论文标题
使用准线性化和Fenchel二元性的非线性最佳控制问题解决方案
Solutions of Nonlinear Optimal Control Problems Using Quasilinearization and Fenchel Duality
论文作者
论文摘要
在本文中，我们考虑了一类特殊的非线性最佳控制问题，其中控制变量受到箱形限制，并且目标函数强烈凸出与控制变量相对应，并且相对于状态变量和控制变量可分离。我们将求解原始非线性问题转换为通过准线性化方法求解一系列约束线性的最佳控制问题。为了有效地解决每个线性季节问题，我们转向研究其双重问题。我们还证明了Fenchel二元性，强双重性能和对应于原始和双重问题的鞍点属性的方案来提出双重问题，这共同确保解决双重问题是有效的。因此，通过求解其双重问题的序列，可以替换解决通过准线性化技术获得的对照限制的线性界面最佳控制问题的序列。我们解决了双重问题的序列，并通过鞍点属性获得了原始控制约束线性季度问题的解决方案。此外，还证明了对每个子问题的解决方案最终将解决方案收敛到原始非线性问题的最佳条件的事实。之后，我们使用当前方法进行数值实验，对于每个子问题，我们在实验中通过Euler离散方案制定了离散的原始和双重问题。本方法的效率通过数值结果验证。
In this paper, we consider a special class of nonlinear optimal control problems, where the control variables are box-constrained and the objective functional is strongly convex corresponding to control variables and separable with respect to the state variables and control variables. We convert solving the original nonlinear problem into solving a sequence of constrained linear-quadratic optimal control problems by quasilinearization method. In order to solve each linear-quadratic problem efficiently we turn to study its dual problem. We formulate dual problem by the scheme of Fenchel duality, the strong duality property and the saddle point property corresponding to primal and dual problem are also proved, which together ensure that solving dual problem is effective. Thus solving the sequence of control constrained linear-quadratic optimal control problems obtained by quasilinearization technique is substituted by solving the sequence of their dual problem. We solve the sequence of dual problem and obtain the solution to primal control constrained linear-quadratic problem by the saddle point property. Furthermore, the fact that solution to each subproblem finally converges to the solution to the optimality conditions of original nonlinear problem is also proved. After that we carry out numerical experiments using present approach, for each subproblem we formulate the discretized primal and dual problem by Euler discretization scheme in our experiments. Efficiency of the present method is validated by numerical results.
