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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

##### MSC:
 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
##### Keywords:
viable trajectories; Heavy viable trajectories