MILP modelling for the time optimal control problem in the case of multiple targets

Abstract

SUMMARYOptimal guidance for a dynamical system from a given point to a set of targets is discussed. Detecting for the best target is done in such a way that the capture time is minimized and desirability of targets is maximized. By extending measure theoretical approach for the classical optimal control problem to this case, the nearly optimal control is constructed from the solution of a mixed integer linear programming problem. To find the lower bound of the optimal time a search algorithm is proposed. Numerical examples are also given.

Quantitative Approximability of Optimal Control by Linear Programing Model for Asymptomatic Dual HIV - Pathogen Infections

“…Finally, the solution of this problem is then approximated by the solution of a finite dimensional LP of sufficiently large dimension. This classical measure theory for the computation of optimal control problem was first adopted by [24], applied by [25,26] and extensively improved by [27]. Of note, model [18] recently applied the method in the calculation of time optimal control problem (TOCP), which led to the approximation of linear programing model.…”

Optimal Control of HIV Dynamic Using Embedding Method

“…Then, the method has been extended for approximating the time optimal problems by an LP model [36]. Here, this approach is used.…”

“…$\begin{array}{cc}right{J}_s=\left[t_0+\frac{\left(s-1\right)\mathrm{\Delta }T}{S-1},t_0+\frac{s\mathrm{\Delta }T}{S-1}\right),& left\\ rights=\mathrm{1,2},\dots ,S-1,\hspace{1em}{J}_S=\left[t_l,t_f\right),& left\end{array}\hspace{0.17em}$ where t l is a lower bound for optimal time t f , which can be obtained by using a search algorithm based on golden section [36] or Fibonnaci search method [37]. Let $\bar{S}$ be the largest number such that $J_{\bar{S}}\subseteq false[t_0,t_0+\eta false]$.…”

“…$t^i{x}_2^j,\mathrm{\hspace{0.17em}\hspace{0.17em}}{x}_2^j{x}_h^i,\hspace{1em}i\in \left\{\mathrm{0,1}\right\},\hspace{0.17em}\hspace{0.17em}j\in \left\{\mathrm{1,2},\mathrm{\hspace{0.17em}\hspace{0.17em}}\dots \right\},\hspace{0.17em}\hspace{0.17em}h\in \left\{\mathrm{1,3},4\right\}.$ In addition, we choose some functions with compact support in the following form [36, 37]: …”

A numerical method for solving optimal control problems using state parametrization

“…Direct methods can be applied using a discretization technique [5] or a parametrization method [1,4,11]. A review on control parametrization for constrained optimal control problems is given in [7].…”