Differential Guidance Game with Incomplete Information on the State Coordinates and Unknown Initial State

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1 ISSN , Differential Equations, 2015, Vol. 51, No. 12, pp c Pleiades Publishing, Ltd., Original Russian Text c V.I. Maksimov, 2015, published in Differentsial nye Uravneniya, 2015, Vol. 51, No. 12, pp CONTROL THEORY Differential Guidance Game with Incomplete Information on the State Coordinates and Unknown Initial State V. I. Maksimov Ural Federal University, Yekaterinburg, Russia Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, Russia maksimov@imm.uran.ru Received April 29, 2015 Abstract We study the problem of guaranteed positional guidance of a linear partially observable control system to a convex target set at a given time. The problem is considered in the case of incomplete information. More precisely, it is assumed that the system is subjected to some unknown disturbance; in addition, the initial state is unknown as well. But the sets of admissible disturbances and the set of admissible initial states are known. The latter is assumed to be finite. We construct an algorithm for solving this problem. DOI: /S INTRODUCTION. STATEMENT OF THE PROBLEM We study the differential game of guidance (the problem of guaranteed positional guidance) [1 4] to a given target set at a fixed terminal time. We present an algorithm for solving this problem under the assumption that part of the state coordinates are measured at discrete sufficiently frequent times. The algorithm is based on the extremal shift method well known in the theory of positional differential games [1, pp ; 2, pp ], the dynamic inversion theory [5 7], and the control package method [8 11]. Consider the control system ẋ(t) =A(t)x(t)+Bu Cv + f(t); (1) here t T =[,ϑ]isthetimevariable, <ϑ< ; x(t) R n is the state of the system at time t; u(t) R m is a control of the ally player at that time moment; v(t) R q is a control of the enemy player; A( ) andf( ) are given Lipschitz functions on T ranging in the space of n n matrices and in R n, respectively; and B and C are constant n m and n q matrices, respectively. We assume that the controller (the ally player) knows a priori that the actual value ˆx 0 of the initial system state lies in a given finite set X 0 R n of admissible initial states. The actual initial state of the system itself is unknown. In addition, the set V of admissible controls of the enemy player is known. In the present paper, for such a set we take some (fixed) set of absolutely continuous functions defined on T and ranging in a convex compact set V R q. Below enemy player s controls are referred to as disturbances. A preset control (a program) of the ally player is understood as any Lebesgue measurable function u( ) : T U. HereU R m is a convex compact set describing the instantaneous control resource. The set of all preset controls is denoted by U. The motion of system (1) corresponding to an admissible initial state x 0 X 0, a program u( ) U, and a perturbation v( ) Vis a (Carathéodory) solution of the system of differential equations (1) defined on the interval T and satisfying the initial condition x( )=x 0 ; this motion is denoted by x( x 0,u( ),v( )). 1656

2 DIFFERENTIAL GUIDANCE GAME WITH INCOMPLETE INFORMATION 1657 Let M R n be a given nonempty convex closed target set, and let a continuous observation matrix Q(t) ofdimensionr n be defined for each t T. The controller (the ally player) has to solve the problem of guaranteed positional guidance at time ϑ: bring the state x(t) of the system at time t ϑ into a small neighborhood of the target set M. In the course of motion, the controller forms its control positionally by observing the current signal y(t) = Q(t)x(t) on the system state x(t). In accordance with the formalization accepted in positional control theory [1, pp ; 2, pp ], a positional control strategy permits the controller to correct the values of the control u( ) at fixed times <t 1 < <t m = ϑ. At each time t j (j =0,...,m 1), the values of the control on the half-open interval t [t j,t j+1 ) are determined on the basis of the prehistory t y(t) of the observation on the interval [,t j ] and the prehistory t u(t) of the control on the half-open interval [,t j ). (For j = 0 the control prehistory is absent.) Therefore, the problem of guaranteed positional guidance is as follows: for an arbitrary given ε>0, choose a positional control strategy such that, for an arbitrary admissible initial state x 0 X 0 and for any disturbance v( ) V,themotionx( ) of system (1) issuing from this state with the chosen positional strategy gets at time t = ϑ into the ε-neighborhood of the target set M. Forthe case in which the initial state of the system is known, a similar problem was studied in [12 14] without using the control package approach. Throughout the following, we assume that system (1) has the structure ż(t) =A 1 (t)z(t)+b 1 u(t) C 1 v(t)+f 1 (t), (2) ṗ(t) =D 1 (t)z(t)+e 1 (t)p(t)+b 2 u(t) C 2 v(t)+f 2 (t). (3) ( ) ( ) ( ) A Here x = {z,p}, z R n1, p R n n1, A(t) = 1 (t) 0 B, B = 1 C, C = 1 ( ) D 1 (t) E 1 (t) B 2 C 2,and f(t) = f 1 (t) f 2 (t). The set X 0 of initial states has the form X 0 = {z 0,P 0 },wherep 0 R n n1 is a finite set and z 0 is a given vector known to the ally player. In this case, we assume that the ally player knows the initial state of the subsystem (2) but does not knows the initial state of the subsystem (3). Let n 1 <r, and let the matrix Q(t) havetheformq(t) = diag(e,g(t)), where E is the n 1 ( n 1 identity ) matrix and G(t) is a given continuous (r n 1 ) (r n 1 ) matrix. Consequently, y(t) = z(t) Gp(t). The results of measurements of the variables y(t j )atthetimest j are given by the vectors {ξj h,ψh j },where ( ξ h j z(τ j ) 2 n 1 + ψ h j Gp(τ j ) 2 r n 1 ) 1/2 h. (4) Here h (0, 1) is the measurement error, and n1 is the Euclidean norm on the space R n1. We everywhere assume that the following conditions are satisfied, the first of which is the domination of resources of the ally player over the resources of the enemy player. Condition 1. There exists a set W R n such that BU = W + CV. Condition 2. The relations q n 1 and rank C 1 = q hold. Condition 3. The set V of admissible disturbances consists of functions v(t) : T V whose variations are bounded by some number d. 2. AUXILIARY RESULTS Before proceeding to the description of the algorithm for solving the problem, we present some results in [10, 11] in a form convenient to us. Consider a control system of the form ẋ(t) =A(t)x(t)+w(t)+f(t). (5) A preset control (a program) [for system (5)] is defined as an arbitrary Lebesgue measurable function w( ) : T W. Let us introduce the fundamental matrix F (, ) of the homogeneous system

3 1658 MAKSIMOV ẋ(t) =A(t)x(t). For each x 0 X 0,let g x0 (t) =Q(t)F (t, )x 0 (t T ); the function g x0 ( ) is called the homogeneous signal corresponding to an admissible initial state x 0. The homogeneous signal corresponding to some admissible initial state is simply called a homogeneous signal. The set of all admissible initial states x 0 corresponding to a homogeneous signal g( ) up to time τ [,ϑ] will be denoted by X 0 (τ g( )); thus, X 0 (τ g( )) = {x 0 X 0 : g( ) [t0,τ] = g x0 ( ) [t0,τ]}; here and in the following, g( ) [t0,τ], whereτ [,ϑ], is the restriction of the homogeneous signal g( ) totheinterval[τ 0,τ]. A family of programs (w x0 ( )) x0 X 0 is called a control package if it satisfies the following nonanticipation condition: for any homogeneous signal g( ), any time τ (,ϑ], and arbitrary initial states x 0,x 0 X 0 (τ g( )), the relation w x 0 (t) =w x 0 (t) holdsforallt [τ 0,τ). A package (w x0 ( )) x0 X 0 is said to be guiding [for system (5)] if the inclusion x(ϑ x 0,w x0 ( )) M holds for any x 0 X 0. If there exists a guiding program package, then we say that the package guidance problem is solvable. Let G be the set of all homogeneous signals. Since this set is finite, it follows that, for each homogeneous signal g( ), there exists an index k g( ) 1 such that τ kg( ) (g( )) = ϑ. For each homogeneous signal g( ), we introduce the set T (g( )) = {τ j (g( )) : j =1,...,k g( ) } of all its branching times and write T = T (g( )). g( ) G Then [10] the set T can be represented in the form T = {τ 1,...,τ K },whereτ j <τ j+1 (i = j,...,k 1). Set τ 0 =. For each k =1,...,K, we introduce the set X 0 (τ k )={X 0 (τ k g( )) : g( ) G}. (6) The elements X 0,k of the set X 0 (τ k ) are referred to as the clusters of initial states at time τ k. For each k =0,...,K, the clusters of initial states at time τ k form a partition of the set X 0 of all admissible initial states; i.e., X 0 = X 0,k X 0(τ k ) X 0,k, X 0,k X 0,k = (X 0,k,X 0,k X 0 (τ k ), X 0,k X 0,k). (7) Let W be the set of all families (w x0 ) x0 X 0 of vectors in W. Any (Lebesgue) measurable function t (w x0 (t)) x0 X 0 : T Wis called an extended program. Just as in [9 11], it is natural to identify a family of programs (w x0 ( )) x0 X 0 with the extended program t (w x0 (t)) x0 X 0.Foreach k = 0,...,K, we introduce the set W k of all families (w x0 ) x0 X 0 W such that the relation w x 0 = w x 0 holds for any cluster X 0,k X 0 (τ k ) and for arbitrary initial states x 0,x 0 X 0,k. An extended program (w x0 ( )) x0 X 0 is said to be admissible if, for each k =0,...,K, the inclusion (w x0 (t)) x0 X 0 W k holds for all t (τ k 1,τ k ] provided that k>1andforallt [,τ 1 ]provided that k =1. Remark 1. If we consider a family of programs parametrized by the initial states of system (1), that is, a family (w x0 ( )) x0 X 0, then we mean the set of functions w x0 ( ) : T R n, x 0 X 0.But if we speak of an extended program, then we mean the function t (w x0 (t)) x0 X 0 W consisting of all functions of the family (w x0 ( )) x0 X 0. Here the dimension of the set W is equal to r N X0, where N X0 is the number of vectors in the set X 0. For j =1, 2,..., we define the extended space R j as the finite-dimensional Hilbert space of all families l =(l x0 ) x0 X 0 of vectors in R j with inner product, Rj of the form l,l Rj = x 0 ) (l =(l x 0 ) x0 X 0 R j, l =(l x 0 ) x0 X 0 R j ). x 0 X 0 (l x 0,l Here and in the following, (, ) is the inner product on a finite-dimensional Euclidean space. The values of extended programs are treated as elements of the space R n.

4 DIFFERENTIAL GUIDANCE GAME WITH INCOMPLETE INFORMATION 1659 Consider the extended system, which consists of copies of system (5) parametrized by the initial states x 0 X 0 ; the copy of the system corresponding to the parameter x 0 issues from the initial state x 0 under the action of the preset control w x0 ( ). We rewrite the extended system in the form ẋ x0 (t) =A(t)x x0 (t)+w x0 (t)+f(t), x x0 ( )=x 0 (x 0 X 0 ). (8) We define the state space of the extended system to be R n. We choose the control in the extended system from the class of all admissible extended programs. For each admissible extended program t (w x0 (t)) x0 X 0, the corresponding motion of the extended system is understood as the function t (x(t x 0,w x0 ( ))) x0 X 0 : T R n. The extended target set is defined as the set M of all families (x x0 ) x0 X 0 R n such that x x0 M for all x 0 X 0. We say that an admissible extended program t (w x0 (t)) x0 X 0 is guiding for the extended system if the motion (x( x 0,w x0 ( ))) x0 X 0 of the extended system corresponding to t (w x0 (t)) x0 X 0 satisfies the condition (x(ϑ x 0,w x0 ( ))) x0 X 0 M. We say that the extended problem of program guidance is solvable if there exists an extended guiding program for the extended system. The following assertion was proved in [9, 10]. Theorem 1. (i) The extended program t (w x0 (t)) x0 X 0 is a control package if and only if it is admissible. (ii) An admissible extended program is a guiding control package if and only if it is guiding for the extended system. (iii) The package guidance problem is solvable if and only if the extended package guidance problem is solvable. The following solvability criterion was obtained in [10] for the extended program guidance problem: sup (l x0 ) x0 X 0 L γ 1 ((l x0 ) x0 X 0 ) 0, (9) where γ 1 ((l x0 ) x0 X 0 )= ( ϑ ) l x0,f(ϑ, )x 0 + F (ϑ, τ)f(τ) dτ x 0 X 0 + K τ k k=1 τ X 0,k X (τ k ) k 1 D(τ) =B (τ)f (ϑ, τ), ) ϱ (D(τ) l x0 U dτ p + (l x0 M), (10) x 0 X 0,k x 0 X 0 ϱ (l U) =inf{(l, x) : x U} and ϱ + (l M) =sup{(l, x) : x M} are the upper and lower support functions, respectively, the prime stands for transposition, and L is some set in the extended state space R n, whose properties can be found in [10]. As was shown in [11], if condition (9) holds for a function γ 1 of the form (10) and the zero extended program t (wx 0 0 (t)) x0 X 0 W (wx 0 0 (t) =0 for almost all t T and for all x 0 X 0 ) is not guiding for the extended system, then there exists anumbera (0, 1] such that max γ a ((l x0 ) x0 X 0 )=0. (11) (l x0 ) x0 X 0 L The following assertion was proved in [11]. Theorem 2. Let W be a strictly convex set containing the zero element, let condition (9) be satisfied, and let (lx 0 ) x0 X 0 L be a vector maximizing the expression (11); moreover, let the vector D(τ) x 0 X 0,k lx 0 be nonzero for all τ T. In addition, let the zero extended program be not guiding (for the extended system), let the extended program t (w x0 (t)) x0 X 0 satisfy the condition w x0 (t) aw (for all x 0 X 0 ), and let the relation ( ) ( ) D(τ) l x 0,w X0,k (τ) = ϱ D(τ) l x 0 aw x 0 X 0,k x 0 X 0,k (τ [τ k 1,τ k )) (12)

5 1660 MAKSIMOV hold for an arbitrary k =1,...,K and an arbitrary cluster X 0,k X(τ k ). Then the extended program t (w x0 (t)) x0 X 0 is guiding for system (8). Remark 2. Let (w x0 ( )) x0 X 0 be the guiding control package for system (5) corresponding to an extended program for system (8). (The latter is defined in accordance with the rule described in Theorem 2.) Assume that a cluster X 0,j+1 = X 0,j+1 (τ j+1 g x0 ( )) [see (7)] in the cluster position X 0 (τ j+1 ) [see (6)] containing the initial state ˆx 0 can somehow be identified at the times τ j + γ, γ (0,τ j+1 τ j ), j =0,...,K 1. Then, as follows from the definition of a control package, all controls in the guiding control package corresponding to distinct elements of the cluster X 0,j+1 differ from each other at most on sets of zero measure on the interval [τ j,τ j+1 ). Therefore, one can assume that, to all initial states of system (8) in one cluster X 0,j+1 on [τ j,τ j+1 ), there corresponds a single control in the restriction of the control set (w x0 ( )) x0 X 0 to the half-open interval [τ j,τ j+1 ); this control will be denoted by w X0,j+1( ), where X 0,j+1 = X 0,j+1 (τ j+1 gˆx0 ( )). Theorem 2, together with Remarks 1 and 2, implies the following assertion. Corollary 1. The control package corresponding to the extended program defined by Theorem 2 is guiding for system (5). We will use the following assertion proved in [7, pp ]. Lemma 2. Let u( ) L (T ; R n ),v( ) W (T ; R n ),T =[a, b], <a<b<+, and t u(τ) dτ ε, v(t) n K, t T. Then the inequality a t a n (u(τ),v(τ)) dτ ε(k +var(t ; v( ))) holds for all t T, where var(t ; v( )) is the variation of the function v( ) on the interval T and W (T ; R n ) is the set of functions y( ) : T R n of bounded variation. 3. SOLUTION ALGORITHM Take a family of partitions of the interval T : Δ h = {t i,h } m h i=0, t i+1,h = t i,h + δ(h),,h =, t mh,h = ϑ, h (0, 1), and a function α = α(h) : (0, 1) (0, 1). Fix a number γ>0. By t i(j),h we denote the point of the partition Δ h that coincides with τ j ; i.e., t i(j),h = τ j.let t i(j,γ),h =max{t i,h : t i,h <τ j + γ}, and let ˆx 0 X 0 be the actual (unknown) initial state. For brevity, below we set t i,j,γ = t i(j,γ),h. Along with system (1), we introduce three auxiliary systems. The first system is described by the vector equation ġ(t) =A 1 (t i,h )ξ h i + B 1u h (t) C 1 v h (t)+f 1 (t i,h ) for t [t i,h,t i+1,h ), i {0,...,m h 1}, (13) with the initial state g( )=z 0, the second is described by the vector equation ġ 1 (t) =A(t)g 1 (t)+bu h (t) Cv h (t)+f(t), t T, (14) and the third system is described by the vector equation ġ 2 (t) =A(t)g 2 (t)+bũ h (t) Cv h (t)+f(t), t T, (15)

6 DIFFERENTIAL GUIDANCE GAME WITH INCOMPLETE INFORMATION 1661 with the initial states g 1 ( )=g 2 ( ) = 0. Here and in the following, g R n1, g 1 = {g 11,g 12 }, g 2 = {g 21,g 22 }, g 11,g 21 R n1,andg 12,g 22 R n n1. Before starting the algorithm operation, we fix h (0, 1), a partition Δ h, and a number α = α(h). We split the algorithm operation into m h 1 similar steps. The following actions are performed on the interval [τ j,τ j+1 )(j =0,...,K 1). For t [t i,h,t i+1,h ), we set where u h i U(ψh 1i,ψh 2i,ηh 1i,ηh 2i ), vh i V(g(t i,h ),ξ h i ) if t [t i,h,t i+1,h ), (16) V(g(t i,h ),ξ h i )={v i V : 2(g(t i,h ) ξ h i, C 1 v i )+α v i 2 q min{2(g(t i,h ) ξ h i, C 1 v)+α v 2 q : v V } + h}, U(ψ1i,ψ h 2i,η h 1i,η h 2i) h ={u i U : (η h 1i ψ1i,b h 1 u i )+(η h 2i ψ2i,b h 2 u i ) min{(η h 1i ψh 1i,B 1u)+(η h 2i ψh 2i,B 2u) : u U} + h}, ( ψ h 1i g 21 (t i,h ) 2 n 1 + ψ h 2i g 22 (t i,h ) 2 n n 1 ) 1/2 h, ( η h 1i g 11 (t i,h ) 2 n 1 + η h 2i g 12 (t i,h ) 2 n n 1 ) 1/2 h. (17) For the same values of t, thecontrol u h (t) =u h i (18) of the form (16) (18) is fed to the input of system (1), and two controls, u h (t) and v h (t) =v h i, (19) are fed to the input of system (13). At time t i,j,γ, we find the vector φ(t i,j,γ )=π(t i,j,γ ) Q(t i,j,γ ) t i,j,γ F (t i,j,γ,τ){bu h (τ) Cv h (τ)+f(τ)} dτ, where π(t i,j,γ )={ξi(j,γ),h h,ψh i(j,γ),h }. In the set of values of homogeneous signals g x 0 (t i,j,γ )(x 0 X 0 ), we find the vector g x0,i,j =argmin{ g x0 (t i,j,γ ) ϕ(t i,j,γ ) r : x 0 X 0 }. (20) For the vector g x0,i,j, we find some cluster X 0,j+1 such that the corresponding value of the homogeneous signal coincides with g x0,i,j at time t = t i,j,γ. Next, for the cluster X 0,j+1 at time t = t i,j,γ, we compute the function w X0,j+1 (t), t [t i,j,γ,τ j+1 ), that is, the corresponding control in the guiding control package, which is found by the rule (12), for system (8). In turn, the control v h (t) of the form (19) and the control u h (t) of the form (18) are fed to the input of system (14), and the controls v h (t) andũ h (t), ũ h (t) =ũ h i (t) if t [t i,h,t i+1,h ) [t i,j,γ,τ j+1 ), ũ h (t) =ũ h j if t i(j),h t<t i,j,γ, are fed to the input of system (15); here ũ h j U is an arbitrary vector, and ũh j (t) is a function such that Bũ h i (t) =Cv h i + w X0,j+1 (t) if t [t i,h,t i+1,h ) [t i,j,γ,τ j+1 ). The operation of the algorithm is terminated at time t = ϑ. Let x( ˆx 0,u h ( ),v( )) be the solution of system (1) induced by an unknown disturbance v( ) V and a control u h ( ), and let M ε be the ε-neighborhood of the set M. Justasabove,ˆx 0 is the actual (unknown) initial state of the system. Condition 4. One has α(h) 0and(δ(h)+h)α 1 (h) 0ash 0.

7 1662 MAKSIMOV Theorem 3. Let Conditions 1 4 be satisfied, and let the assumptions of Theorem 2 hold. Then there exists a number γ > 0 with the following property : for each ε>0, there exists an h = h (ε, γ) (0, 1) such that, for any v( ) V, the inclusion holds for all h (0,h ). x(ϑ x 0,u h ( ),v( )) M ε (21) Proof. From the results in the monograph [7] [see (1.2.26) and (1.2.27)], we obtain the inequalities v h ( ) 2 L 2(T ;R q ) v( ) 2 L 2(T ;R q ) + c 1(δ + h)α 1, (22) max t T g(t) 2 n 1 α v( ) L2(T ;R q ) + c 2 (δ + h). (23) To prove these inequalities, we use inequalities (4). Let We have β h (t i ) n1 = β h (t i )= k=1 t i C 1 [v h (t) v(t)] dt, t i = t i,h. i t k {ġ(t) ż(t) A 1 (t k 1)ξ h k + A 1 (t)z(t)} dt. (24) t k 1 n 1 Relation (24), together with inequality (23), implies the estimate β h (t i ) n1 c 3 (α + δ + h) 1/2 ; (25) in addition, β h (t i ) β h (t) n1 c 4 δ for t [t i,t i+1 ]. (26) In this case, by taking into account Condition 2 and the estimates (25) and (26), for all t T, we obtain the inequality μ t [v h (τ) v(τ)] dτ q By virtue of relation (22), we have the inequality ϑ v( ) v h ( ) 2 L 2(T ;R q ) 2 v( ) 2 L 2(T ;R q ) 2 which, together with Lemma 1 and Condition 3, implies that ϑ v( ) v h ( ) 2 L 2(T ;R q ) 2 t C 1 [v h (τ) v(τ)] dτ c 5 (α + δ + h) 1/2. n 1 (v(τ),v h (τ)) dτ + c 1 (δ + h)α 1, (27) (v(τ) v h (τ),v(τ)) dτ + c 1 (δ + h)α 1 μ(h) =c 5 (α + δ + h) 1/2 (d(v )+d )+c 1 (δ + h)α 1, (28) where d(v )=sup{ v q : v V }.

8 DIFFERENTIAL GUIDANCE GAME WITH INCOMPLETE INFORMATION 1663 Let Φ(t i,j,γ )=y(t i,j,γ ) Q(t i,j,γ ) t i,j,γ F (t i,j,γ,τ){bu h (τ) Cv(τ)+f(τ)} dτ. Then, by virtue of the estimates (4) and (28), we have the inequality ϕ(t i,j,γ ) Φ(t i,j,γ ) r h + d 1 t i,j,γ v(τ) v h (τ) q dτ where d 1 = h + d 1 (ϑ ) 1/2 v( ) v h ( )) L2([,t i,j,γ];r q ), (29) max τ t ϑ F (t, τ) C max Q(t). t T Here is the Euclidean norm of a matrix. One can readily see that Q(t i,j,γ )F (t i,j,γ, )ˆx 0 =Φ(t i,j,γ )=gˆx0 (t i,j,γ ). (30) By following [11] (see Lemma 5) and by using Theorem 2, one can readily show that the inclusion F (ϑ, )ˆx 0 + g 2 (ϑ) M Kγ (31) holds for the above-mentioned choice of the control ũ h (t), where the constant K can be written out in closed form. This constant depends only on the sets X 0, U, andv, the matrices A( ), B, C, andq( ), and the function f( ) but is independent of h, δ, v h ( ), and ũ h ( ). Since the set X 0 is finite and the homogeneous signals are continuous functions of t, it follows that, for each γ>0, there exists a number ν γ > 0 such that g x (τ j + γ) g x (τ j + γ) r ν γ, j =0,...,K, τ j + γ ϑ, (32) provided that x X 0 and x X 0 are elements such that τ j is a branching point of the signals g x ( ) andg x ( ); i.e., g x (t) =g x (t) fort [,τ j ]andg x (t) g x (t) fort (τ j,τ j + γ]. Let Kγ < ε/3, and let h 1 (0, 1) be a number such that the inequality h + d 1 (ϑ ) 1/2 μ 1/2 (h) 0.5ν γ holds for all h (0,h 1 ). The existence of such a number follows from Condition 4. By virtue of inequalities (28) and (29), we have the estimate ϕ(t i,j,γ ) Φ(t i,j,γ ) r 0.5ν γ. (33) Next, by virtue of relations (20), (32), and (33), in this case, we have the relation g x0,i,j = gˆx0 (t i,j,γ ). Since τ j <t i,j,γ and the homogeneous signal gˆx0 ( ) has no branching moment up to time τ j+1, we have the inclusion ˆx 0 X 0,j+1 = X 0,j+1 (τ j+1 gˆx0 ( )) = {x 0 X 0 : g x0 ( ) [t0,τ j+1] = gˆx0 ( ) [t0,τ j+1]}. Therefore, at time t = t i,j,γ, on the basis of g x0,i,j, wedeterminetheclusterx 0,j+1 that contains the unknown initial state ˆx 0 of system (1). Let us estimate the change in the variable ε(t) = g 1 (t) g 2 (t) 2 n.

9 1664 MAKSIMOV We have ( ε(t i+1,h ) ε(t i,h )+2 ν(t i,h ), t i+1,h {A(τ)(g 1 (τ) g 2 (τ)) + Bu h (τ) Bũ h (τ)} dτ + c 6 δ 2, ) t i,h where ν(t i,h )=g 1 (t i,h ) g 2 (t i,h ). Hence we obtain the inequality where ε(t i+1,h ) ε(t i,h )+c 7 g 2 (t i,h ) g 1 (t i,h ) 2 n δ + c 8δ(δ + h)+2 t i+1,h t i,h π i (τ) dτ, (34) π i (τ) =(Ξ h i Ψ h i,bu h (τ) Bũ h (τ)), Ξ h i = {η h 1i,η h 2i}, Ψ h i = {ψ h 1i,ψ h 2i}. Then, by taking into account relations (16) (19), we obtain From the estimates (34) and (35), we obtain the inequality π i (τ) c 9 h. (35) ε(t i+1,h ) (1 + c 7 δ)ε(t i,h )+c 9 δ(δ + h). By using the lemma in [15] and by taking into account inequality (28), we obtain ε(t i+1,h ) {ε( )+c 9 (t i+1,h )(δ + h)} exp{c 7 (t i+1,h )} μ 1 (h) c 10 (h + δ). Let h 2 (0,h 1 )beanumbersuchthat μ 1/2 1 (h) ε/3 (36) for all h (0,h 2 ). Then, by virtue of relations (31) and (36) and the inequality Kγ < ε/3, the inclusion F (ϑ, )ˆx 0 + g 1 (ϑ) M (2/3)ε (37) holds for all h (0,h 2 ). Next, we have F (ϑ, )ˆx 0 + g 1 (ϑ) x(ϑ ˆx 0,u h ( ),v( )) n ϑ F (ϑ, τ) C v h (τ) v(τ) q dτ d 0 (ϑ ) 1/2 v h ( ) v( ) L2(T ;Rq), (38) where d 0 = C max t0 τ ϑ F (ϑ, τ). Leth 3 (0,h 2 )beanumbersuchthat d 0 (ϑ ) 1/2 μ 1/2 (h) ε/3 for all h (0,h 3 ). Then, by virtue of relations (37) and (38), the inclusion (21) holds for all h (0,h 3 ). The proof of the theorem is complete. ACKNOWLEDGMENTS The research was supported by the Russian Scientific Foundation (project no ) and the Russian Fund for Humanities (project no a).

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