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In this paper, the use of time‐varying piecewise quadratic functions is investigated to characterize the finite‐time stability of state‐dependent impulsive dynamical linear systems. Finite‐time stability defines the behavior of a dynamic system over a bounded time interval. More precisely, a system is said to be finite‐time stable if, given a set of initial conditions, its state vector does not exit a predefined domain for a certain finite interval of time. This paper presents new sufficient conditions for finite‐time stability based on time‐varying piecewise quadratic functions. These conditions can be reformulated as a set of Linear Matrix Inequalities that can be efficiently solved through convex optimization solvers. Different numerical analysis are included in order to prove that the presented conditions are able to improve the results presented so far in the literature. [ABSTRACT FROM AUTHOR]

In this study, the authors address the problem of reaching a given percentage of an optimal cost, so as to guarantee a known reserve with respect to the optimum. This reserve aims to counteract fast changes in the operating conditions. The authors assume that the analytical expression of the cost function is not available. To tackle this problem, they propose a novel extremum seeking scheme, the sub‐optimal extremum seeking, that is able to keep a given margin with respect to the estimated optimal cost. Stability properties of the control scheme are proved and the effectiveness of the approach is validated in simulation. [ABSTRACT FROM AUTHOR]