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7.4.1 Hybrid optimal control problem
We begin by defining the class of hybrid control systems and associated optimal control problems that we want to study.
The first ingredient of our hybrid control system is a collection of (timeinvariant) control systems

(7.32) 
where
is a finite index set; for simplicity, we assume that all the systems in the collection (7.32) share the same state space
and control set
. The second ingredient is a collection of switching surfaces (or guards)
, one for each pair
. A function
is an admissible trajectory of our hybrid system corresponding to a control
if there exist time instants
and indices
such that
satisfies

(7.33) 
and
Here
and
are the values of
right before and right after
, respectively, and the value
is taken to be equal to one of these onesided limits, depending on the desired convention. At each possible discontinuity
, a discrete transition (or switching event) is said to occur.
The function
defined by
for
describes the evolution of
along the trajectory;
is often called the discrete state of the hybrid system. We can use it to rewrite (7.33) more concisely as
for
,
.
We consider cost functionals of the form

(7.34) 
where
is the usual running cost and
is the switching cost, for
.
For simplicity, we do not include a terminal cost (this is no loss of generality, as terminal cost can be easily incorporated into the above runningplusswitching cost along the lines of Section 3.3.2).
We also introduce
an endpoint constraint, characterized by a set
for each pair
, according to which the trajectory
must satisfy

(7.35) 
(Here
plays the same role as
at the end of Section 4.3.1.)
Then, the hybrid optimal control problem consists in finding a control that minimizes the cost (7.34) subject to the endpoint constraint (7.35). We emphasize that the choice of a control
is accompanied by the choice of two finite sequences
and
, to which we henceforth refer as the time sequence and switching sequence, respectively.
Next: 7.4.2 Hybrid maximum principle
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Daniel
20101220