Optimization of dynamical systems that are quadratic with respect to the state (Q1413073)

The paper is concerned with the finite-dimensional optimization problem \[ \Phi(u)=\varphi(x(t_1))+\int_T F(x(t),u,t)\,dt \to \min_{u \in U}, \] \[ {\dot x}(t)=f(x(t),u,t), \quad x(t_0)=x^0, \quad t \in T=[t_0,t_1], \] where \(x(t)=(x_1(t),\dots,x_n(t))\), \(u=(u_1,\dots,u_m) \in R^m\), and \(U \subset R^m\) is a compact subset. The functions \(f(x,u,t)\) and \(F(x,u,t)\) are quadratic with respect to \(x\) with coefficients continuous in \(u, t\); the function \(\varphi(x)\) is quadratic on \(R^n\). With the use of exact (i.e., without remainder terms) formulas for the increment of \(\Phi(u)\), the author suggests and studies an algorithm of nonlocal improvement of a current controlling parameter \(u\).