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, if tisfied everywhere except possibly a set of zero volume, guarantees strictly monotonic decrease of Lebesque volume of sets of positive volume. This, however, does not guarantee stability. For example, the ODE does not have a stable equilibrium, while volumes of sets are strictly decreasing with its However, it is possible to make an opposite statement that a system for which a positively weighted volume is strictly monotonically increasing cannot have a stable equi librium Theorem 11.3 Let f: R"H R and p: R"b R be continuously differentiable functions such that p-wveighted volume of every ball in r" is positive, and div(p)(a) 0 for allI E R. Then system(11.1) has no asymptotically stable equilibria and no asymptotically stable limit cycles Proof Assume to the contrary that o: RHR is a stable equilibrium or a stable limit cycle solution of (11.1). Then there exists e>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 theorem4 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, if satisfied everywhere except possibly a set of zero volume, guarantees strictly monotonic decrease of Lebesque volume of sets of positive volume. This, however, does not guarantee stability. For example, the ODE x˙ 1 = −2x1, x˙ 2 = x2 does not have a stable equilibrium, while volumes of sets are strictly decreasing with its flow. However, it is possible to make an opposite statement that a system for which a positively weighted volume is strictly monotonically increasing cannot have a stable equi￾librium. Theorem 11.3 Let f : Rn ∞� Rn and � : Rn ∞� R be continuously differentiable functions such that �-weighted volume of every ball in Rn is positive, and div(f �)(¯x) → 0 for all x¯ ≤ Rn. Then system (11.1) has no asymptotically stable equilibria and no asymptotically stable limit cycles. Proof Assume to the contrary that x0 : R ∞� Rn is a stable equilibrium or a stable limit cycle solution of (11.1). Then there exists � > 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