  In this paper, we present a control synthesis framework for a general class
of nonlinear, control-affine systems under spatiotemporal and input
constraints. First, we study the problem of fixed-time convergence in the
presence of input constraints. The relation between the domain of attraction
for fixed-time stability with respect to input constraints and the required
time of convergence is established. It is shown that increasing the control
authority or the required time of convergence can expand the domain of
attraction for fixed-time stability. Then, we consider the problem of finding a
control input that confines the closed-loop system trajectories in a safe set
and steers them to a goal set within a fixed time. To this end, we present a
Quadratic Program (QP) formulation to compute the corresponding control input.
We use slack variables to guarantee feasibility of the proposed QP under input
constraints. Furthermore, when strict complementary slackness holds, we show
that the solution of the QP is a continuous function of the system states, and
establish uniqueness of closed-loop solutions to guarantee forward invariance
using Nagumo's theorem. We present two case studies, an example of adaptive
cruise control problem and an instance of a two-robot motion planning problem,
to corroborate our proposed methods.
