Higher approximation in the small parameter method for weakly controlled systems
The author considers the control problem \[ \text{P}_\varepsilon:\qquad J_\varepsilon(u)= \int^T_0(F^{(0)}(x,t)+\varepsilon F^{(1)}(x,u,t))dt\to \min, \] \[ \dot x(t)=f^{(0)}(x,t)+ \varepsilon f^{(1)}(x,u,t),\quad x(0)=x^0, \] where \(\;x(t)\in R^m,\;u(t)\in R^r\) and \(\varepsilon\) is a small parameter. A solution to the perturbed problem is sought in the asymptotic form \(\;x(t)=\sum_{j\geq 0} \varepsilon^j x_j(t),\quad u(t)=\sum_{j\geq 0} \varepsilon^j u_j(t).\) The functional being minimized is written as \(J_{\varepsilon}(u)= \sum_{j\geq 0} \varepsilon^j J_j.\) The trajectory \(x=x_0(t)\) is a solution of the Cauchy problem \(\dot x(t)=f^{(0)}(x,t)\), \(x(0)=x^0.\) The first approximation \((x^1, u_0)\) was obtained in the book by \textit{N. N. Moiseev} [``Asymptotic methods of nonlinear mechanics'' (2nd ed. 1981; Zbl 0527.70024; 1st ed. 1969; Zbl 0193.24801)] using the two-point boundary value problems arising from necessary optimality conditions. This paper solves the problems of higher approximations. It formulates and solves successive extremum problems \(\text{P}_i\) for functionals \(J_i\) through the derivatives of the Hamilton functions. Theorems about the existence and uniqueness of solutions of \(\text{P}_i\) and \(\text{P}_\varepsilon\) are verified.
scientific article; zbMATH DE number 1050564