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9.7. STABILITY OF DISSIPATIVE SYSTEMS 237
9.7 Stability of Dissipative Systems
Throughout this section we analyze the possible implications of stability (in the sense of
Lyapunov) for dissipative dynamical systems. Throughout this section, we assume that
the storage function 4) : X -> R+ is differentiable and satisfies the differential dissipation
inequality (9.5).
In the following theorem we consider a dissipative system z/i with storage function 0
and assume that xe is an equilibrium point for the unforced systems ?P, that is, f (xe) _
f(xe,0)=0.
Theorem 9.3 Let be a dissipative dynamical system with respect to the (continuously
differentiable) storage function 0 : X -* IR+, which satisfies (9.5), and assume that the
following conditions are satisfied:
(i) xe is a strictly local minimum for 0:
(ii) The supply rate w = w(u, y) is such that
4)(xe) < 4)(x) Vx in a neigborhood of xe
w(0, y) < 0 Vy.
Under these conditions xe is a stable equilibrium point for the unforced systems x = f (x, 0).
Proof: Define the function V (x) tf 4)(x) - 4)(xe). This function is continuously differentiable,
and by condition (i) is positive definite Vx in a neighborhood of xe. Also, the time
derivative of V along the trajectories of 0 is given by
V (x) = aa(x) f (x, u)
thus, by (9.5) and condition (ii) we have that V(x) < 0 and stability follows by the Lyapunov
stability theorem.
Theorem 9.3 is important not only in that implies the stability of dissipative systems
(with an equilibrium point satisfying the conditions of the theorem) but also in that it
suggests the use of the storage function 0 as a means of constructing Lyapunov functions.
It is also important in that it gives a clear connection between the concept of dissipativity
and stability in the sense of Lyapunov. Notice also that the general class of dissipative
systems discussed in theorem 9.3 includes QSR dissipative systems as a special case. The