in different settings in [5]- [7].
The method of boundary function that was developed in [4,10] allows effectively constructing an asymptotics of solutions for problems with an open control area and smooth controlling actions.
The solving of problems with a closed and bounded control area meets certain difficulties.That is why the problems with fast and slow variables and closed constraints for control have been studied to a less extent.A significant contribution to solving these problems was made by Dontchev and Kokotovic.Problems of fast operation and terminal control with constraints for control in the form of a polygon are dealt with in [5,7].The structure of such optimal control is a relay function with values in the apexes of the polygon.No optimal control with constraints in the form of a sphere, which is a continuous function with a finite and countable number of discontinuity points, has been considered so far.
The asymptotics of solutions of the perturbed control problem was formulated differently in papers [7,9].
In the present work, the basic equation for searching for the asymptotics of the initial vector of the conjugated state of the problem under consideration and optimal control is obtained.General relationships are applied to the case of the optimal control with a point of a small mass in an n-dimensional space under the action of a bounded force.

General statement of problem and condition for optimality
Let us consider a problem that belongs to the class of piecewise continuous controls -optimal control problem for a linear stationary system with a convex integral quality index: where Note that in the considered convex integral quality index J, where the first term can be interpreted as a fine for the control error at a finite time instant T , whereas the second, as an account of an energy spent for the realization of control.
Under the conditions stated, the Pontryagin maximum principle in the problem (1.1) is the necessary and sufficient criterion of optimality.In this case, the problem has the unique solution [3, p. 3.5, Theorem 14]: if z, η is the unique solution to (1.1) and then the optimal control u o is determined from the maximum principle Here •, • is the scalar product in R r .Calculating maximum in (1.3), we find , where S(ξ) := 2, 0 ξ 2, ξ, ξ > 2. (1.4) Note that the determination of function S(•) leads to the validity of inequality (1.5) Let λ := η(T ).Then At a finite time instant t = T we have Replacing the variable τ := T − s, we obtain Thus, the following is valid: Let condition 1 be valid, z(t), u(t) be a solution of the system from Problem (1.1), and η(t) be a solution of the system (1.2).Then z(t), η(t), u(t) is the solution of the maximum principle problem (1.1), (1.2), (1.3) if and only if when η(T ) = λ, u(t) is determined by the formula (1.4), and a vector λ is the unique solution of equation Besides u(t) is the unique optimal control in the problem (1.1).
The vector λ that satisfies the equation (1.6) will be called as a vector determining the optimal control in the problem (1.1).
Statement 2. Let u o (t) be the optimal control in (1.1).Then u o (t) is continuous on [0; T ] and infinitely differentiable at points t such that B * e A * (T − t ) λ = 2.Here λ is a vector determining the optimal control in problem (1.1).P r o o f.The validity of statement follows from (1.4) and analytical form of the matrix exponent e A * t .

Optimal control problem with fast and slow variables
Consider a particular case of problem (1.1), when the system under control contains fast and slow variables and the terminal part of the quality index depends only on slow variables: where are the constant matrices of the corresponding dimensions, and σ(•) is the convex function that is continuously differentiable in R n .
Condition 2. All eigenvalues of matrix A 22 have negative real parts.
Note that since ∇ϕ(z ε ) = ∇σ(x ε ) 0 , then the vector λ ε , which determines the optimal control in the problem (2.1), has the form The vector l ε also will be called as determining the optimal control in problem (2.1).Let then, by virtue of (2.3) the equation (1.6) transforms into (2.4) Note that the optimal control u o ε (t) in the problem (2.1) is expressed through the vector l ε as follows: Theorem 1.Let the Conditions 2 and 3 be valid.Then l ε → l 0 as ε → +0, where l ε is the unique solution of the equation (2.4), and l 0 is the unique solution of the equation P r o o f.It is known that the attainability set for the controllable system under control from (2.1) is uniformly bounded by the time instant T at ε ∈ (0; ε 0 ] (see, for example, [6, theorem 3.1]).Hence, by virtue of (2.4) vectors {l ε } are also bounded at ε ∈ (0; ε 0 ].Therefore, to prove the theorem, it is sufficient to show that all partial limits {l ε } as ε → +0 are equal to l 0 . As follows from the A. B. Vasil'eva's results (see, for example [10, Chapter 3]) there is γ > 0 such that (2.7) Moreover, asymptotic estimates are uniform in t ∈ [0; T ].
Hence, by virtue of (2.2) which determines the matrices A 0 and B 0 and by formulas (2.7) the expression standing ∇σ for the formula (2.4) has the form Let us divide the integral from (2.8) into two terms Then, taking into account that the expression under integral is uniformly constrained and that O(e −γ/ √ ε ) = O(ε α ) as ε → 0 for any α > 0, we obtain from (2.4) and (2.8) (2.9) Let l be a partial limit of the vectors {l ε } as ε → +0, i.e. l ε k → l for a certain {ε k } so that ε k → +0.Going to the limit as k → ∞ in (2.9) we obtain that l is the solution of (2.6).Because of the uniqueness of such a solution we have l = l 0 .
The main problem for (2.1) is to find the complete asymptotic expansion in powers of small parameter ε of the optimal control, optimal values of the quality index, and the optimal process.Formulas (2.5) and (1.5) show that if one manages to gain the complete asymptotic expansion of vector l ε , which determines the optimal control in problem (2.1), this vector can be used for the asymptotic expansions of the above values as well.

Construction of complete asymptotic expansion of vector l ε for an optimal control problem with fast and slow variables
Consider a partial case of problem (2.1): where x ε , y ε , u ∈ R n .Problem (3.1) simulates a motion of a material point of small mass ε > 0 with the coefficient of the medium resistance equals to 1 in the space R n under action of the constrained control force u(t).
Here A 11 = 0, A 12 = I, A 21 = 0, A 22 = −I, B 1 = 0, B 2 = I, and 0 and I are the zero and the identity matrices of dimensional n × n, respectively.For the limit problem we have A 0 = 0, B 0 = I and thus, Conditions 2 and 3 are valid.
From (3.2) we obtain x 0 and l 0 > 2. (3.3) 1. Consider first the case: By virtue of (3.3) and Theorem 1 the inequality l ε < 2 is valid for any sufficiently small ε.Taking into account that (1 − e −t/ε ) 1 at any t 0 and ε > 0, from (3.2) we obtain for l ε the equation Calculating the integral It follows from this representation that l ε is expanded in the asymptotic series in powers of ε.
2. Now consider the case: x 0 > T + 2. By virtue of (3.3) and Theorem 1, the inequality l ε < 2 is valid for all sufficiently small ε.Since for a fixed ε the function (1 − e −t/ε ) l ε increases monotonically from 0 at t = 0 into (1 − e −t/ε ) l ε at t = T (which for sufficiently small ε gives the inequality (1 − e −t/ε ) l ε > 2), there is the unique t Therefore, the equation (3.2) takes the form Calculating the integrals in (3.6) and transposing (−l ε ) into the right part, we obtain Theorem 2. Let x 0 > T + 2. Then the vector l ε which determines the optimal control in problem (3.1) is expanded into a power asymptotic series (for ε → 0) P r o o f.Consider the equation 0 = F (ε, l), where F (•, •) is defined in (3.7).Additionally predetermine e −T /ε at the point ε = 0 as zero.Then we obtain that 0 = F (0, l 0 ) and F (•, •) is infinitely differentiable in ε and l in a certain neighborhood of the point (0; l 0 ).Since In this way, the theorem of implicitly specified function is applicable, which means that l ε (as a function of ε) is infinitely differentiable in ε for all small ε and, therefore, l ε is expanded into the asymptotic series.The coefficients of this series can be found via the standard procedure: substituting the series into the equation (3.7), expanding values dependent on ε into asymptotic series in power of ε, and equaling terms of the same order of smallness with respect to ε, we obtain an equation of the Fl k = g k with the known right parts.Then, by the formula (3.8) we find l k .
In particular, for l 1 we obtain the equation Hence, by virtue of (3.8) we obtain l 1 = g 1 + T l 0 , g 1 l 0 l 0 3 l 0 T + l 0 .