  In this paper, we generalize the definition of backward reachable tube. The
classic definition of backward reachable tube is a set of system state that can
be driven into the target set within a given time horizon. Sometimes, the
concern of researchers is not only the time consumption, but some other forms
of cost of driving the system state toward the target set. Under this
background, the definition of cost-limited backward reachable tube is put
forward in this paper, where the cost is the time integral of a running cost.
And the running cost is a scalar function of system state and control input. A
method to compute the cost-limited backward reachable tube is proposed. In this
method, a cost-limited backward reachable tube is characterized by a non-zero
level set of a value function, which is approximated using recursion and
interpolation. At the end of this paper, some examples are taken to illustrate
the validity and accuracy of the proposed method.
