There is a difference between the definition of H here and its 
definition in the previous section. This is only an apparent difference 
in the sign of F, which occurs because the lower limit of the integral 
is used in the definition of J here rather than the upper limit as used 
earlier. Otherwise there is complete agreement with the results of the 
indirect method. 
CONSTRAINTS ON THE CONTROL AND STATE VARIABLES 
In most applications, the control or the state variables cannot be 
chosen arbitrarily but are subject to constraints. In the problem of a 
ship moving in a current, ship speed is limited by the maximum power 
available. The constraints can generally be expressed in terms of 
inequalities of the form 
g(x, u) < 0 Ca» 
where the vector inequality simply means that the components satisfy the 
inequality. The number of components in the vector ¢ is the number of 
constraints on the system. The analysis does not depend on whether both 
x and u occur implicitly in the inequality; one can have constraints on 
the controls and not on the state of the system or vice versa without 
affecting the analysis. 
In this presentation, the variables in the optimal control problem 
with constraints are the state variable x(t) and the control variable 
u(t) defined on an interval 0 < t < T. The process being controlled is 
described by the dynamic equation (1.1): 
a(S ECE, ze WW) 
with initial condition x = x5 the state and control variables are 
constrained by the inequality (3.1). For simplicity, the terminal cost 
is taken as zero, G = 0, and the cost function is given by the equation: 
T 
C,(@) = J (5 285 wy) Glo Ge) 
26 
