In this article, a novel multidimensional penalty-free method is proposed for the optimal control problem of switched control systems with path constraints. First, by introducing a Boolean vector, the switched control system and path constraints are transformed into affine forms. Typically, the complementary constrained method and piecewise constant function are used to relax and approximate the Boolean vector. Moreover, numerical methods are used to solve the approximated constrained optimal control problem (COCP), which relies on a discrete grid of the control function by using a piecewise linear function. Furthermore, the approximated COCP of affine control systems is equivalently transformed into a constrained dynamic parameter optimization problem by a refined time-scaling transformation. Finally, the global convergence of the proposed multidimensional penalty-free algorithm is proved and the feasibility of the algorithm is verified by numerical examples.

This article investigates the generalized Nash equilibrium (GNE) seeking for the game with equality constraints. Each player cannot directly access all the other player's actions and the gradients of all players' payoff functions are unknown. In these scenarios, an interesting question is under what distributed algorithm the GNE can be found. To address such games, we first design a two‐time‐scale distributed algorithm based on the extremum seeking method and consensus protocol. Then, by utilizing singular perturbation techniques and Lyapunov robust analysis, we show that the players' decisions can be regulated to an arbitrarily small neighborhood of the GNE. Moreover, we further consider the ideal case in which the gradients are known. In this case, the proposed strategy can be degenerated to a gradient‐based algorithm and the players' decisions exponentially converge to the GNE. Finally, two examples along with simulation results are used to illustrate the effectiveness of the proposed algorithms.