Constrained optimal control in terminal systems with prediction (Q1189946)

A controlled system is modelled by the equations: \(\dot x=f(w,x,t)\), where \(x\in\mathbb{R}^ n\) is the state, \(w\in G\subset\mathbb{R}^ \ell\) the control, \(t\) represents time. A probability measure \(P(x_ 0)\) is defined on the set of initial states. The control intends to take \(x\) from the initial state \(x_ 0\) to the manifold \(S\subset\mathbb{R}^{n+1}\) defined by a set of independent conditions: \(s(x,t)=0\), \(s\in S\), while incurring minimal loss: \[ Q=\int_{t_ 0}^ T f_ 0(w,x,\tau)d\tau=\int_{t_ 0}^ T f_ 0(w(u(\tau)),x(\tau))d\tau\to\min, \qquad u\in\mathbb{R}^ m, \] where \(u\) is a vector of control parameters. There exist several computational schemes dealing with this problem. Criteria for solvability are known. Their main problem is the complexity of the iterative procedures. It depends primarily on the dimension \(m\) of the control parameters vector. The author observed that for a class of linear problems the complexity is sharply reduced with a relatively small dimension \(m\). The author proceeds with a program of ``freezing'' some parameters. An optimal class of restrictions imposed on the original set of control parameters is formulated as a variational problem. This problem is very complex as well and approximate numerical techniques are suggested. To improve the efficiency of the proposed restrictions the author partitions the sets of initial and final constraints. It appears that an approach of Monte Carlo type may be very well suited for the implementation of these ideas. Examples of the author's technique are given in controlling consumption of fuel in a rocket and in the spacecraft rendezvous project.