Optimal control for boundary value problems (Q1902857)

The authors consider the problem of minimization of the functional \[ J(u)= \int^{t_1}_{t_0} F(x, u, t) dt+ \varphi_0(x(t_0), x(t_1)) \] on motions of the system \(\dot x= f(x, u, t)\), \(u(t)\in U\subset \mathbb{R}^n\), when initial and terminal states satisfy the given equation \(\varphi(x(t_0), x(t_1))= 0\). The necessary optimality conditions in the form of the maximum principle are obtained under the assumption that the boundary value problem is solvable. Under suitable assumptions the authors substantiate an iterative process for finding the solution of a canonical system of differential equations in the maximal principle satisfying necessary boundary conditions. Conditions ensuring convergence and conditions when the algorithm generates the minimizing sequence of controls are obtained. Thus, the article derives a new technique to solve the problem.