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Solution Bounds, Stability and Attractors' Estimates of Some Nonlinear Time-Varying Systems

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Solution Bounds, Stability and Attractors' Estimates of Some Nonlinear Time-Varying Systems 1 Solution Bounds, Stability and Attractors’ Estimates of Some Nonlinear Time-Varying Systems Mark A. Pinsky and Steve Koblik Abstract. Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous applied and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for a broad class of the corresponding systems. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and estimates the stability basins and trapping regions for the corresponding systems. Finally, we conform the theoretical results in representative simulations. Key Words. Comparison principle, Lipschitz condition, stability criteria, trapping/stability regions, solution bounds AMS subject classification. 34A34, 34C11, 34C29, 34C41, 34D08, 34D10, 34D20, 34D45 1. INTRODUCTION. We are going to study a system defined by the equation n (1) xAtxftxFttt , , [0 , ), x , ft ,0 0 where matrix A nn and functions ft:[ , ) nn and F : n are continuous and bounded, and At is also continuously differentiable. It is assumed that the solution, x t, x0 , to the initial value problem, x t0, x 0 x 0 , is uniquely defined for tt0 . Note that the pertained conditions can be found, e.g., in [1] and [2]. We also examine the solutions to a homogeneous counterpart to (1) (2) x A t x f t, x Development of efficient stability criteria for x 0 solution to (2) is essential in numerous applied problems. There are two main approaches to this problem: the Lyapunov functions method, see for instance [2] and [3], and the first approximation methodology, see e.g. recent reviews [4] and [5] as well as [14] and [15] for additional references and historical perspectives. The former approach, for instance, is widespread in control literature, see [2] and [6]- [13] and additional references therein. However, adequate Lyapunov functions are rare, especially for time dependent and nonlinear systems. The latter approach delivers sufficient stability criteria under two conditions, see [14], and [15]. The first is the Lipschitz condition (3) f x,,, t l t x x t t0 where is a neighborhood of x 0 and llt ˆ be continuous function. The second condition, ()tt0 (4) W t, t00 Ne , t t , 0 N , const 1 bounds the growth rate of the transition matrix W t, t00 Wt W t , where W t is the fundamental solution matrix of M.A. Pinsky is with the Department of Mathematics & Statistics, University of Nevada. Reno, Reno NV 89521 USA (e-mail: [email protected]). Steve Koblik is independent consultant in scientific computing (e-mail: [email protected]) . 2 (5) x A t x Inequality (4) comprises necessary and sufficient conditions for asymptotic/exponential stability of (5), e.g. [3] and [14]. Consequently, it was shown that x 0 solution to (2) is exponentially stable if (3), (4), and the following condition, (6) Nl 0 are satisfied [14], [15]. A more flexible sufficient condition, t (7) W( t , t N exp p ( s ) ds , t t 0 00 t0 which reduces to (4) for pt const was introduced in [16], see also [4] and [5]. In turn, (3) and (7) provide asymptotic stability of (2) if [4], t (8) limsup 1/t p s ds Nl 0 t t0 While the existence of (4) is acknowledged under some broad conditions [14], to our knowledge, there were no attempts to apply bound (4) or (7) to stability analysis of the systems of practical importance. Furthermore, it was shown, e.g. in [17], that the time-histories of different estimates of the bounds of the Euclidian norms for the second order fundamental matrix, i.e. W t exp At , A const , can drastically diverge from each other and the exact values of exp At . This raises concern of the practical value of the listed above sufficient stability criteria. An attempt to escape the utility of prior bounds on W t, t0 in stability analysis of (2) was undertaken in [18]. However, authentication of the developed stability conditions for relatively complex systems can present a challenging task for this approach as well. The problem of estimating the norms of solutions to (1) subject to (3) and (4) was reviewed in [14]and [15]. This problem has some connection to input-to-state stability, which was mainly studied in the context of the Lyapunov function methodology in control literature, see [3] and [19] for further details and references. This paper derives a novel scalar differential inequality for the norms of solutions to (1) or (2), which collapses the dimension of the original estimation problem to one. Due to the comparison principle [3], this inequality further reduces to the analysis of solutions to the auxiliary first order scalar nonlinear equation with variable coefficients. Utility of prior bounds on W t, t0 is naturally voided in our study, which splits in two parts. The first adopts (3) which linearizes the auxiliary equation and yields the solution bounds and corresponding stability criteria. The second part introduces a nonlinear extension of the Lipschitz inequality. This sharpens the bounds of solutions and estimates the trapping/instability regions for nonlinear equations with time dependent coefficients. Our inferences are validated in simulations of the Van der- Pol like model, which includes a time dependent linear block and oscillatory external force. This paper is organized as follows. The next section derives the pivotal differential inequality, the subsequent section develops solution bounds and some stability criteria via utility of the Lipschitz inequality, section 4 introduces a nonlinear extension of the Lipschitz inequality and develops its applications, section 5 includes the simulation results, and section 6 concludes this study. 2. Differential Inequality for Solution Norms The application of variation of parameters lets us write (1) as [3] t xtxWtWtxWtW(,)()()()()11f( , x ( )) F ( ) d 0 0 0 t0 where W t is frequently normalized to satisfy the condition, W t0 I , I is the identity matrix. In section 5 we present normalization of Wt , which is more natural for our studies and, hence, used in our simulations. Presently, we only assume that Wt 0 1 . The last equation yields the following inequality, 3 t (9) xtx(,) WtWtx ()11 () WtW () ()(,()) fx F d 0 0 0 t0 Now we attempt to match (9) with a first order differential inequality and the associate initial condition as follows, Dx ptxk t ftx , Ft (10) x t X 0 |tt o where D is Dini’s upper right-hand derivative in t [3]. In turn, the application of variation of parameters to (10) yields, t x e X e k d 0 t0 t where ps ds and fF, x . Comparison of (9) with the last formula returns, t0 t W expps ds , X W1 t x t 0 0 0 t0 Then, we can write that tt e/ k d W W1 W d tt00 1 Hence, kt W t W t max W / min W is the running condition number of W t , and max and min are maximal and minimal singular values of . We assume that, tt0 , supmax const and ˆ inf min const and is unique. This, in particular, assures that k t is continuous and k t k const . We also write that (11) p t dln W t / dt Hence, pt is continuous since we assumed that is unique. To write (10) in the standard form, we introduce a nonlinear extension of Lipschitz inequality, (12) fxt, L t,,, x x t t0 where is a neighborhood of x 0 and L is a function continuous in t and x . In section 4 we review how to devise (12) for some broad sets of functions. Clearly, (12) reduces to (3) if L is linear in x . Finally, we write (10) in the following form D x p t x ktt L t, x F (13) 1 x t0 W t 0 x 0 Interestingly, the sensitivity of (5) to perturbations is scaled by k t and, hence, varies in time. Due to the comparison principle [3], solutions of (13) are bounded by the corresponding solutions to the associated first order auxiliary differential equation, which is analyzed in section 4. In the following section we linearize (13) through utility of (3). This lets us establish some upper bounds on x t, x0 and derive the corresponding stability criteria. 3. Linearized Auxiliary Equation Using (3), we write (13) as a linear inequality, 4 D x p t ktt l x ktt F ,, x t t0 (14) 1 x t0 W t 0 x 0 Due to the comparison principle [3], solutions of (14) are bounded by the consistent solutions of the corresponding auxiliary differential equation. We mention that our previous assumptions assure that the initial value problem for this equation possesses a unique solution. This leads to the following, Theorem 1. Assume that (3) is intact and, due to our previous assumptions, k and F are continuous and bounded, and p is continuous. Then, (15) xtx ,,,,,0 xtxh 0 xtx nh 0 x tt 0 where t (16) x WtWtx1 exp k lsds h 00 s t0 and t (17) x k F d nh t, t0 t where transition function, t, exp ps k s l sds . Clearly, due to the assumptions made in previous section, the integrals in (16) and (17) exists for tt0 , which assures this statement.
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