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Symmetries and analytical solutions of the Hamilton-Jacobi-Bellman equation for a class of optimal controlproblems

机译：一类最优控制问题的Hamilton-Jacobi-Bellman方程的对称性和解析解

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The main contribution of this paper is to identify explicit classes of locally controllable second-order systems and optimization functionals for which optimal control problems can be solved analytically, assuming that a differentiable optimal cost-to-go function exists for such control problems. An additional contribution of the paper is to obtain a Lyapunov function for the same classes of systems. The paper addresses the Lie point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation for optimal control of second-order nonlinear control systems that are affine in a single input and for which the cost is quadratic in the input. It is shown that if there exists a dilation symmetry of the HJB equation for optimal control problems in this class, this symmetry can be used to obtain a solution. It is concluded that when the cost on the state preserves the dilation symmetry, solving the optimal control problem is reduced to finding the solution to a first-order ordinary differential equation. For some cases where the cost on the state breaks the dilation symmetry, the paper also presents an alternative method to find analytical solutions of the HJB equation corresponding to additive control inputs. The relevance of the proposed methodologies is illustrated in several examples for which analytical solutions are found, including the Van der Pol oscillator and mass-spring systems. Furthermore, it is proved that in the well-known case of a linear quadratic regulator, the quadratic cost is precisely the cost that preserves the dilation symmetry of the equation. Copyright (c) 2015 John Wiley & Sons, Ltd.
机译：本文的主要贡献是，在假定存在针对此类控制问题的可区分的最优成本函数的情况下，确定可以通过解析解决最优控制问题的局部可控制二阶系统和优化函数的显式类。本文的另一个贡献是为相同类别的系统获得Lyapunov函数。本文讨论了Hamilton-Jacobi-Bellman（HJB）方程的Lie点对称性，用于最优控制二阶非线性控制系统，该系统在单个输入中是仿射的，其输入成本为二次方。结果表明，对于此类最优控制问题，如果存在HJB方程的扩张对称性，则该对称性可用于获得解。结论是，当状态的代价保持扩张对称性时，求解最优控制问题就简化为寻找一阶常微分方程的解。对于某些状态代价破坏了扩张对称性的情况，本文还提出了另一种方法来找到与加性控制输入相对应的HJB方程的解析解。在几个可以找到分析解决方案的示例中，说明了所提出方法的相关性，其中包括Van der Pol振荡器和质量弹簧系统。此外，证明了在线性二次调节器的公知情况下，二次成本恰好是保持方程的扩张对称性的成本。版权所有（c）2015 John Wiley＆Sons，Ltd.

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