We deal with an application of the fixed point theorem for nonexpansive mappings to a class of control systems. We study closed-loop and open-loop controllable dynamical systems governed by ordinary differential equations (ODEs) and establish convexity of the set of trajectories. Solutions to the above ODEs are considered as fixed points of the associated system-operator. If convexity of the set of trajectories is established, this can be used to estimate and approximate the reachable set of dynamical systems under consideration. The estimations/approximations of the above type are important in various engineering applications as, for example, the verification of safety properties.

This paper addresses an application of the well-known fixed point theorem for

Assume that for every such feedback control function

The

The reachable sets

This is for example the case if

Given

While the main topic of our paper is the estimation of reachable sets for closed-loop systems of type (

The paper is organized as follows. In Section

We first provide some relevant definitions and facts. Let

It holds that

If

We now consider the concept of a

Let

Now, we return to the given control system (

We now consider the set of admissible feedback controls

The set

The set

Because of

By Lemma

We next state and prove our main result concerning the operator

Assume that

We claim that

Note that Theorem

Under the assumption of Theorem

Since

In fact,

We now deal with the reachable set

Under the assumption of Theorem

Theorem

Note that under the conditions of Theorem

We now present two illustrative examples of control systems (

Let us consider an

Consider the following two-dimensional control system:

In this section we will discuss a special class of closed-loop and open-loop systems (

Let

Assume

Returning to control systems of type (

We now describe an abstract approach for estimating convex reachable sets. Our main idea is as follows: under the assumption of convexity for the reachable set of a given closed-loop control system, we formulate an auxiliary feedback optimal control problem with a linear cost functional. A solution of this problem makes it possible to construct a tangent hyperplane (supporting hyperplane) to the reachable set under consideration. Considering a sufficiently “rich” set of these hyperplanes and their intersections, one can approximate the reachable set with arbitrary accuracy.

Let

We now recall the Rademacher Theorem (see, e.g., [

The right-hand side

It hold that

Clearly, the optimal control problem (

Consider now an interior point

Let us now apply the main convexity result of Theorem

Let

Under the conditions of Theorem

Since

In this paper, we proposed a new convexity criterion for reachable sets for a class of closed-loop control systems. This sufficient condition is based on a general convexity result for solution sets of the corresponding nonlinear dynamical systems. Convexity of the set of trajectories makes it also possible to study some constrained feedback optimal control problems. For some families of closed-loop and open-loop control systems, we construct an overestimation of the examined reachable set, that is, we provide sets that contain the reachable sets of the dynamical system under consideration.