  This paper presents a method for control synthesis under spatio-temporal
constraints. First, we consider the problem of reaching a set $S$ in a
user-defined or prescribed time $T$. We define a new class of control Lyapunov
functions, called prescribed-time control Lyapunov functions (PT CLF), and
present sufficient conditions on the existence of a controller for this problem
in terms of PT CLF. Then, we formulate a quadratic program (QP) to compute a
control input that satisfies these sufficient conditions. Next, we consider
control synthesis under spatio-temporal objectives given as: the closed-loop
trajectories remain in a given set $S_s$ at all times; and, remain in a
specific set $S_i$ during the time interval $[t_i, t_{i+1})$ for $i = 0, 1,
\cdots, N$; and, reach the set $S_{i+1}$ on or before $t = t_{i+1}$. We show
that such spatio-temporal specifications can be translated into temporal logic
formulas. We present sufficient conditions on the existence of a control input
in terms of PT CLF and control barrier functions. Then, we present a QP to
compute the control input efficiently, and show its feasibility under the
assumptions of existence of a PT CLF. To the best of authors' knowledge, this
is the first paper proposing a QP based method for the aforementioned problem
of satisfying spatio-temporal specifications for nonlinear control-affine
dynamics with input constraints. We also discuss the limitations of the
proposed methods and directions of future work to overcome these limitations.
We present numerical examples to corroborate our proposed methods.
