连续系统的动态规划问题

系统的状态方程\(\dot x=f(x,u,t)\)
初态满足\(x(t_0)=x_0\)
性能泛函为\(J(x,t)=\int_{t_0}^{t_f}L(x,u,t) dt+\phi [x(t_f)]\)
问题：求解最优控制u使得性能泛函J(x,t)最小，对应的解为\(u^*,J^*(x,t)\)

对应的Bellman方程为\(-\frac{\partial J^*(x,t)}{\partial t}=\min_{u\in U}\{L(x,u,t)+[\frac{\partial J^*(x,t)}{\partial x}]^T f(x,u,t)\}\)

令哈密顿函数\(H(x,u,\lambda ,t)=L(x,u,t)+[\frac{\partial J^*(x,t)}{\partial x}]^Tf(x,u,t)\)
则对应的Bellman方程为\(-\frac{\partial J^*(x,t)}{\partial t}=\min_{u\in U}\{H(x,u,\lambda ,t)\}\)

解的过程为:
1.构造哈密顿函数
2.从哈密顿函数取极值求得\(u'\)
3.将u'带入Bellman方程中\(-\frac{\partial J^*}{\partial t}=H\)
4.将\(J^*\)带回u'，得到\(u^*\)
5.将\(u^*\)带回状态方程得到\(x^*\)
6.将\(x^*\)带回性能泛函得到\(J^*\)