
Viability methods for sustainable management of fisheries: Michel De Lara, Lue Doyen
Thèrése Guilbaud, MarieJoëlle
Let us consider a nonlinear control system described in discrete time by the difference equation
xt+1 = f(xt; ut); " t ? N;
x0 given, (1)
where the state variable xt belongs to the finite dimensional state space X = RnX, the control variable ut is an element of the control set U = RnU while the dynamics f maps X x U into X.
A controller or a decision maker describes \desirable configurations of the system" through a set
D Ì X x U termed the desirable set (xt; ut) ? D; " t ? N; (2)
where D includes both system states and controls constraints. Typical instances of such a desirable set are given by inequalities requirements: D = (x; u) ? X x U / " i = 1; : : : ; p ; gi(x; u) ? 0 .
The state constraints set associated with D is obtained by projecting the desirable set D onto the state space X:
V0 = ProjX(D) = x ? X j 9u ? U; (x; u) ? D . (3)
Such problems of dynamic control under constraints refers to viability [2] or invariance [10] framework. Basically, such an approach focuses on intertemporal feasible paths. It has been applied for instance to models related to the sustainable management of resource and bioeconomic modeling as in [4, 5, 6, 11, 16, 17, 20]. From the mathematical viewpoint, most of viability and weak invariance results are addressed in the continuous time case. However, some mathematical Works deal with the discretetime case. This includes the study of numerical schemes for the approximation of the viability problems of the continuous dynamics as in [2, 18]. Important contributions for the discrete time case are also captured by the study of the positivity for linear systems as in [1], or by the hybrid control as in [3, 21]. In the control theory literature, problems of constrained control lead to the study of positively invariant sets, particularly ellipsoidal and polyhedral ones for linear systems (see [8, 12, 14] and the survey paper [9]); reachability of target sets or tubes for nonlinear discrete time dynamics is examined in [7].
Viability is defined as the ability to choose, at each time step t ? N, a control ut ? U such that the system configuration remains desirable.

