| Viability methods for sustainable management of fisheries: Michel De Lara, Lue Doyen
Thèrése Guilbaud, Marie-Joëlle
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  or invariance  framework. Basically, such an approach focuses on inter-temporal feasible paths. It has been applied for instance to models related to the sustainable management of resource and bio-economic 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 discrete-time 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 , 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 ); reachability of target sets or tubes for nonlinear discrete time dynamics is examined in .
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.