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| We consider a control problem where the state must approach asymptotically a target $C$ while paying an integral cost with a {\em non-negative} Lagrangian $l$. The dynamics $f$ is just continuous, and no assumptions are made on the zero level set of the Lagrangian $l$. Through an inequality involving a positive number $\bar p_0$ and a Minimum Restraint Function $U=U(x)$ --a special type of Control Lyapunov Function-- we provide a condition implying that {\bf (i)} the system is asymptotically controllable, and {\bf (ii)} the value function is bounded by $U/\bar p_0$. The result has significant consequences for the uniqueness issue of the corresponding Hamilton-Jacobi equation. Furthermore it may be regarded as a first step in the direction of a feedback construction. |
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