Asymptotically stable globally

If 0 and the system of fast motion has a unique and asymptotically stable global steady state at every fixedy,-, we can apply Tikhonov s theorem and, starting from a certain value of e, use a QSS approximation. 

Fig. 2. Global bifurcation for equations (21)—(23). (a) Hamiltonian reference system, (b) Stable separ-atrix loop arising from the effect of the dissipative" perturbation I2. (c) Asymptotically stable...

So far we have defined the local stability ("there exists such <3 as. . . ). Now let us define the global stability for rest points. The rest point c0 is called globally asymptotically stable (as a whole) within the phase space D if it is stable according to Lyapunov, and for any initial conditions d0e D the solution c(t, k, cLa) tends to approach c 0 at t - oo. 

Definition 5—Global Asymptotic Stability. The equilibrium point 0 at time ko of Eq. (39) is said to be globally asymptotically stable if... 

Meadows et al. (1995) showed that the following MPC nonlinear program, corresponding to a moving horizon of length 3, results in a closed loop that is globally asymptotically stable around the equilibrium point Xe = (0, 0).(Recall that an equilibrium point Xe is globally asymptotically stable if x[fc] Xe as for any initial point x[0].)... 

Remark. If Ai-I-A2>l then there is no positive solution of (3.2) and hence no interior equilibrium. In this case E2 is a globally asymptotically stable rest point. If A1-I-A2 < 1, there exists a unique interior rest point and E2 is unstable. 

With our standing hypotheses that all rest points are hyperbolic, Proposition 4.1 implies that if E does not exist then either E2 is asymptotically stable (all eigenvalues of the Jacobian are negative) or Ei exists and is asymptotically stable (all eigenvalues are negative). The results to follow establish that if , i—, 2, is asymptotically stable then it attracts all solutions (is globally asymptotically stable for positive initial conditions). Therefore, when E does not exist, one of the rest points E or E2 attracts all solutions of (3.2). 

Theorem 7.2. Suppose that system (3.2) has no limit cycles. Then Ec is globally asymptotically stable. 

The rest point corresponds to a rest state without the v competitor. When it is asymptotically stable, the v competitor will become extinct (will wash out of the system) for nearby initial conditions. If the stability is global (which will turn out to be the case when stability is local), then v becomes extinct for all positive initial conditions. At Ei, J takes the form... 

Proof of Theorem 7.1 (cont.). Suppose that the hypothesis of (c) holds and that Ej is locally asymptotically stable. (There is a similar proof if E is locally asymptotically stable, and by Theorem E.l both are not.) To establish the conclusion in case (c), we must show that Ej is globally asymptotically stable. Since f fl [ 2> fi iJx is positively invariant for (2.4), and belongs to the basin of attraction of E2 as a consequence of Lemma 7.2, one need only show that... 

A surprising consequence of the analysis was that, when the interior rest point existed, it was unique and globally asymptotically stable. 

Then x is globally asymptotically stable for all initial conditions, x(r) —> x as r oo. In particular the system has no closed orbits. (For a proof, see Jordan and Smith 1987.)... 

Hence E<0, with equality only if e = 0. Therefore is a Liapunov function, and so e = 0 is globally asymptotically stable. ... 

The stationary state p is globally asymptotically stable if r] can be arbitrarily large. 

Fig. 7. Phase plane for h > 1.207. When p is large, the steady state, y = 0, z = 0, is globally asymptotically stable.

In the limit - 0, y(T) changes much more rapidly than x(t) Except near Q = 0 the vector field (x,y) is everywhere nearly horizontal. The two falling sections of the one-dimensional manifold Q 0 are stable, but the middle section is unstable. (We referred to this fact earlier.) For 0 < 6 < 6 and 6 < 6 < /e find that the steady state is globally asymptotically stable (as -> 0). However, under these conditions the system is excitable in the sense described in Chapter IV (pp. 76f) For 6q < 6 < 6 we find an orbitally as3nmptoti-cally stable periodic solution illustrated in Fig. 4. 

Now let us define die global stability of rest points. The rest point Co is called a global asymptotically stable rest point within the phase space D if it is Lyapunov stable and for any condition do QD the solution c(t,k,do) approaches Cq at (x>. An analysis of the problem of... 

Constant characteristics ii = K results in a linear time invariant (LTI) model which is globally asymptotically stable. This case is the basis of all the following models, containing only the effect of the differential variables. 

A steady state into which all initial states of the system will evolve for t—>00 is called globally and asymptotically stable. Such a situation is a particularly simple form of the stability behaviour of a dynamical system. It includes cases where the steady state is not unique but there are multiple steady states among which, however, only one is globally and asymptotically stable. The other steady states are then instable any arbitrarily small fluctuation drives the system out of them into the stable state. 

When discussing the model for the control of metabolic reaction chains in Section 2.7 we have become acquainted with a third type of stability of a steady state. In this model, there always exists a unique steady state, cf. (2.62), (2.63). For concentrations S of the external substrate below a critical value S, i.e. S < S, the steady state is globally and asymptotically stable, whereas for S > S the state of the system leaves the steady state in a spiral motion in the phase space approaching an undamped periodically oscillating trajectory, a so-called limit cycle, cf. (2.75), (2.76). 

The considerations throughout this chapter will be devoted to networks which produce either multiple steady states or limit cycles. The leading idea for these considerations will be the experience that such a stability behaviour or, from the point of view of systems with one globally and asymptotically stable steady state, such an instability behaviour is caused by a feedback coupling within the network. 

In the language of network topology, feedback manifests itself as a closed loop of bonds, junctions and elements. The reverse conclusion, however, does not hold the networks of the pore and carrier models and that of active transport in Sections 5.1 to 5.5 actually involve closed loops, but nevertheless show a unique globally and asymptotically stable steady state. The simplest case of a nontrivial feedback loop, i.e., a feedback loop which really leads to multiple steady states or limit cycles, is an autocatalytic reaction... 

Fig. 10 (a). Despite of the autocatalytic feedback loop which is present in (6.1) even for v = 1 we thus have a unique steady state in X > 0 which is globally and asymptotically stable. 

In this section we shall construct networks involving Al-elements as "integrated circuits". Let us stress once more that the Al-element placed between to external reservoirs A and B is globally and asymptotically stable, cf. Fig. 10. 

The arrows in Fig. 12 result from combining the signs of dX/dt and dY/dt and thus indicate the momentary direction of evolution of the state of the system. From Fig. 12 we conclude that for the steady state (C) is globally and asymptotically stable whereas (A), (Bl) and (B2) are instable. For Xj> < the... 

What actually can be derived on the basis of a network representation is a negative criterion for limit cycles which says that loop-free networks not only have a unique steady state as argued in Section 7.5 but that this steady state also is globally and asymptotically stable. This statement evidently excludes the possibility of a limit cycle. The proof of this criterion has an analytic and a network topological part. In the analytic part one shows that the system of differential equations... 

Apply conditions 1), 2) and 3) as expressed by (7.72) to (7.76) as a criterion for global and asymptotical stability to the networks throughout this book and show that those networks which have already been identified as globally and asymptotically stable indeed satisfy the conditions whereas those showing multiple steady states or limit cycles violate conditions 2) or 3) or even both. 

A system represented by a set of equations describing a physical system is said to be stable around a steady state if the transients of the system are bounded when the system is subjected to small perturbations from the steady state. The system is said to be asymptotically stable if it is stable and it eventually returns to the steady state. If in addition, the steady state is approached exponentially, it is called exponentially asymptotically stable. If any magnitude of perturbation is allowed, it is said to be globally stable. The stability problem as related to reactor startup and control is concerned with the following questions (Luss 1977) ... 

---