Statement of the Necessary Condition for Mayer Problems of Optimal Control

Abstract

We consider here Mayer problems of optimization. Precisely, we are concerned with the problem of the minimum of a functional

$$I\left[ {x,\mu } \right]=g\left( {{t_1},x\left( {{t_1}} \right),{t_2},x\left( {{t_2}} \right)} \right)$$

(4.1.1)

with differential equations, constraints, and boundary conditions

$${{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = f\left( {t,x\left( t \right),u\left( t \right)} \right),{\text{ }}t \in \left[ {{t_1},{t_2}} \right]\left( {a.e.} \right)$$

(4.1.2)

$$\left( {t,x\left( t \right)} \right) \in A,{\text{ }}t \in \left[ {{t_1},{t_2}} \right]$$

(4.1.3)

$$u\left( t \right) \in U\left( t \right),{\text{ }}t \in \left[ {{t_1},{t_2}} \right]\left( {a.e.} \right)$$

(4.1.4)

$$ e\left[ x \right] = \left( {{t_1},x\left( {{t_1}} \right),{t_2},x\left( {{t_2}} \right)} \right) \in B$$

(4.1.5)

in the class Ω of all admissible pairs x(t) = (x¹, … , xⁿ), u(t) = (u¹, … , u ^m ), t _l ≤ t ≤ t ₂ Again, f(t, x, u) = (f ₁ , … , f _n)is a given vector function, and the system (4.1.2) can be written equivalently in the form

$${{d{x^i}} \mathord{\left/ {\vphantom {{d{x^i}} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = {f_i}\left( {t,x\left( t \right),u\left( t \right)} \right),{\text{ t}} \in \left[ {{t_1},{t_2}} \right]\left( {a.e.} \right),i = 1, \ldots ,n$$