Recent zbMATH articles in MSC 26B10 https://www.zbmath.org/atom/cc/26B10 2021-04-16T16:22:00+00:00 Werkzeug Implicit parametrizations and applications in optimization and control. https://www.zbmath.org/1456.49008 2021-04-16T16:22:00+00:00 "Tiba, Dan" https://www.zbmath.org/authors/?q=ai:tiba.dan The subject is the characterization (with numerical applications in mind) of the manifold $$V$$ of solutions of the nonlinear system $F_j(x_1, x_2, \dots, x_d) = 0 \quad (1 \le j \le l )\quad l \le d - 1 \tag{1}$ in the vicinity of $$x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \in V,$$ under the Jacobian assumption $\frac{\partial (F_1, F_2, \dots, F_l)}{\partial (x_1, x_2, \dots, x_l)} \ne 0 \quad \hbox{in} \ x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \, .$ The first step involves the underdetermined linear system $v(x) \cdot \nabla F_j(x) = 0 \quad (1 \le j \le l)$ which is used to obtain bases $$(v_1(x), v_2(x), \dots, v_{d - l}(x))$$ for the tangent spaces of $$V.$$ Next, the chain of differential equations \begin{align*} \frac{\partial y_1(t_1)}{\partial t_1} &= v_1(y_1(t_1)), y_1(0) = x^0 \cr \frac{\partial y_2(t_1, t_2)}{\partial t_2} &= v_2(y_2(t_1, t_2)),\quad y_2(t_1, 0) = y(t_1) \cr & \hskip 2em \dots \dots \dots \dots \cr \frac{\partial y_{d - l}(t_1, t_2, \dots, t_{d - l})}{ \partial t_{d - l}} &= v_{d - l}(y_{d - l}(t_1, t_2, \dots, t_{d - l})) \, , \cr & \hskip 2.7em y_{d - l}(t_1, \dots , t_{d - l - 1}, 0) = y_{d - l - 1}(t_1, t_2, \dots , t_{d - l - 1}) \end{align*} is set up, thus constructing a parametrization of $$V$$ which may be considered as an explicit form of the implicit function theorem. The result is used to construct an algorithm for the solution of the problem of minimizing a function $$g(x_0, x_2, \dots , x_d)$$ subject to (1). Some generalizations are covered, such as the case where (1) includes inequality constraints and/or regularity is relaxed. In the last section the results are applied to the control problem of minimizing $$l(x(0), x(1))$$ among the trajectories of the system $$x'(t) = f(t, x(t), u(t))$$ subject to the state-control constraint $$h(x(t), u(t)) = 0.$$ There are several numerical implementation of the algorithms and the author notes that computations can be carried out using standard Matlab routines. Reviewer: Hector O. Fattorini (Los Angeles)