Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 11: Volume Evolution And System Analysis Lyapunov analysis, which uses monotonicity of a given function of system state along trajectories of a given dynamical system, is a major tool of nonlinear system analysis It is possible, however, to use monotonicity of volumes of subsets of the state space to predict certain properties of system behavior. This lecture gives an introduction to suc methods 11.1 Formulae for volume evolution This section presents the standard formulae for evolution of volumes 11.1.1 Weighted volume bounded on every compact subset of U. For every hypercube sureable function which is Let U be an open subset of R", and P: U HR be a me Q(z,r)={x={x1;x2;……;rn]:|rk-k≤r} contained in U, its weighted volume with respect to p is defined by 1+r r Ve(Q(i, r)) P( 1,T 1-T n-1-T Without going into the fine details of the measure theory, let us say that the weighted volume of a subset X CU with respect to p is well defined if there exists M>O such that for every e>0 there exist(countable) families of cubes Qk and ( Q2)(all contained in Version of October 31. 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 11: Volume Evolution And System Analysis1 Lyapunov analysis, which uses monotonicity of a given function of system state along trajectories of a given dynamical system, is a major tool of nonlinear system analysis. It is possible, however, to use monotonicity of volumes of subsets of the state space to predict certain properties of system behavior. This lecture gives an introduction to such methods. 11.1 Formulae for volume evolution This section presents the standard formulae for evolution of volumes. 11.1.1 Weighted volume Let U be an open subset of Rn, and � : U ∞� R be a measureable function which is bounded on every compact subset of U. For every hypercube Q(¯x, r) = {x = [x1; x2; . . . ; xn] : |xk − x¯k| ∀ r} contained in U, its weighted volume with respect to � is defined by � x xn−1+r ¯ � � � � ¯ ¯ 1+r � x2+r � ¯ � xn+r V�(Q(¯x, r)) = . . . �(x1, x2, . . . , xn)dxn dxn−1 . . . dx2 dx1. x¯ ¯ 1−r x2−r x¯n−1−r x¯n−r Without going into the fine details of the measure theory, let us say that the weighted volume of a subset X � U with respect to � is well defined if there exists M > 0 such that for every � > 0 there exist (countable) families of cubes {Q1 k k} and {Q } (all contained in 2 1Version of October 31, 2003
U)such that X is contained in the union of Qi, the union of all Qk is contained in the union of X and Qk, and ∑W(Q) 1) under a continuous map could cover a cube (positive Lebesque volume)
� � � � � 2 U) such that X is contained in the union of Qi k, the union of all Q2 is contained in the k union of X and Q1 k, and k) 1) under a continuous map could cover a cube (positive Lebesque volume)
11.1.3 Volume change under a differential fow Let consider the case when the map F= St is defined by a smooth differential How Remember that, for a differential function 9: R"HR", div(g) is the trace of the Jacobian of g Theorem 11. 2 Let u be an open subset of R". Let f: U HR and P: U HR be continuously differentiable functions. For t>o let ur be the set of vectors TE U such that the ODE i(t)=f(a(t)) has a solution r: 0, T-U such that a(0)=i. Let Sr: UrHU be the map defined by Sr(r(0)=a(T). Then, if X is contained in a compact subset of Ur and has a p-qeighted volume, the map tH Ve(St(X)) is well defined, differentiable, and its derivative at t=0 zs gu (S(X) dt Proof According to theorem 11.2 v(S(x)=/p(S(2)det(d.s()/dr()d Note that dSt(D dt ∫(亚), and dS(a)/dr(a)=A(t, I),where ),△(0,) Hence det(dSt(a)/d r)>0, and, at t=0, d rIde(dSt(a)/dx(a)) d dt det(△(t,z) d△(t,x where the equality da(T) det(a(t))=det(a(T)trace(a(T)dr was used. Finally, at t=0 d dp(S()det(ds)dr()=(p()()+p()div()=dvp()
� � � � � � � � 3 11.1.3 Volume change under a differential flow Let consider the case when the map F = St is defined by a smooth differential flow. Remember that, for a differential function g : Rn ∞� Rn, div(g) is the trace of the Jacobian of g. Theorem 11.2 Let U be an open subset of Rn. Let f : U ∞� Rn and � : U ∞� R be continuously differentiable functions. For T > 0 let UT be the set of vectors x¯ ≤ U such that the ODE x˙ (t) = f(x(t)), has a solution x : [0, T] ∞� U such that x(0) = x¯. Let ST : UT ∞� U be the map defined by ST (x(0)) = x(T). Then, if X is contained in a compact subset of UT and has a �-weighted volume, the map t ∞� V�(St(X)) is well defined, differentiable, and its derivative at t = 0 is given by dV�(St(X)) = Vdiv(�f) (X). dt Proof According to Theorem 11.2, V�(St(X)) = �(St(¯x))| det(dSt(x)/dx(¯x))|dx. ¯ X Note that � dSt(¯x)� � = f(¯x), dt t=0 and dSt(x)/dx(¯x) = �(t, x¯), where d�(t, x¯) df � = � �(t, x¯), �(0, x¯) = I. dt dx x=St(¯x) Hence det(dSt(x)/dx) > 0, and, at t = 0, d d | det(dSt(x)/dx(¯x))| = det(�(t, x¯)) dt dt d�(t, x¯)� = trace dt t=0 = div(f)(¯x), where the equality d det(A(� )) = det(A(� ))trace A(� ) −1 dA(� ) d� d� was used. Finally, at t = 0, d �(St(¯x))| det(dS x))| = (∈�)(¯ x) + �(¯ x) = div(�f)(x). t(x)/dx(¯ x)f(¯ x)div(f)(¯ dt
11.2 USing volume monotonicity in system analysis Results from the previous section allow one to establish invariance(monotonicity) of weighted volumes of sets evolving according to dynamical system equations. This section discusses application of such invariance in stability analysi 11.2.1 Volume monotonicity and stability Given an ode model i(t)=f(r(t)), where f: RHR is a continuously differentiable function, condition div(f)0 such that lim min a(t)-Io(T)I=0 for every solution c=r()of(11. 1)such that a(0) belongs to the ball B0={:|-r00) Let u(t)=Ve(St(Bo)), where St is the system flow. By assumption, v is monotonically non-increasing, u(0)=0, and v(t)-0 as t-0o. The contradiction proves the theorem
4 11.2 Using volume monotonicity in system analysis Results from the previous section allow one to establish invariance (monotonicity) of weighted volumes of sets evolving according to dynamical system equations. This section discusses application of such invariance in stability analysis. 11.2.1 Volume monotonicity and stability Given an ODE model x˙ (t) = f(x(t)), (11.1) where f : Rn ∞� Rn is a continuously differentiable function, condition div(f) 0 such that lim min |x(t) − x0(� )| = 0 t�� � for every solution x = x(·) of (11.1) such that x(0) belongs to the ball B0 = {x¯ ¯ : |x − x0(0)| ∀ �. Let v(t) = V�(St(B0)), where St is the system flow. By assumption, v is monotonically non-increasing, v(0) = 0, and v(t) � 0 as t � ⊂. The contradiction proves the theorem
11.2.2 Volume monotonicity and strictly invariant sets Let us call a set X C R strictly invariant for system(11.1)if every maximal solution x=x(t)of(1) with a(0)∈ x is defined for all t∈ R and stays in X for all t∈ R. Obviously, if X is a strictly invariant set then, for every weight p, Ve(St(X)) does not change as t changes. Therefore, if one can find a p for which div(pf)>0 almo everywhere, the strict invariance of X should imply that X is a set of a zero Lebesque volume. i.e. the following theorem is true Theorem 11.4 Letu be an open subset of R". Let f: U HR andP: UHR be continuously differentiable functions. Assume that div(p)>0 for almost all points ofU Then, if X is a bounded closed subset of U which is strictly invariant for system(11.1), the Lebesque volame of X equals zero As a special case, when n=2 and p= l, we get the Bendiron theorem, which claims that if, in a simply connected region U, >0 almost everywhere, there exist no non-equilibrium periodic trajectories of (11. 1)in U. Indeed, a non-equilibrium periodic trajectory on a plane bounds a strictly invariant set 11.2.3 Monotonicity of singularly weighted volumes So far, we considered weights which were bounded in the regions of interest. A recent ol servation by A. Rantzer shows that, when studying asymptotic stability of an equilibrium, it is most beneficial to consider weights which are singular at the equilibrium In particular, he has proven the following stability criterion Theorem 11.5 Let f: R"HR" and P: R /10 bR be continuously differentiable functions such that f(0)=0, P(a)f(a)/l is integrable over the set r|> 1, and div(p)> 0 for almost all E R. If either p20 or0 is a locally stable equilibrium of(11.1) then for almost all initial states a(0) the corresponding solution x=a(t)of(11.1) converges to zero as t→∞ To prove the statement for the case when a =0 is a stable equilibrium >0 consider the set Xr of initial conditions (0) for which sup r(t>r VT> The set X is strictly invariant with respect to the How of (11.1), and has well defined e-weighted volume. Hence, by the strict weighted volume monotonicity, the Lebesque measure of X equals zero. Since this is true for all r>0, almost every solution of (11.1) converges to the origin
5 11.2.2 Volume monotonicity and strictly invariant sets Let us call a set X � Rn strictly invariant for system (11.1) if every maximal solution x = x(t) of (11.1) with x(0) ≤ X is defined for all t ≤ R and stays in X for all t ≤ R. Obviously, if X is a strictly invariant set then, for every weight �, V�(St(X)) does not change as t changes. Therefore, if one can find a � for which div(�f) > 0 almost everywhere, the strict invariance of X should imply that X is a set of a zero Lebesque volume, i.e. the following theorem is true. Theorem 11.4 Let U be an open subset of Rn. Let f : U ∞� Rn and � : U ∞� R be continuously differentiable functions. Assume that div(f �) > 0 for almost all points of U. Then, if X is a bounded closed subset of U which is strictly invariant for system (11.1), the Lebesque volume of X equals zero. As a special case, when n = 2 and � ≥ 1, we get the Bendixon theorem, which claims that if, in a simply connected region U, ÷(f) > 0 almost everywhere, there exist no non-equilibrium periodic trajectories of (11.1) in U. Indeed, a non-equilibrium periodic trajectory on a plane bounds a strictly invariant set. 11.2.3 Monotonicity of singularly weighted volumes So far, we considered weights which were bounded in the regions of interest. A recent observation by A. Rantzer shows that, when studying asymptotic stability of an equilibrium, it is most beneficial to consider weights which are singular at the equilibrium. In particular, he has proven the following stability criterion. Theorem 11.5 Let f : Rn ∞� Rn and � : Rn/{0} ∞� R be continuously differentiable functions such that f(0) = 0, �(x)f(x)/|x| is integrable over the set |x| → 1, and div(f �) > 0 for almost all x ≤ Rn. If either � → 0 or 0 is a locally stable equilibrium of (11.1) then for almost all initial states x(0) the corresponding solution x = x(t) of (11.1) converges to zero as t � ⊂. To prove the statement for the case when x = 0 is a stable equilibrium, for every r > 0 consider the set Xr of initial conditions x(0) for which sup |x(t)| > r � T > 0. t�[T,�) The set X is strictly invariant with respect to the flow of (11.1), and has well defined �-weighted volume. Hence, by the strict �-weighted volume monotonicity, the Lebesque measure of Xr equals zero. Since this is true for all r > 0, almost every solution of (11.1) converges to the origin
6 Example 11.1(Rantzer)The system +x-x2 +2: cic. satisfies conditions of Theorem 11.5 with p()=ll
6 Example 11.1 (Rantzer) The system −2x1 + x1 − x2 x˙ 1 = 2 2 x˙ 2 = −6x2 + 2x1x2 satisfies conditions of Theorem 11.5 with �(x) = |x| −4