Problem of dynamical reconstruction of an unknown coefficient for a system of second order differential equations

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 6 (2017), pp Research India Publications Problem of dynamical reconstruction of an unknown coefficient for a system of second order differential equations Marina Blizorukova, Vyacheslav Maksimov Ural Federal University, Institute of Mathematics and Mechanics, Ural Branch, Acad. Sci. of Russia, Russia. Abstract The problem of dynamical reconstruction of an unknown coefficient for a system of nonlinear ordinary differential equations of second order through results of observations of the phase state is considered. Solving algorithm that is stable with respect to informational noises and computational errors is suggested. This algorithm is based on the principle of auxiliary models with feedback controls. AMS subject classification: Keywords: differential equations, reconstruction. 1. Introduction Problems of reconstructing unknown characteristics of dynamical systems through results of measurements are well known in engineering and scientific research [1, 2, 6, 11, 12]. These problems are embedded into the theory of inverse problems of dynamics. One of the approaches to solving such problems was suggested in [3, 8] and developed in [4, 7, 9, 10]. The goal of this paper following the researches in the field is to describe an algorithm for reconstructing an unknown coefficient for systems of second order differential equations. The algorithm should be dynamical. The information on initial data is uncertain and in general time-varying. The analogous problems are considered in [5] (see also the bibliography in this paper). Consider a system described by the equations: ẋ 1 (t) = k(t)x 2 (t) + x 1 (t)(λx 2 (t) ν), ẋ 2 (t) = k(t)x 2 (t) (λx 1 (t) + µ)x 2 (t) + γ(t), (1.1)

2 1846 Marina Blizorukova, Vyacheslav Maksimov t T =[t 0,ϑ], ϑ = const < +, ϑ > t 0, x 1 (t 0 ) = x 10, x 2 (t 0 ) = x 20. Let constants λ>0, ν>0, µ>0, and a function k( ) be known and let a function (an input) γ(t) be uncertain. We assume that the input γ(t) (a Lebesque measurable function satisfying the condition γ(t) P =[ γ,γ], t T ) acts upon the system. Here, γ = const (0, + ). At discrete time moments τ i ={τ i } m i=0, τ i+1 = τ i + δ, τ 0 = t 0, τ m = ϑ, the phase states x 1 (τ i ) and x 2 (τ i ) of system (1.1) are inaccurately measured. Results of measurements (vectors {ξ h 1i,ξh 2i } R2 ) satisfy the inequalities x 1 (τ i ) ξ h 1i h, x 2(τ i ) ξ h 2i h, (1.2) where h (0, 1) is the level of informational noise and x is the modulus of a number x. The problem under consideration consists in designing an algorithm for reconstructing the unknown input γ( ). This is the meaningful statement of the problem investigated in the present paper. Hereinafter, we assume that the following condition is fulfilled. Condition. a) The real input γ = γ(t)generates the solution x(t) = x(t,γ)of equation (1.1) such that inf k(t) + λx 1(t, γ ) c>0. t T b) The function k(t) is Lebesque measurable and bounded. The algorithm for solving the problem consists in the following. An auxiliary control dynamical system M is introduced. This system functioning on the time interval T has an unknown input (control) u h ( ) and an output (a solution) w h ( ). The problem of reconstructing the input γ( ) is substituted by the problem of forming the control u h ( ) in the system M (by the feedback principle) in such a way that the deviation of γ( ) from u h ( ) in the L 2 -metric is small if the measurement accuracy h is small enough. The process of synchronous feedback control of systems (1.1) and M is decomposed into m 1 identical steps. First, at the i-th step carried out on the time interval δ i =[τ i,τ i+1 ), according to the chosen rule u h, the control u h (t) = u h i V h (τ i,ξ h 1i 1,ξh 2i 1,ξh 2i,wh (τ i )), t [τ i,τ i+1 ), (1.3) is calculated. Then, the control u h = u h (t), τ i t<τ i+1, is fed onto the input of the system M. The phase state w h (τ i+1 ) is the output result at the i-th step. Let us proceed to the rigorous statement of the problem in question. Fix a family of partitions of the interval T : h ={τ i,h } m h h=0, τ i+1,h = τ i,h + δ(h), τ 0,h = 0, τ mh,h = ϑ. (1.4)

3 Problem of dynamical reconstruction 1847 We assume for simplicity δ(h) (0, 1) for all h (0, 1). Problem. It is required to specify differential equations of the system M ẇ h (t) = f 1 (,ξ h 1i 1,ξh 2i 1,uh i ), t δ i,h =[τ i,τ i+1 ), (1.5) τ i = τ i,h, i [1 : m h 1], w h (τ 1 ) = w h 0, wh (t) R, and the rule for forming controls u h i at the times τ i as some mapping such that the convergence V h :{τ i,ξ h 1i 1,ξh 2i 1,ξh 2i,w h(τ i )} u h i R (1.6) ϑ t 0 u h (t) γ(t) 2 dt 0 (1.7) takes place whereas h tends to 0. Here, u h (t) = u h i, t δ h,i. 2. The Solving Method From now on, it is assumed that we know numbers d 1, d 2, and k such that x 1 (t) d 1, x 2 (t) d 2, (2.1) k(t) k for a.a. t T. (2.2) In virtue of (2.1) and (2.2), the inequalities k(t) k 1 = k(t 0 ) +k(ϑ t 0 ), (2.3) ẋ 1 (t) d 3 = k 1 d 2 + λd 1 d 2 + νd 1, (2.4) ẋ 2 (t) d 4 = k 1 d 2 + λd 1 d 2 + µd 2 + γ (2.5) are valid. Besides, for t [,τ i ] the inequalities t x 1 (t) ξ h i 1 h + ẋ 1 (τ) dτ h + d 3 δ, (2.6) x 2 (t) ξ h 2i 1 h + d 4δ hold. Write over the right-hand part of equation (1.5) in the form f 1 (t, x 1,x 2,u)= f(t,x 1,x 2 ) + u,

4 1848 Marina Blizorukova, Vyacheslav Maksimov where f(t,x 1,x 2 ) = (k(t) + λx 1 + µ)x 2. Fix a family h of partitions of the interval T of form (1.4) and choose a linear system M described by the following equation: { u h 0, for a.a. t δ 0 = δ 0,h, ẇ h (t) = f(,ξ h 1i 1,ξh 2i 1 ) + uh i, for a.a. t δ i = δ i,h =[τ i,τ i+1 ), (2.7) i [1 : m 1], τ i = τ i,h, m = m h, with the initial condition w h 0 = ξ h 20. Consequently, the equation (1.5) takes the shape of (2.7). Further we assume that the solution of equation (2.7) is constrained by the value d 5 > 0, i.e. Let x ={x 1,x 2 }, w h (t) d 5 t T, h (0, 1). (2.8) Q T ( ) ={v( ) L 2 (T ; R) : v(t) γ for a.a. t T }, V(x( )) ={v( ) Q T ( ) : ẋ 1 (t) = (k(t) + λx 1 (t) + µ)x 2 (t) + v(t) for a.a. t T }. Introduce a family of sets V h i = V h i (τ i,ξ h 1i 1,ξh 2i 1,ξh 2i,w h(τ i )) ={v R : v γ, {(w h (τ i ) ξ h 2i 1 )[ (k() + λξ h 1i 1 + µ)ξ h 2i 1 + v] where (w h (τ i ) ξ h 2i )(ξ h 2i ξ h 2i 1 )δ 1 } σ h,δ }, i [1 : m h 1] K 1 = (1 + d 2 + d 5 )K 3 + d 2 4, K 2 = (1 + d 2 + d 5 )K 4 + 2d 4, δ = δ(h), σ h,δ = 2h(3 + d 4 + d 5 + d 2 )δ 1 + K 1 δ + K 2 h, K 3 = (µ + λ(1 + d 1 ) + k 1 )d 4 + (k + λd 3 )d 2, K 4 = λ(1 + d 1 ) + k 1 + λd 2 + µ. Let the mapping V h (see, (1.3) and (1.6)) be defined by the rule V h (τ i,ξ h 1i 1,ξh 2i 1,ξh 2i,w h(τ i )) = u h i = (2.9) = arg min{ v :v Vi h i,ξ h 1i 1,ξh 2i 1,ξh 2i,w h(τ i ))} for i [1 : m h 1]. Let, in addition, u h (t) = u h i for t δ i,h, i [1 : m h 1], u h (t) = 0 for t δ 0,h. (2.10)

5 Problem of dynamical reconstruction 1849 Lemma 2.1. Let v( ) V(x( )). Then the inclusion takes place. δ 1 τ i v(t) dt Vi h, i [1 : m] Proof. First, we estimate the variation of the value We have where Consider a value ν i = (w h (τ i ) ξ h 2i 1 ) {f(,ξ h 1i 1,ξh 2i 1 ) + v(t)} dt (w h (τ i ) ξ h 2i ) ν 1i = (w h (τ i ) ξ h 2i 1 ) ν 2i = (ξ h 2i ξ h 2i 1 ) {f(t,x 1 (t), x 2 (t)) + v(t)} dt. ν i = ν 1i + ν 2i, (2.11) {[f(,ξ h 1i 1,ξh 2i 1 ) + v(t)] [f(t,x 1 (t), x 2 (t)) + v(t)]} dt, {f(t,x 1 (t), x 2 (t)) + v(t)} dt. λ(t) = f(t,x 1 (t), x 2 (t)) f(,ξ h 1i 1,ξh 2i 1 ), t δ i 1. From (2.1) (2.6), we have the estimation λ(t) k( )ξ h 2i 1 k(t)x 2(t) + (2.12) +λ ξ h 1i 1 ξ h 2i 1 x 1(t)x 2 (t) +µ ξ h 2i 1 x 2(t) k(t) k( ) x 2 (t) +k( ) ξ h 2i 1 x 2(t) + λ x 1 (t) ξ h 1i 1 x 2(t) +(λ ξ h 1i 1 +µ) x 2(t) ξ h 2i 1 δkd 2 + k 1 (h + d 4 δ) + λ(h + d 3 δ)d (µ + λ(1 + d 1 ))(h + d 4 δ) K 4 h + K 3 δ,

6 1850 Marina Blizorukova, Vyacheslav Maksimov In virtue of (2.8), (2.1), and (2.5) we get w h (τ i ) ξ h 2i 1 w h(τ i ) + x 2 ( ) ξ h 2i 1 + x 2( ) d 5 + h + d 2, (2.13) ξ h 2i ξ h 2i 1 2h + x 2(τ i ) x 2 ( ) 2h + d 4 δ. (2.14) Using (2.12), and (2.13), we derive ν 1i (1 + d 2 + d 5 )(K 3 δ + K 4 h)δ, h (0, 1). Also, from (2.5), and (2.14), we obtain ν 2i (2h + d 4 δ)d 4 δ. Consequently ν i (K 1 δ + K 2 h)δ. (2.15) Taking into account the equality we have τ i {f(t,x 1 (t), x 2 (t)) + v(t)} dt = x 2 (τ i ) x 2 ( ), ξ h 1i ξ h 2i 1 {f(t,x 1 (t), x 2 (t)) + v(t)} dt 2h, (2.16) Due to (2.13), and (2.14), we conclude that Therefore, by using (2.16), we get (w h (τ i ) ξ h 2i ) { ξ h 2i ξ h 2i 1 w h (τ i ) ξ h 2i 3h + d 4δ + d 2 + d 5. } {f(t,x 1 (t), x 2 (t)) + v(t)} dt (2.17) 2(3 + d 2 + d 4 + d 5 )h. The statement of the lemma follows from (2.15), and (2.17). The lemma is proved. Lemma 2.2. The following inequalities ε(τ i+1 ) w h (τ i+1 ) x 2 (τ i ) 2 (ϑ t 0 )(σ h + (2c 1 + c 2 ) h δ + 4δ{(c 1 + d 4 ) 2 + d4 2 }), i [0 : m 1], are valid. Here c 2 = 2d d 2 + d 4, c 1 = γ + (k 1 + µ + λ(d 1 + 1))(d 2 + 1).

7 Problem of dynamical reconstruction 1851 Proof. It is easily seen that the inequalities ε(τ i+1 ) ε(τ i ) + 2(w h (τ i ) x 2 ( )){f(,ξ h 1i 1,ξh 2i 1 ) + uh i }δ + (2.18) +2(w h (τ i ) x 2 ( ))(x 2 (τ i ) x 2 ( )) + 4 ( τ i+1 τ i ) 2, { ẋ 2 (t) + ẇ h (t) } dt ξ h 2i d 4 (2.19) are true. Using (2.19), and the inequality ẇ h (t) = f(,ξ h 1i 1,ξh 2i 1 ) + uh i γ + ( k() +λ ξ h 1i 1 +µ) ξ h 2i 1 c 1, as well as (2.5) and (2.18), we derive ε(τ i+1 ) ε(τ i ) + 2(w h (τ i ) ξ h 2i 1 ){f(,ξ 1i 1,ξ h 2i 1 ) + uh i }δ + (2.20) + 2hc 1 δ + 2(w h (τ i ) x 2 (τ i ))(x 2 (τ i ) x 2 ( )) + +2 x 2 (τ i ) x 2 ( ) 2 + 4δ 2 (c 1 + d 4 ) 2. Further, by (2.8) and (2.5), we have (w h (τ i ) x 2 (τ i ))(x 2 (τ i ) x 2 ( )) (w h (τ i ) ξ h 2i )(ξ h 2i ξ h 2i 1 ) (2.21) 2h w h (τ i ) +h x 2 (τ i ) x 2 ( ) +2h ξ h 2i 2hd 5 + hδd 4 + 2h(1 + d 2 ) c 2 h. Taking into account (2.20), (2.21), and x 2 (τ i ) x 2 ( ) 2 d 2 4 δ2,weget ε(τ i+1 ) ε(τ i ) + 2(w h (τ i ) x 1 ( )){f(,ξ h 1i 1,ξh 2i 1 ) + uh i }δ (2.22) (w h (τ i ) ξ h 2i )(ξ h 2i ξ h 2i 1 ) + (2c 1 + c 2 )h + 4δ 2 {(c 1 + d 4 ) 2 + d 2 4 }. In virtue of the rule of choosing the control u h i inequality (see (2.10)) and (2.22) we obtain the ε(τ i+1 ) ε(τ i ) + σ h δ + (2c 1 + c 2 )h + 4δ 2 {(c 1 + d 4 ) 2 + d 2 4 }. The statement of the lemma follows from this inequality. The lemma is proved. Taking into account lemmas 2.1 and 2.2 (see, for example, [8]) we derive the following theorem. Theorem 2.3. The mapping V h of the form (2.9) solves the reconstruction problem provided the model M (see, (1.5)) is given by (2.7). Acknowledgements The work was supported by the Program of Ural Branch of the Russian Academy of Sciences (project no ).

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