Viability, Optimality and Stability of Dynamical Systems and Estimation of Convergence of Numerical Schemes

AMS subject classification: 49N35,49N55,65Lxx.

We present in this paper some recent developments dealing with dynamical controlled systems with state constraints. After recalling the basic frame of Viability Theory and it numerical aspects, we estimate the convergence of numerical schemes for computing the optimal time for target problem. We give also a relaxation result for decomposable problems. These properties are enhanced through the study of the Norvegian Fishermen problem arising in Dynamic of Population. Another interesting application of this approach is shortly presented when considering the approximation of the so-called the minimal time of crisis. This appears for problems where some constraints are “soft” (reversibility) and others are “hard” (irreversibility).