zbMATH — the first resource for mathematics

Trajectoires lourdes de systèmes contrôlés. (French) Zbl 0561.49011
The authors consider the following control problem: For all \(x_ 0\in K\subset R^ n\) find \(T>0\) and an absolute continuous function x(\(\cdot)\) satisfying for almost all \(t\in [0,T]\) the relations (1) \(x'(t)=f(x(t),u(t))\); (2) u(t)\(\in F(x(t))\), where \(F:K\to R^ p\); (3) \(x(0)=x_ 0\). Here \(f:F(K)\to R^ n\) is supposed to be continuous. The x(t) are called viable trajectories, if x(t)\(\in K\) for all \(t\in [0,T]\). Heavy viable trajectories are the ones which are associated with the controls regulating viable trajectories whose velocity has at each \(t\in [0,T]\) the minimal norm. Differential equations governing the evolution of the controls associated to heavy viable trajectories are derived and the existence of heavy trajectories is proved.
Reviewer: K.Zimmermann

49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49J15 Existence theories for optimal control problems involving ordinary differential equations
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations