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| Compact formulation of the augmented evolution equation for optimal control computation |
| ShengZhang1,JiangtaoHuang1,GangLiu1,FeiLiao1,FangfangHu1 |
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| (China Aerodynamics Research and Development Center, Mianyang 621000, Sichuan, China) |
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| 摘要: |
| The augmented evolution equation is established under the framework of the Variation Evolving Method (VEM) that seeks
optimal solutions by solving the transformed Initial-Value Problems (IVPs). To improve the numerical performance, its
compact form is developed herein. Through replacing the states and costates variation evolution with that of the controls, the
dimension-reduced Evolution Partial Differential Equation (EPDE) only solves the control variables along the variation time to get the optimal solution, and the initial conditions for the definite solution may be arbitrary. With this equation, the scale of the resulting IVPs, obtained via the semi-discrete method, is significantly reduced and they may be solved with common Ordinary Differential Equation (ODE) integration methods conveniently.Meanwhile, the state and the costate dynamics share consistent stability in the numerical computation and this avoids the intrinsic numerical difficulty as in the indirect methods. Numerical examples are solved and it is shown that the compact form evolution equation outperforms the primary form in the precision, and the efficiency may be higher for the dense discretization. Actually, it is uncovered that the compact form of the augmented evolution equation is a continuous realization of the Newton type iteration mechanism. |
| 关键词: Optimal control · Lyapunov dynamics stability · Variation evolution · Evolution partial differential equation · Initial-value problem |
| DOI：https://doi.org/10.1007/s11768-025-00276-4 |
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| 基金项目:This work was supported by the National Nature Science Foundation of China under Grant No. 11902332. |
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| Compact formulation of the augmented evolution equation for optimal control computation |
| Sheng Zhang1,Jiangtao Huang1,Gang Liu1,Fei Liao1,Fangfang Hu1 |
| (China Aerodynamics Research and Development Center, Mianyang 621000, Sichuan, China) |
| Abstract: |
| The augmented evolution equation is established under the framework of the Variation Evolving Method (VEM) that seeks
optimal solutions by solving the transformed Initial-Value Problems (IVPs). To improve the numerical performance, its
compact form is developed herein. Through replacing the states and costates variation evolution with that of the controls, the
dimension-reduced Evolution Partial Differential Equation (EPDE) only solves the control variables along the variation time to get the optimal solution, and the initial conditions for the definite solution may be arbitrary. With this equation, the scale of the resulting IVPs, obtained via the semi-discrete method, is significantly reduced and they may be solved with common Ordinary Differential Equation (ODE) integration methods conveniently.Meanwhile, the state and the costate dynamics share consistent stability in the numerical computation and this avoids the intrinsic numerical difficulty as in the indirect methods. Numerical examples are solved and it is shown that the compact form evolution equation outperforms the primary form in the precision, and the efficiency may be higher for the dense discretization. Actually, it is uncovered that the compact form of the augmented evolution equation is a continuous realization of the Newton type iteration mechanism. |
| Key words: Optimal control · Lyapunov dynamics stability · Variation evolution · Evolution partial differential equation · Initial-value problem |
