14、贝尔曼方程粘性解与脉冲控制动态规划

贝尔曼方程粘性解与脉冲控制动态规划

1. 贝尔曼方程粘性解相关内容

1.1 粘性解不等式推导

从不等式 (0 \geq - \varepsilon (r - t) + \int_{t}^{r} g(y(s), U(s)) ds - \phi(t, x) + \phi(r, y(r))) 开始，经过一系列推导：
- 首先，(0 \geq - \varepsilon (r - t) + \int_{t}^{r} g(y(s), U(s)) ds - \phi(t, x) + \phi(r, y(r)) \geq - \varepsilon (r - t) + \int_{t}^{r} g(y(s), U(s)) ds + \int_{t}^{r} \left(\frac{\partial \phi}{\partial s} (s, y(s)) + \langle \phi_x(s, y(s)), f(y(s), U(s)) \rangle \right) ds)
- 然后，(\geq - \varepsilon (r - t) + \int_{t}^{r} \left(\frac{\partial \phi}{\partial s} (s, y(s)) + \inf_{u} \left{ g(y(s), u) + \langle \phi_x(s, y(s)), f(y(s), u) \rangle \right} \right) ds)
- 最终得到 (\varepsilon \geq \frac{1}{r - t} \int_{t}^{r} \left(\frac{\partial \phi}{\partial s} (s, y(s)) + H \left( y(s