Implicit parametrizations and applications in optimization and control
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.
implicit function theorem: optimal control with constraints
scientific article; zbMATH DE number 7293640
scientific article