Differential Dynamic Programming

a₁ B₁ bang-bang control Bellman boundary conditions calculate Calculus of Variations change in cost Chapter computational constraints continuous-time system control function control law cost function D. H. Jacobson D. Q. Mayne d₁ defined denote difference equations differential dynamic programming differential equations discrete-time Dreyfus Equa error estimate f(x+dx fi(xi first-order algorithm given by Equation initial condition integration interval Li(xi linear minimize Mitter Monte-Carlo N₁ nominal control nominal trajectory nonlinear obtained open-loop control optimal control optimal cost optimal trajectory parameters positive-definite procedure random variables realizations reduction in cost Riccati equation satisfied second-order algorithm Section sequence side of Equation solution stochastic Sufficient conditions sufficiently small switch point t₁ Taylor series tion u₁ unspecified arguments V₁ variance variate method vector Vx(t Vxx(d Vxx(t Vxxx w₁ x₁ Xi+1 yields zero นน