Sequential adaptive switching time optimization technique for maximum hands-off control problems

In this paper, we consider maximum hands-off control problem governed by a nonlinear dynamical system, where the maximum hands-off control constraint is characterized by an $ L^{0} $ norm. For this problem, we first approximate the $ L^{0} $ norm constraint by a $ L^{1} $ norm constraint. Then, the control parameterization together with sequential adaptive switching time optimization technique is proposed to approximate the optimal control problem by a sequence of finite-dimensional optimization problems. Furthermore, a smoothing technique is exploited to approximate the non-smooth maximum operator and an error analysis is investigated for this approximation. The gradients of the cost functional with respect to the decision variables in the approximate problem are derived. On the basis of these results, we develop a gradient-based optimization algorithm to solve the resulting optimization problem. Finally, an example is solved to demonstrate the effectiveness of the proposed algorithm.