# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019023

## Minimizing almost smooth control variation in nonlinear optimal control problems

 1 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China 2 Department of Mathematics, Shanghai University, Baoshan 200444, Shanghai, China 3 Xingzhi College, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

* Corresponding author: Changjun Yu

Received  January 2018 Revised  March 2018 Published  March 2019

Fund Project: This paper is supported by NSFC grant 11871039, 11771275, the Scientific Research Project of Zhejiang Provincial Department of science and technology in China (Grant No. LGN19C040001), and the Scientific Research Project of Zhejiang Provincial Department of Education in China (Grant No. Y201329106)

In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach.

Citation: Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019023
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##### References:
Optimal control $u_{1}(t)$.
Optimal control function $u_{2}(t)$.
Optimal state trajectory $x_{1}(t)$.
Optimal state trajectory $x_{2}(t)$.
Optimal control $x_{3}(t)$.
Optimal state trajectory $x_{4}(t)$.
Optimal state trajectory $x_{5}(t)$.
Optimal control $x_{6}(t)$.
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