Derivation of locally optimal controls of weakly controllable systems from Bellman's equation (Q677352)

The author studies the optimal control problem \[ F\bigl( x^\varepsilon (T), T \bigr) \to\min, \quad g\bigl(x^\varepsilon (T),T\bigr)=0; \] \[ \dot x^\varepsilon (\tau)= f_0 (x^\varepsilon, \tau)+ \varepsilon f_1 \bigl(x^\varepsilon, \tau,u(x^\varepsilon, \tau)\bigr), \] \[ x^\varepsilon(t) =x, \quad u\in V, \quad x\in X. \] Let \(B^\varepsilon (x,t)\) be the solution for the corresponding Bellman equation. The main theorem states that under an appropriate assumption one has \(B^\varepsilon (x,t)= B^{(1)} (x,t)+ O(\varepsilon^2)\) where \(B^{(1)} (x,t)\) is defined by the following chain of equations \[ {\partial \over \partial t} B^{(0)} (x,t)+ \bigl\langle \nabla B^{(0)} (x,t), f_0(x,t) \bigr\rangle =0, \] \[ B^{(0)} (x,T)= F(x,T), \quad g(x,T)=0, \] \[ u^{(0)} (x,t)= {\underset u \in {V}{\arg\min}}\bigl\langle \nabla B^{(0)} (x,t), f_1(x,t,u) \bigr\rangle, \] \[ {\partial \over \partial t} B^{(1)}(x,t)+ \bigl\langle \nabla B^{(1)} (x,t),f_0(x,t) \bigr\rangle +\varepsilon f_1 \bigl( x,t, u^{(0)} (x,t)\bigr) =0, \] \[ B^{(1)} (x,T)=F(x,T). \]