The method of complete quadratic approximation in optimal control problems (Q2387074)

The following optimal control problem is considered \[ \dot{x}=f(x,u,t),\; t\in T=[t_0,t_1],\;u(t)\in U, \tag{1} \] \[ x(t_0)=x_0,\tag{2} \] \[ \varphi(x(t_1))+ \int_{T}F(x(t),u(t),t)\,dt\rightarrow\min,\tag{3} \] where the vector-functions \(f(x,u,t)\) and \(F(x,u,t)\) are continuous on \(\mathbb R^{n}\times U\times T\) and doubly continuously differentiable with respect to \((x,u)\); the scalar function \(\varphi (x)\) is doubly continuously differentiable with respect to \(x\); \(U\subset \mathbb R^{n}\) is a convex compact set; admissible controls \(u(t)\) are piecewise continuous functions. Within the problem (1)--(3) the bilinear and biquadratic problems are investigated. For these problems an effective approximation method is proposed based on biquadratic approximation.