Dynamical Input Reconstruction Problem for a Quasi-Linear Stochastic System⁎

17th IFAC Workshop on Control Applications of Optimization 17th on Control Applications of Optimization 17th IFAC IFAC Workshop Workshop onOctober Control 15-19, Applications Optimization Yekaterinburg, Russia, 2018 of 17th IFAC Workshop onOctober Control 15-19, Applications of online Optimization Yekaterinburg, Russia, 2018 Available at www.sciencedirect.com Yekaterinburg, Russia, October 15-19, 2018 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018

ScienceDirect

IFAC PapersOnLine 51-32 (2018) 727–732

Dynamical Input Reconstruction Problem Dynamical Input Reconstruction Problem Dynamical Input Reconstruction Problem ⋆⋆ for a Quasi-Linear Stochastic System Dynamical Input Reconstruction Problem for a Quasi-Linear Stochastic System for a Quasi-Linear Stochastic System ⋆⋆ for a Quasi-Linear Stochastic System ∗ Valeriy L. Rozenberg ∗

∗ Valeriy L. Rozenberg Valeriy Valeriy L. L. Rozenberg Rozenberg ∗∗ ∗ Valeriy L. Rozenberg Krasovskii Institute Branch of ∗ ∗ Institute of of Mathematics Mathematics and and Mechanics, Mechanics, Ural Ural Branch of Krasovskii Institute of Mathematics and Mechanics, Ural Branch of ∗ Krasovskii the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, ∗ the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Krasovskii Institute of Mathematics andul. Mechanics, Ural Branch of Yekaterinburg, 620990, Russia (e-mail: [email protected]) the Russian Academy of Sciences, S. Kovalevskoi 16, Yekaterinburg, 620990, Russia (e-mail: Yekaterinburg, 620990, (e-mail: [email protected]) the Russian Academy of Sciences, ul. [email protected]) S. Kovalevskoi 16, Yekaterinburg, 620990, Russia Russia (e-mail: [email protected]) Yekaterinburg, 620990, Russia (e-mail: [email protected]) Abstract: The problem of reconstructing unknown inputs in quasi-linear stochastic system Abstract: The problem of reconstructing reconstructing unknown unknown inputs inputs in in aaa quasi-linear quasi-linear stochastic stochastic system system Abstract: The problem of with diffusion depending on the phase state is investigated by means of the approach of Abstract: The problem of reconstructing unknown inputs in a quasi-linear stochastic system with diffusion depending on the phase state is investigated by means of the approach of with diffusion depending on the phase state is investigated by means of the approach of Abstract: The problem inversion. ofonreconstructing unknown inputsthe in asimultaneous quasi-linear stochastic system the theory of dynamic The statement when reconstruction of with diffusion depending the phase state is investigated by means of the approach the theory of dynamic inversion. The statement when the simultaneous reconstruction of the theory of dynamic inversion. The statement when the simultaneous reconstruction of with diffusion depending on the and phase state is terms investigated by means of reconstruction the approach of disturbances in the deterministic stochastic of the system is performed from the the theory of dynamic inversion. The statement when the simultaneous of disturbances indynamic the deterministic deterministic and stochastic terms ofthe thesimultaneous system is is performed performed from the the disturbances in the and stochastic terms of the system from the theory of inversion. The statement when reconstruction of discrete information on number of realizations the stochastic process is considered. disturbances in the deterministic stochastic of terms the system is performed from The the discrete information information on aa a number number and of realizations realizations of the of stochastic process is considered. considered. The discrete on of of the stochastic process is disturbances in the deterministic and stochastic terms of the system is performed from The the problem is reduced to an inverse problem for ordinary differential equations describing discrete information on a number of realizations of the stochastic process is considered. The problem information is reduced reduced to to an inverse of problem for ordinary ordinary differentialprocess equations describing The the problem is an inverse problem for differential equations describing discrete on a number realizations of the stochastic is considered. mathematical expectation and covariance matrix of the process. A finite-step software-oriented problem is reduced to an and inverse problem for ordinary differential equations describing the the mathematical expectation covariance matrix of the process. A finite-step software-oriented mathematical expectation and covariance matrix of the process. A finite-step software-oriented problem is reduced to anonand inverse problem for ordinary differential equations describing the solving algorithm based the method of auxiliary is proposed. The key mathematical expectation matrix of thecontrolled process. Amodels finite-step software-oriented solving algorithm algorithm based on onand thecovariance method of of auxiliary controlled models is proposed. proposed. The key key solving based the method auxiliary controlled is The mathematical expectation covariance matrix of the process. Amodels finite-step software-oriented result of the paper is an estimate for the convergence rate of the algorithm with respect to the solving algorithm based on the method of auxiliary controlled models is proposed. The key result of ofalgorithm the paper paper based is an an estimate estimate for the the convergence convergence rate of the the models algorithm with respect to key the result the is for of algorithm respect to the solving on the method of auxiliary rate controlled is with proposed. The number of measurable realizations. result of the paper is an estimate for the convergence rate of the algorithm with respect to the number of measurable realizations. number of measurable realizations. result of of the paper is an estimate for the convergence rate of the algorithm with respect to the number measurable realizations. © 2018, IFAC (International Federation of quasi-linear Automatic Control) Hosting by Elsevier Ltd. All rights reserved. number of measurable realizations. Keywords: Dynamical reconstruction, stochastic system, auxiliary model. Keywords: Dynamical Dynamical reconstruction, reconstruction, quasi-linear quasi-linear stochastic stochastic system, system, auxiliary auxiliary model. model. Keywords: Keywords: Dynamical reconstruction, quasi-linear stochastic system, auxiliary model. Keywords: Dynamical reconstruction, quasi-linearknown stochastic system, auxiliary model. 1. INTRODUCTION of 1. INTRODUCTION known deterministic deterministic disturbances disturbances acting acting in in aaa system system of 1. INTRODUCTION known deterministic disturbances acting in system of linear stochastic differential equations (SDEs), see Rozen1. INTRODUCTION knownstochastic deterministic disturbances acting in a see system of linear differential equations (SDEs), Rozenlinear stochastic differential equations (SDEs), see Rozen1.reconstructing INTRODUCTION known deterministic disturbances acting in aproblem system of berg (2007, 2016). In Rozenberg (2007), the linear stochastic differential equations (SDEs), see RozenThe necessity of unknown parameters berg (2007, 2016). In Rozenberg (2007), the problem of The necessity of reconstructing unknown parameters berg (2007, 2016). In Rozenberg (2007), the problem of Thecontrolled necessity systems of reconstructing reconstructing unknown parameters linear stochastic differential equations (SDEs), see Rozendynamic reconstruction of a disturbance entering into berg (2007, 2016). In Rozenberg (2007), the problem of of in real time based on inaccuThe necessity of unknown parameters dynamic reconstruction of a disturbance entering into of controlled systems in real time based on inaccudynamic reconstruction of a disturbance entering into of controlled systems in real time based on inaccuberg (2007, 2016). In Rozenberg (2007), the problem ofa the Ito integral and characterizing the amplitude of The necessity systems ofinformation reconstructing unknown parameters dynamic reconstruction of a disturbance entering into rate/incomplete on the phase state arises of controlled in real time based on inaccuthe Ito integral and characterizing the amplitude of a rate/incomplete information on the phase state arises the Ito integral and characterizing the amplitude of a rate/incomplete information on the phase state arises dynamic reconstruction of a disturbance entering into random noise was considered for a linear SDE on the of controlled systems in real time based on inaccuthe Ito integral and characterizing the amplitude of a in many scientific studies and rate/incomplete information onapplications. the phase Reconstrucstate arises random noise was considered for a linear SDE on the in many scientific studies and applications. Reconstrucrandom noise was considered for a linear SDE on the in many scientific studies and applications. Reconstructhe Ito integral and characterizing the amplitude of a basis of the measurements of realizations of the whole rate/incomplete information on the phase state arises random noise was considered for a linear SDE on the tion problems fall into the range of inverse problems of in many scientific studies and applications. Reconstrucof the measurements of realizations of the whole tion problems fall into the range of inverse problems problems of basis of the measurements of realizations of the whole tionmany problems fall into into the range of ainverse inverse of basis random noise was considered for awas linear SDE on the phase vector. Then, the statement supplemented by in scientific studies and applications. Reconstrucbasis of the measurements of realizations of the whole dynamics of controlled systems; as rule, they are illtion problems fall the range of problems of phase vector. Then, the the statement statement was supplemented supplemented by dynamics of controlled systems; as rule, they are illvector. Then, was by dynamics ofrequire controlled systems; as rule, they they are illbasis of the measurements of realizations of the of whole the possibility of measuring aa part of coordinates the tion problems fall into the rangeprocedures. of aaainverse problems of phase phase vector. Then, the statement was supplemented by posed and regularizing One of the dynamics of controlled systems; as rule, are illthe possibility of measuring part of coordinates of the posed and require regularizing procedures. One of the the possibility of measuring a part of coordinates of the posed and and require regularizing procedures. One are of the the phase vector. Then, the statement was supplemented by stochastic process (Rozenberg (2016)). The applicability dynamics ofrequire controlled systems; as a rule, they illthe possibility of measuring a part of coordinates of the solution approaches named the method of dynamic inposed regularizing procedures. One of stochastic process (Rozenberg (2016)). The applicability solution approaches named the method of dynamic instochastic process (Rozenberg (2016)). The applicability solution approaches named the method of dynamic inthe possibility of measuring a part of coordinates of the of the algorithms for reconstructing unknown parameters posed and require regularizing procedures. One of the stochastic process (Rozenberg (2016)). The applicability version was proposed by Kryazhimskii, Osipov, and their solution approaches named the methodOsipov, of dynamic in- of the algorithms for reconstructing unknown parameters version was proposed by Kryazhimskii, and their of the for reconstructing unknown parameters version was was proposed by Kryazhimskii, Kryazhimskii, Osipov, and their processfor (Rozenberg (2016)). The applicability developed earlier observed (partially observed) systems solution approaches named the method of dynamic in- stochastic of the algorithms algorithms for reconstructing unknown parameters colleagues, see Kryazhimskii and Osipov (1984); Osipov version proposed by Osipov, and their developed earlier for observed (partially observed) systems colleagues, see Kryazhimskii and Osipov (1984); Osipov developed earlier for observed (partially observed) systems colleagues, see Kryazhimskii and Osipov (1984); Osipov of the algorithms for reconstructing unknown parameters ODEs, see Kryazhimskii and Osipov (1984); Osipov and version was proposed by Kryazhimskii, Osipov, and their developed earlier for observed (partially observed) systems and Kryazhimskii (1995); Osipov et al. (2011). It is based colleagues, see Kryazhimskii and Osipov (1984); Osipov of ODEs, see Kryazhimskii and Osipov (1984); Osipov and and Kryazhimskii (1995); Osipov et al. (2011). It is based of ODEs, see Kryazhimskii and Osipov (1984); Osipov and and Kryazhimskii (1995); Osipov et al. (2011). It is based developed earlier for observed (partially observed) systems Kryazhimskii (1995); Osipov et al. (2011), is substantiated colleagues, see Kryazhimskii and Osipov (1984); Osipov of ODEs, see Kryazhimskii and Osipov (1984); Osipov and on the combination of principles ofetthe the theory ofItpositional positional and Kryazhimskii (1995); Osipovof al.theory (2011).of is based Kryazhimskii (1995); Osipov et al. (2011), is substantiated on the combination of principles Kryazhimskii (1995); Osipov et al. (2011), is substantiated on the combination of principles of the theory of positional of ODEs, see Kryazhimskii and Osipov (1984); Osipov and and corresponding modifications are proposed. The inverse and Kryazhimskii (1995); Osipov et al. (2011). It is based Kryazhimskii (1995); Osipov et al. (2011), is substantiated control (Krasovskii and Subbotin (1988)) and ideas of on the combination ofand principles of the theoryand of positional corresponding modifications are proposed. The inverse control (Krasovskii Subbotin (1988)) ideas of of and and corresponding modifications are proposed. The inverse control (Krasovskii and Subbotin (1988)) and ideas Kryazhimskii (1995); Osipovscalar et al. (2011), is with substantiated problem for a quasi-linear equation diffusion on the combination ofand principles of(Tikhonov the theoryand of positional and corresponding modifications are proposed. The inverse the theory of ill-posed problems and Arsenin control (Krasovskii Subbotin (1988)) ideas of problem for aa quasi-linear scalar equation with diffusion the theory of ill-posed problems (Tikhonov and Arsenin for quasi-linear scalar equation with diffusion the theory theory of ill-posed ill-posed problems (Tikhonov and Arsenin and corresponding modifications are proposed. The inverse depending phase state in the statement assuming control (Krasovskii andproblems Subbotin(Tikhonov (1988)) and ideas of problem problem foron a the quasi-linear scalar equation with diffusion (1981)). A reconstruction problem is reduced to aa feedback the of and Arsenin depending on the phase state in the statement assuming (1981)). A reconstruction problem is reduced to feedback depending on the phase state in the statement assuming (1981)). A reconstruction problem is reduced to a feedback problem for a quasi-linear scalar equation with diffusion the simultaneous reconstruction of disturbances both the theory of ill-posed problems (Tikhonov and Arsenin depending on the phase state in the statement assuming control problem for an auxiliary dynamical system called aa the simultaneous reconstruction of disturbances both in (1981)). A reconstruction problem is reduced to a feedback in control problem for an auxiliary dynamical system called the simultaneous disturbances both in control problem for an auxiliary dynamical system called a depending on the reconstruction phasestochastic state in of the statement assuming deterministic and terms was discussed (1981)). A reconstruction problem is reduced a feedback the simultaneous reconstruction of disturbances both in model. The adaptation of the model controls to the results control problem for an auxiliary dynamical system called a the deterministic and stochastic terms was discussed in model. The adaptation of the model controls to the results the deterministic and stochastic terms was discussed in model. The adaptation of the model controls to the results the simultaneous reconstruction of disturbances both in Rozenberg (2017). The novelty of the present paper concontrol problem for an auxiliary dynamical system a the deterministic and stochastic terms was discussed in of current observations provides an approximation (in an model. The adaptation of the model controls to thecalled results (2017). The novelty of the present paper conof current observations provides an approximation (in an an Rozenberg Rozenberg (2017). The novelty of the present paper conof current observations provides an approximation (in the deterministic and stochastic terms was discussed in sists in considering the input reconstruction problem for a model. The adaptation of the model controls to the results (2017). The novelty of the present paper for conappropriate sense) to the unknown inputs. The method of current observations provides an approximation (in an Rozenberg sists in considering the input reconstruction problem a appropriate sense) to the unknown inputs. The The method method sists in considering the input reconstruction problem for a appropriate sense) to the unknown inputs. Rozenberg (2017). The novelty of the present paper constochastic diffusion system that causes essential technical of current observations provides an approximation (in an sists in considering the input reconstruction problem for a dynamic inversion systems appropriate sense) to was the realized unknownfor inputs. Thedescribed method stochastic diffusion system that causes essential technical of dynamic inversion was realized for systems described stochastic diffusion system that causes essential technical of dynamic inversion was realized for systems described sists in considering the input reconstruction problem for a difficulties. appropriate sense) to the unknown inputs. The method stochastic diffusion system that causes essential technical by ordinary differential equations (ODEs), functional difof dynamic inversion was realized(ODEs), for systems described difficulties. by ordinary differential equations functional difdifficulties. by ordinary differential equations (ODEs), functional difstochastic diffusion system that causes essential technical of dynamic inversion was realized for systems described ferential equations, equations and variational inequalities by ordinary differential equations (ODEs), functional dif- difficulties. ferential equations, equations and variational inequalities ferential equations, equations andothers, variational inequalities difficulties. 2. PROBLEM STATEMENT by ordinary differential equations (ODEs), functional difwith distributed parameters, and see Osipov et al. ferential equations, equations and variational inequalities with distributed parameters, and others, see Osipov et al. 2. PROBLEM PROBLEM STATEMENT with distributed parameters, and others, see Osipov et al. 2. ferential equations, equations and variational inequalities (2011) and its bibliography. with distributed parameters, and others, see Osipov et al. 2. PROBLEM STATEMENT STATEMENT (2011) and its bibliography. (2011) and its bibliography. with distributed parameters, and others, see Osipov et al. Consider a quasi-linear 2. PROBLEM STATEMENT (2011) and its bibliography. (according As to the application of the theory of dynamic inversion Consider aa quasi-linear quasi-linear (according (according to to the the terminology terminology of of Consider to the terminology of (2011) andapplication its bibliography. As to the of the theory of dynamic inversion Rumyantsev and Khrustalev (2006) and other works As to the application of the theory of dynamic inversion Consider a quasi-linear (according to the terminology of to stochastic objects, the problem of positional simulation Rumyantsev and Khrustalev (2006) and other works of As to the application of the theory of dynamic inversion Rumyantsev and Khrustalev (2006) and other works of Consider a quasi-linear (according to the terminology of to stochastic objects, the problem of positional simulation these authors) system of SDEs with diffusion depending to stochastic objects, the problem of positional simulation Rumyantsev and Khrustalev (2006) and other works of As to unknown the application of the theory of dynamic inversion these of an stochastic control in a system described authors) system of SDEs with diffusion depending to stochastic objects, the problem of positional simulation these authors) system of SDEs with diffusion depending Rumyantsev and Khrustalev (2006) and other works of of an unknown stochastic control in a system described on the phase state: of an unknown stochastic control in a system described these authors) system of SDEs with diffusion depending to stochastic objects, the problem ofinpositional simulation by an ODE was considered for the first time in Osipov on the phase state: of an unknown stochastic control a system described on the phase state: these authors) system of SDEs with diffusion depending by an ODE was considered for the first time in Osipov by an anKryazhimskii ODE was wasstochastic considered for present thein first first time in in Osipov on thedx(t, phase state: ω) = (A(t)x(t, ω) + B(t)u of unknown control a system described 1 (t) + f (t)) dt+ and (1986). The paper continues by an ODE considered for the time Osipov ω) = (A(t)x(t, ω) + B(t)u (t) + f (t)) dt+ 1 on thedx(t, phase state: and Kryazhimskii (1986). The The present paper continues dx(t, ω) = (A(t)x(t, ω) + B(t)u 1 (t) + f (t)) dt+ and Kryazhimskii (1986). present paper continues by an ODE was considered for the first time in Osipov dx(t, ω) = (A(t)x(t, ω) + B(t)u += f (t)) dt+ (1) the investigations of the problems of reconstructing un1 (t)ω) and Kryazhimskii (1986). The present paper continues + U (t)x(t, ω) dξ(t, ω), x(0, x 2 0. the investigations of the problems problems of reconstructing reconstructing un(t)x(t, ω) dξ(t, x(0, x . (1) the investigations investigations of the of undx(t,+ ω)U = (A(t)x(t, ω) +ω), B(t)u += f (t)) 2 0 + U ω) dξ(t, ω), x(0, ω) = x (1) 1 (t)ω) and Kryazhimskii of (1986). The present paper continues 2 (t)x(t, 0 . dt+ the the problems of reconstructing un+ U2 (t)x(t, ω) dξ(t, ω), x(0, ω) =nx0 . (1) ⋆ the investigations of the problems of reconstructing unHere, t ∈ T = [0, ϑ], x = (x , x , . . . , x ) ∈ R is a column n 1 2 n + U (t)x(t, ω) dξ(t, ω), x(0, ω) = x . (1) This work was supported by the Program of the Ural Branch of 2 0 n ⋆ This work was supported by the Program of the Ural Branch of Here, t ∈ T = [0, ϑ], x = (x , x , . . . , x ) ∈ R is a column 1 2 n ⋆ Here, t ∈ T = [0, ϑ], x = (x , x , . . . , x ) ∈ R is a column n or random 1 2 deterministic n This work was supported the Program of the Ural of vector, ξ ∈ R; x a known ⋆ Russian Academy of Sciencesby “Estimation of Dynamics of Branch Nonlinear Here, t ∈ T = [0, ϑ], x = (x , x , . . . , x ) ∈ R is a column 0 is 1 2 n This work was supported by the Program of the Ural Branch of vector, ∈ =R; R;[0,x x00 is is known deterministic ora random random Russian Academy of Sciences “Estimation of of Nonlinear vector, aa (x known deterministic Russian Academy ofand Sciences “Estimation of Dynamics Dynamics of18-1-1-9). Nonlinear ⋆ This work Here, t ∈ξξξ T∈ x= . . , xn )conditions; ∈ Rn isor column (normally distributed) initial ω ∈ Controlled Systems Routing Optimization” 1 , xof 2 , .deterministic was supported by the Program of (project the Ural Branch of vector, ∈ R; xϑ], known or random Russian Academy ofand Sciences “Estimation of Dynamics of18-1-1-9). Nonlinear 0 is avector (normally distributed) vector of initial conditions; ω ∈ Ω, Ω, Controlled Systems Routing Optimization” (project (normally distributed) vector of initial conditions; ω Controlled Systems and Routing Optimization” (project 18-1-1-9). vector, ξ ∈ R; x is a known deterministic or Russian Academy Sciences “Estimation of Dynamics Nonlinear 0 (normally distributed) vector of initial conditions;random ω∈ ∈ Ω, Ω, Controlled Systemsofand Routing Optimization” (projectof18-1-1-9). (normally distributed) vector of initial conditions; ω ∈ Ω, Controlled Systems and Routing Optimization” 18-1-1-9). 2405-8963 © © 2018, IFAC IFAC (International Federation(project of Automatic Control) Copyright 2018 727 Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2018 727 Copyright © under 2018 IFAC IFAC 727 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 727 10.1016/j.ifacol.2018.11.460 Copyright © 2018 IFAC 727

IFAC CAO 2018 728 Valeriy L. Rozenberg / IFAC PapersOnLine 51-32 (2018) 727–732 Yekaterinburg, Russia, October 15-19, 2018

(Ω, F, P ) is a probability space; ξ(t, ω) is a standard scalar Wiener process (i.e., a process starting from zero with zero mathematical expectation and dispersion equal to t, see Shiryaev (1974)); A(t) = {aij (t)}, B(t) = {bij (t)}, and f (t) = {fi (t)} are continuous matrix functions of dimension n × n, n × r, and n × 1, respectively. Two external disturbances act on the system: a vector u1 (t) = (u11 (t), u12 (t), . . . , u1r (t)) ∈ Rr and a diagonal matrix U2 (t) = {u21 (t), u22 (t), . . . , u2n (t)} ∈ Rn×n . The action u1 enters into the deterministic term and influences the mathematical expectation of the desired process. Since U2 xdξ = (u21 x1 dξ, u22 x2 dξ, . . . , u2n xn dξ), we can assume that the vector u2 = (u21 , u22 , . . . , u2n ) regulates the amplitude of random noises. Let the vectors u1 and u2 take values from given convex compact sets Su1 and Su2 , and let their elements belong to the space L2 (T ). A solution of system (1) is defined as a stochastic process satisfying (for any t with probability 1) the corresponding integral identity containing the stochastic Ito integral on the right-hand side. As is known, under the above assumptions, there exists a unique solution, which is a normal Markov process with continuous realizations, see Oksendal (1985). Note that equations of form (1) describe simplest linearized models, for example, of changing the size of a biological population in a stochastic medium or of the price dynamics on a commodity market under the influence of random factors. The problem consists in the following. At discrete, frequent enough, times τi ∈ T , τi = iδ, δ = ϑ/l, i ∈ [0 : l], the information on some number N of realizations of the stochastic process x(τi ) is received. We assume that l = l(N ) and there exist an estimate mN i of the mathematical expectation m(t) = M x(t) and an estimate DiN of the covariance matrix D(t) = M (x(t) − m(t))(x(t) − m(t))′ (the prime means transposition) such that       N m − m(τi ) , DN − D(τi ) ≤ h(N ) P max i i i∈[1:l(N )]

= 1 − g(N ), (2) and h(N ), g(N ) → 0 as N → ∞. The standard statistical procedures (Korolyuk et al. (1985)) of constructing the N estimates mN i and Di admit modifications providing the validity of (2).

It is required to design an algorithm for the dynamical reconstruction of the unknown disturbances u1 (·) and u2 (·) generating the stochastic process x(·) from the discrete information on its realizations. The probability of an arbitrarily small deviation of approximations from the desired inputs in the metric of the spaces L2 (T ; Rr ) and L2 (T ; Rn ) should be close to 1 for sufficiently large N and the time discretization step δ = δ(N ) = ϑ/l(N ) concordant with N in an appropriate way. The specific properties of quasi-linear system (1) admit the reduction (by analogy with the method of moments, see Chernous’ko and Kolmanovskii (1978)) of the problem formulated for a SDE to a problem for the system of ODEs describing the mathematical expectation and covariance matrix of the desired process. This allows us to organize a procedure of the simultaneous reconstruction of the disturbances both in the deterministic and stochastic terms of the right-hand side. To solve the problem, we use ideas 728

of the theory of dynamic inversion, see Kryazhimskii and Osipov (1984); Osipov and Kryazhimskii (1995); Osipov et al. (2011); namely, we construct a finite-step softwareoriented solution algorithm based on the method of auxiliary controlled models. The formulated inverse problem can be interpreted as the dynamical reconstruction of the external control action and of the amplitude of random noises in the case when simultaneous measurements of a sufficiently large number of trajectories (for example, of the motion of one-type particles) are possible. 3. PROPERTIES OF THE STATISTICAL ESTIMATES Lemma 1. The standard estimates mN i of the mathematical expectation m(τi ) and DiN of the covariance matrix D(τi ) constructed from N (N > 1) realizations x1 (τi ), x2 (τi ), . . . , xN (τi ) of the random variables x(τi ) by the rules: N 1  r = x (τi ), (3) mN i N r=1 N

DiN =

1  r r N ′ (x (τi ) − mN i )(x (τi ) − mi ) , N − 1 r=1

(4)

provide the validness of relation (2).

Proof. The vector variable x(τi ) has the n-dimensional normal distribution with the mathematical expectation m(τi ) and the covariance matrix D(τi ). For these values, we consider separately the relations composing (2). Let us show that estimate (3) has the following property:      P ∀i ∈ [1 : lm (N )] mN i − m(τi ) ≤ hm (N ) = 1−gm (N ). (5) Here, lm (N ) = ϑ/δm (N ); δm (N ), hm (N ), gm (N ) → 0 as N → ∞. We obtain an estimate corresponding to (5) for each coordinate j ∈ [1 : n] of the vector m(τi ). When deriving confidence estimates, it is sufficient to use the normal distribution of the coordinates without taking into account their covariance (generally speaking, its usage can sharpen confidence domains). First, we prove that ∀j ∈ [1 : n] ∀i ∈ [1 : lm (N )]    ∗ ∗  P mN ij − mj (τi ) ≤ hm (N ) = 1 − fm (N ),

h∗m (N ),

∗ fm (N )

(6)

→ 0 as N → ∞.

As is known from Korolyuk  et al. (1985), the random N N (D N is the estimate variable ξ = (mij − mj (τi )) N/Dijj ijj

of the diagonal element Djj (τi ) of the covariance matrix, i.e., of the dispersion of the value xj (τi )) has t-distribution (Student’s distribution) with (N − 1) degrees of freedom, which is close to the standard normal law N (0, 1) for N > 30. Our aim is to modify the procedure of estimating the mathematical expectation in the case of unknown dispersion in such a way that relation (6) is fulfilled. Consider the following expression for 0 < ǫ < 1/2:    ǫ N P −N ǫ ≤ (mN ij − mj (τi )) N/Dijj ≤ N     N /N ≤ mN − m (τ ) ≤ N ǫ N /N = P −N ǫ Dijj Dijj j i ij    ǫ ∗ N /N Dijj = 1 − fm (N ). = P |mN ij − mj (τi )| ≤ N

IFAC CAO 2018 Valeriy L. Rozenberg / IFAC PapersOnLine 51-32 (2018) 727–732 Yekaterinburg, Russia, October 15-19, 2018

We can assume that h∗m (N ) = C1 /N 1/2−ǫ in (6) (here and below, by Ci we denote auxiliary constants, which can be written explicitly). On the other hand,    ǫ N P −N ǫ ≤ (mN ij − mj (τi )) N/Dijj ≤ N = 2(Ft,N −1 (N ǫ ) − Ft,N −1 (0)) = 2Ft,N −1 (N ǫ ) − 1    = 2 Ft,N −1 (N ǫ ) − Φ(N ǫ 1 − 2/N )   + 2Φ(N ǫ 1 − 2/N ) − 1 ≥ 2Φ(N ǫ 1 − 2/N ) − 1    − 2 Ft,N −1 (N ǫ ) − Φ(N ǫ 1 − 2/N )    = 1 − 2 1 − Φ(N ǫ 1 − 2/N )    − 2 Ft,N −1 (N ǫ ) − Φ(N ǫ 1 − 2/N ).

Here, Ft,N −1 (x) is the probability function of the tdistribution with N − 1 degrees of freedom, Ft,N −1 (0) = 1/2; Φ(x)  x is the function of the normal distribution, 2 1 √ e−y /2 dy. Using the inequality describΦ(x) = 2π −∞ ing the closedness of the functions Ft,N −1 (·) and Φ(·) for large N in the form (see Johnson et al. (1995))    Ft,N −1 (x) − Φ(x 1 − 2/N ) ≤ C2 /N, (7) as well as the asymptotics of the normal distribution as x → ∞ in the form (see Korolyuk et al. (1985))  e−x2 /2  2 1 , (8) 1 − Φ(x) = √ e−x /2 + o x x 2π we obtain    ǫ N P −N ǫ ≤ (mN ij − mj (τi )) N/Dijj ≤ N   2ǫ 2ǫ ≥ 1 − 2C2 /N − C3 e−N /2 /N ǫ + o e−N /2 /N ǫ . Since the asymptotics of the normal distribution (8) is suppressed by the error of the normal approximation (7), ∗ we can assume that fm (N ) ≤ C4 /N in (6).

Now, let us show that it is possible to make a concordance of the number N of measurable realizations and the step δm (N ) = ϑ/lm (N ) in such a way that relation (6) implies relations similar to (5) for each coordinate of m(τi ). Define    ∗ ∗  Amij = mN ij − mj (τi ) ≤ hm (N ) , P (Amij ) = 1 − fm (N ),   N  ∗ (N ). A¯mij = mij − mj (τi ) > h∗m (N ) , P (A¯mij ) = fm Then,     ∗  P ∀i ∈ [1 : lm (N )] mN ij − mj (τi ) ≤ hm (N )   = P Am1j Am2j . . . Amlm (N )j   = 1 − P A¯m1j + A¯m2j + . . . + A¯ml (N )j m

≥ 1 − P (A¯m1j ) − P (A¯m2j ) − . . . − P (A¯mlm (N )j ) ∗ (N ). = 1 − lm (N )fm ∗ −α Set lm (N ) = (fm (N )) , 0 < α < 1. Then, we can assume ∗ ∗ that lm (N )fm (N ) = (fm (N ))1−α = C5 /N 1−α . Note that α lm (N ) is of order N . Thus, ∀j ∈ [1 : n] we have     1/2−ǫ  − m (τ ) ≤ C /N P ∀i ∈ [1 : lm (N )] mN j i 1 ij = 1 − C5 /N 1−α . Then, by analogy with the reasoning above, we get     P ∀j ∈ [1 : n] ∀i ∈ [1 : lm (N )] mN ij − mj (τi )  ≤ C1 /N 1/2−ǫ ≥ 1 − nC5 /N 1−α .

729

729

Passing to the norm of the vector, we derive    √   P ∀i ∈ [1 : lm (N )] mN nC1 /N 1/2−ǫ i − m(τi ) ≤

≥ 1 − nC5 /N 1−α , this actually proves (5) with hm (N ) = C6 /N 1/2−ǫ and gm (N ) = C7 /N 1−α . Now, we prove the similar property of estimate (4):     P ∀i ∈ [1 : ld (N )] DiN − D(τi ) ≤ hd (N ) = 1−gd(N ), (9) ld (N ) = ϑ/δd (N ); δd (N ), hd (N ), gd (N ) → 0 as N → ∞. First, we show that, ∀j, k ∈ [1 : n] and ∀i ∈ [1 : ld (N )],    N (10) − Djk (τi ) ≤ h∗d (N ) = 1 − fd∗ (N ), P Dijk h∗d (N ), fd∗ (N ) → 0 as N → ∞.

In Johnson et al. (1995), it is proved that the random N variable Dijk has the same distribution as the variable σij σik ((1 + ρijk )Zi1 − (1 − ρijk )Zi2 ) , ξi = 2(N − 1) where σij and σik are mean square deviations of the variables xj (τi ) and xk (τi ), ρijk is the correlation coefficient, Zi1 and Zi2 are two independent variables distributed by the law χ2 with (N − 1) degrees of freedom. Note that we have the representation N −1 N −1   σij σik ξir = ξi = 2(N − 1) r=1 r=1   r 2 r 2 ) , × (1 + ρijk )(zi1 ) − (1 − ρijk )(zi2 r r are independent variables distributed by and zi2 where zi1 the standard normal law. All the terms in the sum have the same mathematical expectation and dispersion: M ξir = σij σik ρijk /(N − 1) = Djk (τi )/(N − 1), 2 2 σik (1 + ρ2ijk ). Dξir = Ljk (τi )/(N − 1)2 , Ljk (τi ) = σij Then, M ξi = Djk (τi ), Dξi = Ljk (τi )/(N − 1). As is known, see Korolyuk et al. (1985), the  distribution of the N − Djk (τi )) (N − 1)/Ljk (τi ), normalized variable (Dijk being the sum of identical independent variables, is close to the standard normal law N (0, 1) for large N . Our aim is to modify the procedure of estimating the covariance in such a way that relation (10) is fulfilled. Consider the following expression for 0 < ǫ < 1/2:    N P −N ǫ ≤ (Dijk − Djk (τi )) (N − 1)/Ljk (τi ) ≤ N ǫ    N −Djk (τi )| ≤ N ǫ Ljk (τi )/(N − 1) = 1−fd∗ (N ). = P |Dijk

We can assume that h∗d (N ) = C8 /N 1/2−ǫ in (10). On the other hand,    N − Djk (τi )) (N − 1)/Ljk (τi ) ≤ N ǫ P −N ǫ ≤ (Dijk

= Fi,N −1 (N ǫ ) − Fi,N −1 (−N ǫ ), where Fi,N −1 (x) is the distribution function of the variable N (Dijk − Djk (τi )) (N − 1)/Ljk (τi ). Using the improved Berry–Esseen inequality (see Korolyuk et al. (1985)) describing the closedness of the function Fi,N −1 (x) and the function of the normal distribution √ in the form3 |Fi,N −1 (x) − Φ(x)| < C9 /(2 N − 1(1 + |x| )), (11) as well as the specified above asymptotics of the normal distribution (see (8)), we derive Fi,N −1 (N ǫ ) − Fi,N −1 (−N ǫ ) = Fi,N −1 (N ǫ ) − Φ(N ǫ )

IFAC CAO 2018 730 Valeriy L. Rozenberg / IFAC PapersOnLine 51-32 (2018) 727–732 Yekaterinburg, Russia, October 15-19, 2018

+ Φ(−N ǫ ) − Fi,N −1 (−N ǫ ) + Φ(N ǫ ) − Φ(−N ǫ ) ≥ 2Φ(N ǫ ) − 1 − |Fi,N −1 (N ǫ ) − Φ(N ǫ )| √ 2ǫ − |Fi,N −1 (−N ǫ ) − Φ(−N ǫ )| > 1 − 2e−N /2 /(N ǫ 2π)   √ 2ǫ − C9 /( N − 1(1 + N 3ǫ )) + o e−N /2 /N ǫ .

steps δm (N ) and δd (N ). In order to consider the same (the roughest) partition in these estimates, we set δ(N ) = C16 /N min{α,α(1/2+3ǫ)} , l(N ) = ϑ/δ(N ),

Since the asymptotics of the normal distribution (8) is suppressed by the error of the normal approximation (11), we can assume (see (10)) that fd∗ (N ) ≤ C10 /N 1/2+3ǫ . Relation (10) holds. Now, let us show that it is possible to make a concordance of the number N of measurable realizations and the step δd (N ) = ϑ/ld (N ) in such a way that relation (10) implies relations similar to (9) for each element of the covariance matrix D(τi ). Introducing the notation   N  − Djk (τi ) ≤ h∗d (N ) , P (Adijk ) = 1−fd∗ (N ), Adijk = Dijk    N A¯dijk = Dijk − Djk (τi ) > h∗d (N ) , P (A¯dijk ) = fd∗ (N ), by analogy with the events Amij , we conclude that   N   − Djk (τi ) ≤ h∗d (N ) P ∀i ∈ [1 : ld (N )] Dijk   = P Ad1jk Ad2jk . . . Adld (N )jk   = 1 − P A¯d1jk + A¯d2jk + . . . + A¯dld (N )jk ≥ 1 − P (A¯d1jk ) − P (A¯d2jk ) − . . . − P (A¯dld (N )jk ) = 1 − ld (N )fd∗ (N ). Considering ld (N ) = (fd∗ (N ))−α (α is introduced earlier, ld (N ) is of order N (1/2+3ǫ)α ), we can assume that ld (N )fd∗ (N ) = (fd∗ (N ))1−α = C11 /N (1/2+3ǫ)(1−α) . Thus, ∀j, k ∈ [1 : n], we have    N  − Djk (τi ) ≤ C8 /N 1/2−ǫ P ∀i ∈ [1 : ld (N )] Dijk = 1 − C11 /N (1/2+3ǫ)(1−α) . Then, by analogy with the reasoning above, we get   N  P ∀j, k ∈ [1 : n] ∀i ∈ [1 : ld (N )] Dijk − Djk (τi )  ≤ C8 /N 1/2−ǫ ≥ 1 − n2 C11 /N (1/2+3ǫ)(1−α) .

Passing to the norm of the matrix, we derive     P ∀i ∈ [1 : ld (N )] DiN − D(τi ) ≤ nC8 /N 1/2−ǫ

≥ 1 − n2 C11 /N (1/2+3ǫ)(1−α) , this actually proves (9) with hd (N ) = C12 /N 1/2−ǫ and gd (N ) = C13 /N (1/2+3ǫ)(1−α) .

Thus, taking into account constants written explicitly, we obtain the relations for the values characterizing estimates (5) and (9):    Em = {∀i ∈ [0 : lm (N )] mN i − m(τi ) ≤ hm (N )},

P (Em ) = 1 − gm (N ), δm (N ) = ϑ/lm (N ) = C14 /N α , (12) hm (N ) = C6 /N 1/2−ǫ , gm (N ) = C7 /N 1−α ;  N    Ed = {∀i ∈ [0 : ld (N )] Di − D(τi ) ≤ hd (N )},

P (Ed ) = 1 − gd (N ), δd (N ) = ϑ/ld (N ) = C15 /N α(1/2+3ǫ) , hd (N ) = C12 /N 1/2−ǫ , gd (N ) = C13 /N (1−α)(1/2+3ǫ) , (13) where 0 < ǫ < 1/2, 0 < α < 1.

Note that the structure of estimates (12) and (13) is such that they are valid with the same hm (N ), gm (N ), hd (N ), and gd (N ), when decreasing the number of the nodes τi in the partition of the interval T , i.e., when increasing the 730

h(N ) = C17 /N 1/2−ǫ , g(N ) = C18 /N min{1−α,(1−α)(1/2+3ǫ)} . (14) The values δ(N ), h(N ), and g(N ) coincide by the order of smallness either with the triple δm (N ), hm (N ), and gm (N ) or with the triple δd (N ), hd (N ), and gd (N ). Then, “roughening” a more exact estimate (by the order with respect to 1/N ) from (12) and (13) and using the fact that statistical sample estimates (3) and (4) are independent (Korolyuk et al. (1985)), we can write      N mi − m(τi ) , DiN − D(τi ) P (Em Ed ) = P max i∈[1:l(N )]

 ≤ h(N ) = (1 − g(N )/2)2 ≥ 1 − g(N ).

Actually, this proves relation (2); the corresponding values are found from formulas (14). 4. REDUCTION OF THE PROBLEM Following Rozenberg (2007, 2016), we reduce the reconstruction problem for an SDE to a problem for a system of ODEs. Let us introduce the notation: m0 = M x0 , D0 = M (x0 − m0 )(x0 − m0 )′ . Since the original system is quasi-linear and the mathematical expectation of an Ito integral is zero, the value m(t) does not depend on u2 (t); its dynamics is described by the equation m(t) ˙ = A(t)m(t) + B(t)u1 (t) + f (t), m ∈ Rn , m(0) = m0 . The covariance matrix D(t) does not depend on u1 (t) explicitly. To describe its dynamics, we use the scheme for deriving the equation of the method of moments (Chernous’ko and Kolmanovskii (1978)). By the multidimensional Ito formula, see Oksendal (1985), we have d(x(t) − m(t))(x(t) − m(t))′

= (dx(t))(x(t) − m(t))′ + (x(t) − m(t))(dx(t))′ + (dx(t))(dx(t))′ = ((A(t)(x(t) − m(t)) + A(t)m(t) + B(t)u1 (t) + f (t))dt + U2 (t)x(t)dξ(t))(x(t) − m(t))′ + (x(t) − m(t))((A(t)(x(t) − m(t)) + A(t)m(t) + B(t)u1 (t) +f (t))dt+U2 (t)x(t)dξ(t))′ +U2 (t)((x(t)−m(t))(x(t)−m(t))′ +(x(t)−m(t))m′ (t)+m(t)(x(t)−m(t))′ +m(t)m′ (t))U2′ (t)dt. Integrating the latter expression over the segment T , taking the mathematical expectation and then differentiating, we obtain ˙ D(t) = A(t)D(t)+D(t)A′ (t)+U2 (t)(D(t)+m(t)m′ (t))U2′ (t), (15) D ∈ Rn×n , D(0) = D0 . Matrix equation (15) is rewritten in the form of a vector equation, which is more traditional for the problems under consideration. By virtue of the symmetry of the matrix D(t), its dimension is defined as nd = (n2 + n)/2. We introduce the vector d(t) = {ds (t)}, s ∈ [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. The coordinates of this vector are found from the elements of the matrix D(t) = {dij (t)}, i, j ∈ [1 : n]: ds (t) = dij (t), i ≤ j, s = (n − i/2)(i − 1) + j. (16)

IFAC CAO 2018 Valeriy L. Rozenberg / IFAC PapersOnLine 51-32 (2018) 727–732 Yekaterinburg, Russia, October 15-19, 2018

Transformations similar to described in detail in Rozenberg (2016) allow us to rewrite system (15) in the form ˙ = A(t)d(t) ¯ ¯ d(t) + B(d(t), m(t))u3 (t), d(t0 ) = d0 . ¯ Here, the matrix A(t) : T → Rnd ×nd can be written ex¯ plicitly, the diagonal matrix B(d(t), m(t)) : T → Rnd ×nd is defined by the formulas ¯bss = ds + mi mj , i ≤ j, s = (n − i/2)(i − 1) + j, ¯bsr = 0, s �= r, the vector u3 (t) = {u3s (t)}, s ∈ [1 : nd ] is found as u3s = u2i u2j , i ≤ j, s = (n − i/2)(i − 1) + j. ¯ are Note that the elements of the matrices A¯ and B continuous, the vector u3 (t) takes values from some convex compact set Su3 ∈ Rnd and has elements belonging to the space L2 (T ). We assume that the initial state d0 and N measurements dN i are obtained from D0 and from Di according to (16). It is the vector function u3 (·) that we reconstruct through the information on d(·). Then, limiting ourselves to considering the coordinates that are equal to u22i , i ∈ [1 : n] (such a subvector is denoted by u4 (·)), we show that the value u2 (·) can be reconstructed under additional constraints. Thus, we have the system of equations of dimension n + nd : m(t) ˙ = A(t)m(t) + B(t)u1 (t) + f (t), m(0) = m0 , (17) ˙ = A(t)d(t) ¯ ¯ d(t) + B(d(t), m(t))u3 (t), d(0) = d0 . (18) Now, we can reformulate the original problem. During the process, at the discrete times τi ∈ T , τi = iδ, δ = ϑ/l(N ), i ∈ [0 : l(N )], the inaccurate information on the phase state of system (17), (18) is received:       N mi − m(τi ) , dN  ≤ h(N ) P max i − d(τi ) i∈[1:l(N )]

= 1 − g(N ), (19) where h(N ), g(N ) → 0 as N → ∞. It is required to design an algorithm of dynamical reconstruction of the unknown disturbances u1 (·) and u3 (·) (or u4 (·)) from information (19). The probability of an arbitrarily small deviation of approximations from the desired inputs in the metric of the spaces L2 (T ; Rr ) and L2 (T ; Rnd ), respectively, should be close to 1 for sufficiently large N and the time step of measurements δ = δ(N ) = ϑ/l(N ) concordant with N in an appropriate way. System (17), (18) is nonlinear in phase variables but is linear in control. The problem formulated for this system corresponds to the problem considered, for example, in Kryazhimskii and Osipov (1984). In the next section, it is shown that the finite-step solution algorithm proposed in that paper for ODEs can be applied and it is proved that this algorithm admits a constructive concordance of its parameters with the number of measurable realizations of the original stochastic process. 5. RECONSTRUCTION ALGORITHM Let us adapt the solving algorithm from Kryazhimskii and Osipov (1984) for system (17), (18). At the initial time τ0 = 0, we fix a value N ; then, determine the values lN = l(N ), hN = h(N ), and g N = g(N ) (see (14)) and construct the uniform partition of the interval T with the step δ N = ϑ/lN : τi ∈ T, τi = iδ N , i ∈ [0 : lN ]. The dynamics of the discrete model and its initial state are defined by the relations N N wm (τi+1 ) = wm (τi ) + (A(τi )mN i + B(τi )v1i + f (τi ))δ , (20) 731

731

¯ i )dN + B(d ¯ N , mN )v N )δ N , (21) wd (τi+1 ) = wd (τi ) + (A(τ 3i i i i wm (0) = m0 , wd (0) = d0 . N N , v3i are control actions calculated Here, i ∈ [0 : lN − 1], v1i at the time τi by rules that are specified below. The work of the algorithm is decomposed into lN identical steps. At the ith step performed on (τi , τi+1 ], the input N data for calculations are the estimates mN i , di , and the model state wm (τi ), wd (τi ) obtained by this moment. The N N model controls are found as follows: v1i and v3i are unique solutions of the corresponding extremal problems N v1i = arg min   N 2�wm (τi ) − mi , B(τi )v1 � + αN �v1 �2Rr : v1 ∈ Su1 ,

(22)

N = arg min v3i

  2 N ¯ N N 2�wd (τi ) − dN i , B(di , mi )v3 � + α �v3 �Rnd : v3 ∈ Su3 ,

(23) where αN = α(hN ) is a regularization parameter. After the calculation of controls (22) and (23), the model state wm (τi+1 ), wd (τi+1 ) is recomputed by formulas (20) and (21). The process stops at the terminal time ϑ.

Let U∗ = U∗ (m(·), d(·)) be the set of all disturbances u(·) = (u1 (·), u3 (·)) ∈ L2 (T ; Rr+nd ) generating the pair (m(·), d(·)). An element of the set U∗ that has the minimal L2 (T ; Rr+nd )-norm is denoted by u∗ (·). Its existence and uniqueness follow from the convexity of U∗ and the strict convexity of the norm in L2 (T ; Rr+nd ), see Kryazhimskii and Osipov (1984); Osipov et al. (2011); Vdovin (1989). N The function uN (t) = (v1N (t), v3N (t)), v1N (t) = v1i , v3N (t) = N N v3i , t ∈ [τi , τi+1 ), i ∈ [0 : l − 1] is found from relations (22) and (23). Theorem 2. Let the following conditions of concordance of the parameters hold for N → ∞: δ N + hN hN → 0, g N → 0, δ N → 0, αN → 0, → 0. (24) αN Then, the sequence {uN (·)} is compact in L2 (T ; Rr+nd ) and we have the convergence as N → ∞:   P �uN (·) − u∗ (·)�L2 (T ;Rr+nd ) → 0 → 1. (25) Under the assumption that the variation of real disturbances is bounded, the following estimate for the accuracy of the algorithm with respect to the number of measurable realizations of the process is valid:   P �uN (·) − u∗ (·)�L2 (T ;Rr+nd ) ≤ D1 /N 2/13 = 1 − D2 /N 2/13 , (26) where D1 and D2 are some constants independent of the values under estimation.

Proof. The compactness of the sequence of controls and convergence (25) immediately follows from the application of results of Kryazhimskii and Osipov (1984) to system (17), (18) and from relations (19) and (24). Then, we use the fact that the probability of measuring both components of system (17), (18) with accuracy hN is equal to 1 − g N . Hence, rewriting the convergence rate estimate of the reconstruction algorithm obtained in Vdovin (1989) under the assumption that the variation of real disturbances is bounded for the case of measuring all phase coordinates in the form

IFAC CAO 2018 732 Valeriy L. Rozenberg / IFAC PapersOnLine 51-32 (2018) 727–732 Yekaterinburg, Russia, October 15-19, 2018

� � � N � N �2 �4 2 ¯ v2i (s) − u2i (s)) ds ≤ C15 v2i (s) − u2i (s) ds, (

�1/2 (hN + δ N )2 N N ¯ +α �u (·) − u∗ (·)�L2 (T ;Rr+nd ) ≤ C1 (αN )2 and setting δ N ≤ C¯2 hN and αN = C¯3 (hN )2/3 , we have � � P �uN (·) − u∗ (·)�L2 (T ;Rr+nd ) ≤ C¯4 (hN )1/3 = 1 − g N . �

(27) Here and below, we denote by C¯i auxiliary constants, which are independent of the estimated values and can be written explicitly.

The structure of estimate (27) and formulas (14) admit certain arbitrariness in the formation of the pair (hN , g N ). We orientate to the coincidence of the orders of smallness for these values. Consider two cases. 1. Let 1/2 + 3ǫ < 1, 0 < ǫ < 1/6. Then, formulas (14) take the form δ N = C¯5 /N α(1/2+3ǫ) , hN = C¯6 /N 1/2−ǫ , g N = C¯7 /N (1−α)(1/2+3ǫ) . (28) N N ¯ To obtain the inequality δ ≤ C2 h , it is sufficient to set 1/2 − ǫ = α(1/2 + 3ǫ), which implies ǫ = (1 − α)/(6α + 2), 1/3 < α < 1. For this ǫ, the power exponent of the value 1/N in relations (28) is 2α/(3α + 1) for hN and δ N , and (2 − 2α)/(3α + 1) for g N . To obtain estimate (26), taking into account formula (27), we assume 2α/3(3α + 1) = 2(1 − α)/(3α + 1), which implies α = 3/4 and ǫ = 1/26; consequently, the power exponents of the value 1/N in (26) are equal to 2/13. 2. Let 1/2 + 3ǫ ≥ 1, 1/6 ≤ ǫ ≤ 1/2. Then, formulas (14) take the form hN = C¯12 /N 1/2−ǫ , δ N = C¯13 /N α , g N = C¯14 /N 1−α . Obviously, the largest power exponent of the value 1/N in the approximation part of estimate (26) is 1/9; this is worse than the value obtained in the preceding case. The theorem is proved. It is possible to obtain different modifications of estimate (26). So, if we choose ǫ = 1/6 (α = 1/3, respectively), then the power exponent of the value 1/N in the approximative part of (26) is equal to 1/9, whereas the similar exponent in the probabilistic part is equal to 2/3. Proposition 3. In the case when the vector u2 (·) is unique and its coordinates are nonnegative, i.e., Su2 is such that u2i (t) ≥ 0 ∀t ∈ T ∀i ∈ [1 : n] (naturally for noises with zero mean value), algorithm (20)–(24) admits a modification reconstructing u2 (·) in the L2 (T ; Rn )-metric. Proof. Consider the n-dimensional subvector u4 (·) introduced above and consisting of the coordinates of the vector u3 (·) that are equal to u22i , i ∈ [1 : n], and the subvector v4N (·) consisting of the corresponding coordinates of the � N N (t) that is possible (t) = v4i vector v3N (·). Setting v2i N (t) ≥ 0 ∀t ∈ T ∀i ∈ [1 : n], from (25) we obtain since v4i for N →  ∞  � � n � N �2 P v4i (s) − u4i (s) ds → 0 → 1, T

i=1

  � � n � N �4 P v2i (s) − u2i (s) ds → 0 → 1. T

T

T

i=1

Using the inequalities

732

we derive P

 � 2 N v2i (s) − u2i (s) ds → 0 → 1.

� � n �

T

i=1

Thus, we get the convergence corresponding to (25) and proving the approximation of u2 (·). It is easily seen that the order of the estimate for the accuracy of the algorithm with respect to the value 1/N (see (26)) is preserved. REFERENCES Chernous’ko, F.L. and Kolmanovskii, V.B. (1978). Optimal Control under Random Perturbation. Nauka, Moscow. Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 2. Wiley, New York. Korolyuk, V.S., Portenko, N.I., Skorokhod, A.V., and Turbin, A.F. (1985). Handbook on Probability Theory and Mathematical Statistics. Nauka, Moscow. Krasovskii, N.N. and Subbotin, A.I. (1988). GameTheoretical Control Problems. Springer, New York. Kryazhimskii, A.V. and Osipov, Y.S. (1984). Modelling of a control in a dynamic system. Engrg. Cybernetics, 21(2), 38–47. Oksendal, B. (1985). Stochastic Differential Equations: an Introduction with Applications. Springer, Berlin. Osipov, Y.S. and Kryazhimskii, A.V. (1986). Positional modeling of a stochastic control in dynamical systems. Ser. Lecture Notes in Control and Information Sciences, 81, 696–704. Osipov, Y.S. and Kryazhimskii, A.V. (1995). Inverse Problems for Ordinary Differential Equations: Dynamical Solutions. Gordon & Breach, London. Osipov, Y.S., Kryazhimskii, A.V., and Maksimov, V.I. (2011). Some algorithms for the dynamic reconstruction of inputs. Proc. Steklov Inst. Math., 275(1), S86–S120. Rozenberg, V.L. (2007). Dynamic restoration of the unknown function in the linear stochastic differential equation. Autom. Remote Control, 68(11), 1959–1969. Rozenberg, V.L. (2016). Reconstruction of randomdisturbance amplitude in linear stochastic equations from measurements of some of the coordinates. Comp. Math. Math. Phys., 56(3), 367–375. Rozenberg, V.L. (2017). Dynamical reconstruction of disturbances in a quasi-linear stochastic differential equation. In Tikhonov Readings 2017, Abstracts of Scientific Conference, 87–88. Izd. MGU, Moscow, Russia. Rumyantsev, D.S. and Khrustalev, M.M. (2006). Optimal control of quasi-linear systems of the diffusion type under incomplete information on the state. J. of Computer and Systems Sciences Intern., 45(5), 718–726. Shiryaev, A.N. (1974). Probability, Statistics, and Random Processes. Izd. MGU, Moscow. Tikhonov, A.N. and Arsenin, V.Y. (1981). Solutions of Ill-Posed Problems. Wiley, New York. Vdovin, A.Y. (1989). On the Problem of Perturbation Recovery in a Dynamic System, Candidate’s Dissertation in Physics and Mathematics. Izd. IMM UrO AN SSSR, Sverdlovsk.