Asymptotic solution of a singularly perturbed optimal control problem related to the reconstruction of a defective curve

ISSN 0012-2661, Differential Equations, 2013, Vol. 49, No. 11, pp. 1347–1355. c© Pleiades Publishing, Ltd., 2013.Original Russian Text c© M.G. Dmitriev, Yu.L. Sachkov, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 11, pp. 1381–1389.

CONTROL THEORY

Asymptotic Solution of a Singularly PerturbedOptimal Control Problem Related

to the Reconstruction of a Defective Curve

M. G. Dmitriev and Yu. L. SachkovNational Research University Higher School of Economics, Moscow, Russia

Institute for Program Systems, Russian Academy of Sciences, Pereslavl’-Zalesskii, RussiaReceived May 2, 2013

Abstract—The introduction of “cheap” controls for minimizing the simplest energy functionalin an optimal control problem related to the reconstruction of a defective curve necessitates solv-ing a singularly perturbed variational problem with fixed time and fixed ends. The constructionof a uniform zero asymptotic approximation to the optimal control in the latter problem permitsone to conclude that the optimal trajectories in the original optimal control problem combinea uniform motion in the interior of the time interval with rapid transition layers at the bound-aries of the control interval.

DOI: 10.1134/S0012266113110037

1. INTRODUCTION

Reconstruction of defective or partially hidden images is a topical problem in computer graphics,photography, radiography, and painting. A number of approaches were developed for its solution,many of which are based on modern mathematical methods, in particular, on the use of variationalcalculus and optimal control [1, 2].

The problem of reconstructing a curve on the plane is a special case of the image reconstruc-tion problem; some images can be represented as a family of brightness level lines (known asisophots) [3–5].

An approach to the reconstruction of defective or partially hidden curves by solving an optimalcontrol problem, a sub-Riemannian problem on the group of motions of the plane, was developedin the framework of geometric control theory [3].

Neurogeometry, which studies the geometric structures of the human brain that model 3D imagesof the outside world, is a important branch of neurophysiology of vision at the beginning of thecurrent century. The following model of the primary visual cortex V 1 of the brain was suggested(see [6–10]). The primary cortex V 1 of the brain performs the initial perception (preceding anyessential processing) of visual images. Neurons of the primary cortex are joined into orientationcolumns, each of which is sensitive to visual stimuli at a given point of retina for a specific directionon it. The retina is modeled by a two-dimensional plane (or its subdomain); i.e., each of its pointsis represented by a point (x, y) ∈ R

2. The direction at a given point is modeled by a projectiveline; i.e., θ ∈ P 1. (Recall that the projective line P 1 is the set of lines on the plane passing throughthe origin, P 1 = R/ ∼, where θ ∼ θ′ if θ = θ′ + πn, n ∈ Z.) Therefore, the primary visioncortex V 1 is modeled by the so-called projective tangent bundle PTR

2 = R2 × P 1. From the

viewpoint of neurology, the orientation columns are, in turn, arranged into hypercolumns each ofwhich is sensitive to stimuli at a given point (x, y) with an arbitrary direction θ. The orientationcolumns are linked in two different ways. The first way is given by vertical links between orientationcolumns that belong to one hypercolumn and are sensitive to distinct directions. The other way isimplemented by horizontal links between orientation columns that belong to distinct hypercolumnsand are sensitive to one direction.

1347

1348 DMITRIEV, SACHKOV

Therefore, when the visual cortex V 1 of the brain precepts a plane curve (x(t), y(t)), t ∈ [0, T ],it calculates its lift in PTR

2 by adding a new variable θ(t), t ∈ [0, T ], so that the equations

x = u cos θ, (1)y = u sin θ, (2)

θ = σ (3)

are satisfied. Here the new variable θ(t) specifies the direction of the curve (x(t), y(t)).If the curve (x(t), y(t)), t ∈ [0, T ], is defective on a segment [a, b] ⊂ [0, T ], then it is reconstructed

by the visual cortex V 1 on the basis of the following variational principle: the reconstructing arc(x(t), y(t)), t ∈ [a, b], should minimize the energy J of activation of the retina neurons correspondingto that arc. That energy is modeled by the integral

J =

b∫

a

(u2(t) + α2σ2(t)) dt → min, (4)

where the term u2(t) [respectively, α2σ2(t)] is the infinitesimal energy required for the activationof horizontal (respectively, vertical) links. The parameter α > 0 specifies the relative weight ofhorizontal and vertical links. The desired trajectory of the control system (1)–(3) should satisfythe boundary conditions

(x(a), y(a), θ(a)) = (x0, y0, θ0), (5)(x(b), y(b), θ(b)) = (x1, y1, θ1). (6)

By the Cauchy–Schwarz inequality, the minimization of the energy functional is equivalent tothe minimization of the sub-Riemannian length functional

l =

b∫

a

√u2(t) + α2σ2(t) dt → min . (7)

Therefore, to reconstruct defective curves, the primary visual cortex V 1 computes the solutionof the optimal control problem (1)–(3), (5), (7).

This problem was considered in [11–13], where the parametrization of extremal trajectorieswas obtained, their local and global optimality was studied, and the optimal control problem wasreduced to the solution of systems of equations in elliptic functions. As was shown in these papers,the optimal curve (x(t), y(t)) often contains a cusp, which is undesirable from the viewpoint ofapplications to the reconstruction of curves and images. Computer experiments show that cusps candisappear as the weight coefficient α in the functionals J and l is varied. As a rule, the eliminationof cusps occurs as the parameter α approaches the boundary of its range, i.e., as α → 0 or α → +∞.

Thus, we arrive at the problem on the behavior of solutions of the optimal control problem(1)–(3), (5), (7) as α → 0 and α → +∞. In particular, for applications to the reconstruction ofcurves without cusps, it is important to study the asymptotics of these solutions as α → 0 andα → +∞ and the disappearance or preservation of cusps for these asymptotics.

The aim of the present paper is to analyze the asymptotics of solutions of problem (1)–(3), (5),(7) as α → 0. This problem can be reduced to a singularly perturbed optimal control problem andthen studied with the use of the direct scheme [14] of the theory of singularly perturbed controlproblems.

2. STATEMENT OF THE PROBLEM AND SOLUTION METHOD

Thus, the problem of reconstructing a defective curve is formalized as the optimal control prob-lem

x = u1 cos θ, x(0) = x0, x(t1) = x1,

DIFFERENTIAL EQUATIONS Vol. 49 No. 11 2013

ASYMPTOTIC SOLUTION OF A SINGULARLY PERTURBED OPTIMAL . . . 1349

y = u1 sin θ, y(0) = y0, y(t1) = y1, (8)

θ = σ, θ(0) = θ0, θ(t1) = θ1, u1, σ ∈ R,t1∫

0

(u21 + ε2σ2) dt → min, (9)

where x and y are coordinates in the plane of the curve, θ is the curve inclination angle, u1 andσ are controls, and ε > 0 is a sufficiently small number treated as the weight coefficient of theactivation energy of the horizontal and vertical links between neurons in the primary visual cor-tex V 1. It follows from Eqs. (1) and (2) that no constraint is imposed on the controls u1 and σ,and problem (1), (2) becomes the so-called problem with a “cheap” control, for which the solutionmethods are closely related to the singular perturbation theory [15, 16].

Indeed, by making the change of variables εσ = u2 in Eqs. (1) and (2), we obtain the singularlyperturbed optimal control problem Pε:

x = u1 cos θ, y = u1 sin θ, εθ = u2, (10)v(0) = v0, v(t1) = v1, (11)

Iε(u) =

t1∫

0

(u21 + u2

2) dt → min, (12)

wherev = (x, y, θ)T, u = (u1, u2)T. (13)

We seek the asymptotics of the solution of problem (3), (4) with respect to the parameter ε → 0.There are two approaches to the construction of the asymptotics. The first (traditional) oneis related to the asymptotics of the solution of the corresponding boundary value problem forextremals, and the other is related to the so-called direct scheme of construction of asymptoticsfor solutions of variational problems. To construct a uniform zero asymptotic approximation,we use the second approach, which implies the substitution of the proposed asymptotic expansioninto all conditions of the problem, the expansion of these conditions into series in powers of theparameter ε, and the subsequent solution of simpler variational problems. For the ansatz seriesof the approximate solution of problem (3), (4), we take the series of the method of boundaryfunctions,

z(t, ε) = z(t, ε) + Πz(τ0, ε) + Qz(ε1, ε), (14)

where τ0 = t/ε, τ1 = (t− t1)/ε, ε → 0, zT = (x, y, θ, u1, u2)T; z(t, ε) = z0(t)+εz2(t)+ · · · is a regularseries in integer powers of ε with coefficients depending on t, Πz(τ0, ε) = Π0z(τ0)+ εΠ1z(τ0)+ · · · isa left boundary series in powers of ε with coefficients depending on τ0, and Qz(τ1, ε) = Q0z(τ1) +εQ1z(τ1) + · · · is a right boundary series in integer powers of ε with coefficients depending on τ1.

In accordance with the direct scheme, the series (7) is substituted into all conditions of theproblem Pε, and from the corresponding expansions of these conditions, we obtain a sequence ofvariational problems of smaller dimension, because, by a lemma in [14], we have

inf Iε(u) = inf I0 + ε inf I1 + · · · (15)

3. LIMIT PROBLEM

For ε = 0, from problem (3), (4), we obtain the so-called limit degenerate problem P0 :

I0(u, θ) =12

t1∫

0

(u210 + u2

20) dt → min,

˙x0 = u10 cos θ0, ˙y0 = u10 sin θ0, 0 = u20, (x0, y0)Tt=0 = (x0, y0)T, (x0, y0)Tt=t1= (x1, y1)T.



Since here one differential link becomes the equality-type constraint u20(t) ≡ 0, it follows that,along with the controls u10(t) and u20(t), the free state variable θ0(t) can be treated as a controlin the problem P0; in other words, the physical state coordinate θ appears as a control in the limitproblem.

In P0 we introduce the Hamiltonian

H(ψ, x0, y0, u10, u20, θ0) = ψ10u10 cos θ0 + ψ20u10 sin θ0 + ψ30u20 + (ψ0/2)(u210 + u2

20), (16)

where ψ = (ψ0, ψ10, ψ20, ψ30).Let ψ0 = −1 (the normal case); then

∂H

∂u10

= ψ10 cos θ0 + ψ20 sin θ0 − u10,∂H

∂u20

= ψ30 − u20 = 0,

∂H

∂θ0

= u10(−ψ10 sin θ0 + ψ20 cos θ0) = 0.

Since ˙ψ10 = 0 and ˙ψ20 = 0, it follows that ψ10 and ψ20 are constants. It is also obvious thatψ30 = 0, because u20 = 0 and, in addition, ψ10 and ψ20 remain unknown for now.

It follows from the necessary optimality conditions that the extremal control u10 has the form

u10 = ψ10 cos θ0 + ψ20 cos θ0 and u10(−ψ10 sin θ0 + ψ20 cos θ0) = 0.

Consider the following two cases.(a) (ψ10, ψ20) = (0, 0); then u10 ≡ 0, and θ0 cannot be found from necessary conditions. In this

case, ˙x0 = 0 and ˙y0 = 0, which is impossible for (x0, y0) �= (x1, y1). But if (x0, y0) = (x1, y1), thenthe solution of the problem P0 has the form u10 ≡ 0, I∗

0 = 0.(b) (ψ10, ψ20) �= (0, 0). By passing to polar coordinates, we obtain ψ10 = cos ϕ and ψ20 =

sin ϕ, �= 0. Then u10 = (cos ϕ cos θ0 + sin ϕ sin θ0) = cos(θ0 − ϕ). Since

−ψ10 sin θ0 + ψ20 cos θ0 = (− cos ϕ sin θ0 + sin ϕ cos θ0) = − sin(θ0 − ϕ),

we have∂H

∂θ0

= −2 cos(θ0 − ϕ) sin(θ0 − ϕ) = 0,

where we have used the notation θ0 = ϕ and u10 = . Therefore, for the limit trajectories, we obtainthe problem

˙x0 = cos ϕ, ˙y0 = sin ϕ,

x0(0) = x0, x0(t1) = x1, y0(0) = y0, y0(t1) = y1.

We have

x0(t) = x0 + t cos ϕ, y0(t) = y0 + t sin ϕ,

= r/t1, (17)

where r is the distance between the points (x1, y1) and (x0, y0) and ϕ is the angle between the vectorjoining these points and the positive direction of the x-axis; i.e., (x1−x0, y1−y0)T = r(cos ϕ, sin ϕ)T.

Thus, θ0 = ϕ and hence u10(t) ≡ . Note that the above-performed considerations imply therelation −ψ10 sin θ0 + ψ20 cos θ0 ≡ 0.

Therefore, the optimal control in the problem P0 has the form u10 = , u20 = 0, θ0 = ϕ,the optimal value of the functional is equal to I∗

0 = 2t1/2, and the costate variables take the values

ψ10 = cos ϕ, ψ20 = sin ϕ, ψ30 = 0.



If in the problem P0 the matrix of second partial derivatives of the function (16) with respectto the control variables u10, u20, and θ0 is computed, then it turns out to be negative definite, i.e.,the strengthened Legendre–Clebsch condition is satisfied in the problem P0.

We introduce the function

H(τ0) = cos(θ0(0) + Π0θ(τ0))ψ10(0)u10(0) + sin(θ0(0) + Π0θ(τ0))ψ10(0)u10(0)

− 12u2

10(0) −12(u20(0) + Π0u2(τ0))2,

which is the Hamiltonian of the limit problem H except that the fast variables θ and u2 are replacedby the values of regular terms in the fast variables at zero with regard of the corresponding boundaryfunctions. The function H(τ1) is defined in a similar way.

By H(0) and H(t1) we denote the functions H where all variables are computed at t = 0 andt = t1, respectively.

Let us introduce the difference

H(τ0) − H(0) = ψ10(0)u10(0)[cos(Π0θ0(τ0) + θ0(0)) − cos(θ0(0))]+ ψ20(0)u10(0)[sin(Π0θ0(τ0) + θ0(0))] − Π2

0u2(τ0).

By straightforward computations, we obtain

H(τ0) − H(0) = −2[1 − cos(Π0θ(τ0))] − Π20u2(τ0).

In a similar way, one can compute the difference

H(τ1) − H(t1) = −2[1 − cos(Q0θ(τ1))] − Q20u2(τ1).

Following the direct scheme [14], we obtain I∗ε = I∗

0 + εI∗1 + · · · , where I∗

0 is the minimum valueof the functional in P0, and I∗

1 = ΠI∗ + Q0I∗ + D, where D is a known constant determined by the

solution of the problem P0 and the boundary conditions for the fast variable θ(t),

D = ψ30(t1)θ − ψ30(0)θ0 −t1∫

0

ψ30(t)θ0(t) dt = 0,

because ψ30(t) ≡ 0. Here Π0I∗ and Q0I

∗ are the minimum values of the functionals of two optimalcontrol problems in a neighborhood of the boundaries of the interval [0, t1], namely, the prob-lem Π0P :

−∞∫

0

(H(τ0) − H(0)) dτ → min,Π0Q

dτ0

= Π0u2, Π0θ(0) = θ0 − θ0(0) = θ0 − ϕ (18)

and the problem Q0P :

−∞∫

0

(H(τ1) − H(0)) dτ1 → min,dQ0θ

dτ1

= Q0u2, Q0θ(0) = θ(t1) − θ0(t1) = θ1 − ϕ. (19)

Consider the problem Π0P . We introduce the Hamiltonian system

dΠ0Q

dτ0

=∂H

∂p= Π0ux(τ0), Π0θ(0) = θ0 − ϕ,

dp

dτ0

= − ∂H

∂Π0θ= 2 sin(Π0θ),



where H = pΠ0u2 − 2[1 − cos(Π0θ(τ0))] − Π20u2. For Π0u2(τ0) we have the additional condition

∂H

∂Π0u2

= p(τ0) − 2Π0u2(τ0) = 0;

i.e., Π0u2(τ0) = p(τ0). Then we obtain the system

dΠ0θ

∂τ0

=12p(τ0),

dp

dτ0

= u210(0) sin(Π0θ(τ0)). (20)

The solution of this system with the boundary conditions

Π0θ(0) = θ0 − ϕ, limτ0→+∞

Π0θ(τ0) = 0

has the form

Π0θ(τ0) = arccos(1 − 2 tanh2(c + u10τ0/√

2)) − π,

p(τ0) = 2√

2 u10/ cosh(c + u10τ0/√

2 ), (21)

where

c =12

ln1 + cos((θ0 − ϕ)/2)1 − cos((θ0 − ϕ)/2)

.

By taking p(τ0) for the function Π0θ, i.e., by using the feedback control in the boundary layer,we find from [17] under the smoothness conditions, the strengthened Legendre–Clebsch conditionfor the limit problem, and the controllability in a boundary layer, which are obviously satisfied in thecase under consideration, that there exists a conditionally stable manifold along which system (20)can be reduced by the change of variables (Π0θ, p)T = M−1(y, ˜y )T to the form

(y

˜y

)=

(A 0

0 −A

)(y

˜y

)+ χ(s), s =

(y

˜y

), (22)

where χ(s) is a smooth vector function satisfying the following condition: for each ξ > 0, thereexists a δ > 0 such that ‖χ(s) − χ(s1)‖ ≤ ξ‖s − s1‖ provided that max(‖s‖, ‖s1‖) ≤ δ. HereA = −K is a positive solution of the Riccati equation K2 − 2/4 = 0; i.e., K = /2. The matricesM1 and M2 have the form

M =14

(2 K−1

8K −4

), M−1 =

14

(4 K−1

8K −2

).

Obviously, A < 0; i.e., after a nonsingular transformation, for the Hamiltonian system (20), we ob-tain system (22), which, in the linear approximation, splits into two subsystems, one subsystem fory and another for ˜y .

The first linear subsystem is asymptotically stable in direct time, and so is the other in reversetime.

The representation (22) and the nonlinearity properties of the function χ(s), together with theresults in [17], permit one to claim that there exist constants a > 0 and C > 0 such that the solutionof system (20) satisfies the estimates

|Π0θ(τ0)| ≤ Cε−ατ0 , |p(τ0)| ≤ Cε−ατ0 , 0 < τ0 < +∞,

and hence the inequalities |Π0u2| ≤ Ce−ατ0 provided that the initial condition θ0 − θ0(0) lies ina small neighborhood of the origin.



One can readily see that the problem Q0P has a similar solution Q0u2(τ1) = q, where

dQ0θ(τ1)dτ1

= Q0u2, Q0θ(0) = θ1 − ϕ,dq

dτ1

= 2 sin(Q0θ(τ1)), Q0θ(τ1) → 0, τ1 → −∞.

By using the similar change of variables (Q0θ, q)T = M−12 (k, l)T, one can reduce the latter system

to the form (22), where A = −2K but K is now a negative definite solution of the Riccati equation

K2 − 2/4 = 0; i.e., K = −/2 and M2 =14

(2 K−1

8K −4

).

Accordingly, we have the similar estimates

|Q0θ(τ1)| ≤ Ceατ1 , |q(τ1)| ≤ Ceατ1 , |Q0u2(τ1)| ≤ Ceατ1 , −∞ < τ1 < 0,

if the initial condition θ1 − ϕ lies in a small neighborhood of the zero equilibrium.By approximating problems (18) and (19) by linear-quadratic ones, one can readily obtain

approximate representations of solutions of these problems.Indeed, by extracting the first terms in the expansions of the right-hand sides and the integrands,

we obtain the following two problems:

12

∞∫

0

[(Π0θ)2 + (Π0u2)2] dτ0 → min,dΠ0θ

dτ0

= Π0u2, Π0θ(0) = θ0 − ϕ,

and

12

−∞∫

0

[2(Q0θ)2 + (Q0u2)2] dτ1 → min,dQ0θ

dτ1

= Q0u2, Q0θ0(0) = θ1 − θ0(1) = θl − ϕ.

We have

Π0u2(τ0) = −(/2)Π0θ(τ0), Π0θ(τ0) = (θ0) − ϕ)e�τ0/2 → 0, τ0 → +∞,

Q0u2(τ1) = +(/2)Q0θ(τ1), Q0θ(τ1) = (θ1 − ϕ)e�τ1/2 → 0, τ1 → −∞,

and in addition, it follows from [17] that the estimates

|Π0θ(τ0) − Π0θ(τ0)| = O(Π20θ(τ0)), |Π0u(τ0) − Π0u(τ0)| = O(Π2

0θ(τ0)),

|Q0θ(τ1) − Q0θ(τ1)| = O(Q20θ(τ1)), |Q0u(τ1) − Q0u(τ1)| = O(Q2

0θ(τ1))(23)

hold for sufficiently small |θ0 − ϕ| and |θ1 − ϕ|.

4. MAIN THEOREM

We introduce the zero asymptotic approximation

z0(t, ε) = z0(t) + Π0z(τ0) + Q0z(t1),

where the above-constructed boundary functions are nonzero only for the fast variables (the angleand the angular velocity). In this case, the corresponding approximation for the control is equal to

u0(t, ε) =

(u10(t, ε)

u20(t, ε)

)=

(

Π0u2(τ0) + Q0u2(τ1)

). (24)

Since the right-hand side in the equation for the angle θ in system (8) has no term containing θ, forthe correct numerical integration of the closed-loop system, it is desirable to introduce an artificial



incomplete feedback in θ, as was described in [14]. To this end, instead of u20(t, ε), we considerv20(θ, t, ε) = −Dθ + ω20(t, ε) and ω20(t, ε) = u20(t, ε) + Dθ0(t, ε), where D > 0 is some constant.

Thus, instead of u0(t, ε), we use the function

v0(θ, t, ε) =

(

v20(θ, t, ε)

).

There are (see the survey [16]) several approaches to the proof of estimates for the remainder in theasymptotics of solutions of optimal control problems without constraints on the values of controlswith a small parameter. One of them [14] is related to the construction of a set of admissiblecontrols where the functional of the original perturbed problem turns out to be strongly convex.It is the approach to be used below. Since, in the present paper, there are boundary conditionsfor state variables, it follows that the set of admissible controls is extended by the introduction ofε-admissible controls, which bring state variables into the ε-neighborhood of terminal conditions.By using the method in [14, 18], one can readily show that the constructed function v0 is anε-admissible control. In L2 we introduce the ball Ω with center v0 and radius Lε, where L > 0 issome given number. It was shown in [13] that problem (10), (12) has an optimal control u∗(t, ε)for each ε > 0, and this control obviously belongs to the ball Ω. However, for some symmetricboundary conditions described in [11–13], there may be two optimal controls, to each of whichthere obviously corresponds a unique trajectory (x∗(t, ε), y∗(t, ε), θ∗(t, ε)). The asymptotics of thesolution is constructed in a neighborhood of the unique solution of the degenerate problem andgenerates the corresponding asymptotic approximation. In the case of the nonuniqueness of theoptimal control, the constructed asymptotics provides the same uniform zero approximation forboth optimal controls. Now, instead of problem (10), (12), we consider the corresponding problemwith a free right end. In that new problem, one can follow the lines of [14] and justify the strongconvexity of the functional on the set Ω. This implies the existence of a unique optimal control uon the ball Ω in the corresponding approximate problem on the set Ω. In turn, hence we readilyobtain the following main assertion.

Theorem 1. For a sufficiently small ε > 0, the solution of problem (10)–(12) satisfies thefollowing estimates in some neighborhood of the solution of the problem P0 as ε → +0 :

‖x(v0, t, ε) − x0(t)‖ = O(ε), ‖y(v0, t, ε) − y0(t)‖ = O(ε),‖θ(v0, t, ε) − θ(t, ε)‖ = O(ε), ‖v0(θ, t, ε) − u0(t, ε)‖ = O(ε),

‖x∗(t, ε) − x(v0, t, ε)‖ = O(ε), ‖y∗(t, ε) − y(v0, t, ε)‖ = O(ε),‖θ∗(t, ε) − θ(v0, t, ε)‖ = O(ε), ‖u∗(t, ε) − v0(θ, t, ε)‖ = O(ε),

‖u∗(t, ε) − u(t, ε)‖ = O(ε),

(25)

where (x∗(t, ε), y∗(t, ε), θ∗(t, ε)) and u∗(t, ε) are the optimal trajectory and the corresponding controlin problem (10)–(12).

The theorem implies the following assertions.

Corollary 1. For a sufficiently small ε > 0, the control

u0(t, ε) =

(

σ0(θ, t, ε)

)=

(

v20(θ, t, ε)/ε

)(26)

is ε-admissible in problem (8), (9), and the following estimates hold as ε → +0 :

|u∗1(t, ε) − | = O(ε), |σ∗(t, ε) − v20(θ, t, ε)/ε| = O(1).

Corollary 2. The estimate |I(u0) − I∗ε | = O(ε2) holds along the control u0(t, ε) in (25) in

problem (8), (9) for a sufficiently small ε → +0.



Thus, the controls

u1(t, ε) = , σ(θ, t, ε) = (Π0θ(t/ε) + Q0θ((t − t1)/ε))/ε,

where Π0θ(t/ε) and Q0θ((t − t1)/ε) are some elliptic functions—solutions of the correspondingproblems in the boundary layer—form a control vector that is ε2-suboptimal with respect to thefunctional and ε-admissible with respect to the constraints.

Therefore, the representation (26) and the above-proved exponential estimates for the boundaryfunctions imply that the control σ0(θ, t, ε) contains approximations of two pulses on the boundary ofthe interval [0, t1]. The first pulse provides the passage of θ for t = 0 from θ in the ε-neighborhoodof ϕ, and the other for t = t1 brings θ from the ε-neighborhood of ϕ into θ1.

ACKNOWLEDGMENTS

The research was supported by the Federal Targeted Program “Scientific and Pedagogical Staffof Innovational Russia” for 2009–2013 (contract no. 8209) and the Russian Foundation for BasicResearch (projects nos. 12-01-00913 and 13-01-91162-GFEN a).

REFERENCES1. Chan, T., Kang, S., and Shen, J., Euler’s Elastica and Curvature Based Inpainting, SIAM J. Appl.

Math., 2002, vol. 63, no. 2, pp. 564–592.2. Citti, G. and Sarti, A., A Cortical Based Model of Perceptual Completion in the Rototranslation Space,

J. Math. Imaging Vision, 2006, vol. 24, no. 3, pp. 307–326.3. Agrachev, A.A. and Sachkov, Yu.L., Geometricheskaya teoriya upravleniya (Geometric Control Theory),

Moscow, 2005.4. Jurdjevic, V., Geometric Control Theory, Cambridge, 2006.5. Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Providence:

AMS, 2002.6. Petitot, J., Vers une neuro-geometrie. Fibrations conticales, structures de contact et contours subjectifs

modaux, Math. Inform. Sci. Humaines , 1999, vol. 145, pp. 5–101.7. Petitot, J., Neurogeometrie de la vision — Modeles mathematiques et physiques des architectures fonc-

tionnelles , Palaiseau, Les Editions de l’Ecole Polythecnique, 2008.8. Petitot, J., The Neurogeometry of Pinwheels As a Sub-Riemannian Contact Structure, J. of Physiology,

Paris, 2003, vol. 97, pp. 265–309.9. Sanguinetti, G., Citti, G., and Sarti, A., Image Completion Using a Diffusion Driven Mean Curvature

Flow in a Sub-Riemannian Space, Int. Conf. on Computer Vision Theory Applications (VISAPP’08),Funchal, 2008, pp. 22–25.

10. Hladky, R.K. and Pauls, S.D., Minimal Surfaces in the Roto-Translation Group with Applications toa Neuro-Biological Image Completion Model, J. Math. Imaging Vision, 2010, vol. 36, pp. 1–27.

11. Moiseev, I. and Sachkov, Yu. L., Maxwell Strata in sub-Riemannian Problem on the Group of Motionsof a Plane, ESAIM Control Optim. Calc. Var., 2010, vol. 16, no. 2, pp. 380–399.

12. Sachkov, Yu.L., Conjugate and Cut Time Sub-Riemannian Problem on the Group of Motions of a Plane,ESAIM Control Optim. Calc. Var., 2010, vol. 16, no. 4, pp. 1018–1039.

13. Sachkov, Yu.L., Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Group of Motionsof a Plane, ESAIM Control Optim. Calc. Var., 2011, vol. 17, no. 2, pp. 293–321.

14. Belokopytov, S.V. and Dmitriev, M.G., Solution of Classical Optimal Control Problems with a BoundaryLayer, Avtomat. i Telemekh., 1989, no. 7, pp. 71–82.

15. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykhuravnenii (Asymptotic Expansions of the Solutions of Singularly Perturbed Equations), Moscow: Nauka,1973.

16. Dmitriev, M.G. and Kurina, G.A., Singular Perturbations in Control Problems, Avtomat. i Telemekh.,2006, no. 1, pp. 3–53.

17. Lukes, D.L., Optimal Regulation of Nonlinear Dynamical Systems, SIAM J. Control Optim., 1969,vol. 7, no. 1, pp. 75–100.

18. Vasil’eva, A.V. and Dmitriev, M.G., Singular Perturbations in a Nonlinear Optimal Control Problem,in Aktual’nye problemy matematicheskoi fiziki i vychislitel’noi matematiki (Current Problems in Math-ematical Physics and Numerical Mathematics), Moscow: Nauka, 1984, pp. 40–49.


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Asymptotic solution of a singularly perturbed optimal control problem related to the reconstruction of a defective curve