On Computer-Oriented Algorithms Solving Guaranteed Control Problems under Uncertainty for Stochastic Di erential Equations

Abstract

Two problems of guaranteed closed-loop control under incomplete information are considered for a linear stochastic di erential equation (SDE) from the viewpoint of the constructive method of openloop control packages worked out and realized as a software earlier for the guidance of a linear control system of ordinary di erential equations (ODEs) to a convex target set. The problems consist in designing a deterministic control providing (irrespective of a realized initial state from a given nite set) prescribed properties of the solution (being a random process) by a terminal point in time (A) or at this time (B). It is assumed that a linear signal on some number of realizations is observed. By the equations of the method of moments, the problems for the SDE are reduced to equivalent problems for systems of ODEs describing the mathematical expectation and covariance matrix of the original process. The emphasis is on designing computer-oriented solving algorithms based on feasible nite-dimensional optimization procedures. The solvability conditions for the problems are written. An illustrative example is presented.

Keywords

• guidance problem
• computer-oriented algorithm
• linear stochastic di erential equation

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In mathematical control theory and its applications, the problem of constructing optimal strategies of guaranteed feedback control under conditions of uncertainty is evidently actual. We follow the theory of closed-loop control developed by N.N. Krasovskii's school [1] and apply the approach based on the so-called method of open-loop control packages originating from the technique of nonanticipating strategies [2] to solving the guidance problem for a linear SDE. The method tested on the guidance problems for linear controlled systems of ODEs consists in reducing the problems of guaranteed control formulated in the class of closed-loop strategies to equivalent problems in the class of open-loop control packages. The latter class contains the families of open-loop controls parameterized by admissible initial states and possessing the property of nonanticipation with respect to the dynamics of observations, see [3], [4], and [5].

This paper is devoted to the study of the problem of guiding (with a probability close to 1) a trajectory of a linear SDE to some target set. The statements mean that we should form a deterministic control providing (irrespective of the realized initial state from a speci ed nite set) prescribed properties of the solution (being a random process) by a terminal point in time (A) or at this time (B). Here, we observe a linear signal on some number of realizations. Similar problems arise in practical situations, when it is possible to observe the behavior of a large number of identical objects described by a stochastic dynamics. By the equations of the method of moments [6], the problems for the SDE are reduced to equivalent problems for systems of ODEs describing the mathematical expectation and covariance matrix of the original process. The technique of the method of open-loop control packages developed in [3], [4], and [5] is applied to the systems obtained. A similar reduction procedure was used, for example, in [7] for solving the problem of dynamic reconstruction of an unknown disturbance characterizing the level of random noise in a linear SDE. 2

Statement of the problems

Consider a system of linear SDEs of the following form: dx(t; !) = (A(t)x(t; !) + B1(t)u1(t) + f (t)) dt + B2(t)U2(t) d (t; !): ( 1 ) Here, x(t0; !) = x0, t 2 T = [t0; #], x = (x1; x2; : : : ; xn) 2 IRn, = ( 1; 2; : : : ; k) 2 IRk; ! 2 , ( ; F; P ) is a probability space; (t; !) is a standard Wiener process (i.e., a process starting from zero with zero mathematical expectation and covariance matrix equal to It, I is the unit matrix from IRk k); f (t) is a continuous vector function with values in IRn; A(t) = faij (t)g, B1(t) = fb1ij (t)g, and B2(t) = fb2ij (t)g are continuous matrix functions of dimensions n n, n r, and n k, respectively.

Two controls act in the system: a vector u1(t) = (u11(t); u12(t); : : : ; u1r(t)) 2 IRr and a diagonal matrix U2(t) = fu21(t); u22(t); : : : ; u2k(t)g 2 IRk k, which are Lebesgue measurable on T and take values from speci ed instantaneous control resources Su1 and Su2 being convex compact sets in the corresponding spaces. The control u1 enters the deterministic component and in uences the mathematical expectation of the desired process. Since U2d = (u21d 1; u22d 2; : : : ; u2kd k), we can assume that the vector u2 = (u21; u22; : : : ; u2k) characterizes the di usion of the process (the amplitude of random noises).

The initial state x0 belongs to a nite set of admissible initial states X0, which consists of normally distributed random variables with numerical parameters (m0; D0), where m0 = M x0 is the mathematical expectation, m0 2 M0 = fm01; m20; : : : ; m0n1 g, D0 = M (x0 m0)(x0 m0) is the covariance matrix (the asterisk means transposition), D0 2 D0 = fD01; D02; : : : ; D0n2 g. Thus, the set X0 contains n1n2 elements. We assume that the system's initial state belongs to X0 but is unknown. Equation ( 1 ) is a symbolic notation for the integral identity (! is omitted) x(t) = x0 + (A(s)x(s) + B1(s)u1(s) + f (s)) ds +

B2(s)U2(s) d (s): ( 2 ) The latter integral on the right-hand side of equality ( 2 ) is stochastic and is understood in the sense of Ito. A solution of equation ( 1 ) is de ned as a stochastic process satisfying integral identity ( 2 ) for any t with probability 1. Under the assumptions above, there exists a unique solution, which is a normal Markov process with continuous realizations [8].

The problems in question consist in the following. Let a nonempty nite set St T of admissible guidance times and nonempty convex closed target sets M(t) 2 IRn and D(t) 2 IRn n for any moment t 2 St as well as a continuous matrix observation function Q(t) of dimension q n be given. At any time, it is possible to receive the information on some number N of realizations of the stochastic process x(t). The following signal is available: y(t) = Q(t)x(t): ( 3 )

Assume that, for a nite set of some speci ed times i 2 T , i 2 [1 : l], we can construct, using N realizations of the process x(t), a statistical estimate miN of the mathematical expectation m( i) and a statistical estimate DiN of the covariance matrix D( i) such that

P max i2[1:l] miN m( i) IRn ; DiN

D( i) IRn n where h(N ) and g(N ) ! 0 as N ! 1. Standard procedures of obtaining the estimates miN and DN admit modi cations providing the validity of relation ( 4 ) i and the speci ed convergences.

The problems of guaranteed closed-loop "-guidance consist in forming controls (u1( ); u2( )) guaranteeing, whatever the initial state x0 from the set X0, prescribed properties of the process x by or at the terminal time #. Here, we mean that, for an arbitrary small (in advance speci ed) " > 0, the mathematical expectation m(#) and the covariance matrix D(#) reach at some admissible guidance time t 2 St the "-neighborhoods of the target sets M(t) and D(t), respectively. This is Problem A1. If St = f#g, we have Problem B1. In the motion process, the sought controls are formed using the information on N realizations of the signal ( 3 ). By virtue of estimate ( 4 ), it is reasonable to require that the probability of the desired event should be close to 1 for su ciently large N and algorithm's parameters concordant with N in a special way. For ODEs, Problems A1 and B1 were considered in detail in [4] and [5], respectively. 3

Reduction of the original problems

Let us reduce the guidance problems for the SDE to problems for systems of ODEs. By virtue of the linearity of the original system, the mathematical expectation m(t) depends only on u1(t); its dynamics is described by the equation m_(t) = A(t)m(t)+B1(t)u1(t)+f (t); t 2 T = [t0; #]; m(t0) = m0 2 M0: ( 5 )

We assume that N (N > 1) trajectories xr(t), r 2 [1 : N ], of the original SDE are measured; then, we know values of signal ( 3 ), i.e., yr(t) = Q(t)xr(t).

The signal on the trajectory of equation ( 5 ) is denoted by ym(t) = Qm(t)m(t); its estimate formed by the information on yr; r 2 [1 : N ], by ymN(t): ymN(t) = 1 XN yr(t) = Q(t)mN (t); N r=1 mN (t) = 1 XN xr(t): N r=1 Obviously, Qm(t) = Q(t) and, for the nite set of times i 2 T , i 2 [1 : l], in view of relation ( 4 ), it holds that

P 8i 2 [1 : l] ymN( i) ym( i) IRq

C1h(N ) = 1 g(N ): Here and below, constants Ci can be written explicitly.

The covariance matrix D(t) depends only on U2(t); its dynamics is described by the so-called equation of the method of moments [6] in the following form: D_ (t) = A(t)D(t) + D(t)A (t) + B2(t)U2(t)U2 (t)B2 (t); t 2 T = [t0; #];

D(t0) = D0 2 D0:

For our purposes, matrix equation ( 8 ) is conveniently rewritten in the form of a vector equation, which is more traditional for such problems. By virtue of the symmetry of the matrix D(t), its dimension is de ned as nd = (n2 + n)=2. Let us introduce the vector d(t) = fds(t)g, s 2 [1 : nd], consisting of successively written and enumerated elements of the matrix D(t), taken line by line starting with the element located at the main diagonal. Performing standard matrix operations over A(t) and B2(t), we form continuous matrices A(t): T ! IRnd nd and B(t): T ! IRnd k and use them to rewrite system ( 8 ) in the form d_(t) = A(t)d(t) + B(t)v(t); t 2 T = [t0; #]; d(t0) = d0 2 D0: ( 9 )

The initial state d0 is obtained from D0; the notation for the set D0 is the same. The multiplication of the diagonal matrices U2(t)U2 (t) results in the appearance of the control vector v(t) = (u221(t); u222(t); : : : ; u22k(t)) whose elements take values from some convex compact set Sv 2 IRk for all t 2 T .

The signal on the trajectory of equation ( 9 ) is denoted by yd(t) = Qd(t)d(t); its estimate formed by the information on yr; r 2 [1 : N ], by ydN (t). The latter is constructed as follows:

N 1 1 r=1

N X(yr(t) ymN(t))(yr(t) ymN(t)) = Q(t)

1 N 1 ( 6 ) ( 7 ) ( 8 ) N X(xr(t) mN (t))(xr(t) mN (t)) Q (t) = Q(t)DN (t)Q (t); ( 10 ) where DN (t) = fdiNj (t)g; i; j 2 [1 : n] is the standard estimate of the covariance matrix D(t) for an unknown (estimated by mN (t)) mathematical expectation m(t). By means of algebraic transformations using the symmetry of matrix ( 10 ), the expression Q(t)DN (t)Q (t) is transformed into ydN (t) = Qd(t)dN (t), where Qd(t) is a continuous matrix of dimension nq nd, nq = (q2 + q)=2, and dN (t) is the vector of dimension nd extracted from DN (t). Obviously, for the nite set of times i 2 T , i 2 [1 : l], we have the relation of type ( 4 )

P 8i 2 [1 : l] ydN ( i) yd( i) IRnq

C2h(N ) = 1 g(N ): (11)

Original problems of guaranteed closed-loop "-guidance for the SDE can be reformulated as follows. For an arbitrary small (in advance speci ed) " > 0, it is required to choose controls u1( ) in equation ( 5 ) and v( ) in equation ( 9 ) such that, whatever the initial states m0 2 M0 and d0 2 D0, the trajectories of ( 5 ) and ( 9 ) reach at some admissible guidance time t 2 St the "-neighborhoods of the target sets M(t) and D(t), respectively (Problem A2). If St = f#g, we have Problem B2. It is important that the probability of the desired event should be close to 1. The required controls are formed through the estimates of the signals ym and yd satisfying relations ( 7 ) and (11); actually, these controls de ne the control in SDE ( 1 ). The dependence of the number N of measurable trajectories on the value " is given below. The next theorem follows from the aforesaid. Theorem 1. Problems A1 (B1) and A2 (B2) are equivalent.

Thus, to solve the original problems, one should establish some conditions of consistent solvability of the problems of "-guidance for ODEs ( 5 ) and ( 9 ) and should nd the form of concordance of parameters N and " as well. 4

The method of open-loop control packages: a brief review of results for ODE

Let us present brie y the approach by A.V. Kryazhimskii and Yu.S. Osipov to solving the problem of closed-loop guidance for a linear ODE [3], [5].

Consider a dynamical control system x_ (t) = A(t)x(t) + B(t)u(t) + f (t); t 2 T = [t0; #]; x(t0) = x0 2 X0; where x(t) 2 IRn, u(t) 2 P IRm (P is a convex compact set); A( ), B( ), and f ( ) are continuous matrix functions of dimensions n n, n m, and n 1, respectively; X0 is a nite set of possible initial states. The real initial state of the system is assumed to be unknown. A nonempty nite set St T of admissible guidance times and nonempty convex closed target sets M(t) 2 IRn, t 2 St, as well as a continuous observation function Q(t) of dimension q n are given.

The problem of guaranteed closed-loop "-guidance consists in forming by the signal y(t) = Q(t)x(t) a control guaranteeing that the system's state x(t) reaches at some admissible guidance time t 2 St the "-neighborhood of the target set M(t). This is Problem A. If St = f#g, we have Problem B. The solution of the problem is sought in the class of closed-loop control strategies with memory. The correction of the values of a control u( ) is possible at in advance speci ed times. In [3], the equivalence of the formulated problem of closed-loop control to the so-called problem of package guidance is established. Let us brie y present basic notions of the latter problem. Consider the homogeneous system x_ (t) = A(t)x(t), t 2 T = [t0; #], x(t0) = x0 2 X0; its fundamental matrix is denoted by F ( ; ). For any x0 2 X0, the homogeneous signal corresponding to x0 is the function gx0 (t) = Q(t)F (t; t0)x0, t 2 [t0; #]. The set of all admissible initial states x0 corresponding to a homogeneous signal g( ) till a time is denoted by X0( jg( )) = fx0 2 X0: gx0 ( )j[t0; ] = g( )j[t0; ]g, where g( )j[t0; ] is the restriction of the homogeneous signal g( ) onto the interval [t0; ].

A family (ux0 ( ))x02X0 of open-loop controls is called an open-loop control package if it satis es the condition of nonanticipation: for any homogeneous signal g( ), time 2 (t0; #], and admissible initial states x00; x000 2 X0( jg( )), the equality ux00 (t) = ux000 (t) holds for all t 2 [t0; ]. Any family st = (sx0 )x02X0 of elements of St is called a family of admissible guidance times. An open-loop control package (ux0 ( ))x02X0 is guiding with a family of admissible guidance times st if for any x0 2 X0, the motion from x0 corresponding to ux0 ( ) takes a value exactly in the target set M(sx0 ). If there exists an open-loop control package that is guiding with a family of admissible guidance times st, we say that the idealized problem of package guidance corresponding to the original problem of guaranteed closed-loop control is solvable with the family of admissible guidance times st. These constructions suit for both Problems A and B.

Let G be the set of all homogeneous signals. We introduce the set G0(g( )) of all homogeneous signals coinciding with g( ) in a right-sided neighborhood of the initial time t0. The rst splitting moment of the homogeneous signal g( ) is the time 1(g( )) = maxn 2 [t0; #]: g(t)kIRq = 0o: max max kg0(t) g0( )2G0(g( )) t2[t0; ] If 1(g( )) < #, then, by analogy with G0(g( )), we introduce the set G1(g( )) of all homogeneous signals from G0(g( )) coinciding with g( ) in a right-sided neighborhood of the splitting moment 1(g( )). By analogy with 1(g( )), we de ne the second splitting moment of the homogeneous signal g( ) and so on. Finally, we introduce the set of all the splitting moments of the homogeneous signal g( ): T (g( )) = f j (g( )): j = 1; : : : ; kgg, kg 1, kg (g( )) = #. Then, we consider the set (in ascending order) of all the splitting moments of all the homogeneous signals (possible switching moments for the \ideal" guiding openloop control): T = Sg( )2G T (g( )), T = f 1; : : : ; K g, K Pg( )2G kg( ) is the number of elements of the set T . Obviously, the sets T (g( )) and T are nite due to the niteness of the sets X0 and G. For any k = 1; : : : ; K, the set X0( k) = fX0( kjg( )): g( ) 2 Gg is called the cluster position at the time k; each of its elements X0k is called the cluster of initial states at this moment.

The constructions above were used for designing rather cumbersome criteria for the solvability of the original problems (A and B) mathematically based on solving nite-dimensional optimization problems; see [4] and [5] for details.

Properties of the statistical estimates

Lemma 1. For a nite set of some speci ed times i 2 T , i 2 [1 : l], the standard estimates miN of the mathematical expectation m( i) and DN of the covarii ance matrix D( i) constructed through N (N > 1) realizations x1( i); : : : ; xN ( i) of the random variables x( i) by the following rules [9]: miN = 1 XN xr( i); N r=1

DiN =

1 N 1

N X(xr( i) r=1 miN )(xr( i) miN ) ; (12) provide the validness of relation ( 4 ) (consequently, ( 7 ) and (11)).

The proof of the lemma is presented in [10]. Here, we restrict ourselves by the citation that it is possible to choose the same parameters in relations ( 4 ), ( 7 ), and (11), namely, h(N ) = ChN 1=2; g(N ) = CgN maxf 1; 1=2 3 g; (13) where 0 < < 1=2, Ch and Cg are constants. For example, if ! +0, h(N ) and g(N ) have the power exponents of the value 1=N asymptotically equal to 1=2. 6

Criteria for the solvability of the problems

Let us de ne additional notions for ODEs ( 5 ) and ( 9 ). Let G1 = fg1( )g and G2 = fg2( )g be the sets of all homogeneous signals for ( 5 ) and ( 9 ), respectively. The sets of all splitting moments of all homogeneous signals for ( 5 ) and ( 9 ) are denoted by T 1 = f 11; : : : ; K11 g and T 2 = f 12; : : : ; K22 g; the cluster positions and clusters of initial states at the times k1 and k2, by M0( k1) and M0k, D0( k2) and D0k. Recall that K11 = K22 = # and assume that 01 = 02 = t0. Let us introduce the sets of pairs of homogeneous signals splitted at the moments k1, k 2 [0 : K1 1] and k2, k 2 [0 : K2 1]: G1k = f(gi1( k1); gj1( k1))g, G2k = f(gi2( k2); gj2( k2))g, i 6= j. A moment from the interval ( k1; k1 + C"] ( k1 + C" < dk1e+n1o,teCd bisy a k1conasntdanist)c,aalltedwahidchistainllgtuhisehpinagirmsformo mentGf1kor aarlletdhiestsiinggnuailsshsapblliett,eids at the time k1. Similarly, we de ne a distinguishing moment k2 . For all k1 2 T 1, k 2 [0 : K1 1] and k2 2 T 2, k 2 [0 : K2 1], the corresponding moments k1 and k2 are de ned uniquely; at these moments, the signal's values di er in all the pairs from G1k and G2k [10]. The set of all such moments for ( 5 ) and ( 9 ) T = T 1 [ T 2 ;

T 1 = f 01 ; : : : ; K11 1g;

T 2 = f 02 ; : : : ; K22 1g; (14) determines both the aforesaid set of l (l < K1 + K2) times, at which the N trajectories of the original process are measured, and the set of times, which are possible for switching the closed-loop control. As an example, we formulate the solvability conditions for problem B2 [5]: sup (ld0 )d02D0 2S2

X B1 (s)F1 (#; s)lm0 Su1 ds m02M0k lm0 ; F1(#; s)f (s)E ds 2((ld0 )d02D0 ) 0; 2((ld0 )d02D0 ) =

X m02M0 +(lm0 jM(#)); X hld0 ; F2(#; t0)d0i d02D0 B (s)F2 (#; s)ld0 Sv ds

X d02D0 +(ld0 jD(#)): Here, (lm0 )m02M0 and (ld0 )d02D0 are families of vectors parameterized by the corresponding initial states; S1 and S2 are the sets of families (lm0 )m02M0 , P klm0 kIRn = 1 and (ld0 )d02D0 , P kld0 kI2Rnd = 1; F1( ; ) and F2( ; ) are the 2 fundamental matrices of systems ( 5 ) and ( 9 ). We present the key result of [10]. Theorem 2. Let conditions (15) be ful lled, let the information on N trajectories of SDE ( 1 ) be received at the times composing the set T (14), let the constants Ch, Cg, and be taken from Lemma 1 (see (13)) and

N > (2Ch= ("))2=(1 2 ) ; (") = minn k1 2T 1 ; (gi1;gj1)2G1k kgi1( k1 ) min

gj1( k1 )kIRq ; k2 2T 2 ; (gi2;gj2)2G2k kgi2( k2 ) min gj2( k2 )kIRnq o: Then, problem B1 is solvable with the probability 1 CgN maxf 1; 1=2 3 g and there exists an "-guiding control in equation ( 1 ) based on open-loop control packages for systems ( 5 ) and ( 9 ). 7

Illustrative example

Consider the linear SDE of the rst order: dx(t) = x(t)dt + u1(t)dt + u2(t)d (t); t 2 T = [0; 2]; u1; u2 2 [0; 1]; (17) with the unknown initial state x0 2 X0, X0 consists of four normally distributed random variables with numerical parameters (m0; d0), where the mathematical expectation m0 2 M0 = fm10; m20g, m10 = (3 e)e, m20 = e2, and the dispersion d0 2 D0 = fd01; d20g, d10 = e2=2, d20 = e4. We use the signal y(t) = Q(t)x(t); (15) (16) Let us write ODEs for the mathematical expectation and dispersion, as well as the observed signals, using the formulas from Section 3: m_(t) =

m(t) + u1(t); d_(t) = 2d(t) + u22(t); m(0) = m0 2 fm01; m02g; d(0) = d0 2 fd01; d02g;

ym(t) = Q(t)m(t); yd(t) = Q2(t)d(t): (19) (20) Let the set of admissible guidance times St = f3=2; 2g and the target sets M(3=2) = [1 + 2=pe pe; 1], M( 2 ) = f1g, D(3=2) = [1=2; 1], D( 2 ) = f1g be given. The control aim consists in forming, for an arbitrary small " > 0, an open-loop control (u1; u2) guaranteeing, whatever the initial states m0 2 M0 and d0 2 D0, by the information on N trajectories of equation (17), the attainability (with a probability close to 1) of "-neighborhoods of the target sets M(t) and D(t) by the mathematical expectation m and the dispersion d, respectively, at the moment t = 2 (Problem B2) and by the moment t = 2, i.e., at one of the moments of the set St (Problem A2). The example re ects the natural fact that Problem B2 is not solved, whereas Problem A2 is solvable.

The splitting moments of the homogeneous signals for equations (19) and (20) coincide, K = 2, 1 = 1 (then it is possible to distinguish di erent homogeneous signals), 2 = 2. In the case when the controls u1 and u22 are piecewise constant functions (u[0;1], u(1;2] and v[0;1], v(1;2], respectively), we obtain m( 2 ) = e 2m0 + (1 e 1)(e 1u[0;1] + u(1;2]), d( 2 ) = e 4d0 + (1 e 2)(e 2v[0;1] + v(1;2])=2. It follows from the form of m( 2 ) that the solution of equation (19), starting from the greater initial state m20 = e2, reaches (at t = 2) the set M( 2 ) only under the action of zero control u1 on the whole interval [0; 2], i.e., u[0;1] = u(1;2] = 0. Note that this boundary cannot be reached before the time t = 2; so the set M(3=2) is not attainable in case of the action of zero control u1. At the same time, if the real initial state coincides with the smaller possible value m10 = (3 e)e, then, after the necessary action of zero control till the splitting moment t = 1, the choice of u(1;2] = 1 cannot already force the trajectory to reach the set M( 2 ). However, it is easily seen that the lower boundary of the set M(3=2) can be reached at the time t = 3=2. A similar argument is applicable to equation (20). The open-loop controls for equations (19), (20) solving Problem B2 are unique.

We pass to constructing the closed-loop control using the open-loop control package. Note that, since the splitting moments for equations (19) and (20) coincide, it is su cient to perform measurements of N (N > 1) trajectories x1( ); : : : ; xN ( ) of the original SDE at the unique distinguishing moment = 1 + "; the zero controls are fed onto equations (19) and (20) till this time. Then, we construct by (12) the estimates ymN( ) and ydN ( ) of the signals ym( ) and yd( ) satisfying the relations like ( 7 ), (11), and (13):

P max ymN( ) ym( ) ; ydN ( ) yd( ) Let us derive the condition providing the detection at the time of the real initial states of equations (19) and (20) (m10 or m20 and d10 or d02) and, consequently, of equation (17). Actually, taking into account that u1(t) = 0; u22(t) = 0; t 2 [0; 1 + "], and the form of Q, we should distinguish the values ym1( ) = y"ed2( ( 1 )+"=)m"210ean2(d1+y"m2)d(20. )Th=ere"feor(e1,+N")mm20u,stasbewseullchasthyad1t( ) = "2e 2(1+")d10 and h(N ) < min n("e (1+")jm01 (21) Then, only one of the inequalities ymN( ) ymi( ) h(N ), i = 1; 2, holds with the probability 1 g(N ); the same is valid for the inequalities ydN ( ) ydi( ) h(N ), i = 1; 2. In case equation (19) starts from the initial state m10, we decide to apply the control u1(t) = 1 on the interval (1 + "; 3=2); otherwise (from the state m20), the control u1(t) = 0, t 2 (1 + "; 2). In the rst variant, in view of the time delay in switching the control to optimal, m(3=2) takes a value at the "-neighborhood of the set M(3=2). In the second variant, as a result, we have exactly m( 2 ) = 1. By analogy, we proceed with equation (20): if the real initial state is d1, then we apply the control u22(t) = 1 on the interval (1 + "; 3=2); if 0 d20, then u22(t) = 0, t 2 (1 + "; 2). In the rst case, d(3=2) takes a value at the "-neighborhood of the set D(3=2). In the second case, we have exactly d( 2 ) = 1.

Thus, the closed-loop control method described above solves the original "-guidance problem: it guarantees the attaintment of the solution of equation (19) to the "-neighborhood of the target set M(t), t 2 St, and the attaintment of the solution of equation (20) to the "-neighborhood of the target set D(t), t 2 St, with a probability close to 1. The computations by formulas (13) and (21) showed that N = 103 guarantees the guidance accuracy " = 0:1 with a probability P 0:95, whereas N = 105 guarantees " = 0:01 with P 0:995.

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