Controller Synthesis for Nonlinear Systems with Reachability Specifications Using Monotonicity

In this paper, we consider a control synthesis problem for a continuous-time nonlinear system. The problem under consideration consists in driving the state of the system to some target interval at a given time instant. We propose a solution based on candidate under-approximations of the backward reachable sets using multi-dimensional intervals. We show that a suitable controller can be designed by enforcing a monotonicity property of the closed-loop system on these intervals. For this purpose, we utilize the monotonicity conditions for nonlinear systems with inputs in the infinitesimal form. From these differential inequalities on the control strategy, we design some particular controllers which are time-varying, linear with respect to the state. The approach is illustrated by two examples.