Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 9: Local Behavior Near Trajectories 1 This lecture presents results which describe local behavior of ODE models in a neigbor- hood of a given trajectory, with main attention paid to local stability of periodic solutions. 9.1 Smooth Dependence on Parameters In this section we consider an ODE model x˙(t) = a(x(t),t,?), x(t 0 ) = ?x 0 (?), (9.1) where ? is a parameter. When a and ?x 0 are difierentiable with respect to ?, the solution x(t) = x(t,?) is difierentiable with respect to ? as well. Moreover, the derivative of x(t,?) with respect to ? can be found by solving linear ODE with time-varying coe–cients. Theorem 9.1 Let a : R n ￡ R ￡ R k ∞? R n be a continuous function, ? 0 ≤ R k . Let x 0 : [t 0 ,t 1 ] ∞? R n be a solution of (9.1) with ? = ? 0 . Assume that a is continuously difierentiable with respect to its flrst and third arguments on an open set X such that (x 0 (t),t,? 0 ) ≤ X for all t ≤ [t 0 ,t 1 ]. Then for all ? in a neigborhood of ? 0 the ODE in (9.1) has a unique solution x(t) = x(t,?). This solution is a continuously difierentiable function of ?, and its derivative with respect to ? at ? = ? 0 equals ￠(t), where ￠ : [t 0 ,t 1 ] ∞? R n,k is the n-by-k matrix-valued solution of the ODE ˙ ￠(t) = A(t)￠(t) + B(t), ￠(t 0 ) = ￠ 0 , (9.2) x ∞? a(? x at ?where A(t) is the derivative of the map ? x,t,? 0 ) with respect to ? x = x 0 (t), B(t) is the derivative of the map ? ∞? a(x 0 (t),t,?) at ? = ? 0 , and ￠ 0 is the derivative of the map ? ∞? ?x 0 (?) at ? = ? 0 . 1 Version of October 10, 2003 2 Proof Existence and uniqueness of x(t,?) and ￠(t) follow from Theorem 3.1. Hence, in order to prove difierentiability and the formula for the derivative, it is su–cient to show that there exist a function C : R + ∞?R + such that C(r)/r ? 0 as r ? 0 and ? > 0 such that |x(t,?) ?￠(t)(??? 0 ) ?x 0 (t)|? C(|??? 0 |) whenever |??? 0 |? ?. Indeed, due to continuous difierentiability of a, there exist C 1 ,?0 such that |a(? x?x 0 (t)) ?B(t)(??? 0 )|? C 1 (|x? ?x,t,?) ?a(x 0 (t),t,? 0 ) ?A(t)(? ? x 0 (t)|+ |??? 0 |) and |x 0 (?) ? ?? x 0 (? 0 ) ?￠ 0 (??? 0 )|? C 1 (|??? 0 |) whenever |x? ?? x 0 (t)|+ |??? 0 |? ?, t ≤ [t 0 ,t 1 ]. Hence, for –(t) = x(t,?) ?x 0 (t) ?￠(t)(??? 0 ) we have |– ˙ (t)|? C 2 |–(t)|+ C 3 (|??? 0 |), as long as |–(t)|? ? 1 and |??? 0 |? ? 1 , where ? 1 > 0 is su–ciently small. Together with |–(t 0 )|? C 4 (|??? 0 |), this implies the desired bound. Example 9.1 Consider the difierential equation y˙(t) = 1 + ?sin(y(t)), y(0) = 0, where ? is a small parameter. For ? = 0, the equation can be solved explicitly: y 0 (t) = t. Difierentiating y ? (t) with respect to ? at ? = 0 yields ￠(t) satisfying ˙ ￠(t) = sin(t), ￠(0) = 0, i.e. ￠(t) = 1 ?cos(t). Hence y ? (t) = t + ?(1 ?cos(t)) + O(? 2 ) for small ?. 3 9.2 Stability of periodic solutions In the previous lecture, we were studying stability of equilibrium solutions of difierential equations. In this section, stability of periodic solutions of nonlinear difierential equations is considered. Our main objective is to derive an analog of the Lyapunov’s flrst method, stating that a periodic solution is asymptotically stable if system’s linearization around the solution is stable in a certain sense. 9.2.1 Periodic solutions of time-varying ODE Consider system equations given in the form x˙(t) = f(x(t),t), (9.3) where f : R n ￡R ∞?R n is continuous. Assume that a is (?, ?x)-periodic, in the sense that there exist ? > 0 and ?x ≤R n such that f(t + ?,r) = f(t,r), f(t,r + ?x) = f(t,r) ? t ≤R, r ≤R n . (9.4) Note that while the flrst equation in (9.4) means that f is periodic in t with a period ?, it is possible that ?x = 0, in which case the second equation in (9.4) does not bring any additional information. Deflnition A solution x 0 : R ∞? R n of a (?, ? x)x)-periodic system (9.3) is called (?, ? periodic if x 0 (t + ?) = x 0 (t) + ?x ? t ≤R. (9.5) Example 9.2 According to the deflnition, the solution y(t) = t of the forced pendulum equation y?(t) + ˙y(t) + sin(y(t)) = 1 + sin(t) (9.6) ?as a periodic one (use ? = x = 2…). This is reasonable, since y(t) in the pendulum equation represents an angle, so that shifting y by 2… does not change anything in the system equations. Deflnition A solution x 0 : [t 0 ,→) ∞? R n of (9.3) is called stable if for every – > 0 there exists ? > 0 such that |x(t) ?x 0 (t)|? – ? t ? 0, (9.7) whenever x(￠) is a solution of (9.3) such that |x(0) ? x 0 (0)| < ?. x 0 (￠) is asymptotically stable if it is stable and the convergence |x(t) ? x 0 (t)| ? 0 is guaranteed as long as |x(0)?x 0 (0)|is small enough. x 0 (￠) exponentially stable if, in addition, there exist ,C > 0 such that ∈x(t) ?x 0 (t)∈? C exp(? t)|x(0) ?x 0 (0)| ? t ? 0 (9.8) whenever |x(0) ?x 0 (0)| is small enough. 4 To derive a stability criterion for periodic solutions x 0 : R ∞? R n of (9.3), assume continuous difierentiability of function f = f(? ?x,t) with respect to the flrst argument x for |?x ? x 0 (t)| ? ?, where ? > 0 is small, and difierentiate the solution as a function of initial conditions x(0) … x 0 (0). Theorem 9.2 Let f : R n ￡ R ∞? R n be a continuous (?, ?x)-periodic function. Let x 0 : R ∞? R n be a (?, ?x)-periodic solution of (9.3). Assume that there exists ? > 0 such ?that f is continuously difierentiable with respect to its flrst argument for |x ? x 0 (t)| < ? and t ≤ R. For df A(t) = | x=x 0 (t) , (9.9) dx deflne ￠ : [0, ?] ∞? R n,n be the n-by-n matrix solution of the linear ODE ˙ ￠(t) = A(t)￠(t), ￠(0) = I. (9.10) Then (a) x 0 (￠) is exponentially stable if ￠(?) is a Schur matrix (i.e. if all eigenvalues of ￠(?) have absolute value less than one); (b) x 0 (￠) is not exponentially stable if ￠(?) has an eigenvalue with absolute value greater or equal than 1; (c) x 0 (￠) is not stable if ￠(?) has an eigenvalue with absolute value greater than 1. The matrix-valued function ￠ = ￠(t) is called the evolution matrix of linear system (9.10). The proof of Theorem 9.2 follows the same path as the proof of a similar theorem for stability of equilibria, using time-varying quadratic Lyapunov functions. 9.2.2 Stable limit cycles time-invariant ODE Consider system equations given in the form x˙(t) = a(x(t)), (9.11) R n where a : ￡ R ∞? R n is continuous. Let ?x ≤ R n be such that a(? x) = a(? x ≤ R n x + ? x) ? ? (in particular, ?x = 0 always satisfles this condition). Deflnition Let ? > 0. A non-constant (?, ?x)-periodic solution x 0 : R ∞? R n of system (9.11) is called a stable limit cycle if 5 (a) for every ? > 0 there exists – > 0 such that dist(x(t),x 0 (￠)) < ? for all t ? 0 and all solutions x = x(t) of (9.11) such that dist(x(0),x 0 (￠)) < –, where x,x 0 (￠)) = min |?dist(? x? x 0 (t)|; t?R (b) there exists ? > 0 such that dist(x(t),x 0 (￠)) ? 0 as t ? → for every solution of (9.11) such that dist(x(0),x 0 (￠)) < –. Note that a non-constant periodic solution x 0 = x 0 (t) of time-invariant ODE equations is never asymptotically stable, because, as – ? 0, the initial conditions for the solution x – (t) = x 0 (t + –) approach the initial conditions for x 0 (￠), but the difierence x – (t) ? x 0 (t) does not converge to 0 as t ? → unless x – · x 0 . Therefore, the notion of a stable limit cycle is a relaxed version of asymptotic stability of a solution. Theorem 9.3 Let a : R n ∞? R n be a continuous (?x)-periodic function. Let x 0 : R ∞? R n be a non-constant (?, ?x)-periodic solution of (9.11). Assume that there exists ? > 0 such that a is continuously difierentiable on the set X = {? ?x ≤ R n : |x? x 0 (t)| < ? for some t ≤ R. Let ￠ : [0,?] ∞? R n,n be deflned by (9.9),(9.10). Then (a) if all eigenvalues of ￠(?) except one have absolute value less than 1, x 0 (￠) is a stable limit cycle; (b) if one eigenvalue of ￠(?) has absolute value greater than 1, x 0 (￠) is not a stable limit cycle.