Solution of optimal control problems with modified performance functionals (Q1913118)

The authors consider the optimal control problem \[ J(x,u,T) = \Phi \bigl( x(T), T \bigr) + \int^T_{t_0} L \bigl( x(t), u(t), t \bigr) dt \to \min \] \[ \dot x(t) = f \bigl( x(t), u(t), t \bigr),\;x \in R^n,\;u \in R^m,\;m \leq n, \] \(x(t_0) = x_0\), \((x(T), T) \in S\); \(u(t) \in \Omega\), \(t \in [0,T]\), \(T \in (T_1, T_2)\), where \(S\) is a given smooth manifold from \(\mathbb{R}^n \times (T_1, T_2)\). A method of synthesis of optimal control is proposed. The method is based on a modification of the initial functional \(J\) and a study of the Hamilton-Jacobi equations. The control is synthesized on the basis of the principle of feedback system design.