# Formulation of the Time-Optimal Problem and Maximum Principle

• R. V. Gamkrelidze
Part of the Mathematical Concepts and Methods in Science and Engineering book series (MCSENG, volume 7)

## Abstract

We consider the following differential equation in R n ,
$$x = f(t,x,u)$$
(1.1)
. The point
$$x = \left( {\mathop{ \vdots }\limits_{{{x^{n}}}}^{{{x^{1}}}} } \right) \in {R^{n}}$$
will be called the phase point, the parameter (point)
$$u = \left( {\mathop{ \vdots }\limits_{{{u^{n}}}}^{{{u^{1}}}} } \right) \in {R^{r}}$$
will be called the control parameter, and the vector
$$f(t,x,u) = \left( {\mathop{ \vdots }\limits_{{{f^{n}}(t,x,u)}}^{{{f^{1}}(t,x,u)}} } \right) \in {R^{n}}$$
will be called the phase velocity vector. We assume that/is a continuous function on
$${R^{{1 + n + r}}} = \left\{ {(t,x,u):t \in {R^{n}},u \in {R^{r}}} \right\}$$
and that it has a continuous derivative with respect to x:
$$\frac{{\partial f(t,x,u)}}{{\partial x}} = \left( {\frac{{\partial f(t,x,u)}}{{\partial {x^{1}}}}, \ldots ,\frac{{\partial f(t,x,u)}}{{\partial {x^{n}}}}} \right)$$

## Keywords

Phase Velocity Maximum Principle Control Object Admissible Control Maximum Condition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.