# Control-Lyapunov and Control-Barrier Functions based Quadratic Program for Spatio-temporal Specifications

## TL;DR

This paper introduces a novel quadratic programming approach utilizing control Lyapunov and barrier functions to synthesize controllers that satisfy complex spatio-temporal specifications in nonlinear systems.

## Contribution

It proposes the first QP-based method for control synthesis under spatio-temporal constraints using prescribed-time control Lyapunov functions and barrier functions.

## Key findings

- Successfully synthesizes controllers satisfying spatio-temporal specifications.
- Demonstrates feasibility through numerical examples.
- Provides conditions ensuring the existence of such controllers.

## Abstract

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.