Optimal quantile control of dynamic systems (Q1882168)

A discrete stochastic system (1) \(z_{i+1}=f_i (z_i ,u_i ,x_i ), i=1,\dots,N\), is considered, where \(z_i =(z^1_i ,...,z^m_i )\) is the state vector at instant \(i\), \(f_i (\cdot)\) is a continuously differentiable vector function, \(x_i =(x^1_i ,...,x^n_i )\) is a vector of random perturbations, \(u_i =(u^1_i ,...,u^s_i )\) is the vector of control parameters, \(N\) is the number of steps of the motion. Here \(x_i =x_i (\omega_i ), i=1,\dots,N\), where the elements of the vector \(\omega \) are independent Gaussian random variables. It is assumed that at the terminal instant \(N+1\), the primary objective function \(J=F(z_{N+1})=\Phi (u,\omega )\) is given. A second objective function, that does not depend on \(\omega \), is introduced. This function is helpful in statistically assessing the behavior of \(\Phi (u,\omega )\). An example of such a function is \(\Phi_{\alpha }(u)=\min\{ \phi : P_{\phi }(u)\geq \alpha \}\) where \(P_{\phi }(u)=P\{ \omega : \Phi (u,\omega )\leq \phi \}\). The main idea is the minimization of \(\Phi_{\alpha }(u)\) by choosing the controls \(u_i ,i=1,\dots,N\). A numerical algorithm for determing the optional quantile control of the system (1) is designed by a modified Monte Carlo method and computation of the directional derivatives at each step. The method is illustrated by an application of the algorithm for determing the control a solid-propellent rocket.