Global convergent semi-smooth Newton (GCSSN) method for $L^2$-norm control constrained elliptic optimal control problem with $L^1$-control cost is discussed. The first order necessary optimality conditions are analyzed and reduced system is proposed. After finite difference discretization, we propose the global convergent semi-smooth Newton method for discrete reduced system with the nonmonotone line search. The local superlinear convergence rate and convergence are proved theoretically. The demonstrated theoretical properties are verified with numerical results. To illustrate the effectiveness and efficiency, we compare the proposed method with semi-smooth Newton method in the simulation part.

全局收敛半光滑牛顿 (GCSSN) 方法${\mathrm{大号}}^2$-范数控制约束椭圆最优控制问题${\mathrm{大号}}^1$-讨论控制成本。分析了一阶必要最优性条件，提出了约简系统。在有限差分离散化之后，我们提出了具有非单调线搜索的离散缩减系统的全局收敛半光滑牛顿法。理论上证明了局部超线性收敛速度和收敛性。通过数值结果验证了所证明的理论性质。为了说明有效性和效率，我们在模拟部分将所提出的方法与半光滑牛顿法进行了比较。