We propose a numerical approach for solving systems of nonautonomous ordinary di®erential equations under suitable assumptions. This approach is based on expansion of the solutions by Volterra series and allows to estimate the accuracy of the approximation. Also we can solve some ordinary di®erential equations for which the classical numerical methods fail

Kirov, Nikolay

Krastanov, Michail

English

New Bulgarian University Scholar Electronic Repository

NUMERICAL SOLUTIONS OF ODEsUSING VOLTERRA SERIESNikolay Kirov* and Mikhail Krastanov***Department Informatics, New Bulgarian University andInstitute of Mathematics and Informatics, Bulgarian Academy of Sciences,**Institute of Mathematics and Informatics, Bulgarian Academy of Sciencesnkirov@nbu.bg, krast@math.bas.bgAbstractWe propose a numerical approach for solving systems of nonautonomous ordinarydifferential equations under suitable assumptions. This approach is based on expan-sion of the solutions by Volterra series and allows to estimate the accuracy of theapproximation. Also we can solve some ordinary differential equations for which theclassical numerical methods fail.1 IntroductionA numerical approach for solving systems of nonautonomous ordinary differential equa-tions (ODEs) is proposed under suitable assumptions. This approach is based on expansionof the solutions of ODEs by Volterra series and allows to estimate the distance betweenthe obtained approximation and the true trajectory.The approximation of trajectories of nonlinear control systems can be reduced to theproblem of approximation the solutions of nonautonomous systems of ODEs. As a rule,their right-hand sides depend on the time in a discontinuous way. But applying the tradi-tional numerical approximation schemes of higher order (such as Runge-Kutta schemes) isa nontrivial task (cf. for example [2], [8], etc.). An approach for solving this problem foraffinelly controlled systems is proposed in [3] and [4]. It is based on the well known expan-sion of the solution of ODEs systems by Volterra series (cf. [5]). Combining this approachwith the ideas developed in [6], we obtain a method for approximation the trajectories ofanalytic control systems with guaranteed accuracy (cf. [7]).2 Systems of ODEs and Volterra series.For every point y = (y1, . . . , yn)T from Rn we define ‖y‖ := ∑ni=1 |yi| and B = {y ∈ Rn :‖y‖ ≤ 1}. For x0 ∈ Rn, Ω is a convex compact neighbourhood of the point x0.If z ∈ C, the set of the complex numbers, then |z|, Re z and Im z denote the norm,the real and the imaginary part of z, respectively. For some σ > 0 we setΩσ := {z = (z1, . . . , zn) ∈ Cn : (Re z1, . . . ,Re zn) ∈ Ω + σB; |Im zi| < σ, i = 1, . . . , n}.By FσΩ we denote the set of all real analytic functions φ defined on Ω with boundedanalytic extensions φ¯ on Ωσ. We define a norm in the set FσΩ as follows:‖φ‖σΩ = sup{|φ¯(z)| : z ∈ Ωσ} .1If h(x) = (h1(x), h2(x), . . . , hn(x)), x ∈ Ω is a vector field defined on Ω, we identify hwith the corresponding differential operatorn∑i=1hi(x)∂∂xi, x ∈ Ω.Let VσΩ be the set of all real analytic vector fields h defined on Ω such that everyhi, i = 1, . . . , n belongs to FσΩ. We define the following norm in VσΩ:‖h‖σΩ = max{‖hi‖σΩ, i = 1, . . . , n}.An integrable analytic vector field Xt(x) = (X1(t, x), X2(t, x), . . . , Xn(t, x)), x ∈ Ω,t ∈ R (parameterized by t) is a map t→ Xt ∈ VσΩ such that:i) for every x ∈ Ω the functions X1(., x), X2(·, x), . . . , Xn(·, x) are measurable;ii) for every t ∈ R and x ∈ Ω, |Xi(t, x)| ≤ m(t), i = 1, 2, . . . , n, where m is anintegrable (on every compact interval) function.Let t0 be a real number, M be a compact set contained in the interior of Ω andx0 ∈M , and let Xt be an integrable analytic vector field defined on Ω. Then there existsT (M,Xt) > t0 such that for every point x of M the solution y(., x) of the differentialequationy˙(t, x) = Xt(y(t, x)), y(t0, x) = x (1)is defined on the interval [t0, T (M,Xt)] and y(T, x) ∈ Ω for every T from [t0, T (M,Xt)].In this case we denote by exp∫ Tt0Xt dt : M → Ω the diffeomorphism defined byexp∫ Tt0Xt dt (x) := y(T, x).According to Proposition 2.1 from [?], the positive number T (M,Xt) can be chosenin such a way that T (M,Xt) > t0 and for every T from the open interval (t0, T (M,Xt)),for every point x from M and for every function φ from FσΩ, the following expansion ofφ(exp∫ Tt0Xtdt(x))in Volterra series holds true:φ(exp∫ Tt0Xtdt(x))= φ(x)+∞∑N=1∫ Tt0∫ τ1t0. . .∫ τN−1t0XτNXτN−1 . . . Xτ1φ(x)dτNdτN−1 . . . dτ1(2)and the series is absolutely convergent.3 A computational procedureWe consider the following system of ODEs:ddtx(t) = f(t, x(t)), x(t0) ∈M0, (3)where M0 is a convex and compact subset of Rn. We assume that there exist σ > 0 anda convex and compact neighbourhood Ω of the set M0 such that f(t, ·) belongs to VσΩ for2each t from [t0, Tˆ ]. Trajectories are the absolutely continuous functions x(t), t ∈ [0, Tˆ ]satisfying (3) for almost every t in this interval.Let ε > 0. First, using [7], Proposition 1 we can find a positive real T , t0 < T ≤ Tˆ ,such that for every point x0 from M0 the corresponding trajectory x(·, x0) : [t0, T ] → Rnstarting from x0 is well defined on [t0, T ] and x(t) ∈ Ω for every t from [t0, T ]. Next, weshow how to calculate a point y ∈ Ω such that |x(T, x0) − y| < ε. We choose the pointst0 < t1 < . . . < tk = T such that2nσ∫ ti+1ti‖f(s, ·)‖σΩ ds <12for each i = 0, . . . , k − 1. Let α > 1 be so large that1kαexp(4n2σ∫ Tt0‖f(s, ·)‖σΩ ds)< ε.At the end we determine the accuracy of the local approximation by choosing the positiveinteger ω to be so large that the following inequality holds truenω(2nσ)ω−1 (∫ ti+1ti‖f(s, ·)‖σΩ ds)ω<13kα+1.We set y0 := x0 andyi+1 := yi +ω−1∑N=1∫ ti+1ti∫ τ1ti. . .∫ τN−1tif(τN , ·)f(τN−1, ·) . . . f(τ1, ·)E(yi)dτNdτN−1 . . . dτ1for every i = 0, 1, . . . , k − 1. Since the trajectory x(·, x0) : [t0, T ] → Rn is well defined on[0, T ] and belongs to Ω, we obtain according to [7], Proposition 2 that |x(T, x0)− yk| < ε.Our choice of the points t0, t1, . . . , tk, of α > 0 and of ω > 0 guarantee the neededaccuracy of the numerical approximation. But, the used estimates are not precise enoughand in practice the obtained accuracy is much better than the expected one (cf. theexamples from the next section).4 ExamplesWe consider some examples illustrating the applicability of the proposed numerical ap-proach [7]. First, this approach is suitable for solving ODEs and systems of ODEs.Example 1. Let us consider the following ODE:x˙ = γtγ−1x2, x(0) = x0 > 0,where γ ∈ (0, 1]. For T ≥ x1 = x− 1γ0 it can be seen that there does not exist a compact setM1 satisfying the assumptions of [7], Proposition 1. Hence, we can not conclude that thesolution of this ODE exists on [0, T ]. In fact, this ODE has no solution in this interval.But, if we choose an arbitrary T , such that 0 ≤ T < x1, the solution exists. By settingf(t, x) := γtγ−1x2, it can be directly calculated that∫ t0∫ τ10. . .∫ τk−10f(τk, ·)f(τk−1, ·) . . . f(τ1, ·)E(x0)dτkdτk−1 . . . dτ1 = xk+10 tkγ .30501001502000.5 1 1.5 2 2.5 0501001502000.2 0.4 0.6 0.8 1Figure 1: A Runge-Kutta approximation and the true solutionHence, the series (2) for this ODE is:x0 + x20tγ + x30t2γ + · · ·+ xk+10 tkγ + · · · ,which is convergent and tends to its analytic solutionx(t) =x01− x0tγ , t ∈ [0, x1).We made some numerical experiments with MAPLE for γ = 0.2 and x0 = 1. Weused dsolve command with options numeric and method=dverk78 that finds a numericalsolution using a seventh-eighth order continuous Runge-Kutta method (Fig. 1, left). Thismethod calculates a solution in the interval [0, T ) for T > 2 while the analytic solutionx(t) tends to infinity for t tends to T = 1 (Fig. 1, right).The proposed numerical approach is used for approximations of reachable sets of anonlinear control system.Example 2. [1] Let us consider the following system of ODEs:∣∣∣∣∣∣x˙1 = u, |u| ≤ 1, x1(0) = 0,x˙2 = x1 +12x21, x2(0) = 0.The Volterra series of this system is finite, it has four non zero terms. It follows thatfor each integrable control function u with values from the interval [−1, 1], we can computethe exact single trajectory. The most complicated task is the choice of control sets suchthat the trajectory endpoints could outline the reachable set. The reachable set R(t) iscalled the set of all point which can be steered from the origin within the fixed time t.The boundary of R(t) can be derived (cf. [1]) by using switching curves, correspondingto the bang-bang values of the control (u = ±1) and singular arcs (Fig. 2).We obtained the boundary of R(6) as follows. The points from the upper boundary,and the points from the lower boundary in the interval [4, 6], correspond to piecewiseconstant control functions with only one jump and values ±1, i.e.up(t) :={e, t ∈ [0, p),−e, t ∈ [p, 6), (4)401020304050–6 –4 –2 2 4 6Figure 2: The boundary of the set R(6)where e ∈ {−1, 1} (Fig. 3).The most points from the lower boundary of the reachable set are computed by thefollowing family of control functions:up,e(t) :=−1, t ∈ [0, 1),0, t ∈ [1, p),e, t ∈ [p, 6),(5)where e ∈ {−1, 1} and p ∈ (1, 6] (Fig. 4). The true reachable set R(6) and its approxima-tion of second order, with time-step 1 and step 0.05 for p, are shown on Fig. 5.1020304050–6 –4 –2 0 2 4 601020304050–6 –4 –2 2 4 6Figure 3: Trajectory endpoints for (4) and e ≡ 1 (left) and e ≡ −1 (right)5051015–1 1 2 3 4 051015–5 –4 –3 –2 –1Figure 4: Trajectory endpoints for (5) and e ≡ 1 (left) and e ≡ −1 (right)01020304050–6 –4 –2 2 4 6Figure 5: The true reachable set (—) and second order approximation (· · ·)6References[1] U. Boscain, B. Piccoli: A Short Introduction to Optimal Control, Lecture Notes forthe CIMPA School, Tlemcen, Algeria, 2003.[2] A. Dontchev, F. Lempio: Difference methods for differential inclusions, SIAM J. Re-view 34 (1992), 263-294.[3] L. Gru¨ne, P. Kloeden: Higher order numerical schemes for affinely controlled nonlinearsystems, Numer. Math. 89 (2001), 669-690.[4] L. Gru¨ne, P. E. Kloeden: Numerical schemes of higher order for a class of nonlinearcontrol systems, In: “Numerical Methods and Applications”, I. Dimov, I. Lirkov, S.Margenov, Z. Zlatev, (Eds.), Springer Lecture Notes in Computer Science 2542 (2003),213-220.[5] A. Isidori: Nonlinear control systems. An introduction, Springer, 1995.[6] M. Krastanov, N. Kirov: Dynamic Interactive System for Analysis of Linear Differ-ential Inclusions, In: “Modelling techniques for uncertain systems”, A. Kurzhanski, V.Veliov (Eds.), Birkhauser, 123-130, 1994.[7] N. Kirov, M. Krastanov: Volterra Series and Numerical Approximations of ODEs,In: Numerical Analysis and Its Applications, Zhilin Li, L. Vulkov, J. Was’niewski eds.,Lecture Notes in Computer Science 3401, (2005), 337-344.[8] V. Veliov: On the time discretization of control systems, SIAM J. Control Optim. 35(1997), 1470-1486.7

Numerical Solutions of ODEs using Volterra Series

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