The problem of stabilizing a mathematical hybrid system with switchings between the operating modes is solved. Each of these modes is associated with nonlinear differential equations that have control parameters. The switching instances (conditions) are control components. A stabilizer must be designed in positional form that allows the trajectory of the entire nonlinear system to reach the target set in the phase space for a (prescribed) finite time. To solve the problem, k]an apparatus of continuous piecewise-linear Lyapunov functions is used along with the corresponding piecewise-linear control functions. A theorem concerning the sufficient conditions for the stabilizability of a hybrid system in the considered class of controls is proved. An algorithm for constructing the Lyapunov functions and the stabilizer is given.

中文翻译：

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使用分段线性控制系统稳定开关系统的问题

解决了通过操作模式之间的切换来稳定数学混合系统的问题。这些模式中的每一个都与具有控制参数的非线性微分方程相关。切换实例（条件）是控制组件。稳定器必须以位置形式设计，以允许整个非线性系统的轨迹在（规定的）有限时间内达到相空间中的目标集。为了解决该问题，使用了连续的分段线性Lyapunov函数的设备以及相应的分段线性控制函数。证明了在所考虑的控制类别中有关混合系统稳定性的充分条件的一个定理。给出了构造李雅普诺夫函数和稳定器的算法。