We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set \(\mathbf{C}\)with\(({p_{0}},W)\)-regulated cost, for some continuous, state-dependent function W and some constant \({p_{0}}>0\): it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by \(W(z)/{p_{0}}\). Then, we show how the existence of a special, semiconcave Control Lyapunov Function W, called \({p_{0}}\)-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a \({p_{0}}\)-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with \(({p_{0}},W)\)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.

我们还通过考虑相应的成本来扩展与不连续反馈相关的控制系统的采样和Euler解决方案的经典概念。特别是，对于某些连续的状态，我们将样本和Euler稳定性的概念引入到具有\（（{p_ {0}}，W）\）调节成本的封闭目标集\（\ mathbf {C} \）中依赖性函数w ^和一些恒定\（{P_ {0}}> 0 \） ：它大致意味着我们需要一个稳定的反馈的存在ķ使得从一个点开始的所有相应的采样和欧拉溶液ž已经适当定义有限成本，其上限为\（W（z）/ {p_ {0}} \）。然后，我们展示了一个特殊的存在，semiconcave控制Lyapunov函数如何W¯¯，被称为\（{P_ {0}} \） -最小抑制功能，可以让我们明确地构造这样的反馈ķ。当动力学和拉格朗日在状态变量中为Lipschitz连续时，我们证明，如果存在\（{p_ {0}} \）-最小约束函数（仅是Lipschitz连续），仍可以获得上述K。用\（（{p_ {0}}，W）\）调节的非完整积分器控制系统的成本与带拉格朗日有限度的任何成本相关联的稳定性示例说明了结果。